A Guide to Experiments in Quantum Optics
A Guide to Experiments in Quantum Optics Hans-A. Bachor and Timothy C. Ralph ...

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A Guide to Experiments in Quantum Optics

A Guide to Experiments in Quantum Optics Hans-A. Bachor and Timothy C. Ralph

Third Edition

Authors Hans-A. Bachor

The Australian Nat. University Research School of Physics Science Road Canberra ACT 0200 Australia

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

Dr. Timothy C. Ralph

University of Queensland Department of Physics St. Lucia QLD 4072 Australia

Library of Congress Card No.:

Cover

A catalogue record for this book is available from the British Library.

The artwork was prepared by B. Hage and is based on an experimental demonstration of photon number states published in: Chrzanowski H.M., Assad S.M., Bernu J., and Hage B. et al. (2013). Reconstruction of Photon Number Conditioned States Using Phase Randomized Homodyne Measurements, J. Phys. B At. Mol. Opt. Phys. 46: 104009.

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The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliograﬁe; detailed bibliographic data are available on the Internet at . © 2019 Wiley-VCH Verlag GmbH & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microﬁlm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not speciﬁcally marked as such, are not to be considered unprotected by law. Print ISBN: 978-3-527-41193-1 ePDF ISBN: 978-3-527-68394-9 ePub ISBN: 978-3-527-68393-2 oBook ISBN: 978-3-527-69580-5

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v

Contents Preface xv Acknowledgments xix 1 1.1 1.2 1.3 1.4 1.5

2

2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6 2.1.7 2.2 2.2.1 2.2.2 2.2.2.1 2.2.3 2.3 2.3.1 2.3.1.1 2.3.2 2.3.3 2.3.3.1 2.4 2.5

Introduction 1

Optics in Modern Life 1 The Origin and Progress of Quantum Optics 3 Motivation Through Simple and Direct Teaching Experiments 7 Consequences of Photon Correlations 12 How to Use This Guide 14 References 16 Classical Models of Light 19 Classical Waves 20 Mathematical Description of Waves 20 The Gaussian Beam 21 Quadrature Amplitudes 24 Field Energy, Intensity, and Power 25 A Classical Mode of Light 26 Light Carries Information 28 Modulations 30 Optical Modes and Degrees of Freedom 32 Lasers with Single and Multiple Modes 32 Polarization 33 Poincaré Sphere and Stokes Vectors 35 Multimode Systems 36 Statistical Properties of Classical Light 37 The Origin of Fluctuations 37 Gaussian Noise Approximation 38 Noise Spectra 39 Coherence 40 Correlation Functions 44 An Example: Light from a Chaotic Source as the Idealized Classical Case 46 Spatial Information and Imaging 50

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2.5.1 2.5.2 2.5.3 2.5.4 2.5.5 2.5.6 2.5.7 2.5.8 2.5.9 2.5.10 2.6

State-of-the-Art Imaging 50 Classical Imaging 52 Image Detection 55 Scanning 56 Quantifying Noise and Contrast 58 Coincidence Imaging 59 Imaging with Coherent Light 60 Image Reconstruction with Structured Illumination 60 Image Analysis and Modes 61 Detection Modes and Displacement 61 Summary 62 References 63 Further Reading 64

3

65 Detecting Light 65 The Concept of Photons 68 Light from a Thermal Source 70 Interference Experiments 73 Modelling Single-Photon Experiments 78 Polarization of a Single Photon 79 Some Mathematics 80 Polarization States 81 The Single-Photon Interferometer 83 Intensity Correlation, Bunching, and Anti-bunching 84 Observing Photons in Cavities 88 Summary 90 References 90 Further Reading 92

3.1 3.2 3.3 3.4 3.5 3.5.1 3.5.1.1 3.5.2 3.5.3 3.6 3.7 3.8

Photons: The Motivation to Go Beyond Classical Optics

4

Quantum Models of Light 93

4.1 4.1.1 4.1.2 4.1.3 4.1.4 4.2 4.2.1 4.2.2 4.2.3 4.3 4.3.1 4.3.2 4.3.3 4.3.4 4.3.4.1

Quantization of Light 93 Some General Comments on Quantum Mechanics 93 Quantization of Cavity Modes 94 Quantized Energy 95 The Creation and Annihilation Operators 97 Quantum States of Light 97 Number or Fock States 97 Coherent States 99 Mixed States 101 Quantum Optical Representations 102 Quadrature Amplitude Operators 102 Probability and Quasi-probability Distributions 104 Photon Number Distributions 108 Covariance Matrix 111 Summary of Diﬀerent Representations of Quantum States and Quantum Noise 112

Contents

4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.5 4.5.1 4.5.2 4.5.3 4.5.4 4.5.5 4.5.6 4.5.7 4.6 4.6.1 4.6.2 4.6.3 4.7 4.7.1 4.7.2 4.7.3 4.7.4

Propagation and Detection of Quantum Optical Fields 113 Quantum Optical Modes in Free Space 114 Propagation in Quantum Optics 115 Detection in Quantum Optics 117 An Example: The Beamsplitter 118 Quantum Transfer Functions 120 A Linearized Quantum Noise Description 121 An Example: The Propagating Coherent State 123 Real Laser Beams 123 The Transfer of Operators, Signals, and Noise 124 Sideband Modes as Quantum States 126 Another Example: A Coherent State Pulse Through a Frequency Filter 129 Transformation of the Covariance Matrix 130 Quantum Correlations 131 Photon Correlations 131 Quadrature Correlations 132 Two-Mode Covariance Matrix 133 Summary 134 The Photon Number Basis 134 Quadrature Representations 135 Quantum Operators 135 The Quantum Noise Limit 136 References 136 Further Reading 137

5

Basic Optical Components 139

5.1 5.1.1 5.1.1.1 5.1.2 5.1.3 5.1.4 5.1.5 5.1.5.1 5.1.6

Beamsplitters 140 Classical Description of a Beamsplitter 140 Polarization Properties of Beamsplitters 142 The Beamsplitter in the Quantum Operator Model 143 The Beamsplitter with Single Photons 144 The Beamsplitter and the Photon Statistics 146 The Beamsplitter with Coherent States 149 Transfer Function for a Beamsplitter 149 Comparison Between a Beamsplitter and a Classical Current Junction 151 The Beamsplitter as a Model of Loss 152 Interferometers 153 Classical Description of an Interferometer 154 Quantum Model of the Interferometer 155 The Single-Photon Interferometer 156 Transfer of Intensity Noise Through the Interferometer 156 Sensitivity Limit of an Interferometer 157 Eﬀect of Mode Mismatch on an Interferometer 160 Optical Cavities 162 Classical Description of a Linear Cavity 164

5.1.7 5.2 5.2.1 5.2.2 5.2.3 5.2.4 5.2.5 5.2.6 5.3 5.3.1

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5.3.2 5.3.3 5.3.4 5.3.4.1 5.3.4.2 5.3.4.3 5.3.5 5.3.6 5.3.7 5.3.8 5.3.9 5.3.10 5.4 5.4.1 5.4.2 5.4.3 5.4.4 5.4.5 5.4.5.1 5.4.6 5.4.7 5.4.8

The Special Case of High Reﬂectivities 169 The Phase Response 170 Spatial Properties of Cavities 172 Mode Matching 172 Polarization 174 Tunable Mirrors 175 Equations of Motion for the Cavity Mode 175 The Quantum Equations of Motion for a Cavity 176 The Propagation of Fluctuations Through the Cavity 177 Single Photons Through a Cavity 180 Multimode Cavities 181 Engineering Beamsplitters, Interferometers, and Resonators 182 Other Optical Components 184 Lenses 184 Holograms and Metasurfaces 185 Crystals and Polarizers 187 Optical Fibres and Waveguides 188 Modulators 189 Phase and Amplitude Modulators 191 Spatial Light Modulators 193 Optical Noise Sources 195 Non-linear Processes 195 References 196

6

Lasers and Ampliﬁers 199

6.1 6.1.1 6.1.2 6.1.3 6.1.4 6.1.4.1 6.1.4.2 6.1.4.3 6.1.4.4 6.1.4.5 6.1.5 6.1.6 6.2 6.3 6.3.1 6.3.2 6.3.3 6.3.3.1 6.3.4 6.4 6.4.1 6.4.2

The Laser Concept 199 Technical Speciﬁcations of a Laser 201 Rate Equations 203 Quantum Model of a Laser 207 Examples of Lasers 209 Classes of Lasers 209 Dye Lasers and Argon Ion Lasers 209 The CW Nd:YAG Laser 210 Diode Lasers 213 Limits of the Single-Mode Approximation in Diode Lasers Laser Phase Noise 214 Pulsed Lasers 215 Ampliﬁcation of Optical Signals 215 Parametric Ampliﬁers and Oscillators 218 The Second-Order Non-linearity 219 Parametric Ampliﬁcation 220 Optical Parametric Oscillator 221 Noise Spectrum of the Parametric Oscillator 222 Pair Production 223 Measurement-Based Ampliﬁers 224 Deterministic Measurement-Based Ampliﬁers 225 Heralded Measurement-Based Ampliﬁers 228

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Contents

6.5

Summary 230 References 231

7

Photon Generation and Detection 233

7.1 7.1.1 7.2 7.2.1 7.2.1.1 7.2.1.2 7.2.1.3 7.2.1.4 7.2.2 7.3 7.3.1 7.3.1.1 7.3.1.2 7.3.1.3 7.3.1.4 7.3.2 7.3.2.1 7.3.2.2 7.3.2.3 7.3.3 7.3.4 7.3.4.1 7.3.4.2 7.3.4.3 7.3.4.4 7.3.4.5 7.3.4.6 7.3.4.7 7.4

Photon Sources 236 Deterministic Photon Sources 239 Photon Detection 240 Detecting Individual Photons 240 Photochemical Detectors 241 Photoelectric Detectors 241 Photo-thermal Detectors 243 Multipixel and Imaging Devices 243 Recording Electrical Signals from Individual Photons 245 Generating, Detecting, and Analysing Photocurrents 247 Properties of Photocurrents 247 Beat Measurements 247 Intensity Noise and the Shot Noise Level 248 Quantum Eﬃciency 249 Photodetector Materials 250 Generating Photocurrents 251 Photodiodes and Detector Circuit 251 Ampliﬁers and Electronic Noise 252 Detector Saturation 254 Recording of Photocurrents 255 Spectral Analysis of Photocurrents 257 Digital Fourier Transform 257 Analogue Fourier Transform 258 From Optical Sidebands to the Current Spectrum 258 The Operation of an Electronic Spectrum Analyser 259 Detecting Signal and Noise Independently 260 The Decibel Scale 261 Adding Electronic AC Signals 262 Imaging with Photons 263 References 264 Further Reading 267

8

Quantum Noise: Basic Measurements and Techniques 269

8.1 8.1.1 8.1.1.1 8.1.2 8.1.3 8.1.4 8.1.4.1 8.1.5 8.1.5.1 8.2

Detection and Calibration of Quantum Noise 269 Direct Detection and Calibration 269 White Light Calibration 273 Balanced Detection 273 Detection of Intensity Modulation and SNR 275 Homodyne Detection 275 The Homodyne Detector for Classical Waves 275 Heterodyne Detection 279 Measuring Other Properties 280 Intensity Noise 281

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8.2.1 8.3 8.3.1 8.3.2 8.3.2.1 8.4 8.4.1 8.4.2 8.4.3 8.4.4 8.4.5 8.5

Laser Noise 281 The Intensity Noise Eater 282 Classical Intensity Control 282 Quantum Noise Control 285 Practical Consequences 289 Frequency Stabilization and Locking of Cavities 290 Pound–Drever–Hall Locking 292 Tilt Locking 293 The PID Controller 294 How to Mount a Mirror 295 The Extremes of Mirror Suspension: GW Detectors 296 Injection Locking 296 References 299

9

Squeezed Light 303

9.1 9.1.1 9.1.1.1 9.1.1.2 9.1.2 9.1.2.1 9.2 9.2.1 9.2.2 9.2.3 9.3 9.3.1 9.3.2 9.3.3 9.3.4 9.3.5 9.4 9.4.1 9.4.2 9.4.3 9.4.4 9.4.4.1 9.4.4.2 9.4.4.3 9.4.4.4 9.4.4.5 9.5 9.5.1 9.5.2 9.5.3 9.5.4 9.5.5

The Concept of Squeezing 303 Tools for Squeezing: Two Simple Examples 303 The Kerr Eﬀect 304 Four-Wave Mixing 307 Properties of Squeezed States 310 What Are the Uses of These Various Types of Squeezed Light? 312 Quantum Model of Squeezed States 314 The Formal Deﬁnition of a Squeezed State 314 The Generation of Squeezed States 317 Squeezing as Correlations Between Noise Sidebands 319 Detecting Squeezed Light 322 Detecting Amplitude Squeezed Light 322 Detecting Quadrature Squeezed Light 322 Using a Cavity to Measure Quadrature Squeezing 324 Summary of Diﬀerent Representations of Squeezed States 325 Propagation of Squeezed Light 325 Early Demonstrations of Squeezed Light 330 Four Wave Mixing 330 Optical Parametric Processes 333 Second Harmonic Generation 339 The Kerr Eﬀect 343 The Response of the Kerr Medium 343 Optimizing the Kerr Eﬀect 345 Fibre Kerr Squeezing 346 Atomic Kerr Squeezing 348 Atomic Polarization Self-Rotation 349 Pulsed Squeezing 349 Quantum Noise of Optical Pulses 349 Pulsed Squeezing Experiments with Kerr Media 352 Pulsed SHG and OPO Experiments 353 Soliton Squeezing 354 Spectral Filtering 355

Contents

9.5.6 9.6 9.7 9.8 9.9 9.9.1 9.9.2 9.9.3 9.10

Non-linear Interferometers 356 Amplitude Squeezed Light from Diode Lasers 358 Quantum State Tomography 360 State of the Art of CW Squeezing 363 Squeezing of Multiple Modes 365 Twin-Photon Beams 365 Polarization Squeezing 367 Degenerate Multimode Squeezers 368 Summary: Quantum Limits and Enhancement 370 References 371 Further Reading 376

10

Applications of Quantum Light 377

10.1 10.1.1 10.1.2 10.1.3 10.1.4 10.2 10.3 10.3.1 10.3.1.1 10.3.2 10.3.2.1 10.3.2.2 10.3.2.3 10.3.3 10.3.3.1 10.3.4 10.3.4.1 10.4 10.4.1 10.4.2 10.5 10.5.1 10.6

Quantum Enhanced Sensors 377 Coherent Sensors and Sensitivity Scaling 377 Practical Examples of Sensors 380 Ultimate Sensing Limits 382 Adaptive Phase Estimation 384 Optical Communication 384 Gravitational Wave Detection 389 The Origin and Properties of GW 389 Concept and Design of an Optical GW Detector 392 Quantum Properties of the Ideal Interferometer 393 Conﬁgurations of Interferometers 396 Recycling 397 Modulation Techniques 398 The Sensitivity of GW Observatories 400 Enhancement Below the SQL 402 Interferometry with Squeezed Light 405 Quantum Enhancement Beyond the SQL 410 Quantum Enhanced Imaging 411 Imaging with Photons on Demand 411 Quantum Enhanced Coincidence Imaging 412 Multimode Squeezing Enhancing Sensors 414 Spatial Multimode Squeezing 414 Summary and Outlook 419 References 419

11

QND 425

11.1 11.2 11.3 11.4 11.4.1

QND Measurements of Quadrature Amplitudes 425 Classiﬁcation of QND Measurements 427 Experimental Results 430 Single-Photon QND 432 Measurement-Based QND 434 References 437

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12

Fundamental Tests of Quantum Mechanics 441

12.1 12.2 12.3 12.3.1 12.3.2 12.3.2.1 12.3.2.2 12.3.3 12.3.3.1 12.3.3.2 12.4

Wave–Particle Duality 441 Indistinguishability 446 Non-locality 453 Einstein–Podolsky–Rosen Paradox 453 Characterization of Entangled Beams via Homodyne Detection 458 Logarithmic Negativity and Two-Mode Squeezing 459 Entanglement of Formation 460 Bell Inequalities 461 Long-Distance Bell Inequality Violations 466 Loophole-Free Bell Inequality Violations 466 Summary 468 References 468

13

Quantum Information 473

13.1 13.1.1 13.2 13.3 13.3.1 13.3.2 13.4 13.5 13.5.1 13.5.2 13.5.3 13.6 13.6.1 13.6.2 13.6.3 13.6.4 13.7 13.7.1 13.7.1.1 13.7.1.2 13.7.1.3 13.7.2 13.7.2.1 13.7.2.2 13.7.3 13.7.3.1 13.7.3.2 13.7.4 13.8

Photons as Qubits 473 Other Quantum Encodings 475 Post-selection and Coincidence Counting 475 True Single-Photon Sources 477 Heralded Single Photons 477 Single Photons on Demand 480 Characterizing Photonic Qubits 482 Quantum Key Distribution 484 QKD Using Single Photons 485 QKD Using Continuous Variables 489 No Cloning 492 Teleportation 492 Teleportation of Photon Qubits 493 Continuous Variable Teleportation 495 Entanglement Swapping 502 Entanglement Distillation 502 Quantum Computation 505 Dual-Rail Quantum Computing 506 Quantum Circuits with Linear Optics 507 Cluster States 511 Quantum Gates with Non-linear Optics 513 Single-Rail Quantum Computation 514 Quantum Random Walks 515 Boson Sampling 516 Continuous Variable Quantum Computation 518 Cat State Quantum Computing 519 Continuous Variable Cluster States 521 Large-Scale Quantum Computation 522 Summary 525 References 526 Further Reading 531

Contents

14

The Future: From Q-demonstrations to Q-technologies 533

14.1 14.2 14.3 14.3.1 14.3.2 14.3.3 14.3.4

Demonstrating Quantum Eﬀects 533 Matter Waves and Atoms 535 Q-Technology Based on Optics 537 Applications of Squeezed Light 537 Quantum Communication and Logic with Photons 539 Cavity QED 542 Extending to Other Wavelengths: Microwaves and Cryogenic Circuits 542 Quantum Optomechanics 542 Transfer of Quantum Information Between Diﬀerent Physical Systems 543 Transferring and Storing Quantum States 544 Outlook 544 References 545 Further Reading 547

14.3.5 14.3.6 14.3.7 14.4

Appendices 549

Appendix A: List of Quantum Operators, States, and Functions 549 Appendix B: Calculation of the Quantum Properties of a Feedback Loop 551 Appendix C: Detection of Signal and Noise with an ESA 552 Reference 554 Appendix D: An Example of Analogue Processing of Photocurrents 554 Appendix E: Symbols and Abbreviations 556 Index 559

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Preface The Idea Behind This Guide Some of the most interesting and sometimes puzzling phenomena in optics are those where the quantum mechanical nature of light is apparent. Recent years have seen a rapid expansion of quantum optics. Beautiful demonstrations and applications of the quantum nature of light are now common, and optics has been shown to be one of the key areas of physics to build quantum technologies based on the concepts behind quantum mechanics. The ﬁeld of photonics has already spawned major industries and changed the way we communicate, gather data, and handle information. Future applications of quantum information will enhance this trend and will introduce even more quantum aspects into our everyday activities. Why should we still write and read books at the age of search engines, blogs, and Wikipedia? This question did not arise when the ﬁrst edition of this book was published in 1998, which is a short step back in history but a long time ago in terms of the changes in information technology and the way we live, learn, and do research. You are likely to read this on a screen and can follow up links within seconds, tapping into an amazing wealth of information. This calls even more on this Guide to Experiment in Quantum Optics to explain the concepts, introduce reliable techniques, and help to select from the enormous amount of information that is available. We intend to guide you through the many experiments that have been published and to present and interpret them in one common style. This book provides a practical basis for exploring the ever-expanding technology in optics and electronics and provides a practical link to the rapidly expanding ﬁeld of quantum information, which is now being used in many diﬀerent platforms. Finally, this book is an attempt to communicate our personal insights and experiences in this ﬁeld of science. Science is driven by creative people, and some insight in the way researchers advance the ﬁeld should ﬁnd interest with you the reader. Many excellent textbooks are available on this topic. In the same way as this ﬁeld of research has been initiated by theoretical ideas, most of these books are written from a theoretical point of view. They provide fascinating, solid, and reliable descriptions of the ﬁeld. However, we found the literature frequently leaves out some of the important simpler pictures and intuitive interpretations of the experiments. In this guide we focus on both the ﬁrst and the most recent

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Preface

experiments. The guide explains the underlying physics in the most intuitive way we can ﬁnd. It addresses questions such as the following: What are the limitations of the equipment? What can be measured and what remains a goal for the future? The answers will prepare the readers to form their own mind map of quantum optics. It gives them the tools to make independent predictions of the outcome of their own experiments. For those who want to use quantum optics in practical ways, it tries to give an understanding of the design concepts of the machines that exploit the quantum nature of light. We assume that most of our readers will already have a fairly good understanding of optical phenomena and the way light propagates and how it interacts with components such as lenses, mirrors, detectors, etc. They can imagine the processes in action in an optical instrument. It is useful in the everyday work of a scientist to have practical pictures of what is going on inside the experiment or the machine. Such pictures have been shaped over many years of systematic study and through one’s own experimentation. Our guide provides such pictures, which can be ﬁlled in with many more details from theory and technology. In quantum optics three diﬀerent pictures, or interpretations, are used simultaneously: light as waves, light as photons, and light as the solution to operator equations. To these we add the picture of ﬂuctuations propagating through an optical system, a description using noise transfer functions. All of them are useful; all of them are correct within their limits. They are based on speciﬁc interpretations of the one formalism, and they are mathematically equivalent. In this guide all these four descriptions are introduced rigorously and are used to discuss a series of experiments. We start with simple demonstrations, gradually getting more complex and include most of the landmark experiments in single-photon and quantum noise detection, photon pair generation, squeezed states, quantum non-demolition, and entanglement measurements published to date. All four pictures – waves, photons, operators, and noise spectra – are used in parallel. In each case the most appropriate and intuitive interpretation is highlighted, and the limitations of all four interpretations are stated. In this way we hope to maintain and present a lot of the fascination of quantum optics that has motivated us and many of our colleagues, young and old, over so many years.

The Expansion of Quantum Optics and the Third Edition of This Guide The development of the ﬁeld has been rapid in recent years, and many new technical advances have been reported: vastly improved sources for photon pairs, squeezed and entangled states. We have better detectors and nonlinear media. With the ability to record and process much more information the era of analogue technology is gone. Post-processing is now the norm, quantum gates and quantum memories have been reported. We can generate and process multiple optical modes in frequency, space and time, demonstrate multimode entanglement and design and build reliable complex optical systems.

Preface

The ﬁeld of quantum information is growing rapidly, and new concepts for quantum communication, quantum logic, and quantum internet are emerging. Optics will play a major role in this. The ﬁeld has broadened to include both more fundamental concepts about the quantum nature of information and new ideas for the engineering of complex optical designs. Measurements using non-classical light are now well established and have become a routine operation in some select ﬁelds such as gravitational-wave detectors and sensors. Entanglement has grown from an abstract concept to a technical tool. At the same time commercial companies using quantum techniques for secure communication have already carved out a large market for themselves. The ideas in this guide, and the examples chosen, draw attention to both the techniques of single-photon detection and single-event logic and the techniques of continuous variable logic and the use of continuous coherent and squeezed beams. There are now far more practitioners of the former, and the latter has become a specialized niche, mainly used for metrology. However, both will play a role in the development of future applications and in the understanding the concepts of the quantum world. In addition, quantum optics based on pairs of photons has been combined with the technology of strong laser beams and continuous variables in a series of exciting hybrid experiments. This third edition focusses on the latest landmark experiments, and examples that are now outdated have been deleted. The coverage of quantum information concepts and protocols has been expanded to keep pace with the evolution of the ﬁeld. Reference is made to the state-of-the-art technologies, hinting at potential future developments. This guide follows the well-tested idea of providing independent building blocks in theory and experimental techniques that are used in later chapters to discuss complete experiments and applications. We found that most of them were still valid and thus remain unchanged. Optics is no longer dominant as the medium of choice for quantum technologies. Technical advances have allowed the same concepts now to be used in areas such as cryogenic electric circuits, superconducting systems, trapped ions and neutral atoms, Bose–Einstein condensates (BECs) and atom laser beams, and optomechanical systems, and other examples yet to be tested. This has posed a challenge: how to deﬁne the limits of this guide? To keep the book compact, we decided to focus entirely on optical systems and to point to links with other systems where required, but not to venture into the technical details of these ﬁelds. It is clear that future quantum technology will be using many complementary platforms. Optics will remain a major contributor and quite likely the most direct approach to the fascinating and sometimes counterintuitive eﬀects of the quantum world. Australia, 2018

Hans Bachor and Timothy Ralph

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Acknowledgments This book would not have been possible without the teaching we received from many of the creative and motivating pioneers of the ﬁeld of quantum optics. We have named them in their context throughout the book. HAB is particularly in debt to Marc Levenson, who challenged him ﬁrst with ideas about quantum noise. From him he learned many of the tricks of the trade and the art of checking one's sanity when the experiment produces results that seem to be impossible. TCR wishes particularly to thank Craig Savage for instilling the importance of doing theory that is relevant to experimentalists. Both of us owe a great deal to Dan Walls and Gerard Milburn who provided such beautiful vision of what might be possible. Over the years we learned from and received help for this book from a large international team of colleagues and students. Too many to name individually. All of you made important contributions to the our understanding and the way we communicate our knowledge and sight. We enjoy the feedback and are keen to discuss the mysteries and the past and future of quantum optics. We like to thank Aska Dolinska for the creative graphics work. Most importantly this Third Edition would not have been possible without the support of our loved ones. TCR wishes to thank his wife Barbara for her support and son Isaac for putting up with a distracted father. And HAB wishes to thank his wife Cornelia for the patience she showed with an absent-minded and many times frustrated writer. Hans Bachor Timothy Ralph 2018

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1 Introduction 1.1 Optics in Modern Life The invention of the laser and the use of the special properties of laser light created one of the great technical revolutions of the twentieth century and has impact on many aspects of our modern lives. It allows us to manufacture the gadgets we love to use, including the computer with which this has been written. It allows us to communicate literally at the speed of light and with an information bandwidth that is unprecedented. Optics has been for a while the technology of choice for data storage, with DVD and Blu-ray still being used as a medium for data and 3D optical storage that might emerge in future. The modern alternatives, downloading and streaming, depend in many ways on photonics. In addition, there is a plethora of other applications of light, from laser welding in manufacturing to laser fusion for power generation and from laser-based microscopes to laser surgery. Imaging and display technologies are in many gadgets. The developments of optical technology have been relentless and will continue to be so for many more years. This has allowed us to access new levels of power, controllability, accuracy, complexity, and low cost that were deemed impossible only a few years ago. This is a convincing reason to study and teach quantum optics, as a scientist or engineer. Whilst all of the techniques in photonics use the quantum concept of stimulated emission and the laser [1], most of these applications still use quantum ideas in a quite simple and peripheral way. Most of the designs are based on conventional optics and can be understood on the basis of wave optics and traditional physical optics [2]. An interpretation of light as a classical wave is perfectly adequate for the understanding of eﬀects such as diﬀraction, interference, or image formation. Non-linear optics, such as frequency doubling or wave mixing, is well described by classical theory. Even most of the properties of coherent laser light can be understood and modelled in this way. There is a wealth or new ideas and applications emerging, exploiting the regimes of wave optics that were previously not accessible, with new theories and devices. This includes the many fascinating new devices made possible through new engineered materials, waveguide structures, photonic bandgap materials, linear and non-linear metamaterials. They all hold the promise of even higher performance, better speciﬁcations, and even new applications, such as cloaking or 3D storage, optical data processing at unheard A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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1 Introduction

speeds, or optical data processing. Conferences and research reviews all document that in 2018 a bright future is ahead in these areas of optics. However, which role will optics play in a future that will use quantum technologies? Firstly, the question of what ultimately limits the performance of optical devices lies outside the classical theory of light. We have to focus on a diﬀerent aspect of optics. We follow the ideas that come directly from the quantum nature of light, the full properties of photons, and the subtle eﬀects of quantum optics [3]. Initially, optics was recognized as a pretty ideal medium to study the ﬁner eﬀects of the quantum world. Photons in the visible spectrum were easily generated, detected with high eﬃciency, and not masked by thermal noise at room temperature, and any correlations could be recorded very well. Non-linear eﬀects allowed the generation of a variety of diﬀerent forms of light. This established optics as the medium of choice to explore the quantum world, and the early improvements in the technology in the 1980s led to great demonstrations, such as the Bell inequalities, signalling the beginning of experimental quantum optics. This exploration of the ‘weirdness’ of the quantum world is still today one of the frontiers in quantum optics and discussed in this guide. It was recognized that quantum eﬀects pose a limit to the sensitivity of many measurements, with the now successful detectors for gravitational waves the pre-eminent example. A closer analysis showed that there are ways of, at least partially, circumventing such limitations. Optical quantum metrology was born and is a major topic in this guide. It will continue to evolve – and we now see the ﬁrst routine applications of quantum-enhanced metrology, with few photons and with laser beams. The ideas are spreading and can be realized in other systems such as coherent groups of atoms or ions or in circuits. Quantum-enhanced metrology in its various forms is becoming more widespread, from the deﬁnition of the SI units to very practical situations. Optics technology and our knowledge of protocols and devices have steadily grown in time. Almost gone are the analogue devices of the 1990s, and we have direct access to digital data. For example, we can now detect, record, process, and store information in our experiments in much better ways, with A/D converters, multichannel analysers, memories of Terabit size, etc. This means that we can access much more information and process and analyse data more thoroughly. The next step might well include the use of artiﬁcial intelligence to optimize data analysis and also the performance of devices. The detection of quantum correlations are a case in hand: we can now directly explore the correlation properties, use diﬀerent approaches and measures to evaluate the quality of entanglement, and post-select data, or use heralded information, to select the best quantum data. This provides a path to optical quantum gates and more elaborate quantum logic. We can post-process the information and make it conditional on other measurements. This allows us to not only describe the state with tomography in great detail but also provide a path for looking inside the statistical properties of a beam of light. An example is the Wigner function shown on the front cover, which shows strong negative values that are contained in one beam from an entangled pair of beam – after

1.2 The Origin and Progress of Quantum Optics

conditional detection and letting information from one beam inﬂuence the reading from the other entangled beam [4]. Furthermore, so-called hybrid detection uses entangled beams, where the information from a single photon detector in one beam is used to post-select the most relevant data from stream of photon recorded from the other beam. Using this form of hybrid detection, it is possible to ﬁlter out states of light with speciﬁc quantum properties, for example, Schrödinger cat states [5]. They represent a state that is a quantum superposition of two orthogonal states and is one of the icons and hallmarks of quantum physics, as recognized in the 2012 Nobel Prize of Physics [6]. On the other hand, the last few years has seen an ever-increasing interest in the information that is contained within a beam of light, whether it be images, analogue pulses, or digital data encoded in the light. The emergence of the concept of quantum information has created a rising interest in the ability to communicate, store, and process more complex information than available in classical digital electronics, all by using beams of light or individual photons. This has created a great deal of interest, an expanding research ﬁeld, and a demand for practical quantum optics-based devices. Optics overlaps more and more with condensed matter and soft matter physics, materials science, and mechanical engineering. Ever-improving technology drives these ﬁelds, as well as now insights into quantum logic and quantum control. Optics is in many cases a precursor, sometimes a testing ground, and increasingly a component of such quantum devices.

1.2 The Origin and Progress of Quantum Optics Ever since the quantum interpretation of the black body radiation by M. Planck [7], the discovery of the photoelectric eﬀect by W. Hallwachs [8] and P. Lennard [9] and its interpretation by A. Einstein [10] has the idea of photons been used to describe the origin of light. For the description of the generation of light, a quantum model is essential. The emission of light by atoms, and equally the absorption of light, requires the assumption that light of a certain wavelength 𝜆, or frequency 𝜈, is made up of discrete units of energy each with the same energy hc∕𝜆 = h𝜈. This concept is closely linked to the quantum theory of the atoms themselves. Atoms are best described as quantum systems; their properties can be derived from the wave functions of the atoms. The energy eigenstates of atoms lead directly to the spectra of light which they emit or absorb. The quantum theory of atoms and their interaction with light has been developed extensively; it led to many practical applications [11] and has played crucial role in the formulation of quantum mechanics itself. Spectroscopy relies almost completely on the quantum nature of atoms. However, in this guide we will not be concerned with the quantum theory of atoms, but concentrate on the properties of the light, its propagation, and its applications in optical measurements. Historically, special experiments with extremely low intensities and those that are based on detecting individual photons raised new questions, for instance, the famous interference experiments by G.I. Taylor [12] where the energy ﬂux corresponds to the transit of individual photons. He repeated Th. Young’s double-slit

3

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1 Introduction

experiment [13] with typically less than one photon in the experiment at any one time. In this case, the classical explanation of interference based on electromagnetic waves and the quantum explanation based on the interference of probability amplitudes can be made to coincide by the simple expedient of making the probability of counting a photon proportional to the intensity of the classical ﬁeld. The experiment cannot distinguish between the two explanations. Similarly, the modern versions of these experiments, using the technologies of low noise current detection or, alternatively, photon counting, show the identity of the two interpretations. The search for uniquely quantum optical eﬀects continued through experiments concerned with intensity, rather than amplitude interferometry. The focus shifted to the measurement of ﬂuctuations and the statistical analysis of light. Correlations between the arrival of photons were considered. It started with the famous experiment by R. Hanbury-Brown and R. Twiss [14] who studied the correlation between the ﬂuctuations of the two photocurrents from two diﬀerent detectors illuminated by the same light source. They observed with a thermal light source an enhancement in the two-time intensity correlation function for short time delays. This was a consequence of the large intensity ﬂuctuations of the thermal light and was called photon bunching. This phenomenon can be adequately explained using a classical theory that includes a ﬂuctuating electromagnetic ﬁeld. Once laser light was available, it was found that a strong laser beam well above threshold shows no photon bunching; instead the light has a Poissonian counting statistics. One consequence is that laser beams cannot be perfectly quiet; they show noise in the intensity that is called shot noise, a name that reminds us of the arrival of many particles of light. This result can be derived from both classical and quantum models. Next, it was shown by R.J. Glauber [15] that additional, unique predictions could be made from his quantum formulation of optical coherence. One such prediction is photon anti-bunching, where the initial slope of the two-time correlation function is positive. This corresponds to greater than average separations between the arrival times of photons; the photon counting statistics may be sub-Poissonian, and the ﬂuctuations of the resulting photocurrent would be smaller than the shot noise. It was shown that a classical theory based on ﬂuctuating ﬁeld amplitudes would require negative probabilities in order to predict anti-bunching, and this is clearly not a classical concept but a key feature of a quantum model. It was not until 1976, when H.J. Carmichael and D.F. Walls [16] predicted that light generated by resonance ﬂuorescence from a two-level atom would exhibit anti-bunching, that a physically accessible system was identiﬁed that exhibits non-classical behaviour. This phenomenon was observed in experiments by H.J. Kimble et al. [17], where individual atoms were observed. This opened the era of quantum optics. It is now possible to track and directly display the evolution of the photon number inside a cavity, thanks to the innovation introduced by S. Haroche and coworkers. They have designed better and better superconducting cavities to store microwave photons and can carry out non-demolition experiments in a cavity with very long decay time. They can now directly observe the evolution of the photon number in an individual mode [18]. See

1.2 The Origin and Progress of Quantum Optics

for details Figure 3.7. This is a most beautiful demonstration of the theoretical concepts. The concept of modes will be a central feature in this guide. This concept has evolved in time; it now has several distinct meanings and is playing a pivotal role in quantum optics. For an experimentalist, a single mode can mean a perfect beam shape without any distortion. A single-mode laser means a device that is emitting a beam with exactly one frequency, in one polarization, and with a perfect beam shape. In the theory, a mode is the fundamental entity that can be quantized and be described by simple operators. We have a reliable and direct model for the behaviour of such a single mode, and up to recently most quantum optics were formulated in this single-mode model. At the same time, in optical technology, such as communication or imaging, the concept of a mode is associated with an individual channel of information. This could be a speciﬁc modulation frequency, as we are familiar with from radio transmission, or it could be a speciﬁc spatial image. In the world of optical sensing, we wish to detect and transmit changes of various parameters, such as lengths or position, magnetic or electric ﬁelds, pressure or temperature, to name a few. The evolution of these parameters are then encoded into the light and transmitted to the receiver. In this case, each parameter, and each degree of freedom for the change of the parameter, corresponds to one speciﬁc mode. It has now been recognized that a sensor can be optimized and will show the best possible performance, when the optical mode is fully matched to the information. This can be a simple mode, as mentioned in the case of the traditional single-mode laser, or it can actually be a much more complicated spatial and temporal distribution. The ﬁrst extension in quantum optics has been to multiple beams of light, each containing one mode each. We can have many separate beams, all originating from one complex optical apparatus. However, a single optical beam can also contain multiple modes, for example, spatial modes, temporal modes, or frequency modes. They correspond to separate degrees of freedom or separate information channels within the beam. Such a multimode beam might contain only a few, or the occasional, photon, or it might have the full ﬂux of a laser beam. In either case, the underlying physics is the same. The information transmitted is best described in terms of a set of independent orthogonal modes. Each can be quantized individually and has their own independent statistics. The signals, and the ﬂuctuations, transmitted by these modes do not interact with each other, and the full quantum description of these devices is straightforward using this concept. There has been great progress in the technology to generate individual photons in a controlled way. The technology has evolved to a stage where many of the initial thought experiments dealing with individual atoms can now be performed with real machines. In addition, a whole range of new types of single photon emitters has been developed and studied by many groups around the world. They include quantum dots that show the required quantum properties and are continuously improving in their robustness and reliability. They are now being used regularly in a variety of applications where small size matters. This includes applications for quantum communication with quantum dots placed in speciﬁc places or in the form of nanodiamonds imbedded as emitters in biological samples that respond to the local conditions in living cells.

5

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1 Introduction

Initially, these single photon eﬀects were a pure curiosity. They showed the diﬀerence between the quantum and the classical world and were used to test quantum mechanics. In particular, the critique by A. Einstein et al. and their EPR paradox [19] required further investigation. Through the work of J.S. Bell [20], it became possible to carry out quantitative tests. They are all based on the quantum concept of entanglement. In the simplest case, this means pairs of quantum systems that have a common origin and have linked properties, such that the detection of one allows the perfect prediction of the other. Such systems include not only individual photons or atoms, or coherent states of light, such as laser beams but also physical matter, such as was demonstrated with Bose–Einstein condensates. See Chapter 14. Starting with the experiments by A. Aspect et al. [21] with atomic sources, and later with the development of sources of twin photon pairs and experiments by A. Zeilinger [22] and many others, the quantum predictions have been extremely well tested. The concept of entanglement has now been demonstrated and tested with single photons and also with intense beams of light, over large distances of many kilometres and through optical ﬁbres. There has been a long series of experiments with the aim to demonstrate the validity and necessity of quantum physics beyond any doubt, closing step by step every loophole. It has even been possible to extend the concept of entanglement from two entities, such as two particles, to many particles, most prominently to many trapped ions. Such multipartite entanglement is at the core of plans to build quantum simulators and quantum logic devices. Similarly there has been an extension from two modes of light to several modes. This leads to the idea of multimode entanglement, which is a new topic of research, with both interesting physics and the potential for new applications. In summary, the progress in quantum optics, which initially was concerned with the quality of the quantum demonstrations, has recently led to the conquering of complexity and the study of the robustness, or decoherence, of such complex systems. The peculiarities of single photon quantum processes have led to many new technical applications, mainly in the area of communication. For example, it was found that information sent by single photons can be made secure against listening in; the unique quantum properties do not allow the copying of the information. Any eavesdropping will introduce noise and can therefore be detected. Quantum cryptography and quantum key distribution (QKD) [23] are now viable technologies, based on the inability to clone single photons or quantum modes. There is great anticipation of other future technological applications of photons. The superposition of diﬀerent states of photons can be used to send more complex information, allowing the transfer not only of classical bits of information, such as 0 and 1, but also of qubits based on the superposition of states 0 and 1. The coherent evolution of several such quantum systems could lead to the development of quantum logic connections, quantum gates, and eventually whole quantum computers that promise to solve certain classes of mathematical problems much more eﬃciently and rapidly [24]. Just as astounding is the concept of teleportation [25], the disembodied communication of the complete quantum information from one place to another. This has already been tested in experiments (see Chapter 13).

1.3 Motivation Through Simple and Direct Teaching Experiments

Optics is now a testing ground for future quantum information technologies and for data communication and processing. At the same time, the concepts and ideas of quantum optics are emerging in other related quantum technologies, for example, based on quantum circuits or mechanical systems, which are all quantized and show in principle the same physics. We can see the equivalence of the physics of photons, phonons, polaritons, and other particles that describe the excitation of quantum systems. Technology is moving very fast, and we can expect over the next decade to see machines that utilize any of these individual quantum systems or even combine several of them. The rules of quantum optics, as described in this guide, will become important in many diﬀerent other types of instruments. Optics could become one of the best ways to visualize the broader quantum concepts that apply to many other types of instruments.

1.3 Motivation Through Simple and Direct Teaching Experiments The best starting point for the demonstration of photons in current technology is to start with images that are generated by CCD (charge-coupled device) cameras. It is worthwhile to note that basically all information we have about light and optical eﬀects is based on either some form of photodetector or the human eye. The ﬁrst generates an electric signal, either pulses or a continuous photon current. Whilst the eye produces, in a clever and complex way, a signal in the neurones in the brain, even they can be interpreted as a form of electric signal. It is the characteristics of these signals and their statistics that provides access to the quantum properties of light. In the extreme case we would have a detector that is capable of detecting one individual photon at a time. This is now possible with special CCD cameras, where the sensitivity has been lifted to such a high level and thermal noise has been reduced so much that it hardly interferes with the observation. The number of dark counts, that is, clicks that are generated at random without photons present, can be neglected. In this way we can actually detect and record images at the single photon level as shown in Figure 1.1. The human eye is actually capable of seeing at this level, and many animals exceed our capability. At this level of illumination, the world looks very grainy. We can see individual events. The image is composed out of individual local detection events. An example, taken with a special CCD, is shown in Figure 1.1. At some level this can be regarded as trivial – obviously it will be like this. We have photons. On the other this image demonstrates several subtle points. We can see that the concept of intensity, bright and dark, is directly related to the concept of probability. Areas with a high local photon ﬂux, or high probability, correspond to high intensities, that is, a high amplitude of the electromagnetic ﬁeld. Conversely, dark areas of the image correspond to a low intensity. The central concept in quantum science of a probability amplitude become directly obvious. Two, or many, waves with diﬀerent amplitudes and relative phases can interfere, and this is a central concept of the classical wave picture of light. The example

7

1 Introduction

(a) 300 Number of photons

8

250 200 150 100 50 0 0

(b)

50

100

150

200

Pixel

Figure 1.1 Image of the fringes from a double-slit experiment recorded at the single photon level. (a) Distribution of individual photons. (b) Histogram of photons in each vertical line. The full movie can be seen at https://www.lcf.institutoptique.fr/Groupes-de-recherche/Gazquantiques/Membres/Permanents/Alain-Aspect. Source: After https://www.lcf.institutoptique .fr/Groupes-de-recherche/Gaz-quantiques/Membres/Permanents/Alain-Aspect.

shown in Figure 1.1 is a real interference fringe system, generated with coherent light and a double-slit apparatus. This is one of the simplest arrangements that shows the wave nature of light – and yet in this case it is detected by a number of individual photons. There is no contradiction here – the two explanations, waves and particles, are fully complementary. The shape of the fringes, the overall intensity function, is best described with the wave model. At the same time, the statistics of the light, the visibility that is observed, and thus the quality of the fringes can best be described with the photon model. This very simple example immediately shows the strengths and weaknesses of both models. It demonstrates how we want to use these two models for diﬀerent aspects of the experiment. We should choose the pictures in our mind that is most suitable to understand the speciﬁc aspect of the experiment we wish to investigate. The next level would be to quantify the statistics. What would be an appropriate statistical model for the ﬂuctuations, detected in one speciﬁc area of the image and averaged over many time intervals? Equivalently we could ask: what is the statistics of the number of photons detected in many separate areas of equal size, all receiving the same intensity? In the ﬁrst case one could imagine listening to the number of clicks being generated. It would sound like a random sequence of clicks, as we know them from a Geiger counter detecting 𝛼 or 𝛾 rays.

1.3 Motivation Through Simple and Direct Teaching Experiments

A proper analysis would reveal that we have a Poissonian distribution for the diﬀerent time intervals. The same result applies to the analysis of the number of photons in many image areas of equal size. The graininess we see in Figure 1.1 is a direct representation of the counting statistics, which can be derived from simple statistical arguments, assuming random processes. The randomness in the arrival times of the photons in a weak beam of light is so perfect that it is now used as the gold standard for randomness, exceeding in quality the widely used random number generators based on electronic processes or imbedded in computers. This has led to commercial activities where optically generated random numbers now can be accessed via the internet and the generators are for proﬁt. The demand is substantial – and already a small niche market for quantum random numbers has emerged. The 1960s saw a rapid development of new laser light sources and improvements in light detection techniques. This allowed the distinction between incoherent (thermal) and coherent (laser) light on the basis of photon statistics. The groups of F.T. Arecchi [26], L. Mandel, and R.E. Pike all demonstrated in their experiments that the photon counting statistic goes from super-Poissonian at the lasing threshold to Poissonian far above threshold. The corresponding theoretical work by R.J. Glauber [15] was based on the concept that both the atomic variables and the light are quantized and showed that light can be described by a coherent state, the closest quantum counterpart to a classical ﬁeld. The results are essentially equivalent to a quantum treatment of a classical oscillator. However, it is an important consequence of the quantum model that any measurement of the properties of this state, intensity, amplitude, or phase of the light, will be limited by these ﬂuctuations. This is the optical manifestation of Heisenberg’s uncertainty principle. Quantum noise can impose a limit to the performance of lasers, sensors, and communication systems, and near the quantum limit the performance can be quite diﬀerent. To illustrate this, consider the result of a very simple and practical experiment: use a laser, such as a laser pointer with a few milliwatt of power, and detect all of the light with a photodiode. The average number of detection events is now very large. Listening to the output, we can no longer distinguish between the individual clicks; we will hear a continuous noise. The level of noise will increase with the intensity, as the width of the Poissonian distribution increases with the average number of photons detected. This quantum noise is the direct experimental evidence of the quantum statistics of light. If we record the current from the photodetector on an oscilloscope, we would see a trace with random ﬂuctuations, above and below a mean level, representing the average intensity. The excursions have a Gaussian distribution around the mean value, which is fully consistent with the transition from a Poissonian to a Gaussian distribution. Relatively speaking this eﬀect gets smaller if we average the signal over more photons, that means using a brighter laser or measuring for a longer time. As we will see, this eﬀect cannot be avoided completely. Quantum noise is not an eﬀect that can be eliminated by good traditional engineering. Decades ago quantum noise was called shot noise and was regarded by many as a consequence of the photodetection process, as a randomness of the stream of electrons produced in the photodetector. This view prevailed for a long time,

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Modulator A Laser B

X2

X1

Figure 1.2 Experiment to investigate the correlation between two parts of a laser beam, using a laser pointer, modulator, two detectors, and an x–y oscilloscope.

particularly in engineering textbooks. However, this view is misleading as we ﬁnd in the squeezing experiments described in this book. It cannot account for situations where non-linear optical processes modify the quantum noise, whilst the detector remains unchanged. In the squeezing experiments the quantum noise is changed optically, and consequently we have to assume it is a property of the light, not of the detector. Today it is straightforward to see these eﬀects of quantum noise with simple equipment, accessible to undergraduate students. The light source can be a laser pointer, carefully selected to operate in a single mode. The laser beam is split by a beamsplitter into two beams of equal power that are detected by two individual detectors, A and B, at diﬀerent locations. Each photocurrent is ampliﬁed, and they are both displayed via the x and y axes of one oscilloscope. We want to see both the average (DC) value of the photocurrent, which represents the time averaged intensity of the beams A and B, and the ﬂuctuations (AC) of the two photocurrents, as they represent any modulation and the ﬂuctuations, or noise, of the intensity of the two beams. This can easily be done by looking at the signals at a few megahertz, which is typically the bandwidth of the RF ampliﬁers used. This is shown in Figure 1.2. The laser can be modulated, through periodic variations of the laser current. Classical optics predicts that the intensity of both beams A and B will be modulated, and this will show up clearly as a line at 45∘ on the oscilloscope. That means the two photocurrents change synchronously in time, an increase in one is accompanied by an increase in the other etc. In practice some care is required to achieve a clear 45∘ line: there should be no delay in the signals from the detector to the oscilloscope and the size of the signals has been matched. This result shows a strong correlation between the two beams. A correlation of the photocurrents means a correlation of the ﬂux of photons. We can also look at the ﬂuctuations in more detail. If we look at beams A and B individually, we see ﬂuctuations of the current above and below the long-term average value. Closer inspection would show a Gaussian distribution of the excursions below and above, as we expect it for random noise in the beam. The magnitude of the ﬂuctuations will follow the gain curve of the RF ampliﬁers. With careful calibration of the frequency response of our equipment, we ﬁnd that the noise in the light is not frequency dependent; it is white noise. The magnitude of the ﬂuctuations is the same for a laser beam with or without a small modulation.

1.3 Motivation Through Simple and Direct Teaching Experiments

A

A

C

C

X2

X2

X1

X1

(a)

(b) A C

X2

X1

(c)

Figure 1.3 The noise of beams A and B displayed on an oscilloscope. (a) No beam, technical background noise. (b) The laser beam has no modulation, quantum noise. (c) The laser beam with modulation. The signal from the two detectors is displayed in the coordinates X1 and X2 , correlations appear in the direction C, and uncorrelated signal appears in both directions A and C.

The noise increases with intensity, and careful measurements show that the variance of the noise is proportional to the intensity. Is the noise from A correlated with the noise from B? The oscilloscope image immediately provides the answer. The noise from A and B results in a fuzzy blob on the screen. See Figure 1.3b. The ﬂuctuations in the two photocurrents are independent. The point on the screen of the oscilloscope traces out a random walk independently in both directions. This is in stark contrast to the modulation where the photocurrent ﬂuctuates synchronously, as shown in Figure 1.3c. The lack of correlation is immediately evident. This conﬁrms the special properties of quantum noise; it does not obey the classical rules. One simple explanation is that for small ﬂuctuations we would notice the eﬀect that one photon cannot be split into halves at the beamsplitter. It will appear in one or the other beam, with a random choice which beam it will join. We will see in Chapter 3 how this simple statement leads to an explanation of the properties of quantum noise in this situation. If we have no modulation, the blob is right in the middle of the screen. The blob is circular, given there is no additional noise in the system and the photon noise dominates over the electronic noise generated in the apparatus. For a laser beam with modulation, we will see a fuzzy area tilted at 45∘ , as shown in Figure 1.3c. This is a combination of the two eﬀects, a line

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1 Introduction

in direction C for the modulation and a fuzzy blob for the laser noise. Classical modulations and quantum noise have diﬀerent properties, even when they have similar magnitudes, and this simple experiment can show the diﬀerence.

1.4 Consequences of Photon Correlations It is worthwhile to explore the properties of quantum noise somewhat further. It was found that, in clear distinction to classical noise, no technical trick can eliminate quantum noise. This becomes evident from our experiment. A common conventional idea for noise suppression would be to make two copies of the signal, both containing the noise, and to subtract them for each other. We have all the necessary components in the apparatus used in Figure 1.2. We have made two copies of the beam. We can use one beam, say, A, for our experiment, record it with detector A, and then subtract the electronic ﬂuctuations generated by detector B. Based on our observations we now see that we can subtract the classical modulation, which is common to both beams. However, we cannot subtract the quantum noise, which is independent and uncorrelated. The subtraction works for any modulation, for harmonic modulation demonstrated here, and for random intensity modulations, such as those generated by a low-quality, noisy current source for the laser. That means we can distinguish between classical laser noise and quantum noise that is a consequence of the quantum properties of the photons. The result remains unchanged if the subtraction is replaced by a sum or if the two currents, or if the two currents are added with an arbitrary short time delay, such as introduced be an extra length of cable. The resulting noise is always the quadrature sum of the two-input signal. It is independent of the sign, or phase, of the summation. This is equivalent to the statement that the noise in the two photocurrents is not correlated. At this stage it is not easy to identify the point in the experiment where this uncorrelated noise is generated. One interpretation assumes that the noise in the photocurrents is generated in the photodetectors. A diﬀerent interpretation assumes that the beamsplitter is a random selector for photons and consequently the intensities of the two beams are random and thus uncorrelated. A distinction can only be made by further experiments with squeezed light, which are discussed later in this book. An alternative scheme for noise suppression is the use of feedback control, as shown in Figure 1.4. It achieves equivalent results to diﬀerence detection. The Modulator

Laser

Feedback control

Figure 1.4 The second attempt to eliminate laser noise: an improved apparatus using a feedback controller, or ‘noise eater’.

1.4 Consequences of Photon Correlations

intensity of the light can be controlled with a modulator, such as an acousto-optic modulator or an electro-optic modulator. Using a feedback ampliﬁer with appropriately chosen gain and phase lag, the intensity noise can be reduced, and all the technical noise can be eliminated [27]. It is possible to get very close to the quantum noise limit, but the quantum noise itself cannot be suppressed [28, 29]. This phenomenon can be understood by considering the properties of photons. As mentioned above, the quantum noise measured by the two detectors is not correlated; thus the feedback control, when operating only on quantum noise on one detector, will not be able to control the noise in the beam that reaches the other detector. This can be explained using a full quantum theory. The role of the photon generation process can be explored further. It was found that the noise of the light may be below the standard quantum limit if the pumping process exhibits sub-Poissonian statistics. This is particularly easy to achieve for LEDs, which are high eﬃciency light sources driven directly by electric currents, or with semiconductor lasers. The currents driving these devices are classical, at the level of the ﬂuctuations we are concerned with, and the ﬂuctuations can be controlled with ease to levels well below the shot noise level. For sources with high quantum eﬃciencies, the sub-Poissonian statistics of the drive current is transferred directly to the statistics of the light emitted. Such experiments were pioneered for the case of diode lasers by the group of Y. Yamamoto [30]. They showed that intensity ﬂuctuations can be suppressed in a high impedance semiconductor laser driven by a constant current. Similar work had earlier been carried out with LEDs by several groups. If we use this type of source in our experiment shown in Figure 1.3, the fuzzy blob would be smaller, in both directions, than the blob for a normal laser with the same output intensity. To explain the quantum noise and the correlations fully, we should not restrict ourselves to ﬂuctuations of the intensity of the light or the statistics of the photon arrival time. We will ﬁnd that there is noise both in the magnitude and phase of the electromagnetic wave. Consequently our quantum model requires a two-dimensional description of the light, with properties which we will call quadratures. Fluctuations, or noise, can be characterized by the variance in both the amplitude and phase quadrature. Almost a decade after the observation of photon anti-bunching in atomic ﬂuorescence, another quantum phenomenon of light was observed – the suppression, or squeezing, of quantum ﬂuctuations [31]. For a coherent state the uncertainties in the quadratures are equal and minimize the product in Heisenberg’s uncertainty principle. A consequence is that measurements of both the amplitude or the phase quadrature of the light show quantum noise. In a squeezed state, the ﬂuctuations in the quadratures are no longer identical. One quadrature may have reduced quantum ﬂuctuations at the expense of increased ﬂuctuations in the other quadrature. Such squeezed light could be used to beat the standard quantum limit. After the initial theoretical predictions, the race was on to ﬁnd such a process. A number of non-linear processes were tried simultaneously by several competing groups. The observation of a squeezed state of light was achieved in 1985 ﬁrst by the group of R.E. Slusher at Bell Labs in four-wave mixing in sodium atomic beam [32], soon followed by the group of M.D. Levenson and R. Shelby at IBM [33] the group of H.J. Kimble with an optical parametric oscillator [34]. In recent years a number of other non-linear processes have been used to

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1 Introduction

Non-linear medium Spectrum analyser

Laser

Squeezed light

Local oscillator + _ Homodyne detector

Figure 1.5 A typical squeezing experiment. The non-linear medium generates the squeezed light that is detected by a homodyne detection scheme.

demonstrate the quantum noise suppression based on squeezing [35]. A generic layout is shown in Figure 1.5. The experiments are now reliable, and practical applications are feasible. Quantum information is a key to many applications of quantum optics. Plans for using the complexity of quantum states to code information, to teleport it, to use it for secure communication and cryptography, and to store quantum information and possibly use it for complex logical processes and quantum computing have all been widely discussed and demonstrated to a greater or lesser degree. The concept of entanglement has emerged as one of the key qualities of quantum optics. As it will be shown in this guide, it is now possible to create entangled beams of light either from pairs of individual photons, which is now possible in technical applications and in teaching laboratories [37], or from the combination of two squeezed beams, and the demand and interest in non-classical states of light has sharply risen. We can see quantum optics playing a large role in future communication and computing technologies [23, 36]. For this reason, the guide covers both single photon and CW beam experiments parallel to each other. It provides a uniﬁed description and compares the achievements and tries to predict the future potential of these experiments.

1.5 How to Use This Guide This guide leads us through experiments in quantum optics, experiments that deal with light and demonstrate, or use, the quantum nature of light. It shows the practicalities and challenges of these experiments and gives an interpretation of their results. One of the current diﬃculties in understanding the ﬁeld of quantum optics is the diversity of the models used. On the one hand, the theory and most of the publications in quantum optics are based on a rigorous quantum model that is rather abstract. On the other hand, the teaching of physical optics and the experimental training in using devices such as modulators, detectors, and data acquisition systems are based on classical wave ideas. This training is extremely

1.5 How to Use This Guide

Experiment versus theory | a1>, | v > States

c1| a1> + c2| a2> + c3| v >

H Operators

(a)

|0> Operator and vacuum state

Complex state Prob. of det and correlation

Info SNR

(b)

Beams and modulation Experiment linear and non-linear intensity, phase components Signals and noise at

Loss

α2in, Φin, δX1(Ω), δX2(Ω), V1in (Ω), V2in (Ω) Quantum transfer function

Output beams

Detectors electronics

2

αout, Φout, V1out (Ω), V2out (Ω) Vv = 1 Vacuum beam

Coherent states variances at Ω

(c)

Figure 1.6 Comparison between an experiment (b) and the two theory descriptions for few photon states (a) and laser beams (c).

useful, but frequently does not include the quantum processes. Actually, the language used by these two approaches can be very diﬀerent, and it is not always obvious how to relate a result from a theoretical model to a technical device and vice versa. As an example compare the schematic representation of a squeezing experiment, shown in Figure 1.6, both in terms of the theoretical treatments for photons and laser beams and in an experimental description. The purpose of this guide is to bridge the gap between theory and experiment. This is done by describing the diﬀerent building blocks in separate chapters and combining them into complete experiments as described in recent literature. This guide starts with a classical model of light in Chapter 2. Experiments reveal that we require a concept of photons, Chapter 3, which is expanded into a quantum model of light in Chapter 4. The properties of optical components and devices are given in Chapter 5, followed by a detailed description of lasers and ampliﬁers in Chapter 6. Next is a detailed discussion of photodetection for single photons and beams in Chapter 7. On this basis we build with the discussion of complete experiments. The technical details required for reliable experimentation with quantum noise, including techniques such as cavity locking and feedback controller, are given in Chapter 8. The concept of squeezing is central to most attempts to improve optical devices beyond the quantum noise limit and is introduced and discussed in Chapter 9. It also describes the various squeezing experiments and their results are discussed; the diﬀerent interpretations are compared. Finally, the applications of squeezed light are described in Chapter 10 with a detailed description of the largest application,

15

16

1 Introduction

the detection of gravitational waves. Similar to squeezing Chapter 11 discusses quantum non-demolition experiments. In Chapter 12 more and more complete experiments to test the fundamental concepts of quantum mechanics are discussed. Finally, the concepts of quantum information and the rapidly evolving state of art in using quantum optics, both single photons or CW beams, in quantum information devices are presented in Chapter 13. Quantum optics has helped to build the foundations of many other form of quantum technology, and the links to these many parallel lines of research is brieﬂy outlined in Chapter 14, which tries to look a little into the future. This guide can be used in diﬀerent ways. A reader who is primarily interested in learning about the ideas and concepts of quantum optics would best concentrate on Chapters 2–4, 9, 12, and 13 but may leave out many of the technical details. For these readers Chapter 5 would provide a useful exercise in applying the concepts introduced in Chapter 4. In contrast, a reader who wishes to ﬁnd out the limitations of optical engineering or wants to learn about the intricacies of experimentation would concentrate more on Chapters 2, 5, 6, and 8, and for an extension into experiments involving squeezed light, Chapters 9 and 10 can be added. A quick overview of the possibilities opened by quantum optics can be gained by reading Chapters 3, 4, 6, 9, 11–14. We hope that in this way our book provides a useful guide to the fascinating world of quantum optics.

References 1 Saleh, B.E.A. and Teich, M.C. (2007). Fundamentals of Photonics, 2e. Wiley. 2 Born, M. and Wolf, E. (1959). The Principles of Optics. Pergamon Press. 3 Haroche, S. and Raimond, J.-M. (2006). Exploring the Quantum. Oxford

University Press. 4 Hage, B., Janousek, J., Armstrong, S. et al. (2011). Demonstrating various

quantum eﬀects with two entangled laser beams. Eur. Phys. J. D 63: 457. 5 Ourjoumtsev, A., Jeong, H., Tualle-Brouri, R., and Grangier, P. (2007). Gener-

6 7 8

9 10 11

ation of optical “Schrödinger cats” from photon number states. Nature 488: 784. https://doi.org/10.1038/nature06054. Haroche, S. (2012) Nobel Lecture 2012. http://www.nobelprize.org/ nobelprizes/physics/laureates/2012/haroche-lecture.html. Planck, M. (1900). Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum. Verh. Dtsch. Phys. Ges. 2: 237. Hallwachs, W. (1889). Ueber den Zusammenhang des Elektrizitätzverlustes durch Beleuchtung mit der Lichtabsorption. Ann. Phys. 273 (8): 666. https:// doi.org/10.1102/andp.18892730811. Lenard, P. (1902). Über die lichtelektrische Wirkung. Ann. Phys. 313 (5): 149. https://doi.org/10.1002/andp.19023130510. Einstein, A. (1905). Über einen die Erzeugung und Verwandlung des Lichtes betreﬀenden heuristischen Gesichtspunkt. Ann. Phys. 17: 132. Thorne, A., Litzen, U., and Johansson, S. (1999). Spectrophysics, Principles and Applications, 3e. Springer-Verlag.

References

12 Taylor, G.I. (1909). Interference fringes with feeble light. Proc. Cambridge Phi-

los. Soc. 15: 114. 13 Young, Th. (1807). Course of Lectures on Natural Philosophy and Mechanical

Arts. London: Taylor and Walton. 14 Hanbury-Brown, R. and Twiss, R.Q. (1956). Correlation between photons in

two coherent beams of light. Nature 177: 27–29. 15 Glauber, R.J. (1963). The quantum theory of optical coherence. Phys. Rev.

Lett. 130: 2529. 16 Carmichael, H.J. and Walls, D.F. (1976). A quantum-mechanical master

equation treatment of the dynamical stark eﬀect. J. Phys. B 9: 1199. 17 Kimble, H.J., Dagenais, M., and Mandel, L. (1977). Photon antibunching in

resonance ﬂuorescence. Phys. Rev. Lett. 39: 691. 18 Guerlin, C., Bernu, J., Deleglise, S. et al. (2007). Progressive ﬁeld state collapse

and quantum non-demolition photon counting. Nature 448: 889. 19 Einstein, A., Podolsky, B., and Rosen, N. (1935). Can a quantum-mechanical

description of physical reality be considered complete?. Phys. Rev. A 47: 777. 20 Bell, J.S. (1964). On the Einstein-Podolsky-Rosen paradox. Physics 1: 195. 21 Aspect, A., Grangier, P., and Roger, G. (1982). Experimental realisation of

22 23 24 25

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28 29 30

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Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: a new violation of Bell’s inequalities. Phys. Rev. Lett. 49: 91. Zeilinger, A. (2000). Quantum cryptography with entangled photons. Phys. Rev. Lett. 84: 4729. Kok, P. and Lovett, B.W. (2010). Introduction to Optical Quantum Information Processing. Cambridge University Press. Nielsen, M.A. and Chuang, I.L. (2000). Quantum Computation and Quantum Information. Cambridge University Press. Bennett, C.H., Brassard, G., Crepau, C. et al. (1993). Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70: 1895. Arecchi, F.T., Berne, A., and Sona, A. (1966). Measurement of the time evolution of the radiation ﬁeld by joint photocurrent distributions. Phys. Rev. Lett. 17: 260. Robertson, N.A., Hoggan, S., Mangan, J.B., and Hough, J. (1986). Intensity stabilisation of an Argon laser using an electro-optic modulator: performance and limitation. Appl. Phys. B 39: 149. Mertz, J. and Heidmann, A. (1993). Photon noise reduction by controlled deletion techniques. J. Opt. Soc. Am. B 10: 745. Taubman, M.S., Wiseman, H., McClelland, D.E., and Bachor, H.-A. (1995). Quantum eﬀects of intensity feedback. J. Opt. Soc. Am. B 12: 1792. Machida, S. and Yamamoto, Y. (1989). Observation of amplitude squeezing from semiconductor lasers by balanced direct detectors with a delay line. Opt. Lett. 14: 1045. Walls, D. (1983). Squeezed states of light. Nature 306: 141. Slusher, R.E., Hollberg, L.W., Yurke, B. et al. (1985). Observation of squeezed states generated by four-wave mixing in an optical cavity. Phys. Rev. Lett. 55: 2409.

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1 Introduction

33 Shelby, R.M., Levenson, M.D., Walls, D.F. et al. (1986). Generation of

34 35

36 37

squeezed states of light with a ﬁber-optics ring interferometer. Phys. Rev. A33: 4008. Wu, L.-A., Xiao, M., and Kimble, H.J. (1987). Squeezed states of light from an optical parametric oscillator. J. Opt. Soc. Am. B 4: 1465. (a) Kimble, H.J. and Walls, D.F. (ed.) (1987). Squeezed states of the electromagnetic ﬁeld. J. Opt. Soc. Am. B 4 (10 Special Issue); (b) Loudon, R. and Knight, P.L. (ed.) (1987). Squeezed light. J. Mod. Phys. Opt. 34 (Special Issue); (c) Yamamoto, Y. and Haus, H.A. (1986). Preparation, measurement and information capacity of optical quantum states. Rev. Mod. Phys. 58: 1001. Kok, P., Munro, W.J., Nemoto, K. et al. (2007). Linear optical quantum computing. Rev. Mod. Phys. 79: 135. Galvez, E.J. (2014). Resource Letter SPE-1: Single-photon experiments in the undergraduate laboratory. Am. J. Phys. 82 (11): 1019. https://doi.org/10.1119/1 .4872135.

19

2 Classical Models of Light The development of optics was driven by the desire to understand and interpret the many optical phenomena that were observed. Over the centuries more and more reﬁned instruments were invented to make observations, revealing ever more fascinating details. All the optical phenomena investigated during the nineteenth century were observed either directly or with photographic plates. The spatial distribution of light was the main attraction. The experiments measured long-term averages of the intensity; no time-resolved information was available. The formation of images and the interference patterns were the main topics of interest. In this context the classical optical models, namely, geometrical optics, dating back to the Greeks and developed ever since, and wave optics, introduced through Huygens’ ideas and Fresnel’s mathematical models, were suﬃcient. Excellent technology emerged, largely in the form of imaging devices of any kind, from microscopes through photography to telescopes. In parallel spectroscopy emerged as a tool for atomic and molecular physics. A revolution came with the invention of the laser. This changed optics in several ways. It allowed the use of coherent light, bringing the possibilities of wave optics to their full potential, for example, through holography. Lasers allowed spectroscopic investigations with unprecedented resolution, testing the quantum mechanical description of the atom in even more detail. The fast light pulses generated by lasers allowed the study of the dynamics of chemical and biological processes. In addition, a range of unforeseen eﬀects emerged, for example, the high intensities resulted in non-linear optical eﬀects. One fundamental assumption of the wave model, namely, that the frequency of the light remained unchanged, was found to be seriously violated in this new regime. The sum, diﬀerence, and higher harmonic frequencies were generated. Finally, optics was found to be one of the best testing grounds of quantum mechanics. The combination of sensitive photoelectric detection with reliable, stable laser sources meant not only that interesting quantum mechanical states could be produced but also that quantum mechanical eﬀects had to be accounted for in technical applications. Light is now almost exclusively observed using photoelectric detectors, which can register individual events or produce photocurrents. A description of these eﬀects requires the ﬁeld of statistical optics. We need a description of the beam of light that emerges from a laser. A laser beam is a well-conﬁned, coherent electromagnetic ﬁeld oscillating at a A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

20

2 Classical Models of Light

frequency 𝜈 or angular frequency 𝜔 = 2π𝜈. This frequency is constant for all times and locations and can only be changed by the interaction of light with non-linear materials. Furthermore, the beam propagates and maintains its shape unperturbed by diﬀraction. The beam carries information in the form of modulation, spatial in the form of images or temporal in the form of signals or pulses. Transmitting information in various forms, such as intensity, phase, polarization, and spatial changes, is the biggest use of optics. In this chapter we develop classical models suited to describing light. In the following chapters we will see how these must be modiﬁed to account for quantum mechanical eﬀects.

2.1 Classical Waves 2.1.1

Mathematical Description of Waves

An electromagnetic wave in an isotropic insulating medium is described by the wave equation 1 𝜕2 E(r, t) = 0 c 𝜕t 2 with a solution for the electric ﬁeld vector E(r, t), given by ∇2 E(r, t) −

(2.1)

E(r, t) = E0 [𝛼(r, t) exp(i2π𝜈t) + 𝛼 ∗ (r, t) exp(−i2π𝜈t)] p(r, t)

(2.2)

The wave oscillates at a frequency 𝜈 in Hz. For now we consider only a single-frequency component, but in general the solution can be an arbitrary superposition of such components. The electric ﬁeld vector is perpendicular to the local propagation direction. The orientation of the electric ﬁeld on the plane tangential to the local wave front is known as the polarization and is described by the vector p(r, t). The dimensionless complex amplitude 𝛼(r, t) of the wave can be written as a magnitude 𝛼0 (r, t) and a phase 𝜙(r, t): 𝛼(r, t) = 𝛼0 (r, t) exp{i𝜙(r, t)}

(2.3)

The magnitude 𝛼0 (r, t) will change in time when the light is modulated or pulsed. It will also vary in space, describing the extent of the wave. The wave will typically be constant in space for dimensions of the order of several wavelengths. The direction and shape of the wave front are determined by the phase term 𝜙(r, t), and the spatial distribution of the 𝜙(r, t) describes the curvature of the waves. Simple examples of speciﬁc solutions are monochromatic plane waves and spherical waves. These represent the extreme cases in terms of angular and spatial conﬁnement. The monochromatic plane wave is the ideal case in which the complex amplitude magnitude 𝛼0 is a constant and the phase is given by 𝜙(r) = −k ⋅ r

(2.4)

The wave propagates in a direction and with a wavelength 𝜆 represented by the wave vector k, where k∕k is a unit vector that provides the direction of the local

2.1 Classical Waves

wave front and k = 2π∕𝜆. The wavelength is related to the frequency via 𝜆 = c∕n𝜈, where n is the refractive index of the medium in which the wave propagates and c is the speed of light in the vacuum. Thus we can write 𝛼(z) = 𝛼0 exp{−ikz}

(2.5)

where we have assumed that the wave propagates in the z direction. The wave has a constant amplitude in any plane perpendicular to this optical axis. Substituting Eq. (2.5) into Eq. (2.2) and assuming 𝛼0 as real, we obtain the familiar solution E(z, t) = E0 cos(𝜈t − kz) p(z, t)

(2.6)

Linear polarization means that p(z, t) has a ﬁxed direction perpendicular to k. Circular polarization means that in any one location p(z, t) the direction is rotating in time in the plane perpendicular to k. Equation (2.6) represents a wave that has no angular spread. In contrast the spherical wave originates from one singular point. All its properties are described by one parameter r, the distance to its point of origin. The amplitude magnitude reduces with 1∕r, and the phase is constant on spheres around the origin: 𝛼 (2.7) 𝛼(r, t) = 𝛼(r) = 0 exp{ikr} r 2.1.2

The Gaussian Beam

Both of these models are not physical, they are only crude approximations to a beam of light emitted by a real laser. Such a beam is conﬁned to a well-deﬁned region and should be as close as possible to the ideal of a spatially localized but non-divergent wave. It has to be immune to spreading by diﬀraction and therefore has to be a stable solution of the wave equation. An example of such solutions is the Gaussian modes. The simplest of these is circularly symmetric around the optical axis, thus depending only on the radial distance 𝜌 from the beam axis: 𝜌 2 = x2 + y 2 This special solution is the Gaussian beam or fundamental mode, with a complex amplitude distribution of ( ) i𝛼 −ik𝜌2 𝛼(r, t) = 0 exp (2.8) exp(ikz), q(z) = z + iz0 q(z) 2q(z) where z0 is a constant. The Gaussian beam is a good spatial representation of a laser beam. Laser light is generated inside a cavity, constructed of mirrors, by multiple interference of waves and simultaneous ampliﬁcation. These processes result in a well-deﬁned, stable ﬁeld distribution inside the cavity, the so-called cavity modes [1–3]. The freely propagating beam is the extension of this internal ﬁeld. It has the remarkable property that it does not change its shape during propagation. Both the internal and external ﬁelds are described by Eq. (2.8). By taking

21

22

2 Classical Models of Light

the square of the complex amplitude 𝛼(r, t), the intensity is found, resulting in a Gaussian transverse intensity distribution: ( ( ) ) W0 2 −2𝜌2 exp I(𝜌, z) = I0 W (z) W 2 (z) √ ( )2 z (2.9) and W (z) = W0 1 + z0 The properties of this Gaussian beam are the following: The shape of the intensity distribution remains unchanged by diﬀraction. The size of the beam is given by W (z). The beam has its narrowest width at z = 0, the so-called waist of the beam. It corresponds to the classical focus, but rather than pointlike, the Gaussian beam has a size of W0 . This value gives the radial distance where the intensity has dropped from I0 = I(0, 0) at the centre to I(W0 , 0) = I0 ∕e. At the waist the phase front is a plane, and 𝜙(𝜌) is a constant. Moving away from the waist, the beam radius W (z) expands, the intensity at the axis reduces, and, at the same time, the wave front curves. Along the optical axis the Gaussian beam scales with the parameter z0 , which is known as the Rayleigh length: z0 =

πW02

𝜆 It describes the point where the intensity on axis has dropped to half the value √ at the waist, I(0, z0 ) = 12 I0 , and the beam radius has expanded by a factor of 2. This can be regarded as the spatial limit to the focus region, and the value 2z0 is frequently described as the depth of focus or as the confocal parameter. Far away from the waist, for z ≫ z0 , the Gaussian beam has essentially the wave front of a spherical wave originating at the waist. Obviously, the Gaussian beam is still centred on the beam axis and has only a ﬁnite extent W (z). At these large distances, the size of the beam W (z) depends linearly on z; the beam has a constant divergence angle of Θ=

2𝜆 2πW0

(2.10)

In addition to the wave front curvature, a gradual phase shift occurs on the beam axis between a uniform plane wave with identical phase at z = 0 and the actual Gaussian beam. The phase diﬀerence increases to π∕4 at z = z0 and asymptotically approaches π∕2 (Table 2.1). Table 2.1 Properties of a Gaussian beam. z=0

z = z0

Beam waist

W = W0

√ W = 2W0

W (z) = z∕z0 W0

Intensity

I0

I0 ∕2

I0 (z0 ∕z)2

Wave front

Plane wave

Curved wave front

Spherical wave

Phase retardation

0

π∕4

π∕2

Location

z ≫ z0

2.1 Classical Waves

Figure 2.1 Gaussian beam propagation along the z axis.

I(y) I(x)

z W(z) W0 = W(z = 0) x x

As known from experiments with laser beams, the Gaussian beam is very robust in regard to propagation through optical components. Any component that is axially symmetric and the eﬀect of which can be described by a paraboloidal wave front will keep the Gaussian characteristic of the beam. This means the emerging beam will be Gaussian again, however, now with a new position z = z′ of the waist. All spherical lenses and spherical and parabolic mirrors transform a Gaussian beam into another Gaussian beam. Another important consequence of this transformation is the fact that Gaussian beams are self-reproducing under reﬂection with spherical mirrors (Figure 2.1). However, the Gaussian beam (Eq. (2.8)) is not the only stable solution of the paraxial wave equation. There are many other spatial solutions with non-Gaussian intensity distributions that can be generated by a laser and propagate without change in shape. One particular set of solutions is the Hermite–Gaussian beams. These beams have the same underlying phase, except for an excess term 𝜁l,m (z), which varies slowly with z. They have the same paraboloidal wave front curvature as the Gaussian beam, which matches the curvature of spherical optical components aligned on the axis. Their intensity distribution is ﬁxed and scales with W (z). The distribution is two-dimensional (2D), with symmetry in the x and y directions and usually not with a maximum on the beam axis. The intensity on the beam axis decreases with the same factor W (z) as the Gaussian beam along the z axis. The various solutions can be represented by a product of two Hermite–Gaussian functions, one in the x and the other in the y direction, which directly leads to the following notation: (√ ) ( ) (√ ) W0 2y 2x Gm Gl 𝛼l,m (x, y, z) = 𝛼0 Al,m W (z) W (z) W (z) )} { ( kr2 for l, m = 0, 1, 2, … − 𝜁l,m (z) × exp −i kz + 2R(z) (2.11) Here Al,m is a constant and Gl (u) and Gm (u) are standard mathematical functions, the Hermite polynomials of order l, m, respectively, which can be found in tables [2, 4]. In particular, G0 (u) = exp{−u2 ∕2}, and thus the solution 𝛼0,0 (x, y, z) is identical to the Gaussian beam described above. This set of solutions forms a complete basis for all functions with paraboloidal wave fronts. Any such beam can be described by a superposition of Hermite–Gaussian beams. All these components propagate together, maintaining the amplitude distribution. The only

23

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2 Classical Models of Light

00

10

20

30

01

11

21

31

02

12

22

33

Figure 2.2 Intensity distributions 𝛼l,m (x, y)2 of Hermite–Gaussian beams. These modes are labelled transverse excited mode TEMlm with the parameters l and m describing the horizontal and vertical order of the modes. Source: https://www.wikipedia.org/.Licensed Under CC BY SA 3.0.

component with circular symmetry is the Gaussian beam 𝛼0,0 (x, y, z). However, it is possible to describe other circularly symmetric intensity distributions. Consider, for example, the superposition of two independent beams with 𝛼1,0 (x, y, z) and 𝛼0,1 (x, y, z) and equal intensity; if these beams have random phase, their interference can be neglected and the sum of the intensities results in a donut-shaped circularly symmetric function, occasionally observed in lasers with high internal gain (Figure 2.2). 2.1.3

Quadrature Amplitudes

The description of a wave as a complex amplitude (Eq. (2.3)), with amplitude 𝛼0 (r, t) and phase distribution 𝜙(r, t), is useful for the description of interference and diﬀraction. As shown above, the phase distribution 𝜙(r, t) describes the shape of the wave front as well as the absolute phase, in respect to a reference wave. However, it is useful to separate the two to consider the absolute phase independently from the phase distribution or wave front curvature. An equivalent form of Eq. (2.2) can be given in terms of the quadrature amplitudes, X1 and X2, the amplitudes of the associated sine and cosine waves: E(r, t) = E0 [X1(r, t) cos(2π𝜈t) + X2(r, t) sin(2π𝜈t)] p(r, t)

(2.12)

The quadrature amplitudes are proportional to the real and imaginary parts of the complex amplitude: X1(r, t) = 𝛼(r, t) + 𝛼 ∗ (r, t) X2(r, t) = −i[𝛼(r, t) − 𝛼 ∗ (r, t)]

(2.13)

2.1 Classical Waves

Figure 2.3 Phasor diagram (a) for a constant single wave and (b) the representation of interference of several coherent waves.

X2

X2

E

α total

ϕ (a)

E

X1

(b)

X1

In this notation the absolute phase 𝜙0 of the wave is associated with the distribution of the amplitude between the quadratures X1 and X2: ( ) X2 (2.14) 𝜙0 = tan−1 X1 For X2 = 0 the wave is simply a cosine wave (Eq. (2.6)). We can regard this wave arbitrarily as the phase reference. A common description of classical waves whenever interference or diﬀraction is described is phasor diagram, a 2D diagram of the values X1 and X2 as shown in Figure 2.3a. In such a diagram, which represents the ﬁeld at one point in space and time, each wave corresponds to one particular vector of length 𝛼 from the origin to the point (X1, X2), and this vector could be measured as the complex amplitude averaged over a few optical cycles of length 1∕𝜈. The magnitude of the wave is given by the distance 𝛼 of the point from the origin, and the relative phase 𝜙0 is given by the angle with the X1 axis, since the absolute phase is arbitrary. It is useful to choose 𝛼 as real (𝜙 = 0), which means the vector is parallel to the X1 axis. If several waves are present at one point at the same time, they will interfere. Each individual wave is represented by one vector; the total ﬁeld is given by the sum of the vectors, as shown in Figure 2.3b. These types of diagrams are a useful visualization of the interference eﬀects. After the total amplitude 𝛼total has been determined, we can again select a phase for this total wave and choose 𝛼total as being real. All forms of interference can be treated this way and we obtain a value for the average, or DC, value of the optical amplitude. In addition, the wave can have ﬂuctuations or modulations in both amplitude and phase. Such variations correspond to ﬂuctuations in the quadrature values. We can describe the time-dependent wave with the complex amplitude 𝛼(t) = 𝛼 + 𝛿X1(t) + i 𝛿X2(t)

(2.15)

In the limit 𝛼 ≫ 𝛿X1(t), which means for waves that change or ﬂuctuate only very little around a ﬁxed value 𝛼, we see that 𝛿X1(t), 𝛿X2(t) describe the changes in the amplitude and phase, respectively. This is the regime that we use to describe beams of light detected by detectors that produce a photocurrent. We will call them the amplitude and phase quadratures. The direct link between variations in the quadrature values and those of the ﬁeld magnitude and phase is shown diagrammatically in Figure 2.4. 2.1.4

Field Energy, Intensity, and Power

The energy passing through a small surface area element dx, dy around the point x, y in a short time interval dt around the time t is obtained by integrating the

25

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2 Classical Models of Light

Δϕ

X1 ΔX1

E

ΔE

ΔX2

t

X2

Figure 2.4 Fluctuations shown in phase space: the ﬁeld changes in time and moves from point to point in the phasor diagram. This corresponds to waves with diﬀerent amplitude and phase.

square of Eq. (2.2). Note that because the ﬁeld is propagating in the z direction, this is equivalent to the energy in the volume element dx, dy, c dt. We get (x, y, t) = E02 2 𝛼 ∗ 𝛼 dt dx dy

(2.16)

where we have assumed that dx dy dt is suﬃciently small that 𝛼 does not vary over the interval, but that dt ≫ 1∕Ω is satisﬁed and the time average will be over a large number of optical cycles. As a result only the phase-independent terms survive. The intensity of the beam is I(x, y) =

∫one second

E02 2 𝛼 ∗ 𝛼 dt

(2.17)

What is normally measured by a photodetector is the power P of the light, the total ﬂux of energy reaching the detector in a second, integrated over the area of the detector: P=

∫

I(x, y) dx dy

(2.18)

The energy in a small volume element (Eq. (2.16)) can also be expressed in terms of the quadratures as (x, y, t) = E02 (X12 + X22 ) dt dx dy where we have used Eq. (2.13). The parameter E02

(2.19) is proportional to the frequency

of the light, 𝜈, speciﬁcally

ℏ2π𝜈 (2.20) 2 where ℏ is Planck’s constant. It is the optical power P(t) in Eq. (2.18) that is measured in most experiments, and from this we infer the amplitude, the phase, and the ﬂuctuations of the beam. E02 =

2.1.5

A Classical Mode of Light

The idea of a phasor diagram can now be combined with the concept of spatial modes as described in Eq. (2.3). Let us concentrate on the simplest type of beam, a Gaussian beam. Equation (2.8) relates the relative amplitude and phase at any one point back to a point at z = 0 at the centre of the waist. Once the size W0 is given, the entire spatial distribution is well deﬁned. We can assign one overall optical

2.1 Classical Waves

P

W0

z ν, ϕ0

I0

Figure 2.5 Graphical representation of the parameters of a Gaussian mode of light.

power, or peak intensity I0 , or one overall phase to the entire spatial distribution. Thus the entire laser beam can be described by just six parameters. The spatial shape of the laser beam with its very narrow focus, see Figure 2.5 is the key for many applications of laser light. It allows us to deliver very high power to a very small volume, which is the key to most applications for cutting, ablating, or ionizing materials and many of the medical applications. The small size of the focus, limited by diﬀraction to a about half of a wavelength, allows high spatial resolution in microscopy or the coupling of light into a single-mode optical ﬁbre. Even more magically the high gradients in intensity in a laser focus can be used to exert forces onto particles that move inside the laser beam, and this can be used to trap particles, alas individual atoms or ions inside the focus, the physics behind optical tweezers invented by 2018 Nobel laureate A. Ashkin [5, 6]. This itself is the foundation of many applications in biology and also of many new quantum experiments with laser-cooled atoms, single atoms, and ions. Such a mode of light is strictly speaking constant in time, at least for short times. It corresponds to an idealized laser that has a perfectly stable, time-independent output, with an intensity distribution that is not changing at all in time. More realistically the laser output will not be perfectly constant, but there will be ﬂuctuations of the average intensity, the amplitude and phase, and all parameters listed in Table 2.2. Experiments conﬁrm that there are always ﬂuctuations (see Figure 1.2). They can be stabilized to some degree by technical means, but ultimately quantum mechanics limits how small the ﬂuctuations can be. This description of the light as a mode, which can be quantized, will be the key to an elegant quantum model. Table 2.2 The parameters of a mode of light. Parameters required to describe one mode of light

Intensity of the mode

I0

Frequency of optical ﬁeld

𝜈

Absolute phase of this mode

𝜙0

Direction of propagation

z

Location of the waist (z = 0)

r0

Waist diameter of the beam

W0

Direction of polarization

P

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2 Classical Models of Light

2.1.6

Light Carries Information

Light is used to carry information. In many cases we think about images, where the information is encoded in the spatial domain, transverse to the propagation of light. In this case the magnitude and the phase of the light are a function of location r, the image is in the transverse x − −y plane, and the image can change along the z axis of propagation. Quite frequently this is constant in time – until the input image is changed. In many other situations we want to use light to measure some parameter or to communicate information from one place to another. In both cases we need a beam of light that can carry information, a simple constant beam of light that can be modulated in time. The physical characteristics of a freely propagating laser beam deﬁne such a mode of light. The shape of this mode is well determined. In all these cases the propagation of the light, be it in free space, through a complex apparatus, through an optical ﬁbre, etc., can be described by the mode of light, the simplest case being the Gaussian beam. We will select the appropriate degree of freedom that is being modulated, and it is useful to describe the entire beam, including all its time and frequency components, as one entity, as one mode. The beam can change in its appearance while it propagates through the apparatus – but it remains one entity. We will ﬁnd in Chapter 4 that this is the most convenient and practical way to quantize the light. However, the temporal dependence is most important, since it contains the information in the form of modulations. We describe this with a time-varying amplitude 𝛼(t) for this mode. The various aspects of the measurement and manipulation of laser beams occur at diﬀerent timescales and at their corresponding frequencies. Let us introduce the nomenclature for a set of natural and technical frequencies and timescales (Figure 2.6).

Δtusp

Δtsamp

Δtmeas

One sample Integrate interval for variance and average

Δtosc = 1/ Ωmod

Period of oscillation in periodic signal

Δtrec

T

Interval between recordings

Envelope s(t)

2π νopt

Ωsamp

Ωmod, Ωsens

Ωenv

High

Sampling of photon flux frequency

Modulator and signal carrier frequencies

Signal content frequency

CW

Long

Low

Figure 2.6 Deﬁnition of diﬀerent timescales and frequencies used in measuring and manipulating light.

2.1 Classical Waves

The natural starting point is the optical frequency 2π𝜈L of a laser beam. This is the highest frequency, typically 1015 Hz, and all the modulation frequencies Ωmod will be lower. We can sample the photon ﬂux at very high rates with the sampling frequency Ωsamp , which is between 2π𝜈L and Ωmod . The shortest technical timescale is Δtsamp = 1∕Ωsamp , which in many situations is achieved through the fast gating of either a photon counter or an analogue-to-digital converter. We require a number of samples to evaluate a variance for the ﬂuctuations or an average value for the photon ﬂux, which requires the time Δtmeas . Many such data points are required to impose a periodic signal at Ωmod or measure a periodic signal Ωsens from an external source. This requires a time interval Δtosc = 1∕Ωmod . In a sensing application we might want to take a whole series of readings, taken at time intervals Δtread , and we will have a full evaluation of the signal and noise for a considerable period of time T. In most cases we can assume that the performance of the instrument remains constant over a long time, that the noise is stationary and we can use samples of the noise to describe the performance, and that the laser parameters, such as optical power and optical frequency, are well stabilized. In this case we can record the envelope function from many readings and analyse the behaviour of a parameter we are sensing. We might ﬁnd periodic behaviour and typical envelope frequencies Ωenv . The absolute values for these timescales and frequencies can vary by several orders of magnitude and cannot be stated in general terms. The instruments can contain anything from femtosecond pulses to a constant laser. We can have short pulses of light where each pulse carries one piece of information. This will be particularly important when we think about individual photons in each pulse. Here the typical timescale Δtmeas for each measurement can be shorter than the time between two pulses, and with modern equipment this can be as short as a few picoseconds. In the other extreme we can have a continuous wave (CW) laser that is steady, and we measure for a time interval T and take many readings. The modulation and sensing frequencies can vary from single hertz for sensors such as environmental or astronomical detectors to gigahertz in communication systems. Still, there is a great deal of commonality that we will use in describing the properties of light, classical and quantum. Finally, there is the option of quasi-CW communication where we use trains of pulses with high repetition frequency, in particular to achieve high peak powers, and the measurement averages over a number of pulses. For the modulation and sensing, we can choose several degrees of freedom, such as phase, frequency, polarization, position and mode shape, etc., and in this way pack more information into one beam. Technically this is described as multiplexing. One example is the modulation of the polarization, which has 2 degrees of freedom, and we require two orthogonal polarization axes for one single beam. More degrees of freedom are possible, for example, additional optical frequencies or several additional co-propagating spatial modes. In addition we will encounter situations where we have several beams within one apparatus, for example, created by a series of beamsplitters. In all these cases we can deﬁne the available degrees of freedom, ﬁnd the corresponding orthogonal functions that describe

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these optical ﬁelds and their modulations, and describe this apparatus with a set of multiple modes. 2.1.7

Modulations

Information carried by a beam of light can be described in the form of modulation sidebands. Note that the equations used here are identical to those describing radio waves where amplitude modulation (AM) and frequency modulation (FM), or technically speaking phase modulation (PM), are the familiar, standard techniques. Many of the technical expressions, such as carrier for the fundamental frequency components and sidebands, are taken directly from this much older ﬁeld of technology. A beam of light that is amplitude modulated by a fraction M at one frequency Ωmod can be described by ( ) M 𝛼(t) = 𝛼0 1 − (1 − cos(2πΩmod t)) exp(i2π𝜈L t) 2) ( M exp(i2π𝜈L t) = 𝛼0 1 − 2 M + 𝛼0 [ exp(i2π(𝜈L + Ωmod )t) + exp(i2π(𝜈L − Ωmod )t) ] (2.21) 4 We see that the eﬀect of the modulation is to create new frequency components at 𝜈L + Ωmod and 𝜈L − Ωmod . These are known as the upper and lower sidebands, respectively, whilst the component remaining at frequency 𝜈L is known as the carrier (see Figure 2.7a). For completely modulated light, which means M = 1, α X2

νL − Ω2 νL − Ω1 νL + Ω1 (a)

X2

=

AM

νL + Ω2 ν

X2

Ω1

+

Ω2

FM

α

α X1

X1

(b)

δX2

δX1

X1 Ω1

Ω2 (c)

Ω

Figure 2.7 Three diﬀerent graphical representations of amplitude (AM) and frequency (FM) modulation at frequencies Ω1 and Ω2 , respectively. Part (a) shows the sidebands of the optical frequency at 𝜈L ± Ω1 and 𝜈L ± Ω2 , (b) shows the same modulations in the time domain and in a phasor diagram, and (c) shows the complete phasor diagram in frequency space.

2.1 Classical Waves

the sidebands have exactly 12 the amplitude of the fundamental component. The sidebands are perfectly correlated, and they have the same amplitude and phase, which means the two beat signals, generated by the sidebands at 𝜈L + Ωmod and 𝜈L − Ωmod beating with the carrier, are in phase. A second representation is in the time domain through oscillating vectors 𝛿𝛼 in the phasor diagram Figure 2.7b. The two vectors, 𝛿𝛼+Ω and 𝛿𝛼−Ω , rotate in opposite directions with angular frequency 2πΩ. A third representation is in frequency space in the phasor diagram shown in Figure 2.7c. Notice that for a given modulation depth M, the sideband amplitudes scale with the amplitude of the carrier, 𝛼0 . Thus even very small modulation depths, M, can result in signiﬁcant sideband amplitude if the carrier amplitude is large. The detected intensity is given by ( )2 M (2.22) I(t) = I0 |𝛼(t)|2 = I0 1 − (1 − cos(2πΩmod t)) 2 and oscillates at the frequency Ωmod around the long-term average I0 . The resulting photocurrent has two components, a long-term average iDC and a constant component at frequency Ωmod , labelled i(Ωmod ). A common (and useful) situation is when the modulation depth is very small, M ≪ 1. Then the resulting intensity modulation is linear in the modulation depth: I(t) ≈ I0 [1 − M(1 − cos(2πΩmod t)) ]

(2.23)

A light beam with modulated phase can be described in a similar way. The amplitude can be represented by 𝛼(t) = 𝛼0 exp[iM cos(2πΩmod t)] exp(i2π𝜈L t)

(2.24)

Clearly the intensity will not be modulated as |𝛼(t)|2 = |𝛼0 |2 . Expanding 𝛼t we obtain ) ( M2 2 𝛼(t) = 𝛼0 1 + iM cos(2πΩmod t) − cos (2πΩmod t) + · · · exp(i2π𝜈L t) 2 [( ) M2 = 𝛼0 1 − + · · · exp(i2π𝜈L t) 4 ) ( M + · · · (exp(i2π(𝜈L + Ωmod )t) + exp(i2π(𝜈L − Ωmod )t)) +i 2 ( 2 ) M − + · · · (exp(i2π(𝜈L + 2Ωmod )t) + exp(i2π(𝜈L − 2Ωmod )t)) 8 +· · · ]

(2.25)

Suppose we measure the quadrature amplitudes (X1 and X2) of this ﬁeld relative to a cosine reference wave (i.e. we take 𝛼 to be real). The X2 quadrature will have pairs of sidebands separated from the carrier by ±Ωmod , ±3Ωmod , … The X1 quadrature on the other hand will have sidebands separated from the carrier by ±2Ωmod , ±4Ωmod , … Again a simple, and useful, situation is for the modulation depth to be very small, M ≪ 1. The amplitude is approximately [ ] M 𝛼(t) ≈ 𝛼0 exp(i2π𝜈L t) + i [exp(i2π(𝜈L + Ωmod )t) + exp(i2π(𝜈L − Ωmod )t)] 2 (2.26)

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and the sidebands will only appear on measurements of the X2 quadrature. Exact expressions for larger M involve nth-order Bessel functions [7]. The phasor diagram corresponding to this double sideband approximation is shown in Figure 2.7b. The sidebands have the same magnitude. The relative phase of the amplitudes is such that the two resulting beat signals have the opposite sign and the beat signals cancel. Thus, as we have seen, phase modulation cannot be detected with a simple intensity detector. Another way of saying this is that the sidebands for phase modulation are perfectly anti-correlated. The treatment of frequency modulation proceeds in a similar way. Any information carried by the mode can be expressed as a combination of amplitude and phase modulation. Any modulation, in turn, can be expressed in terms of the corresponding sidebands.

2.2 Optical Modes and Degrees of Freedom 2.2.1

Lasers with Single and Multiple Modes

A single mode is only the simplest option for a laser beam output. The resonator inside the laser will have several resonance conditions. Optical waves with diﬀerent frequencies can be ampliﬁed, and also diﬀerent spatial amplitude and phase distributions can be in resonance. Each of these waves, with their speciﬁc set of parameters, can be considered as a diﬀerent mode, with distinctively diﬀerent values of the parameters listed in Table 2.2, and in the case of spatial modes diﬀerent amplitude and phase distributions. The details depend crucially on the geometry of the laser resonator [2] and the spectral response of the gain medium, and some of the details are summarized in Chapters 5 and 6. In many cases the laser might produce an output in several of these modes simultaneously – with all of these modes competing for gain. Examples of other laser modes are beams with the same shape but diﬀerent carrier frequencies, spaced in regular intervals and all linked by a common phase. This concept, known as a frequency comb, won the inventors a Nobel prize [8] and is the basis of many new applications in optical metrology of time signals and of new forms of spectroscopy and non-linear optics. Short laser pulses can be thought of as a frequency comb with a speciﬁc envelope function determining the number and strength of the individual components. These envelope functions are shaped to be the eﬀects inside the laser, or they can also be modiﬁed outside by instruments that have a frequency-dependent response, or dispersion, or with active means such as electro-optic pulse shapers. Frequently the aim of engineering is to produce a single-mode laser, which can be described by the simple set of parameters, with a waveform such as stated in Eq. (2.8). However, not every single-mode beam has to be this simple. A single-mode beam can be transformed, for example, with lenses or beamsplitters, or we could delete a part the beam, and during propagation diﬀraction would change the intensity distribution, possibly quite dramatically. The underlying idea of the mode remains the same in such a mode transformation, and the relevant question remains: can we ﬁnd one single mathematical description that contains all properties of the laser beam? Clearly, yes we can.

2.2 Optical Modes and Degrees of Freedom

This is correct for the spatially transformed single-mode beam. Another example is a carefully crafted pulsed laser system where techniques such as mode locking produce trains of pulses as short as femtoseconds, with correspondingly wide frequency combs, where all components of the comb are phase and amplitude linked. This entire ensemble of spectral components can be described as a single mode with one simple set of parameters. In this case the mode looks complex, in space or frequency, but we can still speak of one mode since all the information is still contained as a modulation of this speciﬁc entity. In each case we have to consider the number of degrees of freedom that are available, not the apparent complexity of the beam of light. 2.2.2

Polarization

Let us consider beams that require a description by two or modes. One clear example is the polarization of an otherwise single-mode beam. The polarization properties can be described by separately considering the amplitude of the ﬁeld oscillating in two transverse directions; let us call them horizontal H and vertical V . These are two orthogonal coordinates that could be in any physical directions. Mathematically we require two orthogonal basis vectors, H and V. A 2D complex vector 𝛼 p(r, t) with magnitudes |𝛼H |, |𝛼V | and phases 𝜙H , 𝜙V for the two directions provides the full description: 𝛼 p(r, t) = |𝛼H (r, t)| ei𝜙H H + |𝛼V (r, t)| ei𝜙V V = (𝛼H (r, t), 𝛼V (r, t))

(2.27)

The important point is that the properties, and also the information, in the two coordinates are completely independent. For example, a beam can have all its intensity in one or the other coordinate V and H, or it can have a shared amplitude in both. We would call a beam with 𝛼V = 0 horizontally polarized and with 𝛼H = 0 vertically polarized. A beam with 𝛼V = 𝛼H is diagonally polarized, D, while a beam with 𝛼V = −𝛼H is anti-diagonally polarized, A, orthogonal to D. We can make beams with the amplitude 𝛼V = i𝛼H , which we call right-hand circularly polarized (R), and with 𝛼V = −i𝛼H , which are left-hand circularly polarized (L). All of these descriptions, or projections on diﬀerent basis vectors, are equivalent, and we can transform the projection between any of them without gaining or losing information. We can choose the basis vectors that match best to the experiment, resulting in the simplest description. In all cases the projection is based on a 2D vector, and we have two orthogonal modes, both travelling within the same laser beam. Any transformation of the polarization in the apparatus, for example, through polarizers, wave plates as shown in Figure 2.8, through ﬁlters, glass plates, etc., can be described by 2D matrices Ξ, also known as Jones matrices, described in most textbooks about optics, such as [3]: 𝛼 p (r, t)out = Ξ 𝛼 p (r, t)in = Ξ1 Ξ2 Ξ3 … 𝛼 p (r, t)in

(2.28)

Here each component in the apparatus is described by a speciﬁc matrix Ξi , i = 1, 2, …, and light propagating through a sequence of such components can be described by multiplication of the individual matrices Ξ1 ,Ξ2 ,Ξ3 , … Examples for such Jones matrices include Ξprop for simple propagation, ΞH for a linear

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λ 2

V

PBS

R L

D

H

(a)

λ 4

A

(b)

PBS

(c)

PBS

Figure 2.8 Three examples of the polarization state of a single laser beam. The diﬀerent projections used are (a) horizontal H and vertical V, (b) diagonal D and anti-diagonal A, and (c) left-hand circular L and right-hand circular R polarization. In each case two detectors are used, each detecting one of the polarization components. The transformation from one basis to another is achieved by a combination of polarizing beamsplitter (PBS) and either quarter-wave or half-wave plate.

polarizer in direction H, ΞV for a linear polarizer in direction V , and Ξ𝜙 for a wave retarder by a phase angle 𝜙: ( ( ( ( ) ) ) ) 1 0 1 0 0 0 1 0 , ΞH = , ΞV = , Ξ𝜙 = Ξprop = 0 1 0 0 0 1 0 ei𝜙 (2.29) Note that the order of the matrices is important and has to be the same as the order of components in the experiment. Interchanging components, such as placing a polarizer in a diﬀerent section of the apparatus, can have dramatic eﬀects. A very important point is that the beam can be modulated diﬀerently in the coordinates H and V , and as long as we detect the light in this basis, the information streams that we receive in these coordinates are completely independent. The mathematical statement of orthogonality between H and V is fully equivalent to having no crosstalk between the coordinates. We can use this fact to optimize our experiment. For example, by modulating only 𝛼H and not 𝛼V and by using two detectors, set for the H and V polarization, respectively, we can ﬁnely tune the optics until we ﬁnd no signal coming from the V detector. We then change the modulation to V and should ﬁnd no signal coming from the H detector. We now have established two independent communication channels. The laser beam can carry twice as much information using polarization modulation than in amplitude modulation of an unpolarized beam. This shows the distinction of a two-mode beam from a single-mode beam: it has 2 degrees of freedom, it contains more information, and we can consider the question of correlation between the two modes. For example, if we take a beam with pure horizontal modulation, 𝛼V = 0. That means we only use one single mode and we have no correlation between the information in H and V. If we detect the same beam in the D and A coordinates rotated at 45∘ , we share the modulation in the two components and we have a strong correlation between 𝛼D and 𝛼A . However, there is no additional information, and the beam can still be described by a single mode. No information is added in the choice of a diﬀerent projection. In contrast, if we have independent modulations in both H and V, and that means no correlation between 𝛼H and 𝛼V , then we require a two-mode description. Detection in H and V would give us two independent channels and no

2.2 Optical Modes and Degrees of Freedom

crosstalk. In this case the detection in the basis D and A would create a correlation and would also mix up the information from the two channels. We still have a two-mode case, just but not an optimal description. 2.2.2.1

Poincaré Sphere and Stokes Vectors

It is useful to mention an alternative description of polarization. With the complex Jones vector, we have four components, and we can describe the state of the polarization by two real parameters: the magnitude ratio 𝛼H ∕𝛼V and the phase diﬀerence 𝜙H − 𝜙V . Alternatively, we might characterize the state of polarization by two angles, 𝜒 and 𝜓, which represent the orientation and ellipticity of the polarization ellipse, see Figure 2.9. We can now describe the polarization in the form of a three-dimensional (3D) diagram, the Poincaré sphere where each state is given by a point on the surface of the sphere of unit radius, with the spherical coordinates given by (r = 1, Θ = 90 − 2𝜒, Φ = 2𝜓). Each point on the sphere is a diﬀerent polarization state. For example, points on the equator, that is with 𝜒 = 0, represent linear polarization. The two points on the equator, u1 with 2𝜓 = 0, and u2 with 2𝜓 = 90 degree, represent linear polarization in the H and V axis. The north pole u3 with 2𝜒 = 90 and south pole with 2𝜒 = −90 degree represent left-handed and right-handed circular polarized light respectively. Other states on the sphere represent the various forms of elliptical polarization. The coordinates u1 , u2 , u3 of a point on the Poincaré sphere in Cartesian coordinates are also known as the ﬁrst three Stokes parameters (S1 , S2 , S3 ). The fourth parameter S0 is simply proportional to the intensity of the beam. For the full details, see Ref. [3]. This four-component vector, introduced by Stokes, is the original description of polarization. It contains the same information as the Jones vector with its two complex components. We ﬁnd that the latter is mathematically more elegant and ﬁts to the two-mode description, which is so powerful in describing quantum optics systems. However, the Poincaré sphere is a practical way to illustrate the eﬀects of polarizing components. It will appear explicitly when we discuss the eﬀect of polarization squeezing, where this description u3

y χ ψ x

1 1

2χ Polarization ellipse (a)

2ψ u1 (b)

u2

Poincaré sphere

Figure 2.9 Graphical representation of polarization (a) through the two angles, 𝜒 and 𝜓, which represent the orientation and ellipticity of the polarization ellipse and (b) the Poincaré sphere, in both spherical and Cartesian coordinates. Source: From B.E.A. Saleh and M.C. Teich 2007 [3].

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provides the most direct insight in the quantum noise. In addition it allows us to see the correspondence between polarization and other two-mode systems such as the spin of an atom or the time evolution of a two-level system. This is a beautiful generalized graphical representation that has its place in any description of atomic physics or non-linear optics. Finally, it is worthwhile to see the link between the two modes, which exist within the one laser beam, and the two separate beams, which are normally required for the simultaneous detection of the modes. A common example in polarization is the use of the polarizing beamsplitter, which sends all the light including modulation in the H polarization to one output beam and all the light and modulation in the orthogonal V polarization into a separate beam, as shown in left-hand side of Figure 2.8. In this case we would associate the beam travelling to one detector with one mode and the other beam with the other mode. Similarly we could have a beam with two diﬀerent carrier frequencies and then use a grating to separate the two modes into separate beams and modes.

2.2.3

Multimode Systems

Is it strictly necessary to physically separate the modes to detect them? Not necessarily. A counterexample would be a laser beam that has two separate spatial modes, like two orthogonal Hermite–Gaussian modes (see Figure 2.2), which are detected by a photodetector with many pixels. We can detect the two spatial modes independently by recording the photocurrents from all pixels of the detector and then doing some clever recombination of the currents to represent the individual modes. At no point are the modes spatially separated, but they propagate together. However, the mathematical description is the same as for a set of modes, and beams, travelling completely separate through space. This example shows the importance and need for a multimode description. And there are situations where the number of modes can be much larger than two. It is possible to create beams of light that have N independent modes, or information channels. For example, this could be beams with N diﬀerent carrier frequencies, with N diﬀerent spatial modes, or with N separate beams created by an apparatus with a complex components that are sensitive to polarization, frequency, spatial shape, etc. In all these situations we can create a basis of N orthogonal modes, which we can modulate and detect individually. For the case of spatial modes, two modes described by 𝛼i (r, t) and 𝛼j (r, t) are orthogonal if ∫

𝛼i (r, t)𝛼j∗ (r, t) dr = 0

(2.30)

We can have N independent information channels with no crosstalk between any of them, or we can create, through mode transformation, correlations between the modes. The analysis of a beam using a large set of orthogonal basis vectors allows us to determine the smallest possible number N of modes that is required for a speciﬁc situation: ( 𝛼1 (r, t), 𝛼2 (r, t), … , 𝛼(N−1) (r, t), 𝛼N (r, t) )

(2.31)

2.3 Statistical Properties of Classical Light

In practice this leads to the mathematical challenge of ﬁnding the lowest possible value of N or the lowest possible number of independent degrees of freedom for any given situation. There are many applications of this type of multimode optical physics. In optical communication it allows us to increase the information ﬂux that is communicated. The commonly used terminology is multiplexing. In optical sensing it allows us to isolate exactly the degrees of freedom of the laser beam we wish to detect and suppress other eﬀects on the beam we do not want to detect. In coherent image detection and analysis, we can deﬁne an object that we wish to detect as an optical mode and the changes to this object as orthogonal modes. That means each of the higher-order modes contains information about one speciﬁc degree of freedom, for example, a translation, rotation, or stretching of the object in the image. These changes can be tracked by detecting only the speciﬁc higher-order mode. One interesting example is the translation of a Gaussian beam in the x and y coordinates. The information about this change is stored in the mode that represents the derivative of TEM00 in these two directions, respectively, and corresponds directly to the modes TEM01 and TEM10 , which are shown in Figure 2.2. By recording the intensity changes in exactly these two modes, the position of the centre of the beam can be isolated and tracked with high precision [9]. Many other examples are being developed for precision sensing.

2.3 Statistical Properties of Classical Light 2.3.1

The Origin of Fluctuations

The classical model so far assumes that light consists of a continuous train of electromagnetic waves. This is unrealistic: atoms emitting light have a ﬁnite lifetime; they are emitting bursts of light. Since these lifetimes are generally short compared to the detection interval td , these bursts will not be detected individually. The result will be a well-deﬁned average intensity, which will ﬂuctuate on timescales > td . Similarly in most classical, or thermal, light sources, there will be a large number of atoms, each emitting light independently. In addition, the atoms will move, which leads to frequency shifts, and they can collide, which disrupts the emission process and leads to phase shifts. All these eﬀects have two major consequences: 1. Any light source has a spectral line shape. The frequency of the real light is not perfectly constant; it is changing rapidly in time. The description of any light requires a spectral distribution that is an extended spectrum, not a delta function. 2. Any light source will have some intensity noise. The intensity will not be constant. The amplitude of the light will change in time. There will be changes in the amplitude due to the ﬂuctuations of the number of emitting atoms but also due to the jumps, or discontinuities, in the phase of the light emitted by the individual atoms.

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The ﬁrst point leads to the study of spectral line shapes, an area of study that can reveal in great detail the condition of the atoms. The study of spectra is a well-established and well-documented area of research. However, in the context of this guide, we will not expand on this subject. The relevant results are cited, where required. For the analysis of our quantum optics experiments, we are not concerned as much with the spectral distribution of the light. We are more interested in the noise, and thus the statistical properties, of intensity, amplitude, and phase. Only the intensity ﬂuctuations can be directly measured in the experiments. However, with the homodyne detector, which will be explained in Section 8.1.4, we have a tool to investigate both the phase and amplitude of the light. The measurement of the statistical properties of light can be carried out in two alternative ways: either through a measurement of the ﬂuctuations themselves in the form of temporal noise traces or noise spectra or, alternatively, through the measurement of coherence or correlation functions. While requiring entirely diﬀerent equipment, the two concepts are related, and in this chapter predictions for both types of measurement will be derived. The emphasis here is on the results and the underlying assumptions and not on the details of the derivation. Many of the details can be found in the excellent monographs by Loudon [7], Louisell [10], and Gardiner [11]. In Chapter 8 it will be shown how these statistical properties can be measured and how they have been used to distinguish between classical and quantum properties of the light. 2.3.1.1

Gaussian Noise Approximation

In many of our discussions, we will be assuming that the ﬂuctuations have a Gaussian distribution over time. This assumption is certainly true for classical random systems that obey the central limit theorem. They are generally applicable for systems where the individual ﬂuctuations are small and the overall ﬁeld evolves smoothly in time. For example, the amplitude varies in small steps, and the phase evolves smoothly at the timescales at which we are observing. Thermal noise is a typical example. We will also ﬁnd that the quantum noise of laser beams does obey this limit. In these cases the variance, Eq. (2.34), is the only parameter of importance, and it gives a complete description of the ﬂuctuations. In many applications the Gaussian approximation is valid. However, there are situations in classical and quantum optics where non-Gaussian processes are important. One simple example would be a beam where the amplitude, or phase, is digitally modulated through an on–oﬀ switch or controlled by a function with several discrete values. Here the switching can be fast and abrupt. Or consider an apparatus that contains a component with a non-linear transmission function, such that the ﬂuctuations are clipped or limited at a certain level. In these cases we would see non-Gaussian ﬂuctuations. They require more complex mathematical descriptions, and the statistics would include higher-order moments, not just the ﬁrst-order variance. Non-Gaussian systems are not only interesting but also increasingly important in applications. While they are not common in metrology, we will ﬁnd that they play a major role in digital communication, in optical logic systems, and in digital data storage and data processing. We will have to consider the validity of the

2.3 Statistical Properties of Classical Light

Gaussian approximation when any of these applications are the motivation for our experiments in quantum optics. 2.3.2

Noise Spectra

A very direct way of measuring the ﬂuctuations, or the noise, of the light is to record the time history of the intensity I(t) and to evaluate the variance ΔI 2 (t) for a certain detection time. The variance of a classical value is deﬁned as ΔI 2 (t) = ⟨(I(t) − ⟨I⟩)2 ⟩ = ⟨I 2 ⟩ − ⟨I⟩2 = Var(I(t))

(2.32)

where ⟨I⟩ is the average intensity and the brackets ⟨· · ·⟩ denote averaging over a detection interval of length td . Similarly, we deﬁne the root mean square (RMS) value as √ √ RMS(I(t)) = Var(I(t)) = ΔI 2 (t) (2.33) As we have seen in Section 2.1.7, information can be carried on a light beam as small amplitude or phase modulations. The variance gives limited information about the level of noise at a particular modulation frequency and thus how successfully information can be carried. More useful is the spectral variance or noise spectrum, which gives the noise power as a function of frequency from the carrier. The spectral variance of a variable y is given by the Fourier transform of the autocorrelation function for that variable, ∞

V (Ω) =

1 ei2πΩ𝜏 ⟨y(t)y(t + 𝜏)⟩d𝜏 2π ∫−∞

(2.34)

and can be evaluated experimentally using a spectrum analyser (see Chapter 7). Most commonly we will be evaluating the spectral variances of the intensity, y = 𝛼 ∗ 𝛼, and the quadrature amplitudes, y = 𝛼 + 𝛼 ∗ and y = i(𝛼 − 𝛼 ∗ ). In many practical situations we can write our variable in the form of y(t) = y0 + 𝛿y(t)

(2.35)

where y0 is the average value of the variable and all the signals and noise are ﬂuctuations around the average, carried by the term 𝛿y(t), which is assumed to have zero mean (i.e. ⟨𝛿y(t)⟩ = 0). Under these conditions the spectrum is simply given by V (Ω) = ⟨|𝛿̃y(Ω)|2 ⟩

(2.36)

where the tilde indicates a Fourier transform. A simple example is illustrative at this point. Suppose the ﬂuctuations in our variable are made up of two parts: a deterministic signal, 𝛿ys = g cos(2πΩt), and randomly varying noise that we represent by the stochastic function 𝛿yn = f 𝜁 (t) (f and g are positive constants). The signal and noise are uncorrelated, so we can treat them separately. For the signal we use Eq. (2.36). The Fourier transform of a cosine is a pair of delta functions at ±Ω. Squaring and averaging over a small range of frequencies (say, 𝛿Ω) around ±Ω, we end up with two signals of size g 2 . For the noise it is easier to work from Eq. (2.34). We can take the

39

2 Classical Models of Light

log (ΔI2(Ω))

40

M(Ω2)

Signal SNR

Noise

Ω=0

Ω1

Ω2

Detection frequency, Ω

Figure 2.10 Example of a noise spectrum showing noise at many frequencies and two modulations at Ω1 and Ω2 . This plot has a logarithmic scale and the signal-to-noise ratio can be read oﬀ it directly if M(Ω) ≫ Var(I(Ω)).

autocorrelation of the noise to be ⟨𝜁 (t)𝜁 (t + 𝜏)⟩ = 𝛿(𝜏). That is, the noise is random on all timescales; there is no correlation between how it behaves from instant to instant. The Fourier transform of a delta function is a constant. Thus, after integration over the small bandwidth, the noise contributes 𝛿Ωf 2 to the spectrum at all frequencies. The signal-to-noise ratio (SNR) is deﬁned by SNR + 1 =

g2 𝛿Ωf 2

(2.37)

In Figure 2.10 an actual intensity spectrum with AM modulation is plotted. In this case the coeﬃcient g corresponds to the modulation M𝛼0 . The relative intensity noise RIN is the ratio of the variance to the average intensity. When stated as a function of the detection frequency Ω, the integration time td , or bandwidth, has to be taken into account. For a bandwidth of 1 Hz, it is given by RIN(Ω) = Var(I(Ω))∕⟨I⟩2

(2.38)

In classical theory it is in principle possible to make any noise arbitrarily small in the amplitude and phase such that the amount of information that can be carried by a beam of light is limited only by technical considerations. We shall ﬁnd that this is not true in the quantum theory of light. Strict bounds apply to the amount of information that can be encoded on light of a particular power. 2.3.3

Coherence

Light emitted by a pointlike source has properties closely resembling one single wave. The amplitudes at two closely spaced points are linked with each other, as part of the same wave. In contrast, the amplitudes at two very diﬀerent points or at two very diﬀerent times will not be identical. The question how far the detectors can be separated in space, or in time, before the amplitudes will diﬀer depends on an additional property of the light: its coherence. Any source of light has a coherence length and a coherence time. The amplitude at two points within the coherence length and coherence time is in phase and has the same ﬂuctuations. Interference can be observed. Light at two points separated by more than the

2.3 Statistical Properties of Classical Light

Figure 2.11 Stellar interferometer as proposed and developed by A. Michelson and F.G. Pease [12].

δθ

M3

M4 M2

M1 Lens

Detector

coherence length, or the coherence time, has independent amplitudes that will not interfere. One clear example of this concept is the technique of measuring the size of a star, which has a small angular size but is not a point source, using a stellar interferometer. The size of a distant star can usually not be resolved with a normal optical telescope. The starlight forms an image in the focal plane of the telescope, which is given by the diﬀraction of the light. The image depends on the geometrical shape and the size of the aperture of the telescope and has little to do with the actual shape and size of the object, the star. Diﬀraction limits the resolution. Only a bigger aperture of the telescope, larger than technically feasible, would increase the resolution. To overcome this limitation, A.H. Fizeau proposed the idea of a stellar interferometer, which was ﬁrst used by A.A. Michelson and F.G. Pease [12]. The concept of the interferometer is shown in Figure 2.11. In this apparatus the starlight is collected by two mirrors, M1 and M2 , which direct the light via the mirrors M3 and M4 to a telescope. There the two rays of light are combined in the focal plane. In addition we have optical ﬁlters in the instrument that select only a limited range of optical wavelengths. The mirrors M1 and M2 are separated by a distance d, and the apparatus is carefully aligned such that the distances from the focal plane to M1 and to M2 are identical. As a consequence we have interference of the two partial waves in the focal plane. An interference system will appear due to the fact that the mirrors M1 and M2 are not exactly at 45∘ to the axis of the telescope. This arrangement is equivalent to a Michelson interferometer with an optical wedge in one arm, and it measures the ﬁrst-order coherence of the light. For an extended source we get light from diﬀerent parts of the source reaching the telescope under slightly diﬀerent directions – let us say with a very small angular diﬀerence of 𝛿𝜃. Each wave will produce an interference system, but at a slightly diﬀerent location in the focal plane. This is due to the additional optical path Δl = d sin(𝛿𝜃) ≈ d 𝛿𝜃 inside the interferometer. We assume that the light from the diﬀerent parts of the star is independent. Consequently, it will interfere with light from the same part of the star, but not with light from other parts, and the intensity of the interference patterns from the diﬀerent directions is added. The interference pattern in the focal plane will be visible as long as the path

41

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2 Classical Models of Light

diﬀerence is small compared to an optical wavelength. If the separation d between M1 and M2 is increased, the total interference pattern will eventually disappear, since the individual patterns are now shifted in respect to each other. An estimate for the separation at which fringes are no longer visible is d 𝛿𝜃 = 𝜆. In this situation the two individual patterns from the edges of the star are shifted by one fringe period, and all other patterns lie in between, washing out the total pattern. A more detailed analysis predicts the change in the visibility of the interference patterns as a function of the separation d. The interferometer measures the coherence of the light. In this case the coherence is limited due to the independence of the diﬀerent parts of the star, and the coherence length is the path diﬀerence for the light travelling from diﬀerent edges of the star to the Earth. The corresponding coherence time for the entire starlight is the diﬀerence in travel time. We ﬁnd that interferometry can be used to measure the coherence length and coherence time. Michelson and Pease built an instrument based on this idea and was able to measure star diameters down to about 0.02 arc seconds [12, 13]. There are improved versions of this instrument in operation, in particular an instrument called SUSI (Sydney University Stellar Interferometer), which have achieved resolutions as good as 0.004 arc seconds. However, there are technical limitations. It is very easy to lose the visibility of the fringes, already the wavelength is restricted to an interval of about 5 nm, and even then the path diﬀerence inside the interferometer has to be kept to less than 0.01 mm. A wider spectrum, containing more light, would require an even better tolerance. It is very diﬃcult to maintain such a small path diﬀerence for the entire time of the observation, while the mirrors track the star and the separation between the mirrors is changed by up to several metres. To avoid these complications, R. Hanbury Brown & R.Q. Twiss proposed an entirely diﬀerent scheme to measure coherence. Theirs is a very elegant approach that uses intensity ﬂuctuations rather than interference, an idea that had shortly before been used in radio astronomy. In this way they founded a new discipline based on the statistical properties of light. They set out to measure the correlation between the intensities measured at diﬀerent locations rather than the amplitude and therefore avoided all the complications associated with building an optical interferometer. In their experiment they replaced the mirrors M1 and M2 (Figure 2.11) with curved mirrors, which collected the light onto two diﬀerent photomultipliers, as shown in Figure 2.12 [14]. The photocurrents from these two detectors are ampliﬁed independently and then multiplied with each other, and the time average of the product is recorded. This signal reﬂects the correlation between the intensities. This apparatus does not require the mechanical stability of an interferometer. However, some questions arise: In which way is this experiment related to an interferometer? How can it provide similar information if we have disbanded the crucial phase measurement? To answer these questions, we remind ourselves that stellar interferometers measure the spatial coherence of light, transverse to the direction of observation. At any one point the phase of the light ﬂuctuates wildly, and so does the amplitude of light from a thermal source. Generally the amplitude and the phase are not directly linked, and both ﬂuctuate independently. However, for two points within

2.3 Statistical Properties of Classical Light

d

x Multiplier

Figure 2.12 Intensity interferometer proposed and developed by Hanbury Brown & Twiss. Source: Brown and Twiss 1956 [14]. Reproduced with the permission of Springer Nature.

a coherence length, the phase and the amplitude are linked to each other. The light will interfere and the intensity ﬂuctuations are correlated to each other. For two points separated by more than a coherence length, the interference will disappear, and so will the intensity correlation. Fortunately for Hanbury Brown and Twiss, and for us, it is a characteristic property of thermal light that the coherence length over which the phase is constant is approximately the same length in which the amplitude changes very little. Thus a measurement of the path length over which the intensity correlation disappears is in most cases identical to a measurement of the phase coherence length. Hanbury Brown and Twiss measured this distance and therefore were able to determine the size of the source. Already in 1956 they measured the diameter of the star Sirius. An improved apparatus was installed in Narrabri, Australia, with an extended baseline. The detectors are moved on a circular track with a diameter of 396 m [15], and the star diameters can be determined down to 0.0005 arc seconds. How does a simple multiplier reveal the correlation between the signal? We can describe the intensity at one detector as the sum of a constant average value, ⟨I⟩, and a zero mean ﬂuctuating term, 𝛿I1 (t) by I1 (t) = ⟨I⟩ + 𝛿I1 (t). Similarly the intensity at the other detector can be written as I2 (t) = ⟨I⟩ + 𝛿I2 (t). While both ⟨𝛿I1 (t)⟩ and ⟨𝛿I2 (t)⟩ are equal to zero, for correlated waves, we get ⟨𝛿I1 (t)𝛿I2 (t)⟩ ≠ 0. If the intensities are perfectly correlated, then 𝛿I1 (t) = 𝛿I2 (t) and the time average of the product will be ⟨I1 (t)I2 (t)⟩ = ⟨I 2 + 𝛿I1 (t)2 ⟩. If the separation of the sources exceeds the coherence length, then the intensities are not correlated and the product 𝛿I1 (t)𝛿I2 (t) randomly changes sign and the average ⟨𝛿I1 (t)𝛿I2 (t)⟩ vanishes. That means ⟨I1 (t)I2 (t)⟩ = ⟨I⟩2 , which is distinctly diﬀerent to the correlated case. A reduction in the correlation signal indicates that the coherence length has been reached. A study of the correlation signal for diﬀerent receiver separations allows a measurement of the second-order coherence of the light.

43

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2 Classical Models of Light

2.3.3.1

Correlation Functions

After this introduction and motivation, we will now examine ﬁrst- and secondorder coherence, and the correlation functions that describe them, quantitatively. First-order coherence is important when two light beams are physically superimposed. As a generic example let us consider the case of two beams being combined on a 50 : 50 beamsplitter, e.g. a semi-transparent mirror, followed by direct detection of the intensity of one output beam. We allow a time delay, 𝜏, to be imposed on one of the beams before the beamsplitter. We can describe the input ﬁelds to the beamsplitter by their complex amplitudes 𝛼1 (t) and 𝛼2 (t + 𝜏). We assume the beams are in single identical spatial modes that are completely detected; thus we can ignore the spatial dependence of the amplitudes. For our purposes here we can assume that √ the action of the beamsplitter is simply to produce the output 𝛼out (t) = (1∕ 2)(𝛼1 (t) + 𝛼2 (t + 𝜏)), an equal linear superposition of the inputs. The detector produces a photocurrent proportional to the intensity of the output ﬁeld, which in turn is proportional to the absolute square of the amplitude, |𝛼out |2 . Our measurement result will then be proportional to |𝛼out |2 averaged over some measurement interval, say, td , such that 1 |𝛼out |2 dt td ∫ 1 = ⟨|𝛼1 (t)|2 + |𝛼2 (t + 𝜏)|2 + 𝛼1 (t)𝛼2∗ (t + 𝜏) + 𝛼1∗ (t)𝛼2 (t + 𝜏)⟩ 2 (2.39)

⟨|𝛼out |2 ⟩ =

where we have used ⟨· · ·⟩ to indicate the time average. Suppose that 𝛼1 and 𝛼2 both derive from the same continuous source oscillating at frequency 𝜈 such that 𝛼1 (t) = 𝛼2 (t) = 𝛼(t) exp(i2π𝜈t). Then ⟨|𝛼out |2 ⟩ = ⟨|𝛼|2 ⟩ + cos(2π𝜈𝜏 + 𝜙) ⟨𝛼 ∗ (t)𝛼(t + 𝜏)⟩

(2.40)

There is interference between the waves resulting in a modulation of the photocurrent as a function of the path diﬀerence. This basic eﬀect is the source of fringes in all ﬁrst-order interferometry including double-slit experiments. The temporal coherence of the light is quantiﬁed by the ﬁrst-order correlation function, g (1) (𝜏), deﬁned via g (1) (𝜏) =

⟨𝛼 ∗ (t) 𝛼(t + 𝜏)⟩ ⟨|𝛼|2 ⟩

(2.41)

To see that g (1) (𝜏) measures the coherence of the light, consider the depth of the modulation in the above example. This is given by the visibility , deﬁned by =

Imax − Imin Imax + Imin

(2.42)

Maximum modulation results in = 1, whilst no modulation results in = 0. For the output of Eq. (2.40), we ﬁnd |⟨𝛼 ∗ (t) 𝛼(t + 𝜏)⟩| ⟨|𝛼|2 ⟩ = |g (1) (𝜏)|

(𝜏) =

(2.43)

2.3 Statistical Properties of Classical Light

Thus, for this simple example, the ﬁrst-order correlation is equal to the visibility. If the source was a perfect sinusoid, then 𝛼 would be a constant in time and we would have g (1) (𝜏) = 1 regardless of the time delay. This corresponds to an inﬁnite coherence length. More realistically the source will have some bandwidth. Suppose that the source has a Gaussian spread of frequencies Δ𝜈 around 𝜈. This can be represented in Fourier (or frequency) space by 𝛼(Ω) ̃ = 𝛼c exp(−2π(𝜈 − Ω)2 ∕Δ𝜈 2 )

(2.44)

where 𝛼(Ω) ̃ is the Fourier transform of 𝛼(t) and 𝛼c is a constant. Taking the inverse transform gives Δ𝜈 𝛼(t) = 𝛼c √ exp(−πΔ𝜈 2 t 2 ) 2

(2.45)

for our amplitude in time space. Thus our ﬁrst-order correlation function becomes g (1) (𝜏) =

Δ𝜈 2 1 |𝛼c |2 exp(−πΔ𝜈 2 𝜏 2 ) exp(−πΔ𝜈 2 t 2 )dt td 2 ∫

1 Δ𝜈 2 |𝛼c |2 exp(−πΔ𝜈 2 t 2 )dt td 2 ∫ ) ( π (2.46) = exp − Δ𝜈 2 𝜏 2 2 a Gaussian function of 𝜏 with a width inversely proportional to the frequency spread. For a narrow frequency spread, Δ𝜈 small, we have a large coherence length, whilst for a broad spectrum, Δ𝜈 large, the coherence length becomes very short. It is this ﬁrst-order coherence length that is measured by the stellar interferometer. Now let us consider intensity correlation measurements. As a generic example we consider direct detection of the intensity of two beams that have not been optically mixed, but one of which has a variable time delay imposed on it before detection. The two photocurrents from the detectors are then sent into a correlator, which multiplies the photocurrents together. Suppose ﬁrstly that, as in our previous discussion, the two beams originate from the same source. The output of the correlator will then be proportional to the second-order correlation function, g (2) , which is deﬁned by g (2) (𝜏) =

⟨I(t)I(t + 𝜏)⟩ ⟨𝛼 ∗ (t)𝛼(t)𝛼 ∗ (t + 𝜏)𝛼(t + 𝜏)⟩ = ⟨I(t)⟩2 ⟨𝛼 ∗ (t)𝛼(t)⟩2

(2.47)

where the I are the detected intensities. From the symmetry of this deﬁnition follows the property g (2) (−𝜏) = g (2) (𝜏), and measurements need only to be made for positive values of 𝜏. Whilst the degree of ﬁrst-order coherence g (1) (𝜏) takes on values between 0 and 1 and is unity for 𝜏 = 0, the second-order correlation is not bounded above. Remembering that the variance of I is given by ΔI 2 = ⟨I 2 ⟩ − ⟨I⟩2 , we can write the second-order correlation at 𝜏 = 0 as g (2) (0) =

ΔI 2 +1 ⟨I⟩2

(2.48)

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Given that the variance obeys ΔI 2 ≥ 0, we ﬁnd that the value of the second-order coherence at zero time delay is g (2) (0) ≥ 1

(2.49)

and it is not possible to derive an upper limit. This proof is only valid for 𝜏 = 0 and cannot be applied to other delay times. The only restriction stems from the positive nature of the intensity, resulting in ∞ ≥ g (2) (𝜏) ≥ 0

for all

𝜏≠0

(2.50)

It is possible to relate the degree of second-order coherence at later times to the degree of coherence at zero delay time by using the Schwarz inequality in the form ⟨I(t1 )I(t1 + 𝜏)⟩2 ≤ ⟨I(t1 )2 ⟩⟨I(t1 + 𝜏)2 ⟩

(2.51)

For ergodic or stationary systems, two time averages depend only on the time diﬀerence, not the absolute time. For such systems we then have ⟨I(t1 )2 ⟩ = ⟨I(t1 + 𝜏)2 ⟩. Consequently we have ⟨I(t1 )I(t1 + 𝜏)⟩ ≤ ⟨I(t1 )2 ⟩ yielding g (2) (𝜏) ≤ g (2) (0)

(2.52)

For a perfectly stable wave (i.e. zero variance), the equal sign applies and g (2) (𝜏) = 1 for all delay times. Equation (2.52) means that the degree of coherence for any classical wave should always be less than g (2) (0). This is a result that has been contradicted for quantum states of light and has been used to demonstrate experimentally the quantum nature of light as shown in detail in Chapter 3.

2.4 An Example: Light from a Chaotic Source as the Idealized Classical Case In most practical cases classical light will be noisy. Only a perfect oscillator would emit an electromagnetic wave with perfectly constant amplitude and phase. There are some relatively simple models of realistic light sources. Chaotic light is the idealized approximation for the light generated by independent sources emitting resonance radiation. A practical example is a spectral lamp. In contrast, thermal light is an approximation of the light emitted by many interacting atoms that are thermally excited and together emit a broad, non-resonant spectrum of light. A practical example is a hot, glowing ﬁlament. Thermal light will be discussed in Chapter 3. The source of chaotic light is a large ensemble of atoms that are independent and thermally excited. They all emit light at the same resonance frequency. For simplicity, we assume they are suﬃciently slow so that the eﬀect of Doppler shifts can be neglected. The atoms will collide with each other; thus the waves of radiation emitted by each atom will be perturbed by collisions. The noise properties are best derived by analysing the total complex amplitude 𝛼(t) of the wave, which is the sum of the amplitudes emitted by the individual atoms. We can evaluate the coherence function of the total amplitude, and from there we can determine the

2.4 An Example: Light from a Chaotic Source as the Idealized Classical Case

intensity noise spectrum. For a large number M of atoms, the total wave amplitude is 𝛼(t) =

M ∑

𝛼j (t) = 𝛼0 exp(i2π𝜈0 t)

j=0

M ∑

(exp(i𝜙j )) = |𝛼(t)| exp(i𝜙(t))

(2.53)

j=0

where 𝜙j represents the phase of the individual contributions. For simplicity we assume that all atoms emit into the same polarization. For a large number of atoms, |𝛼(t)| and 𝜙(t) are eﬀectively randomly varying functions that describe the magnitude and the phase of the total amplitude at any time t. If the average time between collisions is 𝜏0 , one can derive [7] a probability (𝜏) that an atom is still in free ﬂight after a time 𝜏 since its last collision which is given by ) ( 𝜏 (2.54) (𝜏)d𝜏 = exp − 𝜏0 The time interval 𝜏0 is usually very short. If we assume that a single measurement takes a time larger than 𝜏0 , the coherence function can be evaluated as ⟨𝛼 ∗ (t)𝛼(t + 𝜏)⟩ = 𝛼02 exp(i2π𝜈0 𝜏) ⟨[exp(−i𝜙1 (t)) + · · · + exp(−i𝜙M (t))] [exp(i𝜙1 (t + 𝜏)) + · · · + exp(i𝜙M (t + 𝜏))]⟩ In multiplying out the large brackets, many cross terms appear, which correspond to waves emitted by diﬀerent atoms. Since the phase between the light from different atoms is random, they cancel and give no contribution to the total average. Thus ⟨𝛼 (t)𝛼(t + 𝜏)⟩ = |𝛼0 | exp(i2π𝜈0 𝜏) ∗

2

M ∑

exp[i(𝜙j (t + 𝜏) − 𝜙j (t))]

j=1

= M⟨𝛼j∗ (t)𝛼j (t + 𝜏)⟩ where 𝛼j (t) is the amplitude emitted by a single atom and thus the entire correlation is completely described by the single-atom correlation function. A single atom emits a steady train of waves until a random phase jump occurs due to a collision. If one neglects the small amount of light that is emitted during the collision (impact approximation), one gets the correlation terms ⟨𝛼j∗ (t)𝛼j (t + 𝜏)⟩ = |𝛼0 |2 exp(−i2π𝜈0 𝜏)P(𝜏) ( ) 𝜏 = |𝛼0 |2 exp −i2π𝜈0 𝜏 − 𝜏0 and the ﬁrst-order correlation function ( ) 𝜏 g (1) (𝜏) = exp −i2π𝜈0 𝜏 − 𝜏0

(2.55)

For all types of chaotic light, it is possible to derive the following relationship between the ﬁrst- and second-order correlation coeﬃcient [7]: g (2) (𝜏) = 1 + |g (1) (𝜏)|2

(2.56)

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The exact form of both g (2) (𝜏) and g (1) (𝜏) depends on the type and the statistical distribution of collisions. For illustration a numerical example is given in Figure 2.13 showing the various properties and descriptions of a ﬂuctuating light beam. Here the chaotic light is simulated by calculating explicitly the emission from 20 independent sources, all with the same amplitude. The initial phase of these sources is selected at random. The initial conditions are given in Figure 2.13a: all twenty partial waves 𝛼i (t = 0) and the total amplitude 𝛼(t = 0) are shown. The phase evolution of the emitters is simulated by adding to each emitter a change in phase that can randomly vary in the range −0.13π to 0.13π for each time step. Thus each emitter has a random development of its phase, occasionally with a phase jump. For each time interval the total amplitude can be described as one point in a polar diagram, which leads to the distribution of points shown in Figure 2.13b. The vector describes a convoluted path; it moves from one point to another seemingly at random. The density of points can be interpreted as a probability of ﬁnding a certain amplitude. The points are scattered around the origin. While the amplitudes would average out, the most probable intensity is not zero. The full time development of the magnitude of the total amplitude |𝛼(t)|, the total phase 𝜙(t), and the scaled intensity I(t) = I0 𝛼 ∗ 𝛼 are all shown in Figure 2.13c–e. The magnitude |𝛼(t)| and the intensity I(t) show dramatic ﬂuctuations, occasionally dropping close to zero. The intensity can be described by the long-term average ⟨I⟩ and the RMS value, both included in Figure 2.13d. For this numerical example, the correlation functions can be calculated directly. Equations (2.41) and (2.47) have been evaluated, with the results shown in Figure 2.13f,g. The results are very close to the predictions for a chaotic source. Over the time interval shown, the ﬁrst-order correlation functions change smoothly from g (1) (0) = 1 down to 0.15. In the same time interval, g (2) (𝜏) declines from g (2) (0) = 1.8–1.0. There are some oscillations in g (2) (𝜏), which is typical for such a short statistical sample. The analysis of the longer time sample would approach the theoretically predicted smooth decline of the values of g (2) (𝜏). The link between g (2) and g (1) given in Eq. (2.56) is clearly reproduced in this example. In order to predict the intensity noise of the chaotic source of M atoms, we consider the instantaneous intensity I(t) and the averaged intensity ⟨I⟩. We ﬁnd 1 ⟨I⟩ = ⟨I(t)⟩ = E02 𝜖0 c|𝛼0 |2 2

⟨| M |2 ⟩ |∑ | 1 | exp(i𝜙 (t))| = E02 𝜖0 c|𝛼0 |2 M j | | 2 | j=1 | | |

(2.57)

since all the cross terms average out due to the randomness of the phase functions. The total average intensity is simply the sum of the average intensity emitted by a single atom. In order to evaluate the variance, we require a value for (

1 2 ⟨I(t) ⟩ = E 𝜖 c|𝛼 |2 2 0 0 0 2

)2

⟨| M |4 ⟩ |∑ | | exp(i𝜙j (t))| | | | j=1 | | |

(2.58)

2.4 An Example: Light from a Chaotic Source as the Idealized Classical Case

5 (a) Vector sum at one time (magnified)

(b) Time development of vector

10

0

0

10 5

5 10

0

5

10

0

10

(c) Magnitude 5

0 100

(d) Intensity

50 Average Int. 0 π

(e) Phase

0 −π 1

2

(g) g(2)(τ)

(f) g(1)(τ)

0.5

1

0 0

τmax

0

0

τmax

Figure 2.13 An example of the statistical properties of chaotic light. The light is simulated as the total ﬁeld emitted by 20 independent emitters, each with constant amplitude but randomly evolving phase. Part (a) shows the individual vectors at t = 0 and (b) the resulting total phase vector for various times. The time history is shown for the magnitude (c), the intensity (d), and the phase of the ﬁeld (e). This leads to the ﬁrst-order (f ) and second-order (g) correlation function.

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The only terms that will contribute to this value are those where each factor is multiplied by its own complex conjugate. These terms result in ⟨( M )2 ( M )2 ⟩ )2 ( ∑ ∑ 1 ei𝜙j e−i𝜙j E2 𝜖 c|𝛼 |2 ⟨I(t)2 ⟩ = 2 0 0 0 j j ⟨M ⟩ M )2 ∑ ( ∑ 1 2 2 i(2𝜙j −2𝜙j ) i(𝜙j +𝜙k −𝜙j −𝜙k ) = e +2 e +··· E 𝜖 c|𝛼 | 2 0 0 0 j j≠k )2 ( 1 2 E0 𝜖0 c|𝛼0 |2 ⟨M + 2M(M − 1) + · · ·⟩ = 2 )2 ( 1 2 E0 𝜖0 c|𝛼0 |2 (M + 2M(M − 1)) = 2 where · · · represents terms that average to zero. With Eq. (2.57) we obtain ( )2 ) ( ⟨I⟩ 1 (2M2 − M) = ⟨I⟩2 2 − ⟨I(t)2 ⟩ = M M and using ⟨I(t)2 ⟩ = 2⟨I⟩2 for large M, the variance is ΔI(t)2 = (2⟨I⟩2 − ⟨I⟩2 ) = ⟨I⟩2

(2.59)

This is a simple and important result: for a chaotic source the variance of the time-resolved intensity is equal to the square of the average intensity. Calculations for more practical situations, which include Doppler broadening and realistic models of the line broadening, show basically the same result. The intensity noise spectrum will be conﬁned to the spectral linewidth. Outside the linewidth this classical description predicts an arbitrarily low intensity noise spectrum.

2.5 Spatial Information and Imaging 2.5.1

State-of-the-Art Imaging

Imaging is a widely used term. It transfers the spatial properties of an object, typically a 2D cross section, into an image that we either see directly or record in the form of paper or canvas and now more likely in an electronic format as a dataset that can be displayed or printed. Viewing something by eye or assisted with a lens, telescope, or microscope involves imaging, so is any form of recording with a camera. The quality and range of applications of imaging is improving and expanding all the time, with better spatial and temporal resolution, higher sensitivity, and the ability to transform a wider array of physical properties into images. Technology is improving at a rapid pace and with lower costs is making it much more widely accessible. A smartphone is an amazing imaging device for more and more purposes. We are surrounded by images. Figure 2.14 shows four examples from very diﬀerent situations. Figure 2.14a is a demonstration of the quantum nature of light using Young’s double-slit fringes recorded with increasing photon ﬂux. This is a teaching demonstration by T.L. Dimitrova and A. Weis for the classroom [16].

2.5 Spatial Information and Imaging

(a)

(b)

(d)

(c)

Figure 2.14 A variety of new imaging techniques: (a) teaching demonstration of Young’s double-slit experiment; (b) 3D image taken with short pulses, photon packets, and a depth-resolving fast detector; (c) scan of a brain tumour using PET-MRI, single-photon detection reconstruction, and false colour display; and (d) image of a live neurone in a rat brain created with scanning two-photon microscopy. Source: From [16–19].

Figure 2.14b shows a 3D reconstruction of an image where not only the position of the photons but also their arrival time is recorded using short light pulses, photon packets, and an optical time-of-ﬂight (TOF) technique. In this example from 2005, the depth resolution is less than a millimetre, and the resolution is 64 × 64 pixels [17]. From the data a 3D image is reconstructed, and this technology is constantly improving. Systematic progress is being made to add more pixels and enhance the speciﬁcations of the single-photon detector arrays, reaching 256 × 64 pixels by 2012 [20] and more since, by creating new applications such as TOF LIDAR imaging systems for autonomous devices. Figure 2.14c shows a ﬁctional slice through the brain of a person who has tumour. The image has been generated by multimodal PET-MRI [18]. This medical diagnostic technique uses time-resolved imaging of individual photons generated in scintillating crystals when hit by gamma rays, which are created in a nuclear decay. The image shows false colour coding, which scales with the local production rate of gamma rays and conversion into detectable photons. This signal is interpreted as an unusual activity in the brain. Finally, Figure 2.14d shows an in vivo scan of a live neurone in a rat brain. The neurone has been ﬁlled with a ﬂuorescent dye, and the dye is stimulated in a two-photon (2P) process with a focussed laser beam. The focal region is scanned in the three spatial dimensions,

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many slices were taken, and a 3D image was reconstructed. This well-known confocal microscopy technique can be used to visualize the neurone, to modify the neurone with lasers beams [19], and in the future to record the electrical activity and the information ﬂow between neurones optically. It is clear from Figure 2.14 that the generation and detection of individual photons plays a major role in these state-of-the-art examples of imaging. Photons are a concept that we require to understand how the image is created, and in particular Figure 2.14a shows explicitly that the statistics of the photons inﬂuences the quality of the images. When the intensity, and thus the photon ﬂux, is low, we can clearly see the graininess of the images. The contrast improves with more light and more photons being detected. This is barely noticeable with our own eye in everyday situations but is rather pronounced when modern detectors with high sensitivity are used.

2.5.2

Classical Imaging

Imaging refers in general to the creation of a 2D intensity and colour distribution I(X, Y , 𝜆), a function that we see with our eyes or record for posterity and for data analysis. Most of these images involve colour, which means a dependence on the spectrum of wavelengths 𝜆. With our well-trained vision system, we can see images in three dimensions (3D) I(X, Y , Z, 𝜆), and most people can distinguish more than 100 000 hues of colour, which are mixtures of wavelengths. And we do this with a reasonable time resolution of a few tens of milliseconds, which can be described by a function with a total of 5 degrees of freedom, I(X, Y , Z, t, 𝜆). We all have the capacity to deal with such a complex situation with our vision system. And we all operate devices that record and process images, nowadays in digital and pixelated format, and that extend the capability of the human eye. Central to the description is an object distribution of complex optical amplitude 𝛼(x, y) = 𝛼0 (x, y) exp(i𝜙(x, t)) with 𝛼0 (x, y) the magnitude of the optical ﬁeld and 𝜙(x, y) the corresponding optical phase across the object plane. These are the important components if we deal with coherent imaging where the phase of the wave front is stable, propagating in well-deﬁned ways, and detectable. In practice this might require working with a very narrow bandwidth Δ𝜆 or a laser beam. Note that it is possible to build special imaging devices using interferometric tricks, which can make the phase distribution visible even for light with low coherence, for example, through phase contrast imaging and microscopy or white light interferometry. The object distribution is created by the illumination of the object plane from a light source, and various eﬀects create the light distribution, including straight transmission of the light, specular reﬂection or scattering of the object, ﬂuorescence of a concentration of dye, etc. We can treat each point in the object plane as a source of radiation, which follows through to the image system according to the rules of wave propagation. For incoherent illumination imaging we only require the intensity distribution Iobject (x, y). In the detection plane we normally form the intensity distribution Iimage (X, Y ) = |𝛼image (X, Y )|2 , the intensity distribution we are going to see. To have a reasonable intensity, we can integrate the intensity

2.5 Spatial Information and Imaging

over a small area in the object plane of the size 𝛿x times 𝛿y: 𝛿x,𝛿y

Iobject (x, y) =

∫−𝛿x,−𝛿y

|𝛼object (xi , yj )|2 dx, dy

(2.60)

Optics is generally concerned with the transfer of the light to see how the amplitude 𝛼object (x, y) in the object plane is transformed into the image intensity Iimage (X, Y ). For the propagation of this image through an optical system, we should keep track of both amplitudes and the optical phase terms. This will allow us to consider both coherent imaging, including speckle, holography, and spatial ﬁltering. It is common to describe the properties of an imaging system by the mathematical link between the object and the image, in particular the image of a single point, or pixel in the object plane, a delta function 𝛿(x0 , y0 ). The mathematical link between object and image is conventionally done in two steps – ﬁrst the scaling function s(x, y) of the image, which includes magniﬁcation, stretching, and distortion, and second a function that images each object point at location (x, y) in the object plane into a point at (X, Y ) in the image plane: Iimage (X, Y ) = Iobject [s(x), s(y)]

(2.61)

In the simplest case this could be a magniﬁcation by a factor M: (X − X0 ) = M(x − x0 ) and (Y − Y0 ) = M(y − y0 )

(2.62)

where (x0 , y0 ) and (X0 , Y0 ) are the origin of the coordinate systems in the two planes and are connected by the central ray, pointing in the direction of the propagation of the wave and perpendicular to the wave front. In many cases there will be distortions, which means that s(x, y) takes on a far more complex form. This can be simulated by ray tracing where we distribute the intensity at the point Iobject (x, y) into several components emitted in diﬀerent directions and each is traced through the imaging system, see Figure 2.15. We now have access to very powerful software routines that can do this for complex lens systems. In the ideal case the rays would meet again in a point. In reality they will form a distributed function, in the simplest form a symmetric Gaussian-like distribution. The image of a single point, or delta function, leads to a point spread function psf(X,Y ) for the amplitude or PSF(X,Y ) for the intensity distribution in the image plane around the point (X, Y ). In most cases we can assume that the point spread function is constant across the image plane and just one function PSF(0,0) (dX, dY ). The complete image can then be evaluated as a convolution of the object with one constant PSF: 𝛿X

Iimage (X, Y ) =

𝛿Y

∫−𝛿X ∫−𝛿Y

Iimage (X, Y )PSF(0,0) (dX, dY )dX, dY

(2.63)

This has the eﬀect of reduced spatial resolution. A great deal of engineering and ingenuity has gone into optimizing the PSF, and this is what we gain when we purchase a superior and expensive lens system, camera, or microscope. A very powerful method to describe the imaging process has been Fourier optics. It is very practical to describe the image quality and in particular the resolution of the image by analysing the object in terms of a series of orthogonal

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y Light source Image = detection plane

x

Image system Object amplitude

Object intensity

a (x, y) Φ (x, y)

I (x, y)

Coherent

Y Yk Xj

Incoherent

X

I (X,Y, Ω)

Wave propagation Ray approximation

Detector pixels digitization N (X j,Y k)

Point spread function Resolution

Figure 2.15 Schematic of classical imaging.

spatial harmonic functions. These are normally given in dimensionless units of oscillation per full frame of the image. The spatial frequencies describe the modulation of the amplitude across the image frame, starting with a constant average amplitude F0 and then contributions Fobj,k and Fobj,k at k = 1, 2, 3, … and l = 1, 2, 3, … oscillations per image frame in both directions. This reaches up to a maximum at the Nyquist limit of k = N∕2, l = M∕2 oscillations per image where N and M are the number of pixels considered. This series of harmonics, added up in both directions and all spatial frequencies, is equivalent description to the object amplitude: 𝛼object (x, y) =

N∕2 M∕2 ∑ ∑ k=0 l=0

( Fobj,l cos

2πk(x − x0 ) xmax

(

) Fobj,l cos

2πl(y − y0 ) ymax

)

(2.64) Alternatively, we can use a continuous Fourier transform. One of the reasons for the usefulness of this transformation in optics is that the Fourier analysis can be done directly by optical means. It is the distribution in far-ﬁeld diﬀraction plane and also in the plane of a perfect lens system. For collimated input beams in the focal plane of the lens, this Fourier description provides direct access to the understanding of the limits of resolution and the impact of diﬀraction on the imaging system. This concept, made popular by scientists such as Ernst Abbe in Germany and A. J. Fresnel in France, in the nineteenth century allowed the progress of modern lens design. They showed that the high spatial frequencies are essential to maintaining the resolution. An imaging system needs large numerical aperture in order to achieve high resolution. The notion of diﬀraction-limited resolution means that the actual PSF is very close to the

2.5 Spatial Information and Imaging

Figure 2.16 Phase distribution described by the low-order Zernike polynomials. 0

Z0

–1

1

Z1

–2

Z2

Z1

0

Z2

2

Z2

PSF given by the aperture size and geometry of the system. This theory was ﬁrst developed for incoherent light where the intensity spread function is considered. There is an extensive area of optics design that describes the propagation of the wave front with a diﬀerent set of functions, known as the Zernike polynomials. This is a diﬀerent set of orthogonal spatial functions that are used to describe the wave front when it propagates through the lens system. It is speciﬁcally chosen to describe the properties of the imaging process in terms of the most common distortions that can be observed. The ﬁrst six terms are shown in Figure 2.16. These polynomials are both a mathematical basis to describe the optics and a tool for the experienced expert to describe the quality and details of the imaging system, and one by one they describe tip in x, tilt in y, defocus, stigmatism, coma, etc. A quantitative description of the Zernike polynomials can be found in many optics textbooks, such as the classic text written by Born and Wolf. There is widespread literature about the use of these mode functions in lens design and vision correction. More recently these modes of light are playing an increasing role in the dynamic compensation for wave front distortions. With the use of fast spatial light modulators, wave front detection methods, and feedback algorithms, it is now possible to compensate dynamically for the ﬂuctuations that occur in the atmosphere above ground-based telescopes. The quality of the image and optical focussing can be improved using adaptive optics techniques. They can also be used to correct distortions that occur when imaging structures inside biological samples, particular for in vivo studies of brain tissue. These techniques have become much more accessible and powerful with the advent of fast multi-pixel spatial light modulators and more and more powerful control algorithms. 2.5.3

Image Detection

Finally, the intensity in the image plane is sampled by the active components of the eye or the electronic detector. In the case of the human eye, these are the receptors, cones, and rods, which are sampling the intensity and convert them into neurological signals to be processed by the optical nerve and the brain. In technical applications we now have multi-pixel detectors, commonly now with thousands of pixels in both directions, N > 1000, M > 1000. In each pixel the amount of light is integrated across a pixel and also in time due to the ﬁnite response time of the detector, and this is converted into a photocurrent, ideally proportional to the intensity or averaged over time and stored as an electric

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charge. The abundance of photodiodes, with fast response time, charge-coupled devices (CCD), or complementary metal–oxide–semiconductor (CMOS) detectors now allows optimization for every imaging task. More than 10 megapixels and less than 10 ms exposure time are now easily achieved. For practical reasons there is a strong demand for image compression techniques; otherwise the information content could not be easily handled, in particular for video recordings. There are several algorithms, and those most widely used, JPEG for still images and MPEG for video images, are based on a projection on a set of 2D basis functions. In the case of JPEG, this is a set of wavelet functions that are related but not identical to the harmonic functions in Eq.(2.64). Normally, only the magnitude of these contributions is stored. The resolution of the stored image depends on the order to which this projection is extended. In most cases a trade-oﬀ is made between the resolution limits due to data storage capacity and the properties of the image system. The technical capacities are increasing, and with this the quality is improving, leading to the developments of new standards such as ultra high deﬁnition (UHD) and 4K. There is now a convergence of the standards that deﬁne detection and displays, resulting in very realistic images. 2.5.4

Scanning

An alternative imaging scheme is to scan a small well-deﬁned spot, for example, to project an image or to record absorption or ﬂuorescence as shown in Figure 2.17. Many diﬀerent projection displays have been developed over the last decade, initially incoherent, using various forms of modulators, lamps, or more recently LEDs. The data from the recorded image are used to control the spatial intensity modulators. As an alternative, coherent laser beam can be used, which are scanned either with mechanical mirror scanning systems, such as galvanometers or spinning prisms as used in some laser printers, or alternatively with spatial phase modulators and optical arrangements using concepts similar to holography. These types of laser projection imaging systems are now becoming widespread in commercial cinema displays with the UHD standard or even higher pixel count. An equivalent imaging technique is laser scanning for imaging in LIDAR, the optical equivalent of RADAR that uses the detection of the reﬂected light synchronized to the scanning to recreate an image. Here the scanning is over a wide angular section, and the detector might contain just one pixel, a bulk detector, or alternatively a chip with many pixels. Short pulses, fast scanning speed, and excellent synchronization are important in order to measure distances with a resolution of centimetres or even millimetres. Suﬃcient contrast is achieved by developing detectors with a high dynamic range. LIDAR used to be a scientiﬁc speciality for atmospheric research or speciﬁc defence applications. However, since 2015, the need for vision systems in autonomous vehicles has created a rapid development and expansion of this imaging technique. At the microscopic scale the equivalent scanning techniques are used in scanning beam microscopy and confocal microscopy, where the ability to focus a laser down to the size of about one wavelength with a high-quality high numerical aperture lens system is used to produce high-resolution images. The basic idea is

2.5 Spatial Information and Imaging

Laser Scanning unit

Laser beam focussed at the mask

Beamsplitter

Time recording, synchronized with scanner

Object plane, with mask

I1 ( t)

or fluorescent dye

Bulk detector with one element

Figure 2.17 Schematic of a scanning imaging system. In this example a mask is illuminated with a scanned focussed laser beam, and all the intensity is recorded with a large single-element bulk detector. Alternatively a structure ﬁlled with a ﬂuorescent dye could be placed in the object plane, and all the light emitted at the focus is recorded by this detector. A time series is recorded and stored as a stack of slices from which a 2D or 3D image can be reconstructed.

that the sample under investigation contains a ﬂuorescent dye that is excited by the laser beam. One dye molecule requires a single photon with matching energy for excitation and emits one other photon, with lower energy, into an arbitrary direction. This ﬂuorescent light is detected with a large numerical aperture lens, in order to increase both sensitivity and resolution. The spatial resolution is given by the size of the laser beam, typically a Gaussian beam with a focus inside the sample. However, the ﬂuorescence can come from many parts of the beam, and this technique works well with a thin sample. In order to produce more localized images, 2P excitation is used. Here the probability of excitation is proportional to the square of the beam intensity, or local photon ﬂux, and thus the ﬂuorescence is strongly concentrated in the location of the laser beam focus. In this way spatial resolution close to the diﬀraction limit can be achieved in the x and y direction, with typically 10 times lesser resolution in the z direction. One elegant technique to scan the beam is holographic projection with a spatial phase modulator that allows fast scanning in all three dimensions. After recording and storing multiple x, y slices, each with a speciﬁc value of z, it is possible to reconstruct a complete 3D representation of the ﬂuorescence in the sample, which can be stored and displayed from any point of view. This technique has widespread use in biology and in particular neuroscience, as shown in ﬁgure 2.14d where one individual neurone is being imaged in a living

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sample of brain tissue. This is very powerful and in future can be extended to functional imaging of the activities inside an organ or a cell. For example, with the help of a dye or other substances, which are sensitive to the local electric ﬁelds, we can not only display the spatial structure of the neurone but also its electrical activity. The action potential and information ﬂow can be imaged. Considerable progress is being made in developing more and more advanced image scanning techniques. 2.5.5

Quantifying Noise and Contrast

Spatial and temporal resolution is one speciﬁcation of image quality. How can we describe the quality of the object and image distributions in terms of clarity, noise, and contrast? One universal approach is to analyse the SNR. Let us consider this ﬁrst for a scanned image, where the entire information is contained in a long time series. The total time signal can be described, after Fourier analysis, by a set of signal components at modulation frequencies Ωj with amplitudes 𝛼j for each component. In order to describe the quality, we also need to consider the random ﬂuctuations or noise in the time series. This could be due to statistical eﬀects in the light intensity or due to noise generated in the detection device. To make the eﬀect of noise visible, we compare recordings of the image, which includes both signal and noise, [S + N], and of the noise alone. Assuming that the noise is Gaussian in its statistical properties, we can quantify the noise by evaluating the variance VN for a long time interval. We might ﬁnd noise that is stronger at certain frequencies Ωj , or we might ﬁnd white noise with the same strength at all frequencies. Then we can apply the same process to the signal and determine [S + N](Ωj ). For a strong high contrast image, each simple harmonic signal component at Ωj contributes a signal that is proportional to the square of the modulation amplitude. For such a large modulation amplitude 𝛼j (Ωj ), we can approximate the SNR as SNR(Ωj ) ≈ Var([S + N](Ωj ))∕Var(N(Ωj )) ∝ |𝛼j (Ωj )|2

(2.65)

This is used as a description of the quality of the time series, and once we place the scans together in an image, it can also describe the quality of the 2D images. For a very simple example, such as the fringes shown in Figure 1.1a, which have a cosine intensity distribution, we can obviously use this process, now replacing the time with the spatial coordinate x and the time frequency Ω with a spatial frequency 𝜈. The SNR(𝜈) for such a single harmonic spatial function describes the quality, or contrast, of this particular image of the modulation. Let us expand this to all other spatial frequencies and consider the SNR separately for each spatial Fourier component 𝜈j resulting in a spectrum SNR(𝜈, 𝜇), where 𝜈 and 𝜇 describe the spatial frequencies along the x and y axes. We can describe the quality of the object function by such a 2D contrast function and is linked directly to the contrast or visibility of the object. The formal process includes an extension of Eq. (2.65) into two dimensions. The equivalent of white noise required is now independent of the position anywhere in the image plane. Towards high spatial frequencies 𝜈, 𝜇, the signal will get smaller, and the contrast will drop oﬀ due to diﬀraction eﬀects. Setting a practical minimum value

2.5 Spatial Information and Imaging

for the SNR(𝜈, 𝜇), we can deﬁne the spatial resolution of the object function. In practice this is tested with a variety of object functions, for example, in the various forms of test charts, which are widely used in photography or optical testing. They include features of smaller and smaller size, corresponding to a series of spatial frequencies. An important speciﬁcation in the design, manufacture, and testing of imaging systems is the transmission of the SNR as a function of spatial frequencies. Low-quality optics lowers the contrast and the resolution. 2.5.6

Coincidence Imaging

There is also an alternative process of imaging where two beams of light correlated in both intensity and direction of propagation are used and the image is formed through coincidence detection. Each beam is directed onto separate detectors, in this case by the use of a beamsplitter and mirror (see Figure 2.18). One detector (2) has spatial resolution. The other is a bulk detector with only one spatial element and records the total intensity of the beam (1). The object to be imaged is inserted into the beam (1). Is it possible to obtain spatial resolution despite the use of a bulk detector and the imaging system being on the wrong beam? By recording only the correlated component of the photocurrents between the two photodetectors, a clear image can be created. By recording the coincidence counts for all outputs, simultaneously or in a scanning sequence, the information from the bulk detector (1) will provide the required information of how much light has been absorbed in any speciﬁc location of the mask. This might appear to be a complex arrangement. However, it has the major technical advantages that it might not be possible to build the imaging system Mirror

Imaging system Detector 2 with spatial resolution Y Yk

Beamsplitter

ed lat re m and r Co bea ns s tio itie ec ns dir inte

Xj

X

I2 (X,Y, Λ1)

Object plane, with mask

I1 ( Λ2)

Detector 1 with one element

Correlation processing

Figure 2.18 Schematic of coincidence imaging using pairs of optical rays with correlated directions, one single-element bulk detector, and one detector with spatial resolution.

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around the mask, for example, because it is inside a small container, in a vacuum system, or an environment that does not allow direct optical access. This type of imaging has been given the popular name of ghost imaging, presumably because there is nothing tangible in front of the conventional imaging system. It has been shown that this system works with coherent light such as laser beams, as well as with light that has low coherence and with light that has thermal ﬂuctuations. The real strength comes once we manage to have beams with diﬀerent wavelengths fully correlated. For example, beam 1 could be at a wavelength, such as infrared, where no detector with spatial resolution is available, while beam 2 can be conveniently imaged. In all these situations we can make use of the rules of classical optics to synthesize an image from the coincidence counts. First, demonstrations have been reported [21], and the goal is to create images of otherwise invisible objects, for example, clouds of gas such as methane or carbon dioxide that have absorption lines only in the infrared where it is diﬃcult to build tunable light sources. 2.5.7

Imaging with Coherent Light

Note that in the case of coherent imaging, for example, with a monochromatic laser beam, the propagation is slightly more complex due to the diﬀraction processes than the predictions from ray optics. This would require full analytic solutions and can also be summarized in the form of propagating closed forms of laser modes, such as the TEM modes shown in Figure 2.2. One frequent situation is the focussing of a laser beam into a second waist at locations that do not simply follow the normal lens equations used in incoherent illumination. A second area where coherence will show dramatic eﬀects is the illumination and imaging of irregular objects, such as rough surfaces or liquids with scatter centres, which will create a very complex diﬀraction pattern known as speckle. This pattern is predictable but also responds to the slightest changes in alignment and thus appears unsteady due to the inﬂuence of mechanical vibrations. Thus speckle is normally seen as an unwanted side eﬀect of coherent illumination. 2.5.8

Image Reconstruction with Structured Illumination

Images can be formed using a single-pixel photodetector by illuminating the object with a range of speciﬁcally chosen patterns and recording the reﬂection from each of these patterns. The general principle is that the total reﬂection will be diﬀerent for each pattern and that with a clever algorithm it should be possible to reconstruct the spatial details of the object. Clearly great care has to be taken in selecting the set of patterns, which should be orthogonal in the sense that the information created by each individual pattern is as independent from the others as possible. Various sets of patterns have been tested over time; starting in the 1970s [22], the algorithms have been optimized, and it is now possible to create useful images. The system can be portable and fast. An impressive demonstration comes from the team led by Miles Padgett at Glasgow of the real-time detection of methane gas leaks using a single-pixel camera [23]. In this case 256 diﬀerent masks are used, and imaging with an equivalent of

2.5 Spatial Information and Imaging

16 × 16 pixels at 25 frames per seconds is demonstrated, all with a single-pixel detector. With improved computing power, many more such applications of image reconstruction will emerge [24]. 2.5.9

Image Analysis and Modes

In many situations the imaging process is not completed when the intensity distribution, or photon ﬂux, has been recorded. The image is analysed for its content. For our own vision we have a very sophisticated processor for this in our optical nerve and the brain. This can analyse images with remarkable speed and accuracy. In technical terms we can analyse one image through its characteristic features, for example, we can enhance or suppress the contrast or certain colours or make edges more visible. This can be done by processing the photocurrents from the various pixels of the detector. It is also possible to ﬁlter out certain shapes in the image, which we could achieve by adding the photocurrents from a group of pixels. The analogue equivalent is that we consider a certain mode of light, which correspond to intensity distribution and wave front shape. We can expand the actual image at the detector plane over a whole set of such modes. A mode can describe a speciﬁc feature we are looking for. It can be a local feature, for example, a face or a feature in a medical image, which can be represented by a speciﬁc weighted sum of pixels. Or it could be a recurrent feature, such as regular set of windows in the image of a high-rise building or a speciﬁc set of tracks in the image of an electronic circuit. The detection mode is a very versatile concept. Once deﬁned we can track it from image to image. Face recognition, ﬁngerprint recognition, and similar biometric schemes are one form of image analysis. It can be implemented directly by combining the outputs of many pixels in a clever way. 2.5.10

Detection Modes and Displacement

This approach allows us to detect spatial changes within the detection plane. We can determine the movement of the mode, how far and how fast. We can measure how it rotates, expands, or shrinks. That means we can detect changes in specifically designed degrees of freedom by comparing images or analysing a whole string of images recorded like a video stream. In this way spatial measurement with beams of light can be seen as a special form of imaging by deﬁning a speciﬁc mode u(X, Y ) and tracking its changes. One of the simplest examples is to use a round beam of light. This is our mode that we place on a detector with two pixels, or two groups of pixels, and we measure the diﬀerence of the photocurrents. The mode that comes from the object is round. The detection mode is a plane cut in half. If the beam moves, we will get a strong signal for the displacement as the symmetry between the top and bottom half is broken. This simple device can measure the displacement with remarkable accuracy and high intensities; this can be much less than the diﬀraction limit, only limited by the SNR. In the language of modes, we can deﬁne a mode that represents the displacement of the beam, with the new mode for displacement in X deﬁned as uΔX (X, Y ) = 𝜕u(X, Y )∕𝜕X.

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2 Classical Models of Light

The device should now have a detection mode udet , which is matched well to uΔX . The split detector mentioned above is a good but not perfect approximation. Let us use the example of a coherent laser beam to derive the optimum detection scheme. The beam produced by a perfect laser can be described by the mode that is shaped inside the laser by the resonator where a set of mirrors build up a powerful internal standing wave selectively ampliﬁed by the gain of the pumped laser medium. The resulting output beam is the TEM0,0 mode (see Figure 2.5), the lowest order of the set of Hermite–Gaussian modes described in Eq. (2.11) and shown in Figure 2.5. A beam perfectly aligned on the centre of the split detector beam is described by 𝛼0,0 (x, y, z), with the beautiful property that it is transformed into another TEM0,0 mode when imaged with a lens, in the limit of the paraxial approximation. We can easily propagate a laser beam through an imaging system and will have a TEM0,0 mode in the detection plane. When we now consider a shift of the beam, using Eq. (2.11), we ﬁnd that the ﬁrst derivative 𝜕𝛼0,0 ∕𝜕x = 𝛼1,0 . This means that all the information of the movement of the beam is contained in a TEM1,0 mode. Measuring this mode is the best and complete way of measuring the shift of the beam. The optimum device is a detector that has the detection mode TEM1,0 . Similarly we can detect shifts in the y direction. We can expand this idea further and ﬁnd that each of the TEMl,m modes, shown in Figure 2.2, represents one independent degree of freedom, such as tilt and expansion. Generalizing this concept further, we can see that imaging can be interpreted as a transformation from one optical mode into another. We start with an illumination mode that enters the object plane and creates the mode uobj that contains the amplitude and phase distribution. This is a new mode, which is projected through the imaging system to the detector. There we have a detector mode udet , which is deﬁned by the shape of the detector and its pixels and the wiring and combination of pixels that is used. The detection process is now a projection, or overlap, of the image of uobj onto udet . With a properly chosen detection mode, we can measure the change of the spatial properties of uobj in one well-speciﬁed degree of freedom. It is quite possible to do this for several degrees of freedom at the same time, by monitoring several detection modes, ideally all deﬁned orthogonal to each other. A straightforward example is displacement and expansion in two orthogonal directions, or rotation in opposite directions [9].

2.6 Summary We have now a set of tools to describe the properties of a beam of light: it is a mode with a set of DC values and we can describe the power that can be measured both in space and in time. In addition, we can communicate information with the beam through modulations using the AC sidebands. Communication of data is now the most widespread use of light; it has led to a whole industry of photonics. Noise limits the quality of communication. We can describe the ﬂuctuations of the light as a spectrum or alternatively through correlation functions. This noise is intrinsic to the way the light is generated, and we distinguish between incoherent light, such as sunlight, coherent light from a laser, and all cases in between.

References

Correlation measurements have opened new ways in determining the properties of the light, with many applications ranging from astronomy to microscopy. Finally, we have so many applications of such a diverse range of techniques in imaging. Initially in the domain of human sight and photography, this is expanding rapidly with the ever-improving technology for detection, image processing, and reconstruction. Optical engineering has found many alternative approaches to image the very large and the very small and many ways to show us the dynamics of objects and eﬀects and processes that were invisible before [25]. The ﬁeld of classical optics is wide, versatile, and beautiful. However, all of this relies on photons and their special properties.

References 1 Kogelnik, H. and Li, T. (1966). Laser beams and resonators. Appl. Opt. 5:

1550. 2 Siegman, A.E. (1986). Lasers. University Science Books. 3 Saleh, B.E.A. and Teich, M.C. (2007). Fundamentals of Photonics, 2e. Wiley. 4 Abramovitz, M. and Stegun, I. (1972). Handbook of Mathematical Functions.

New York: Dover. 5 Askin, A. (1970). Acceleration and trapping of particles by radiation pressure.

Phys. Rev. Lett. 24: 156. 6 Askin, A. (1978). Traping of atoms by resonance radiation pressure. Phys. Rev. 7 8 9

10 11 12 13 14 15 16 17

Lett. 40: 729. Loudon, R. (1973). Quantum Theory of Light. Oxford: Oxford University Press. Hänsch, T. (2005) Nobel Prize Lecture 2005. http://www.nobelprize.org/nobel prizes/physics/laureates/2005/hansch-lecture.html. Delaubert, V., Treps, N., Harb, C.C. et al. (2006). Quantum measurements of spatial conjugate variables: displacement and tilt of a Gaussian beam. Opt. Lett. 35: 1537. Louisell, W.H. (1973). Quantum Statistical Properties of Radiation. New York: Wiley. Gardiner, C.W. (1991). Quantum Noise. Berlin: Springer-Verlag. Michelson, A.A. and Pease, F.G. (1921). Measurement of the diameter of 𝛼 orionis with the the interferometer. Astrophys. J. 53: 249. Pease, F.G. (1931). Interferometer methods in astronomy. Ergeb. Exakten Naturwiss. 10: 84. Brown, R.H. and Twiss, R.Q. (1956). Correlation between photons in two coherent beams of light. Nature 177: 27. Brown, R.H. (1964). The stellar interferometer at Narrabri observatory. Sky Telescope 28: 64. Dimitrova, T.L. and Weis, A. (2008). The wave particle duality of light: a demonstration experiment. Am. J. Phys. 76: 137. Niclass, C. and Charbon, E. (2005). A single photon detector array with 64 × 64 resolution and millimetric time depth accuracy for 3D imaging.

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18 19 20

21

22 23 24 25

In: ISSCC. 2005 IEEE International Digest of Technical Papers. Solid-State Circuit Conference, 364. https://doi.org/10.1109/ISSCC.2005.1494020. Charbon, E. (2014). Single photon imaging in complementary metal oxide semiconductor processes. Trans. R. Soc. A 372: 20130100. Go, M.A., Choy, J.M.C., Colibaba, A.S. et al. (2016). Targeted pruning of neuron’s dendritic tree via femtosecond laser dentrotomy. Sci. Rep. 6: 19078. Niclass, C., Ito, K., Soga, M. et al. (2012). Design and characterisation of a 256×64-pixel sinle-photon-imager in CMOS for a MEMS-based laser scanning time-of-ﬂight sensor. Opt. Express 20: 11863. Kim, C.C. and Kanner, G. (2010). Infrared two-color ghost imaging using entangled beams. Proceedings Volume 7815, Quantum Communications and Quantum Imaging VIII; 781503. Pratt, W.K., Kane, J., and Andrews, H.C. (1969). Hadamard transfer image coding. Proceedings of IEEE57, 58. https://doi.org/10.1109/PROC.1969.6869. Gibson, G.M., Sun, B., Edgar, M.P. et al. (2017). Real-time imaging of methane gas leaks using a single-pixel camera. Opt. Express 25: 002998. Altmann, Y. et al. (2018). Quantum-inspired computational imaging. Science 361: 2298. SPIE International Year of Light (2015). Celebrating Light.

Further Reading In this area is numerous. Highly comprehensive reference books are: Born, M., and Wolf, E. (1959). The Principles of Optics. London: Pergamon Press. Mandel, L., and Wolf, E. (1995). Optical Coherence and Quantum Optics. Cambridge: Cambridge University Press.

65

3 Photons: The Motivation to Go Beyond Classical Optics Although classical optics is very successful in describing many aspects of light, it is not all inclusive. Many fascinating eﬀects are due to the quantum nature of light; they are the foundation of quantum optics. This chapter introduces the concept of the photon as the smallest detectable quantity of light and develops some of the basics required for modelling quantum optics experiments.

3.1 Detecting Light So far we have considered light to be just one form of electromagnetic waves, much like radio waves. The detection of radio waves and of light is fundamentally diﬀerent. Radio waves are detected with an antenna, which is a macroscopic structure made out of many atoms. The waves induce an electric ﬁeld in the antenna, a movement of many electrons. This leads to an electric signal that can be ampliﬁed and processed. The signal is proportional to the magnitude of the electric ﬁeld. In contrast, the detection of light relies on the interaction with individual atoms. The microscopic, or atomic, properties of the material of the detector determine the detection process. A photodetector does not measure the ﬁeld; it cannot follow the frequency 𝜈 of the oscillations, typically of the order of 1015 Hz. It can, however, determine a transfer of energy from the radiation ﬁeld to the atoms in the detector. And from this process we can conclude properties of the light intensity. What are the experimental details of the process of photodetection? One fundamental fact is the resonance-like property of the interaction. Atoms interact only with radiation that matches their eigenfrequencies. The absorption and emission of light results in very speciﬁc spectra. For free atoms, as those in a gas, these spectra have a resonant structure; they consist of several narrow, well-deﬁned lines. Thus, free atoms are not suitable for a general-purpose detector. Consequently, we use a material with many interacting atoms with a broad absorption spectrum. But the fact remains – individual atoms absorb the radiation that matches their individual eigenfrequencies. One well-established detector is the photographic plate. The atoms in the photosensitive emulsion can absorb light. One individual absorption process can lead to the chemical change of an entire grain in the emulsion and therefore A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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3 Photons: The Motivation to Go Beyond Classical Optics

create one dark dot on the plate. This technique has one major technical problem: long exposure times. Not every photon incident leads to the change of a grain. It requires a lot of intensity to expose a plate. It is also far more convenient to deal with electric signals. Thus we use the photoelectric eﬀect. Very early experiments with this eﬀect showed that radiation interacts with such a detector by the absorption of multiples of the energy h𝜈 [1] (Figure 3.1). The probability det per unit time of detecting one individual photoelectron averaged over the detection time interval tmeas scales with the integral of the intensity det =

D h𝜈tmeas ∫0

tmeas

I(t ′ )dt ′

(3.1)

This equation applies to any form of the photoelectric eﬀect. The response time is very similar. The conversion eﬃciency D depends on the frequency of the detected light and on the material used. Each individual detection process results in one individual electron. This is not suﬃcient to produce a signal that can be processed. Several techniques have been invented to turn one electron into a measurable pulse of many electrons. This has led to the development of a range of modern photodetectors, such as photomultipliers, photodiodes, and charge-coupled devices (CCDs) and complementary metal-oxide semiconductor (CMOS) sensors [2]. For technical detail see Chapter 7. Whilst diﬀerent in their technology, they all produce a photocurrent that is proportional to the light intensity, and the detector circuitry averages over a time tmeas , which is much longer than a cycle of the oscillation. The link between the photocurrent i(t) and intensity I(t) is given by i(t) =

D′

tmeas

tmeas ∫0

I(t ′ )dt ′

(3.2)

where D′ describes the sensitivity and the gain of the detector. Detection times as short as 10−11 seconds have been achieved. Such a detector can only record changes of the intensity that occur on timescales longer than tmeas . The electric signal will contain a direct current (DC) term, proportional to the long-term average intensity, and alternating current (AC) components with frequencies up to 1∕tmeas . Please note that in our description so far, there has been no need for the quantization of the radiation. We need a quantized detection process, but Eq. (3.1) could also be used for a classical electromagnetic ﬁeld. And indeed, at high intensities, such a classical interpretation is valid. However, at low intensities, we ﬁnd experimental evidence that the light ﬁeld has to be quantized and that we need the Photo-current i(t)

Beam of light

t Sequence of photons

Figure 3.1 Photoelectric detection.

3.1 Detecting Light

concept of particles of light, of photons. For example, consider a situation where a pulse of light contains energy less than h𝜈. In this case the classical equation (3.1) would predict a probability that is larger than zero, whilst the experimental evidence shows that there is no detection at all. This can be taken into account by choosing a quantized theory of the detection process [3] that replaces Eq. (3.1) with t

meas D′ ′ ̃ † A⟩dt ̃ ⟨A (3.3) h𝜈 tmeas ∫0 ̃ and A ̃ † are operators that describe the complex amplitude of the electric where A ﬁeld and its conjugate. We will ﬁnd that there are some peculiar features of these operators compared with the classical functions. For example, the sequence of the operators in the integral is important. The sequence in Eq. (3.3) corresponds to the normal ordering that has been chosen such that absorption of photons is the detection process and a diﬀerent ordering would lead to completely unrealistic eﬀects. This quantum model is described in Chapter 4. One motivation for introducing quantized radiation stems from the question: how quickly, or with what response time, can low intensity light pulses be detected? In a classical theory there would have to be a delay in order to accumulate suﬃcient energy in one atom to release an electron. The response time tmin for light with a photon ﬂux density of N photons/m2 s and for an atom with cross section A m2 can be estimated as tmin = 1∕(NA). This is the time required for the energy equivalent to one photon to be transmitted through one atom; it would be a minimum for the response time. Is this realistic? Let us consider sunlight as an example. Green light, with a wavelength of 500 nm, is selected with an optical ﬁlter that transmits about 1% of the entire spectrum. This green light has a photon ﬂux density of about 1019 photons/m2 s as can be calculated from Planck’s radiation law using the well-known data for the sun, with a surface temperature of 6000 K and diameter of 32 arc minutes. Assuming an atomic cross section of 10−18 m2 , the detection time is about 0.1 second. For less intense beams the time would increase to fantastic values, whilst we know that our eye can be fairly fast even at intensities much lower than full sunlight. Actually, we can almost detect individual photons with our eyes. The latest studies have shown that we can sense the presence of individual photons and that the actual sensitivity depends on the actual conditions, including what has been seen immediately beforehand [4]. In order to see detectable features, the sensitivity of the human eye for green light can be estimated as about 5 × 10−18 W, which corresponds to a photon ﬂux of about 12 photons/s into the eye or a photon ﬂux density of about 109 photons/m2 s . This low level is suﬃcient to release photoelectrons, and the detection process certainly does not take longer than one second. If the simple classical model was correct, we could not see low intensity images, such as the impressive display of star patterns in a dark night. Furthermore, the vision process would be entirely diﬀerent to our experience. The main diﬀerence between bright and dim objects would be the time it takes to see them; the detected intensity would always appear to be similar. Thus we conclude that our personal experience of vision requires the concept of photons.

det =

67

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3 Photons: The Motivation to Go Beyond Classical Optics

It was pointed out that the simpliﬁed model of the atom interaction overestimates the response time. A faster response can be estimated by assuming that atoms can draw energy from a wider area. This would be the case when the atoms are separated or when only some of the densely packed atoms contribute to the detection process. Estimates for such a situation have been made for classical oscillators by Lorentz [5] and can be combined with the assumptions of modern atomic theories [6, 7]. These calculations show the ability of atoms to draw energy from a wide area. One physical interpretation of this eﬀect is that the incoming ﬁeld induces a dipole moment in the atom, which in turn emits a wave that destructively interferes with the incoming wave. In this way the energy is concentrated in the atom. For the case of sunlight, a response time of about 10−7 seconds, and for the sensitivity limit of the eye, a response of 10−1 seconds can be calculated. Whilst these values are short enough to be comparable with the response time of the human eye, they are still orders of magnitudes too long compared with experiments. Fast response times were measured even before the advent of modern short pulse laser technology. In 1927 Lawrence and Beams [8] used an electric spark to generate a light pulse, which was detected with a photocathode that could be gated using an additional internal grid. The timing of the grid could be delayed by propagating the electric gating pulses along various lengths of cable, and it was observed that the photoelectrons from the detector appeared simultaneously with the light pulse, within an accuracy of 10−9 seconds. Forrester et al. [9] used the detection of beat signals between two monochromatic thermal sources and concluded that the detectors can follow the electromagnetic ﬁeld with accumulation times less than 10−10 seconds. Nowadays even shorter pulses, femtoseconds, and higher modulation frequencies are routinely used in optical instruments. All these observations show the need for a concept of light as a quantized system – the need for photons.

3.2 The Concept of Photons We really need more than one concept of photons. So far we have taken the view of an experimentalist who would talk about detecting photons from a propagating beam of light. Photons have properties quite diﬀerent from those of classical particles. They have energy h𝜈, zero rest mass, momentum ℏk, and spin ℏ, and they are non-interacting particles following the Bose statistics. A laser beam contains a large number of these particles (see Table 3.1), and the measurement of the beam intensity corresponds to a measurement of the ﬂux of photons. The photons that were created by the emission process, that is, the de-excitation of the atoms via stimulated or spontaneous emission, are destroyed in the detection process. This in turn creates an electron–hole pair in the detector material. For the experimentalist photons are particles that can be localized, for example, at the detector. The experimentalist takes the consequence of the detection, the electron created or the grain in a photographic plate that is darkened, as the evidence of the photon. Figure 3.2 shows two examples of images that are typical for the detection of individual photons in one laser beam at very low light intensity. In this case the

3.2 The Concept of Photons

Figure 3.2 Two images from multi-pixel sensors, such as a CCD or CMOS camera sensor, with low intensity illumination. Each image has an array of individual detectors, or pixels, where the individual photons trigger electrical events that are recorded and displayed. The number of photons recorded in each pixel is colour-coded, from no photon as black to several photons as red or organ yellow to even higher numbers as green. These images show graininess or randomness, with no speciﬁc pattern. The images would be similar for thermal and laser light. The information can be used for imaging or for the measurement of photon statistics or could be added up as a total photon ﬂux. Table 3.1 Examples of the photon ﬂux and the photon density of typical optical ﬁelds.

Type of light

Intensity I (W/m2 )

White light (T = 6000 K)

103 4

Spectral lamp

10

5

Electric ﬁeld E (V/m)

Photon density (m−3 )

Photons/mode

103

1013

10−4

3

14

10−2

15

3 × 10

4

10

CW laser

10

10

10

1010

Pulsed laser

1013

108

1023

1018

For simplicity it is assumed that all ﬁelds have the same wavelength of 𝜆 = 500 nm, and thus all photons have the same energy h𝜈 = 2.510−19 J. In the case of the pulsed laser, the photon density inside the pulse has been evaluated. The table shows that we can consider a mode to contain a large number of photons only for laser beams.

detector array has pixels that are smaller than the size of the laser beam, which is in the form a Gaussian single-mode beam. A small faction of the beam illuminates the array of pixels shown here, each creating a number of detection events during the measurement interval chosen. The two images could be representative of two samples in time at the same location with the same average intensity or alternatively of two spatial samples in the one laser beam recorded at the same time. In either case the selection of events is random, and all the photons belong to the same mode. There is one probability distribution for detection for the entire mode. It is tempting from this point of view to envisage a beam of light as a stream of photons travelling at the speed of light. But this is incorrect and, as we shall see in the following sections, cannot successfully explain many experimental results. A diﬀerent concept for photons, perhaps more familiar to a theoretician, is needed to describe the propagation of light. Here we would talk about the photon number of a mode, where ‘mode’ refers to the radiation ﬁeld modes described in Section 2.1. Each mode has energy states like a harmonic oscillator, and the

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3 Photons: The Motivation to Go Beyond Classical Optics

state of excitation of a mode is said to be the number of photons in the mode. In this concept photons are distributed, they do not exist at any speciﬁc location, and they are a property of the entire mode. The only possible processes are the annihilation and creation of a photon in which the entire mode is aﬀected, and to describe this we need the formal rules of quantum mechanics. The wonder of quantum mechanics is that the apparently contradictory points of view – localized, particle-like emission and detection but delocalized ﬁeld-like propagation, often referred to as wave–particle duality – can coexist. A laser beam can be described by a single mode propagating from the laser, with physical properties much like the Gaussian mode in Eq. (2.8). The photon number of the laser mode is linked to the ﬂux of photons in the external laser beam, which can be detected. Laser modes typically contain a large number of photons, and the ﬂux is high. For thermal light, on the other hand, the number of possible modes can exceed the photon number, and on average there will be less than 1 photon in any mode (see Table 3.1). A more detailed review of the concept of photons can be found in books by H. Paul [7] and R. Loudon [10].

3.3 Light from a Thermal Source We will now look at the example of light emitted by a broadband thermal source. We will assume that the emission properties of the atoms are quantized, as ﬁrst proposed by M. Planck [11], who tried to explain new observations of the black-body radiation or Schwarzkörperstrahlung, which had been made at the newly founded Physikalische-Technische Reichsanstalt (PTR) in Berlin, precursor to the modern PTB (Physikalische Technische Bundesanstalt), by O. Lummer and E. Pringsheim in 1900. These beautiful technical and theoretical achievements deserve description in some details. See also [12]. In 1885 W. Wien and O. Lummer followed an idea by G. Kirchhoﬀ and proposed to build an isothermal resonator that should produce inside the cavity the so-called black-body radiation. The light can be observed through a small hole in the cavity and was expected to match the predictions of the new Wien’s law [13]. In 1900 the team of O. Lummer, F. Kurlbaum, and H. Rubens constructed at the PTR a unique electrically heated resonator that could reach a temperature of 1600 ∘ C, and they recorded the radiation spectrum from 1 to 6 μm, as shown in Figure 3.3), using a new bolometer design. With even more technical developments of the radiation detectors, they were able to make remarkable improvements and extended the range of measurements to 18 μm with suﬃcient precision. The data showed a deviation from Wien’s radiation law of up to 50% [14]. The experimentalist reported the result personally to Max Planck, who was Kirchoﬀ ’s successor at the Universität in Berlin. On the very same day, Planck discovered an empirical formulation of the radiation law. He reported his results a few days later at a meeting of the DPG (Deutsche Physikalische Gesellschaft) following lecture of the experimentalists. Together, they found a close ﬁt to the observed radiation data. Two months later, in December 1900, Planck presented a full theoretical deduction of his equation using an atomistic–probabilistic description of entropy introduced by L. Boltzmann,

3.3 Light from a Thermal Source

E 140 130 120 110 1646 K (1653 K)

100 90 80 70 60

1460 K

50 40 30

1259 K

(b)

20 1095 K

10 (a) 0

998 K 904 K 723 K

1

2

3

4

5

6

λ

Figure 3.3 (a) Absolute measurements of the spectral energy distribution carried out in 1900 by O. Lummer and E. Pringsheim on the electrically heated black body over a spectral range of 1–6 μm. The broken line was calculated with Wien’s radiation law. The continuous line represents the measurements that increasingly deviate from Wien’s law with higher temperature and increasing wavelength, but were in good agreement with Planck’s radiation law. (b) The apparatus is also shown. Source: From PTB historical records and summary by J. Hollandt [12].

which he had rejected until then. In an ‘act of desperation’, Planck only allowed certain discrete states of energy. Using the experimental data he was able to determine the free variables in his equation. Whilst now regarded as the hour of birth of quantum mechanics, neither Planck himself nor his audience were aware of the widespread consequences of this discovery of Planck’s constant in the new century ahead. The term photon was not used at the time. We will use it to describe the fact that the light is emitted and absorbed in ‘lumps’ or ‘quanta’ by atoms, molecules, and solid materials. Using a semi-classical approach, light emitted by a thermal source can be analysed by considering a ﬁnite volume of space with zero ﬁeld boundary conditions. The region will support a discrete number of standing waves of the electromagnetic ﬁeld. We ﬁrst derive the number and the spectral density of these modes, and from there we determine the distribution of photons in the light emitted by the source. If the region has the physical dimension L, it can support a variety of modes with the wave vectors kx = 𝜋𝜈x∕L, ky = 𝜋𝜈y∕L, kz = 𝜋𝜈z∕L, resulting in a mode density: 𝜌𝜈 d𝜈 =

8𝜋𝜈 2 d𝜈 c3

(3.4)

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3 Photons: The Motivation to Go Beyond Classical Optics

These modes are thermally excited according to a Boltzmann distribution: exp(−En ∕kB T) (En ) = ∑∞ n′ =0 exp(−En′ ∕kB T)

(3.5)

Assuming the condition En = (n + 12 )h𝜈 for the energy of a mode, we can derive a probability distribution for n photons in one mode: n (𝜈, T) =

1 − exp(−h𝜈∕kB T) exp(n h𝜈∕kB T)

(3.6)

We can now evaluate the average number n(𝜈, T) of photons in a mode: n(𝜈, T) =

∞ ∑ n=0

nn (𝜈, T) =

1 exp(h𝜈∕kB T) − 1

(3.7)

which is the well-known black-body formula. It explains the existence of the black-body spectrum of a solid at temperature T. The ﬂuctuations in a single mode of this radiation can be rather large. For normal thermal conditions and for radiation in the visible or near infrared, the average excitation is very low. For example, at T = 300 K and for 𝜈 = 1015 Hz, the average excitation is about 7 × 10−4 photons in each mode. The probability of ﬁnding a ﬁxed number of photons, for example, n = 0, 1, 2, …, in one mode follows a Bose–Einstein distribution that actually has a maximum at n = 0. However, this calculation is only valid if we count photons in a single spatio-temporal mode. More typically, for the detection of light from a thermal source with measurable power, we have to consider many modes at diﬀerent frequencies. They all contribute to the measured intensity. We have to evaluate the probability m that m photons are emitted in the detection interval td . The interval is chosen so large that m ≫ 1. Further we assume that the average intensity of the light is not time dependent. The light source sends out a stream of photons, and at all times we measure the same average intensity and the same average number of photons. The result from classical statistics for many independent sources is t ⟨I⟩ ⟨m⟩m exp(−⟨m⟩) with ⟨m⟩ = d (3.8) m! h𝜈 which is known as the Poissonian distribution. One important property of this distribution is Δm2 = ⟨m⟩, which states that the width of the distribution, measured by the variance, is equal to the mean value of the distribution. The probability distribution in Eq. (3.8) describes the ﬂuctuations of the intensity emitted by a thermal source. The nature of these ﬂuctuations is such that they are independent of the time intervals from which the photon number data is collected. These time intervals can be close and regularly spaced or widely separated and intermittent; the results will be the same, provided a suﬃciently large sample is taken. Noise that behaves in this way is known as stationary noise. As a consequence g (1) (𝜏) and g (2) (𝜏) are independent of 𝜏. In particular, the noise spectrum I(Ω) (averaged over a small frequency range 𝛿Ω about Ω) is independent of the detection frequency Ω, i.e. all frequency components carry the same intensity (I(Ω) is ﬂat). This type of noise m (td ) =

3.4 Interference Experiments

is referred to as white noise. The value of the variance is ﬁxed for a given average intensity and depends only on the length of the detection interval: Var(I(Ω)) =

⟨I⟩h𝜈 td

(3.9)

This statistical result can be applied to any other situation where events generated by independent processes are counted and the average number of such events is not dependent on time. Examples are the measurement of particles from radioactive decay or the number of raindrops, particles of rain, in a steady downpour. The spectrum of the noise for all these cases is equivalent to the sound of a heavy rain shower on a metal roof. It was this analogy, namely, the dropping of lead shot on a metal surface, that was the origin for the name of this statistical phenomenon: shot noise. The classical description provides the tools for describing the statistical properties of light. Chapter 4 gives a complete quantum mechanical description. The classical description can serve as a reference, and the interesting question will be: how do the classical predictions compare with the more complete results from quantum theory? We will ﬁnd that the predictions for thermal light, that is, light emitted by many independent atoms, are identical in both the classical and the quantum models. The lesson here is that it is sometimes suﬃcient to quantize the emitting (and absorbing) atoms whilst still retaining a classical description of the light (this is often referred to as a semi-classical description). In contrast, the results from a quantum mechanical description of the laser will be quite diﬀerent from the results for a classical oscillator. A detailed comparison between the two models will show up several features that are uniquely due to the quantum nature of the light.

3.4 Interference Experiments The measurement of interference, made accessible through the use of instruments such as a Michelson interferometer, clearly demonstrates the wave nature and the coherence properties of light. However, the same type of interference experiments carried out at very low intensities also documents some of the quantum properties. This was done in very early experiments by Taylor [15]. These experiments used photographic plates and extremely long exposure times – up to three months. The experiments showed that the visibility of the fringes remained constant, at a given total exposure, independent of how long it took and thus how many photons could have been in the instrument at any one time. It should be noted that all these interference experiments are most impressive achievements. The interferometers are very sensitive devices, and nothing would have been easier than wiping out the interference fringes through a small perturbation of the apparatus. More recent experiments [16, 17] using photoelectric detection conﬁrmed these results. In addition, some of these experiments used interferometers with total optical paths far exceeding the coherence length of the light source. For example, in the experiment by Janossy, each arm of the Michelson interferometer

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3 Photons: The Motivation to Go Beyond Classical Optics

was 14.5 m long, requiring a very quiet and temperature-stable (ΔT < 0.001 K) environment, which was achieved in an underground tunnel. Again, the visibility remained unchanged – independent of the actual intensity. Complementary results were obtained with the synchrotron radiation emitted by a storage ring. Here the number of electrons generating the radiation can be determined separately from the intensity of the radiation – and again the visibility was shown to be independent of the photon number. All these experiments support the statement by P. Dirac: The new theory, which connects the wave function with probabilities for one photon, gets over the diﬃculty by making each photon go partly into each of the two components. Each photon then interferes only with itself. Interference between two diﬀerent photons never occurs. [18] The beamsplitter in the interferometer does not divide the light into smaller parcels – any detection of the light shows quanta of the same amount of energy. This result cannot be derived if we think of photons as small local parcels of energy – miniature billiard balls – with the properties of classical particles. It reveals one of the consequences of quantum mechanics, namely, that a conclusion about the speciﬁc outcome of an experiment cannot be drawn before the experiment is carried out. This consequence of quantum mechanics becomes very clear by considering the various possible outcomes of an interference experiment where detectors and counters are used in an attempt to measure the presence of the photon in the one or the other of the interferometer arms. Any such attempt would fail, as it has been so elegantly expressed by R. Feynman et al. in his lecture notes [19]. The duality of the wave–particle model is a clear conclusion from these experiments: light is neither exclusively a wave nor exclusively a stream of particles. The type and condition of the experiment determines which one of these models will dominate and which one will give the simpler interpretation. All the time light is something more inclusive, more complicated than either of these models would suggest. The statement that ‘a photon will only interfere with itself’ should not be taken too literally in the sense that it could be tested through the observation of individual events. In the case of the photographic recording, each detection event produces one clear dark dot on the photographic plate. This dot could be part of an interference pattern or of a random distribution of dots. It is the distribution of many dots – a frequent repeat of the same measurement – that reveals the result of the experiment. Theoretically, one single dot measured exactly at the location on the plate where there should be absolutely no intensity according to the calculation would have an enormous signiﬁcance. Such an event would prove the calculation wrong (Figure 3.4). But we have to note that in reality there will always be some other imperfection, such as an imbalance of the beamsplitter, ﬁnite coherence length, or background noise, which could have resulted in this particular single event. It is the distribution of the dots, or of the counts in the electronic instrument, that is crucial for the test of a model. As is common

3.4 Interference Experiments

4 1

3

2

Figure 3.4 Historical example: interferometer at the turn of the century. Source: From the notes of A.A. Michelson and E.W. Morley [20]).

for all tests of quantum mechanics, a large number of individual experiments are required to allow a valid conclusion. The nature of the light seems to depend on the type of experiment: a single beamsplitter can often be well described by the photon picture, and a complete interferometer requires the wave picture. An interesting extension of the demonstration described above are the so-called delayed choice experiments. Here we are trying to trick the light into the wrong behaviour by delaying the decision of which type of experiment is being performed until it has already passed the ﬁrst beamsplitter. Will the photon be able to predict the type of experiment that is lying ahead and react in the appropriate way? Whilst this view is certainly naive, it poses a valid question: does it matter whether we have a delay built into the interferometer? One example for such an experiment [21], shown in Figure 3.5, uses pulses of light of short duration (150 ps). The intensity was chosen suﬃciently low that on average only every ﬁfth pulse contained one photon. Pulse 150 ps

n = 0.2 photons

Electro-optic shutter

Polarizing beamsplitter

5m Fibre

Det 1

5m

Det 2

Figure 3.5 Schematics of a delayed choice experiment. Source: After Hellmuth et al. [21].

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3 Photons: The Motivation to Go Beyond Classical Optics

Both arms of the interferometer contain 5 m of single-mode ﬁbre. One arm contained an electro-optical switch that allowed this arm to be blocked and the interference to be turned oﬀ through an electronic signal. In order to put the photons in the above-mentioned dilemma, the interferometer was operated such that the switch was operated after the pulse had passed through the ﬁrst beamsplitter. However, the delay had no eﬀect on the visibility of the interference pattern. The outcome of the experiment depended only on the conﬁguration of the experiment that was present at the time of detection, independent of the earlier state of the apparatus. A more thorough discussion of these types of experiments is given in Chapter 12. This raises the question as to whether photons from two completely independent sources can interfere with each other. It is well known that light from diﬀerent sources, or from diﬀerent parts of the same source, does not interfere when observed for a long time. This is a consequence of the fact that the phase of the electromagnetic wave emitted by the two sources has two independent phase terms. In a normal interferometer, which has wave front splitting mirrors, this does not matter as long as the path length diﬀerence is less than the coherence length. Similarly, for independent sources, the fringes should also appear if the measurement takes less time than the coherence time tcoh . During such a short time interval, the phase will not vary enough to reduce the visibility. This has been shown in an experiment with two independent lasers [22], in this early experiment two ruby lasers. Spatial interference patterns from single pulses of the lasers were recorded, clearly showing the interference between photons from diﬀerent sources. An alternative is to measure the beat between two lasers; this was ﬁrst demonstrated for two independent He–Ne lasers [23]. The experiment operated continuously, and the narrow linewidths of the gas lasers made the observations straightforward. The major problem was the drifts in the laser frequency due to mechanical and temperature variations. Once this was overcome, this experiment demonstrated clearly that interference exists between beams from independent sources, provided that the detection time t1 is less than the coherence time tcoh . With modern technology, such as tunable diode lasers and radio wave spectrum analysers, this type of experiment is almost trivial. The eﬀect is common knowledge for a laser physicist. However, the interesting question remains: should this interference eﬀect between independent lasers persist in the limit of very small intensities? The ﬁrst answer might be No, since we cannot be certain of the relative phase during the long measurement time required for the weak beams. However, if we select only data measured at times when the relative phase is within a set small range, one could expect an interference pattern to become visible. Exactly this experiment was carried out by Radloﬀ et al. [24]. Two He–Ne lasers were used that had been built in one mechanical structure in order to minimize the technical noise and the phase ﬂuctuations due to vibrations. The beat frequency between the beams was continuously monitored. A very small fraction of the light was selected and formed a spatial interference pattern. This was gated by an electro-optic shutter and only opened when the frequency was within a preset range. The photon ﬂux was as low as 105 photons/s with a typical shutter time of 10−5 seconds.

3.4 Interference Experiments

After exposures of about 30 minutes, clearly recognizable interference patterns became visible. These patterns were present even at arbitrarily low intensities. The particle picture is not at all suitable for this situation. It is extremely unlikely that one photon from each laser will pass together through the shutter. The wave model on the other hand provides a very simple explanation. The only problem arises when we want to understand the quantization of the energy together with interference. These results for the interference of photons from independent sources are in some contradiction to the statement by P. Dirac, to repeat: ‘Each photon interferes with itself. Interference between two diﬀerent photons does not occur’. Dirac made this statement whilst considering only interference experiments known at his time. It should not be applied to the experiments involving independent sources. Nevertheless, attempts have been made to rescue Dirac’s statement by evoking additional processes that guarantee that photons are emitted simultaneously by both lasers. But these attempts are really not all that helpful in understanding the quantum nature of light. In summary, we can say that the only conditions under which interference can be observed are those where it is impossible to detect which path the photon has taken. Any attempt to measure the path, or even to reconstruct the path after the event, would wipe out the interference pattern. A discussion of these types of experiments has led to the postulate of Heisenberg’s uncertainty principle. For this particular case the principle states that ΔxΔp ≥ h∕4𝜋. It enforces that we can measure either the interference pattern, by determining the position x of the photon, or alternatively the path of the photon, as measured by its momentum p. But we cannot measure both simultaneously [7]. Discussion about these results has played a prominent role in the development of quantum mechanics as a whole. A related idea was behind the design by J.D. Franson, who designed an interferometer that can be used to distinguish between single- and two-photon interference [25]. His design is shown in Figure 3.6. In this apparatus let us consider two coincident photons, emitted by a source, which individually travel in opposite directions towards two identical interferometers. Each interferometer contains θ2

θ1

γ2

D2

Source

D′2

γ1

D1

D′1

Figure 3.6 The Franson interferometer. Source: Franson 1991 [25]. Reproduced with permission of APS.

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a shorter and a longer path, and the diﬀerence Δttrans in transit times over the two paths is taken to be much larger than the coherence time of the photons. In addition, it has individual phase settings Θ1 and Θ2 . That means each interferometer by itself would not produce fringes individually and the visibility of a fringe due to a change of the phase Θ1 and Θ2 would be extremely low for such a large path length imbalance. However, interference between the quantum mechanical amplitudes for the photons to have both travelled the shorter paths and the longer paths produces a modulation in the coincidence counting rate R of photons from the two sets of detectors, given by cos2 [(Θ1 + Θ2 + 2𝜋𝜈 Δttrans )∕2]. This type of interferometer would only show a modulation due to the quantum mechanical wave function of the photons, which is sensitive to indistinguishability of the paths. That means an interference can occur for the quantum mechanical amplitudes and thus for the counting rates. This would be a clear distinction between two types of interference. Moreover, if we illuminate this interferometer with an entangled pair of photons, there is a link between the probability amplitudes on both sides of this interferometer, and we can measure the visibility of the fringes due to the settings Θ1 and Θ2 . This should allow us to learn about the quantum properties of the photon pairs and will be discussed in Chapter 13.

3.5 Modelling Single-Photon Experiments We now turn to a more idealized description of single-photon experiments in order to see how they may be modelled. We will introduce a number of basic optical elements as ‘black boxes’, describing and contrasting their operation for classical light waves and single photons. Descriptions of how these optical elements can be constructed will be found in later chapters. Similarly, the simple quantum mechanical models introduced here will also later be expanded. The average number of photons per second in a beam of light is proportional to its intensity. However, the quantization of the ﬁeld introduces a degree of freedom not present in the classical theory, the photon statistics. Suppose we had a ‘photon gun’, that is, a light source that emits just one photon when we pull the trigger. Such a photon gun can be produced through pulsed excitation of a single molecule, though presently not conveniently. Such a source would share some characteristics with the highly attenuated sources described in Section 3.4. However, whilst an attenuated laser may produce one photon on average in some time interval, there will be ﬂuctuations: sometimes there will be two or more photons, and sometimes there will be none. On the other hand, we will always detect just one photon in the measurement interval coming from our photon gun. There is no way to describe this diﬀerence in the photon statistics with a classical description of light. We must introduce a new concept, that of the quantum state of the light. The quantum state describes all that is worth knowing about a light beam. The photon gun produces light in a single-photon number state, which we will write as |1⟩. Studying the behaviour of such a light state in simple situations illustrates some of the basics of the quantum description of light.

3.5 Modelling Single-Photon Experiments

3.5.1

Polarization of a Single Photon

Let us ﬁrst consider how a single polarized photon behaves. We will describe this behaviour using single-photon polarization states. We will see how it is possible to build a consistent description of light phenomena in spite of wave–particle duality. To do so, however, we will need to introduce some quite diﬀerent concepts to those found in classical theory. Recall that in our classical description the polarization of the ﬁeld was described by the vector p(t), which gave the direction of the electric ﬁeld tangential to the wave front. For linearly polarized light p will be ﬁxed. For circularly polarized light p will rotate at the optical frequency. It is possible to design optics that will decompose a light ﬁeld into two orthogonal modes. The amplitudes of the two modes will be equal to the projection of p onto the chosen axes. For example, a horizontal/vertical polarizing beamsplitter will project the ﬁeld onto a horizontally polarized mode and a vertically polarized mode. These two modes will exit through diﬀerent ports of the beamsplitter. Similarly, one can use a diagonal/anti-diagonal polarizing beamsplitter to decompose the ﬁeld into linearly polarized ﬁelds at 45∘ and −45∘ away from horizontal. With slightly more eﬀort it is also possible to create a left/right circular polarizing beamsplitter that decomposes the ﬁeld into clockwise and anticlockwise rotating circular polarizations. Suppose we ﬁre a string of single photons at a horizontal/vertical polarizing beamsplitter. As we have noted, classically such a device will direct horizontally polarized light in one direction and vertically polarized light in the other. Thus we will deﬁne photons that exit through the horizontal port of this beamsplitter as being either in the horizontal state or the vertical state as |H⟩

or |V ⟩

(3.10)

By ‘exit through the horizontal port’, we mean that a photon detector placed at the output of the horizontal port will detect a photon, or equally a photon detector placed at the vertical port does not detect a photon. These situations are illustrated in Figure 2.8. The symbol for the state, |−⟩, is referred to as a ket. These deﬁnitions are sensible because a photon that exits through the horizontal port of the beamsplitter and is passed through another polarizing beamsplitter will certainly exit through the horizontal port. A similar deﬁnition applies to vertical photons. The diagonal single-photon state, |D⟩, and the anti-diagonal state, |A⟩, can be deﬁned in a similar way by analysing the beams with a diagonal/anti-diagonal polarizing beamsplitter. So far this is straightforward. The physics of classical polarized beams and single photons looks the same. Things become more interesting when we start to mix up the polarizations. If we send a diagonally polarized classical beam of light into a horizontal/vertical polarizing beamsplitter, then half the beam will exit through the horizontal port and half will exit through the vertical port. What will happen if we send single photons in the state |D⟩ through this beamsplitter? As we have discussed in Section 3.4, the photons cannot be divided, in the sense that a

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detector placed at one of the output ports will either detect a whole photon or no photon. Instead they must go one way or the other. To be consistent with the classical result for many photons, it must be that they go one way 50% of the time and the other way the other 50% of the time. But can we tell in which direction an individual photon will go? To answer this ﬁrst remember how |D⟩ photons behave at a diagonal/anti-diagonal polarizing beamsplitter: they all exit through the diagonal port. That is, they all behave perfectly predictably and identically. Yet when these identically behaving photons are sent into a horizontal/vertical beamsplitter, we have argued some must take one path and others take the other. Thus the path an individual photon in the state |D⟩ takes through the horizontal/vertical beamsplitter is not speciﬁed. All that is speciﬁed is that on average half the photons will go one way and half will go the other. This example illustrates the fact that in quantum mechanics it is probabilities of outcomes, not particular outcomes, that are predicted. Even though we can say with certainty that a photon in the state |D⟩ will emerge from the diagonal port of a diagonal/anti-diagonal beamsplitter, its reaction to a horizontal/vertical beamsplitter is as random as a ﬂip of a coin. It is tempting to think that maybe there are other variables (perhaps hidden to us) that do determine in a precise way the polarization behaviour of individual photons under all conditions. However such a possibility can be ruled out experimentally (see Chapter 12). We are forced to accept the intrinsic indeterminacy of the quantum world. Continuing our discussion of polarization states, we can also introduce the right circular single-photon polarization state, |R⟩, and the left circular state, |L⟩, in an analogous way to the other states. A photon in state |R⟩ will randomly take one port or the other when sent into either a horizontal/vertical or a diagonal/anti-diagonal beamsplitter. A photon in state |L⟩ will behave in the same way. Similarly a photon in state |H⟩ will give a random result for both diagonal/anti-diagonal and right/left circular polarizing beamsplitters and so on for the other states. We will now outline the mathematical formalism for treating these situations. 3.5.1.1

Some Mathematics

The state kets we have introduced are vectors. They live in a vector space called Hilbert space. Hilbert space is a complex vector space: when we add vectors, we are allowed to do so using complex coeﬃcients. The dimensions of the vector space are equal to the number of degrees of freedom of the property being described. In the polarization example our vector space is discrete and has dimension 2. That is, in all the experiments we have described, there are only two possible outcomes: the photon comes out from one port or the other. It can only take two discrete values. This is in stark contrast to the classical case in which the amount of the ﬁeld that comes out of one or other port can vary continuously between all and nothing. In general, discrete systems can have any number of dimensions, and continuous systems can also be described. A continuous variable can give any outcome in some range, and their vector space will have inﬁnite dimensions. We will look at such examples in optics in Chapter 4. As for other vector spaces, we can reach any vector in our space through linear superpositions of n orthogonal basis kets, where n is the dimension of the system.

3.5 Modelling Single-Photon Experiments

To deﬁne orthogonality we introduce bra vectors, which are written ⟨x|. Bras and kets are related by the Hermitian conjugate, an operation similar to complex conjugation and indicated by the ‘dagger’ symbol, †. Thus we can write z∗ ⟨x| = (z|x⟩)† .

(3.11)

where z is just a complex number. Vectors |x⟩ and |y⟩ are orthogonal if ⟨x| × |y⟩ ≡ ⟨x|y⟩ = 0

(3.12)

In general the object ⟨x|y⟩ is called a bracket and is similar to the dot product in real vector spaces. A ket is said to be normalized if ⟨x|x⟩ = 1. We will generally assume this property. Simple physical questions can be asked in the following way. If we prepare our system in the state |𝜓⟩ and then use an analyser to ask the question ‘Is my system in the state |𝜎⟩?’, the probability, P, that we get the answer ‘Yes’ is given by P = |⟨𝜓|𝜎⟩|2

(3.13)

The bracket ⟨𝜓|𝜎⟩ is referred to as the probability amplitude and will in general be a complex number. The orthogonality condition then says that if we prepare our system in |𝜎⟩, our analyser will never ﬁnd it in |𝜓⟩ if these two states are orthogonal. Normalization reﬂects the fact that if we prepare our state in |𝜓⟩, we will certainly (P = 1) ﬁnd it there with our analyser. We will try in this book to minimize the mathematics by introducing concepts on a need-to-use basis. However, a deep understanding of the workings of quantum optics requires a good understanding of the mathematical structure of quantum mechanics. Readers unfamiliar with the basics of quantum mechanics are encouraged to study other texts such as Refs. [26] and [27].

3.5.2

Polarization States

The concepts introduced above allow us to identify our polarization states |H⟩ and |V ⟩ as orthogonal states. That is, ⟨H|V ⟩ = ⟨V |H⟩ = 0, which means that a photon in state |H⟩ never comes out the vertical port and vice versa. We will assume also that our states are normalized such that ⟨H|H⟩ = ⟨V |V ⟩ = 1. This means that we should be able to express all our polarization states as linear superpositions of the horizontal and vertical states. Consider ﬁrst the diagonal states. From our discussion in Section 3.5.1, we know that they must have the properties ⟨D|A⟩ = ⟨A|D⟩ = 0 |⟨D|H⟩|2 = |⟨D|V ⟩|2 = 0.5 |⟨A|H⟩|2 = |⟨A|V ⟩|2 = 0.5

(3.14)

or the diagonal states are orthogonal and a diagonal state placed through a horizontal/vertical polarizing beamsplitter has a 50% probability of exiting through

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either port. It is straightforward to check that 1 |D⟩ = √ (|H⟩ + |V ⟩) 2 1 |A⟩ = √ (|H⟩ − |V ⟩) 2

(3.15)

satisfy the conditions of Eq. (3.14). Now consider the circularly polarized states. We need to satisfy ⟨R|L⟩ = ⟨L|R⟩ = 0 along with |⟨R|H⟩|2 = |⟨R|V ⟩|2 = 0.5 |⟨L|H⟩|2 = |⟨L|V ⟩|2 = 0.5 and |⟨R|D⟩| = |⟨R|A⟩|2 = 0.5 2

|⟨L|D⟩|2 = |⟨L|A⟩|2 = 0.5

(3.16)

This looks impossible until we remember that Hilbert space is a complex vector space. Thus we are allowed to form the complex coeﬃcient superpositions 1 |R⟩ = √ (|H⟩ + i|V ⟩) 2 1 |L⟩ = √ (|H⟩ − i|V ⟩) 2

(3.17)

which satisfy the above requirements. It follows that an arbitrary polarization state can be represented by |𝜔⟩ = x|H⟩ + ei𝜙 y|V ⟩

(3.18)

where x2 + y2 = 1 and 𝜙, x, and y are real numbers (absolute phase factors have no physical signiﬁcance). Notice that such a representation is not unique. We could equally well write all our states as superpositions of diagonal and anti-diagonal states or circular states. Any two orthogonal states can span the Hilbert space. We have seen that in order to describe the behaviour of a quantized light ﬁeld, we have had to introduce two new concepts: (i) the state of the light and (ii) that predictions are made of probabilities of events, not actual events. The fact that these concepts lead us to represent, for example, a diagonal photon as an equal superposition of horizontal and vertical photon states may seem relatively benign at this point. That such a superposition has no analogue for classical particles will be demonstrated more graphically in Section 3.5.3. In Chapters 12 and 13, we will discuss in detail photon experiments that use polarization as their degree of freedom. In particular, we will investigate the unique communications and information processing abilities these quantum states can exhibit.

3.5 Modelling Single-Photon Experiments

3.5.3

The Single-Photon Interferometer

We will now describe a simple single-photon interference experiment using a Mach–Zehnder interferometer, similar in concept to the experiments described in Section 3.4. A more general description of this device can be found in Section 5.2. Here we will illustrate the basic physics of the arrangement in the extreme situation of only a single photon present in the interferometer at a time and how the mathematics introduced in the previous sections can describe it. A Mach–Zehnder interferometer consists of a 50/50 beamsplitter, which divides the input light into two equal parts, followed by a series of mirrors, which direct the two beams back together again on a second 50/50 beamsplitter that recombines them. The path length diﬀerence between the two arms of the interferometer can be adjusted. This then corresponds to the situation discussed in Section 5.2, and thus classically we expect the amount of light exiting one port to change as a function of the path length diﬀerence. In particular, if our path length diﬀerence remains well within the coherence length, we will be able to pick a path length such that all the light exits from a single port. This set-up is depicted schematically in Figure 3.7. Let us now describe the behaviour of this interferometer when single photons pass through it. We will assume that the path length diﬀerence is held in the situation described above such that for classical beams all the light enters and exits through a single port. We will also assume that our beamsplitters and mirrors are polarization independent so that the polarization of our single photons is irrelevant, and we can simply label a single photon in a beam by the number state ket |1⟩. There are however two ways in which the photon can enter the interferometer: via the beam path labelled a in Figure 3.7 or by the beam path labelled b. Suppose we in fact send our photon in by beam path a. We can describe this initial situation by the state ket |𝜎⟩ = |1⟩a |0⟩b

(3.19)

meaning there is one photon in beam a and no photon in beam b. There is an even probability of the photon exiting from either port of the ﬁrst beamsplitter. The output state from the beamsplitter can thus be written as 1 |𝜎 ′ ⟩ = √ (|1⟩a |0⟩b + |0⟩a |1⟩b ) 2 Figure 3.7 Schematic of single-photon interferometer.

50/50

(3.20)

Pathlength adjustment

a

b 50/50

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The path length condition means that, apart from an overall phase, this state remains the same as it propagates through the interferometer. The time symmetric nature of light propagation through a beamsplitter means that when the beams reach the second beamsplitter, they must return to their original state of a photon in a and no photon in b (see Eq. (3.19)). Thus we just get the photon equivalent of the classical result – the photons always exit through a single port. Now let us consider the physical signiﬁcance of our description. In particular, what does Eq. (3.20) mean? It says that the photon is in a superposition of being in each path. It is tempting to think that this just means the photon is in one or the other path with a 50% probability. Certainly the probability of ﬁnding the photon in path a Pa = |⟨1|𝜎 ′ ⟩a |2 = 0.5 is 50%. But is the interference eﬀect, i.e. the fact that the photons always exit from the a port (see Figure 3.7), consistent with the photons taking one path or the other? Well, the second beamsplitter is identical to the ﬁrst one so our ﬁrst puzzle is: why should the photon only exit through one port? We might imagine that it picks up some kind of ‘label’ at the ﬁrst beamsplitter that tells it what to do at the second. So suppose we ﬁnd our photon in beam a in the middle of the interferometer, but then we ‘release’ it and let it ﬁnish its journey to the second beamsplitter. Experimentally we ﬁnd that now there is no interference and the photon is just as likely to come out from either of the output ports. Quantum mechanically, because we now know that the photon is in beam a, the state inside the interferometer becomes |𝜎 ′ ⟩m = |1⟩a |0⟩b , and so the photon just behaves as it did at the ﬁrst beamsplitter and has a 50/50 chance of emerging from either port. We might argue that when we ‘caught’ the photon, it forgot its label and so just behaved randomly at the second beamsplitter. So consider another possibility. Suppose we look for the photon in beam b but we do not ﬁnd it. Now quantum mechanically the situation is the same as above; we know that the photon must be in beam a so the state again becomes |𝜎 ′ ⟩m = |1⟩a |0⟩b and the interference should disappear. But any common- sense description would say this cannot be. We have not interacted with the photon, so any label it has must be unaﬀected. Thus we would expect the interference to still appear. Experiments, however, conﬁrm the quantum mechanical result (see Chapter 12). We are forced to conclude that the superposition state of Eq. (3.20) represents in some objective sense the presence of the photon in both paths of the interferometer. Any experiment that causes the photon to manifest itself in one or the other path, whether an actual detection occurs or not, destroys this superposition and in turn wipes out the interference. This is wave–particle duality and we will encounter it in many guises throughout this book.

3.6 Intensity Correlation, Bunching, and Anti-bunching The quantum properties of light can be investigated by using the techniques of photon counting. We will ﬁnd that the statistics of photons are considerably different from the statistics of independent particles. Again, we have to combine

3.6 Intensity Correlation, Bunching, and Anti-bunching

the particle picture with the wave picture. For these statistical experiments it is necessary to measure quantitatively the correlation between diﬀerent beams of light. This approach has already been discussed in Section 2.3.3. In particular the discussion of the Hanbury Brown and Twiss experiment for thermal light, in their case starlight, showed a way for determining the correlation between ﬂuctuating currents ia (t) and ib (t) by recording and analysing the product ia (t)ib (t) of these currents. This technique can be used for any type of light, including laser light, and it also applies to any intensity of light. At very low intensities the photodetector becomes a photon counter. For a photon counter the probability of a count is proportional to the intensity of the light. Now the number n(t, tdec ) of counts recorded at time t within a time interval tdec represents the intensity of the light. The detection time interval tdec should be shorter than the coherence time tcoh of the light source; otherwise the ﬂuctuations are simply averaged out. As before we determine the intensity correlation by using detectors Deta and Detb , measuring the photon counts na (t, tdec ) and nb (t, tdec ) independently, and then forming the product na (t, tdec )nb (t, tdec ). Actually, such medium intensities, ranging from a few to several tens of counts per detection interval, are very challenging. No particularly satisfactory technology for this range has yet emerged, though progress is steadily being made (see Chapter 12). On the other hand, at very low intensities, where the probability of more than a single count per detection interval can be neglected, good and reliable technology exists (see Chapter 7). A moment’s thought shows that the probability of coincidences, which means events where both detectors register a photon within a short time interval, is proportional to the value of na (t, tdec )nb (t, tdec ). It is technically very attractive to build a circuit that only responds to such coincidences. This technique shows an increase of the coincidence rate under the same conditions where big correlations are measured. Going back to the Hanbury Brown and Twiss experiment, the coincidence rate is high if the separation between the detectors is suﬃciently low. This is an excess of coincidences – compared to the numbers one would expect from completely random counting events. In other words, whenever a photon has reached detector Deta , the probability for another photon to be at Detb is enhanced. This could not be understood by regarding photons naively as independent particles, thereby ignoring the wave properties of the light. The photons emitted from the diﬀerent part of the source would not know of each other. Their arrival time would not be correlated. In contrast, the wave picture predicts that the instantaneous intensity on the detector is inﬂuenced by all atoms in the source. Each part of the source is making a small contribution. Thus both detectors see related contributions where light is originated from one atom in the source. Again, this means we cannot consider individual photons, each with one well-deﬁned birthplace. Atoms cannot be localized in that way. These observations are concerned with spatial eﬀects. Intensity information can also be gained from temporal eﬀects within a single beam. To measure the time correlation of a beam of light, we can use a trick that was used by Hanbury Brown and Twiss as a precursor to the stellar observations. We can use a beamsplitter to generate two light beams with each one being detected individually, as

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na (t, tdec)

Deta

nb (t, tdec)

Detb

X

Figure 3.8 Schematic layout of an optical intensity correlator.

Correlator (product)

shown in Figure 3.8. The coincidence rate can now be directly measured using detectors that are gated. Only events that happen within the gate time at both detectors are recorded. By moving one of the detectors either further away or closer to the beamsplitter, we introduce a delay time 𝜏 and can determine the coincidence rate as a function of the delay time. Alternatively, we can leave the detectors stationary and use electronic delays. For thermal light the coincidence rate for 𝜏 < tcoh should be higher than for 𝜏 > tcoh . This is a direct analogy to the properties of the second-order correlation coeﬃcient. It corresponds to g (2) (𝜏 < tcoh ) > g (2) (𝜏 > tcoh )

(3.21)

as shown in Eq. (2.47). Since the coincidence rate for 𝜏 ≫ tcoh can only be those events that are completely random, we simply have to look for excess coincidences. The long delay time measurements serve as a normalization. One practical diﬃculty is that the detection time tmeas has to be less than the coherence time if we are to be within the single-mode approximation. For normal thermal light the linewidth is very large, the coherence time is extremely short, and the response of the detectors is too slow. A direct measurement of the coincidences for thermal light is not possible. In contrast, laser light has a small linewidth and long coherence time, and the experiment is feasible. One clever experimental trick is to simulate thermal light through scattering of laser light on rapidly and randomly moving objects. This was ﬁrst done by Arrecchi et al. [28]. The experiments used light from a gas laser with long coherence time, which was scattered by polystyrene particles of various sizes suspended in water and detected by two detectors. The coincidence rate, or second-order correlation coeﬃcient, was measured. For comparison the laser light itself was investigated, and the results show a clear excess of coincidences. The shape of g (2) (𝜏) ﬁts properly to the prediction of a thermal light source with a Gaussian linewidth, and the diﬀerence between thermal light and coherent laser light is clearly demonstrated. An alternative interpretation of these results is to say that the photons from the thermal light arrive in bunches and the probability for photons to arrive together, in coincidence, is large for short delay time intervals. This phenomenon is known as photon bunching. It is typical for thermal light that has naturally large ﬂuctuations. In contrast, for coherent light there is no preferred time interval between photons. Coherent light is not bunched. The coincidences show random Poissonian statistics on all timescales. Can there be anti-bunched light, where the photons keep a distance, in time, to each other? This would be light that clearly violates the classical conditions. The second-order correlation function would be less than 1, which is not possible for classical statistics, as

3.6 Intensity Correlation, Bunching, and Anti-bunching

Microscope objective

g2 (τ) 3 Optical excitation 2

Timer Start

Phototube Stop

1 Computer

25 (a)

(b)

50 Delay τ (ns)

75

100

Figure 3.9 The experiment by Kimble et al. [29]. (a) Schematic outline. (b) Measured values for g(2) (𝜏) derived from the data. The broken curve shows the theoretically expected form for a single atom, arbitrarily normalized to the same peak.

already shown in Section 2.3. A measurement of anti-bunched light would be a clear demonstration of quantum behaviour, of non-classical light. Well in fact the single-photon quantum state we discussed in the last section would be just such a state. Recall from Eq. (3.20) that such a beam will send a photon in one or other direction at a beamsplitter. Thus, at zero time delay, it is impossible to record a coincidence from such a state in the set-up of Figure 3.8. Hence we have g (2) (0) = 0. A pioneering experiment by Kimble et al. succeeded in 1977 in demonstrating the generation of non-classical light [29]. They used the ﬂuorescence of single atoms to generate light with the statistical properties of anti-bunching. The experiment was carried out with Na atoms, which were ﬁrst prepared into a two-level system and then excited by two resonant continuous-wave (CW) laser beams, as shown in the schematic layout in Figure 3.9a. The ﬂuorescence light from a very small volume, containing on average not more than a single atom, was imaged through a 50/50 beamsplitter on to two photodetectors, D1 and D2 . Upon arrival of a photon, the detectors generated pulses that were processed, via counting logic, and produced the probability function (t, t + 𝜏) for the arrival of photons at diﬀerent detectors with a time delay 𝜏. This information was converted into the second-order correlation function g (2) (0) (see Section 2.3). The excitation of individual atoms is determined by the quantum mechanical evolution of the atomic wave function. After the single atom has been excited by a laser beam, it will emit one photon, let us say at time t = 0. The atom cannot immediately emit another photon; it requires time for re-excitation. Consequently the probability for the generation of pairs of photons with small time separation is very small. The ﬂuorescence light has a value of g (2) (0) close to zero, as shown in Figure 3.9b, which is not possible for classical light. This light clearly shows quantum properties – it is anti-bunched. Since this ﬁrst demonstration, several experiments, for example, reported by J. Cresser et al. [30], J.G. Walker and E. Jakeman [31], and P. Grangier et al. [32], have been able to reveal the properties of similar quantum systems in more and more detail.

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More recently, the observation of anti-bunching has become the method of choice for identifying single quantum emitters. For example, ﬂuorescent light emitted by nitrogen vacancy (NV) centres in diamond can be analysed, and when strong anti-bunching is observed, one can be sure that the observed ﬂuorescence comes from a single NV centre in diamond [33]. See Chapter 13 for details.

3.7 Observing Photons in Cavities Many researchers pointed out that the most direct evidence of the quantization of the radiation ﬁeld, namely, the discrete nature of the photocurrent, could also be explained by a classical description of the radiation ﬁeld, provided that the detector is a linear quantum system. Do we have better evidence of the quantum properties of the radiation ﬁeld? Researchers set out to ﬁnd a more direct proof for the discreteness of the ﬁeld and analysed the light interacting with atoms contained inside a cavity. The atom can be approximated as a system with just two energy levels, a two-level system. The coupling of this system with a single mode of the radiation ﬁeld leads to a dynamical evolution of the atomic excitation in the form of nutations at the Rabi frequency. This simple model for the interaction is referred to as the Jaynes–Cummings model [34], well described in the literature [35], and extensively used in coherent optics with light or microwaves (nuclear magnetic resonance, NMR). In a strong coherent ﬁeld, which means a beam with many photons, the coupling leads to a nutation at a frequency associated with the mean photon number n𝛼 , which is a continuously variable number. However, for a weak ﬁeld (n𝛼 ≈ 1), the Rabi frequencies should be discrete and linked directly to the photon numbers in the ﬁeld. By placing the atoms inside a resonator, interesting properties of the coupled atom–light system can be studied. This idea led to a series of beautiful experiments with the micromaser built at Garching in the group around H. Walther. A similar technology of cavity atom coupling has been perfected by a team in Paris around S. Haroche and coworkers [36]. In these systems so-called Rydberg atoms, that is, atoms in excited states with very high quantum numbers (n > 40) and that have very simple, long-lived states, are placed inside a microwave cavity of extremely high quality. The cavity encloses the atoms almost completely, and it can be cooled to exclude thermal noise, and the coupling between the internal ﬁeld, the photons, and the probe in the form of the atoms can be made to dominate over all other eﬀects. The atoms couple to the microwave ﬁeld, and the coupled systems evolve in an elegant way determined by quantum mechanics. By observing the atoms that leave from the resonator, conclusions can be drawn about the state of the light–atom ﬁeld inside. These systems can show features such as the collapse and revival of the wave function of the atom, as reviewed, for example, early by P. Meystre and M. Sargent III [26] and more recently by S. Haroche and coworkers [36]. We will come back to these experiments when we discuss diﬀerent ways of generating strong non-classical states of light in Chapter 12 (Figure 3.7). Early experiments with Rydberg atoms by G. Rempe et al. [38] in a micromaser built in Garching have revealed an oscillation of the atomic population in a

0.5

76 Ph 5 4 oto nn 3 2 um be 1 0 r, n

0

Probability

7 6 5 4 3 2 1 0

5.0 〈n〉

1.0

Photon number, 〈n〉

3.7 Observing Photons in Cavities

4.5 4.0 0.06 0.07 0.08 Time (s)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 100 Time (s) 80 60 r, N e 40 mb Direct observation of 20 nu m dynamics of photons in a cavity o t A

Figure 3.10 Direct observation of photons in a cavity via QND detection by S. Haroche team. The left side shows the progressive collapse of the ﬁeld into a photon number state in the cavity by observing more and more atoms. The photon number probabilities are plotted versus the photon number n and the atom number N. The histograms evolve, as N increases from 0 to 110 atoms, from a ﬂat distribution into n = 5. The right-hand side shows the dynamics. The mean photon number ⟨n⟩ is continuously monitored via QND over 0.7 s. The inset zooms into the n = 5 to 4 jump, showing that it is detected with a time resolution of about 0.01 s. Source: Haroche and coworkers 2007 [37]. Reproduced with permission of Springer Nature.

thermal ﬁeld with (1.5 < n𝛼 < 3.8) and in the micromaser ﬁeld. Similarly, the series of experiments in Paris led by S. Haroche and coworkers has achieved extreme conditions [36]. Here the Rydberg atoms traverse a cavity where both atom–cavity interaction time and cavity damping time are very long. This requires a cryogenic cavity (T = 0.8 K) of very high quality (Q = 7 × 107 ) and a very stable source of slow atoms. The average delay between successive atoms was set to 2.5 ms, much longer than the cavity decay time. In the cavity the radiation ﬁeld and the atoms are coupled together, resulting in a time evolution that is dominated by oscillations at the Rabi frequency, and this determines the state of the atom at the end of its travel through the ﬁeld in the resonator. After leaving the cavity the atoms were detected by state-selective ﬁeld ionization. From these data, accumulated for many atoms, the dynamics of the atom–cavity evolution was reconstructed, as shown in Fig. 3.10. This showed a mixture of oscillations in time. Taking the Fourier transform of the traces at low average photon numbers, the experiments clearly showed discrete frequency components that correspond to the discrete Rabi oscillations associated with the photon numbers n = 1, 2, 3, 4, …. The discreteness of the radiation ﬁeld is directly evident. The technology has been systematically improved. More recent experiments by the Paris group have reﬁned this approach to the point where it is now possible to probe the ﬁeld inside the cavity repeatedly – in this way ﬁrst preparing and then analysing discrete photon number states in the cavity [37, 39]. It is now possible to carry out continuous quantum non-demolition (QND) measurements by using the coupling of the light ﬁeld to the excitation of atoms in the cavity, and we can

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directly observe the establishment of the photon number. More details of this approach are given in Chapter 11. The major improvement over the earlier experiments is an increase of about 2 orders of magnitude in the quality of the cavity, pushing the damping time up to t = 0.13 s. This means that an average photon will travel over 10 000 km, back and forth within the cavity, before escaping. Given that successive atoms can now be sent across the cavity and measured every 0.5 ms, the trapped cavity ﬁeld can now be probed with thousands of individual atoms before it decays. Quantum optics predicts that if a near stationary cavity ﬁeld is measured to have a particular number of photons, then successive measurements will ﬁnd the same number of photons in the ﬁeld, as described in Chapter 4. This is exactly what is observed, with hundreds of successive atomic measurements returning the same discrete photon number. These experiments unambiguously show that the discreteness of optical energy measurements is a property of the ﬁeld, not of the atoms in the measurement device. S. Haroche shared the Nobel Prize for physics in 2012 for this work, and his lecture provides a fascinating overview of the search for the evidence of photons.

3.8 Summary We have compelling experimental evidence that photons exist and that the classical wave picture has to be extended. There is no doubt that we require quantum description of light for many current experiments and even technologies. Some of these observations are linked directly to the detection process, whilst others show in detail the quantum nature of light. This is introduced formally in Chapter 4. For many years optical experiments in the visible spectrum have had the strong advantage that they combine high eﬃciency light generation and detection, can be operated at room temperature without much impact from thermal noise, and, importantly, are directly convincing to the observer. Optical experiments continue to play an important role of testing and verifying the quantum theory and its many predictions. At the same time optical techniques have been a driver of many, many technologies such as communication, sensing, data storage, and even entertainment. We are now busy in creating new applications based on quantum ideas.

References 1 Lenard, P. (1902). Über die lichtelektrische Wirkung. Ann. Phys. 8: 149. 2 Rieke, G. (2003). Detection of Light: From the Ultraviolet to the Submillimeter,

2e. Cambridge: Cambridge University Press. 3 Glauber, R.J. (1962). Coherent and incoherent states of the radiation ﬁeld.

Phys. Rev. 131: 2766. 4 Tinsley, J.N., Molodtsov, M.I., Prevedel, R. et al. (2016). Direct detection of a

single photon by humans. Nat. Commun. 7: 12172. 5 Lorentz, H.A. (1910). Alte und neue Fragen der Physik. Phys. Z. 11: 1234.

References

6 Paul, H. and Fischer, R. (1983). Light absorption by a dipole. Usp. Fiz. Nauk

141: 375. 7 Paul, H. (1995). Photonen, eine Einführung in die Quantenoptik. Stuttgart:

Teubner. 8 Lawrence, E.O. and Beams, J.W. (1927). The instantaneity of the photoelectric

eﬀect. Phys. Rev. 29: 903. 9 Forrester, A.T., Gudmundsen, R.A., and Johnson, P.O. (1955). Photoelectric

mixing of incoherent light. Phys. Rev. 99: 1691. 10 Loudon, R. (2000). The Quantum Theory of Light, 3e. Oxford: Oxford Univer-

sity Press. 11 Planck, M. (1900). Zur Theorie des Gesetzes der Energieverteilung im Nor-

malspektrum. Verh. Dtsch. Phys. Ges. 2: 237. 12 Hollandt, J. (2012). 125 Years of Research On and with the Black Body,

Metrology Throughout the Ages, vol. 3, no.3, 87. PTB Mitteilungen. 13 Wien, W. (1896). Über die Energievertheilung im Emissionsspectrum eines

schwarzen Körpers. Ann. Phys. 294: 662. 14 Lummer, O. and Pringsheim, E. (1900). Über die Strahlung des schwarzen

Körpers fur lange Wellen. Verh. Dtsch. Phys. Ges. 2: 163–180. 15 Taylor, G.I. (1909). Interference fringes with feeble light. Proc. Cambridge Phi-

los. Soc. 15: 114. 16 Janossy, L. and Naray, Zs. (1958). Investigation into interference phenomena

17 18 19 20 21 22 23 24 25 26 27 28

at extremely low light intensities by means of a large Michelson interferometer. Il Nuovo Cimento 9: 588. Hariharan, P., Brown, N., and Sanders, B.C. (1993). Interference of independent laser beams at the single-photon level. J. Mod. Opt. 40: 113. Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4e. Oxford University Press. Feynman, R., Leighton, R.B., and Sands, M. (1963). The Feynman Lectures on Physics, vol. 3. Addison and Wesley. Michelson, A.A. and Morley E.W. (1927). On the relative motion of the earth and the luminiferous ether. Am. J. Sci. 1887 XXXIV (203). Hellmuth, T., Walther, H., Zajonc, A., and Schleich, W. (1987). Delayed-choice experiments in quantum interference. Phys. Rev. A 35: 2532. Magyar, G. and Mandel, L. (1963). Interference fringes produced by the superposition of two independent maser light beams. Nature 198: 255. Javan, A., Ballik, E.A., and Bond, W.L. (1962). Frequency characteristics of a continuous wave He-Ne optical maser. J. Opt. Soc. Am. 52: 96. Radloﬀ, W. (1971). Zur Interferenz unabhängiger Lichtstrahlen geringer Intensit á́t. Ann. Phys. 26: 178. Franson, J.D. (1991). Violation of a simple inequality for classical ﬁelds. Phys. Rev. Lett. 67: 290. Meystre, P. and Sargent, M. III (1989). Elements of Quantum Optics, 448. Springer-Verlag. Sakurai, J.J. (1985). Modern Quantum Mechanics. Addison-Wesley. Arrecchi, F.T., Berne, A., and Sona, A. (1966). Measurement of time evolution of the radiation ﬁeld by joint photo-current distributions. Phys. Rev. Lett. 17: 260.

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29 Kimble, H.J., Dagenais, M., and Mandel, L. (1977). Photon antibunching in

resonance ﬂuorescence. Phys. Rev. Lett. 39: 691. 30 Cresser, J.D., Haeger, J., Leuchs, G. et al. (1982). In: Dissipative Systems in

Quantum Optics (ed. R. Bonifacio), 21. Springer-Verlag. 31 Walker, J.G. and Jakeman, E. (1985). Photon-antibunching by use of a photo-

electron event triggered optical shutter. Opt. Acta 32: 1303. 32 Grangier, P., Roger, G., Aspect, A. et al. (1986). Observation of photon anti-

bunching in phase-matched multiatom resonance. Phys. Rev. Lett. 57: 687. 33 Scgirhagle, R., Chang, K., Loretz, M., and Degen, C.L. (2014).

34

35 36 37 38 39

Nitrogen-vacancy centers in diamond: nanoscale sensors for physics and biology. Annu. Rev. Phys. Chem. 65: 83. Jaynes, E.T. and Cummings, F.W. (1963). Comparison of quantum and semiclassical radiation theories with application to the beam maser. Proc. IEEE 51: 89. Cohen-Tannoudj, C., Diu, B., and Laloë, F. (1980). Quantum Mechanics. Wiley. Brune, M., Schmidt-Kaler, F., Maali, A. et al. (1996). Quantum Rabi oscillation: a direct test of ﬁeld quantization in a cavity. Phys. Rev. Lett. 76: 1800. Guerlin, C., Bernu, J., Deleglise, S. et al. (2007). Progressive ﬁeld state collapse and quantum non-demolition photon counting. Nature 448: 889. Rempe, G., Walther, H., and Klein, N. (1987). Observation of quantum collapse and revival in a one-atom maser. Phys. Rev. Lett. 58: 353. Deleglise, S., Dotsenko, I., Sayrin, C. et al. (2008). Reconstruction of non-classical cavity ﬁeld states with snapshots of their decoherence. Nature 455: 510.

Further Reading Haroche, S. (2012). Nobel Lecture 2012. www.nobelprize.org/nobel prizes/physics/laureates/2012/haroche-lecture.html. Haroche, S. and Raimond, J.-M. (2006). Exploring the Quantum. Oxford University Press. Kingston, R.H. (1995). Optical Sources, Detectors and Systems. Academic Press. Kwiat, P.G., Mattle, K., Weinfurter, H. et al. (1995). New high-intensity source of polarization entangled photon pairs. Phys. Rev. Lett. 75: 4337. Mandel, L. and Wolf, E. (1995). Optical Coherence and Quantum Optics. Cambridge: Cambridge University Press. Zeilinger, A., Weihs, G., Jennewein, T., and Aspelmeyer, M. (2005). Happy centenary, photon. Nature 433: 230.

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4 Quantum Models of Light 4.1 Quantization of Light 4.1.1

Some General Comments on Quantum Mechanics

In this chapter we will develop quantum mechanical models for light that will help us understand the various experimental results and techniques we will encounter in this book. Let us begin by reviewing the types of predictions one can make about experimental outcomes of quantum mechanical systems and how these are obtained from the mathematics. In classical mechanics, an observable quantity is represented by a variable such as x, the position of a particle. This is just a number that may vary as a function of time and/or other parameters. The outcome of an experiment that measures position will just be the value of x. In general, for quantum mechanical systems, the result of an experiment will not be unique. Instead, a range of possible outcomes will occur with some probabilities. Consequently, a single number is insuﬃcient to describe all the possible outcomes of a particular experiment. Thus, in quantum mechanics, variables are ̂ the position operator, describing the type of meareplaced by operators such as x, surement to be made, and states, represented by kets, |𝜎⟩. As we saw in the last chapter, these contain the information about the possible outcomes. Each operator representing an observable quantity will have a set of eigenstates associated with it whose eigenvalues represent the spectrum of possible results that a measurement of this observable can produce. Typically a system will be in a state that is a superposition of various eigenstates of some observable. A special case is when the system is actually in an eigenstate of the observable, in which case a deﬁnite result is predicted. Most generally the system will be in a statistical mixture of various quantum states. Quantum theory makes predictions about the results of large ensembles of measurements. The average value of an observable is given by the expectation value of the observable’s operator: ̂ = ⟨𝜎|x|𝜎⟩ ̂ ⟨x⟩

(4.1)

Equation (4.1) predicts the average value of measurements performed on a large ensemble of systems all prepared in the same state, |𝜎⟩, of the observable ̂ We can also consider expectation values of higher represented by the operator x. A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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4 Quantum Models of Light

moments such as ⟨x̂ 2 ⟩, which naturally leads us to deﬁne the variance of an observable as ̂ 2 ⟨Δx̂ 2 ⟩ = ⟨x̂ 2 ⟩ − ⟨x⟩

(4.2)

and similarly for higher-order properties. In classical mechanics a non-zero variance, or noise, is associated with imperfect preparation of the system with the desired property. In quantum mechanics even ideally prepared systems can exhibit non-zero variance due to the probabilistic nature of the theory. Such non-zero ﬂuctuations are thus often referred to as quantum noise. The characteristics of a system with respect to a particular observable can be completely characterized by the expectation values of all the moments of the observables’ operator. However, in many cases of interest in this book, the systems will exhibit Gaussian statistics. In such cases, the system is completely characterized by just the ﬁrst- and second-order moments. A powerful link between classical variables and quantum operators exists in the procedure of canonical quantization, ﬁrst pioneered by Dirac. If a system is described classically by the canonically conjugate variables x and p (in the sense of Hamilton’s equations [1]), then the correct quantum mechanical description can be obtained by simply replacing x and p with the operators x̂ and p̂ in the Hamiltonian and replacing the classical Poisson bracket with the commutation ̂ p] ̂ = x̂ p̂ − p̂ x̂ = iℏ. In the following section we will use this procerelation [x, dure to convert the classical description of light introduced in Chapter 2 into a quantum mechanical description. 4.1.2

Quantization of Cavity Modes

The classical description of light in Section 2.1 assumed as a model for a laser beam a single frequency wave. Although an idealization, for classical light it is not an unreasonable one as it would neglect modulations and ﬂuctuations of the light. Anticipating this we start our discussion by considering the discrete optical modes in a ring cavity. A suﬃciently high ﬁnesse cavity will only support ﬁelds with wavelengths 𝜆 according to 𝜆 = L∕n,

n = 0, 1, 2, 3, …

(4.3)

where L is the cavity round-trip length. This discrete frequency approximation is valid in both the classical and quantum domains. Recall ﬁrst from Section 2.1.4 that the energy in a classical light ﬁeld is proportional to the sum of the squares of the quadrature amplitudes X1 and X2. Speciﬁcally for polarized modes in a high ﬁnesse cavity, the energy is given classically by ℏ𝜔k (4.4) (X12k + X22k ) 4 where k labels the diﬀerent frequency modes and 𝜔k = 2π𝜈k is the angular frequency. Moreover we can identify suitably scaled versions of the quadratures H = Σk

4.1 Quantization of Light

as the canonical variables of the classical light wave. Speciﬁcally qk and pk are canonical variables where √ ℏ𝜔k qk = X1k 2 √ ℏ pk = − X2 (4.5) 2𝜔k k We can now apply canonical quantization by introducing the operators q̂ k and p̂ k with [q̂ k , p̂ l ] = iℏ𝛿kl . This immediately implies that the operator equivalents of ̂ k and X𝟐 ̂ k obey the commutator the quadrature amplitudes X𝟏 ̂ l ] = 2i𝛿kl ̂ k , X𝟐 [X𝟏 The energy operator or Hamiltonian for the system is just ∑ ℏ𝜔k ̂ 2) ̂ 2 + X𝟐 ̂ = (X𝟏 H k k 4 k 4.1.3

(4.6)

(4.7)

Quantized Energy

The Hamiltonian is composed of a sum of components with a whole range of frequencies. These diﬀerent waves are independent in the sense that diﬀerent frequency components commute with each other (see Eq. (4.6)) – much like we are used to considering classical waves of diﬀerent frequencies to be independent. Like in a classical cavity the frequency is a discrete variable, and only frequencies 𝜔 associated with the spatial modes are possible. In fact the Hamiltonian is mathematically equivalent to a sum of uncoupled harmonic oscillators, one for each frequency 𝜔. Each term describes one mode. For each mode we can solve the operator equation. Before carrying out the calculations in detail, let us consider the overall properties of the quantized harmonic oscillator. One important consequence of the quantization of the oscillator is the discreteness of the possible energy states of this system. The system has a series of possible energies that are equally spaced as illustrated in Figure 4.1: ) ( 1 ℏ𝜔 (4.8) n = n + 2 Similarly, the quantized ﬁeld has discrete energies, n . The energy of the ﬁeld can only be increased by multiples of the energy quantum of ℏ𝜔. This quantization suggests that the ﬁeld is made up of particles – photons. However, one has to be careful to use only those properties of particles that can be directly derived from the quantized ﬁeld equations. Care has to be taken with this picture to avoid an over-interpretation and to regard photons too much as particles behaving like small classical particles. It is worthwhile to contemplate one peculiar case, the lowest possible energy 0 = 12 ℏ𝜔, known as the zero point energy. The value of 0 is not zero. There is remaining energy. However, this does not upset our measurements since it is not

95

4 Quantum Models of Light

Figure 4.1 Energy levels of a harmonic oscillator.

Energy

96

(n + 1/2) ħω

possible to extract this energy from the system. It is the lowest energy that can be achieved with this particular mode of light. All measurements of the energy of the mode refer back to it. The fact that we can only measure changes in the energy, not the absolute energy, means that the ground state is not directly accessible. Further discussions on this may be found in E.A. Power [2]. It is noteworthy that the value of 0 is unique to the optical frequency. A ﬁeld that consists of many modes has a total zero point energy of 12 ℏ𝜔 per mode. Since in theory there is no upper bound on the frequencies, the energy of the ground state is inﬁnite – this creates a conceptual problem. Mathematically this can be overcome by renormalization [3], and in practice any real experiment will have a ﬁnite detection bandwidth and the inﬁnity is avoided. It is possible to measure the eﬀect of the zero point energy directly. One can create an experimental situation where the number of allowed modes in a multimode system can be changed, and as a consequence the total zero point energy changes. This increase in energy results in a force that tries to change the system towards the lowest total zero point energy. Such eﬀects were investigated by H.B.G. Casimir and D. Polder [4]. Consider a cavity with length d. The cavity supports modes for all wavelengths 𝜆 = 2d∕n = L∕n. The longest supported wavelength is L. The size of the cavity determines the energy of the allowed low frequency modes and thus the total zero point energy. That means that for a smaller cavity, shorter L, the total zero point energy is smaller than for a larger cavity. The change in total zero point energy is proportional to the change in number of modes. H.B.G. Casimir and D. Polder [4] worked this out for a system of two conducting plates. In this case the total zero point energy varies with L−3 , and the resulting force Fcas is given by Aπhc [N] (4.9) 4.810−20 d4 where A is the area of the plates and d = L∕2 is the separation. This force does not depend on the electron charge, and it does not require any electric ﬁeld between the plates. It is a quantum mechanical eﬀect that applies to two plates in vacuum. The forces can be signiﬁcant for very small plate separations. As an example values of A = 10−4 m2 and d = 0.01 mm result in Fcas = 1.3 × 10−11 N. This pure quantum force was demonstrated in careful measurements where all other forces were excluded [5]. Without elaborating, it is worth noting that similar forces are part of the explanation of the well-known van der Waals forces [2]. These forces are essential for the explanation of a range of intermolecular eﬀects and include Fcas =

4.2 Quantum States of Light

such practical eﬀects as the stability of colloids in paint suspensions. The zero point energy can play an important role. 4.1.4

The Creation and Annihilation Operators

Just as, given certain assumptions about the spatial modes, we could describe the essential characteristics of a classical light ﬁeld via its complex amplitude, 𝛼 (see Chapter 2), so its quantized equivalent, â , plays a similar role in a quantum description of light. The operator â and its adjoint, â † , raise and lower the excitation of a speciﬁc mode by one increment, respectively. It follows from Eq. (4.6) ̂ = i(̂a† − â ) that they obey the Boson ̂ = â + â † and X𝟐 and the relationships X𝟏 commutation rules [̂a𝜔 , â †𝜔′ ] = (̂a𝜔 â †𝜔′ − â †𝜔 â 𝜔′ ) = 𝛿𝜔𝜔′

(4.10)

[̂a𝜔 , â 𝜔′ ] = [̂a†𝜔 , â †𝜔′ ] = 0

(4.11)

and We can rewrite the Hamiltonian operator of the radiation ﬁeld of a single mode in terms of these operators as ̂ = 1 ℏ𝜔(̂a† â + â â † ) H 2 ( ) ( ) 1 1 = ℏ𝜔 n̂ + (4.12) = ℏ𝜔 â † â + 2 2 where n̂ = â † â is the photon number operator. The Hamiltonian above represents the number of photons in one mode multiplied by the energy of one photon in that mode. For a multimode ﬁeld the total Hamiltonian would be the sum of several terms such as given in Eq. (4.12), one for each frequency component. Note the speciﬁc ordering of the operators â † and â . The ordering is important since the operators do not commute. ̂ p̂ and the creation Finally, the link between the conjugate variable operators q, and annihilation operators â † , â is given by √ ℏ𝜔k q̂ = (̂a + â † ) 2 √ ℏ p̂ = −i (̂a† − â ) (4.13) 2𝜔k It is again worth noting that the operator algebra of the quantized electromagnetic ﬁeld is identical to that of the quantized simple harmonic oscillator.

4.2 Quantum States of Light 4.2.1

Number or Fock States

The Hamiltonian for a single mode of the radiation ﬁeld (Eq. (4.12)) has the eigenvalues ℏ𝜔(n + 12 ). The corresponding eigenstates can be represented by the

97

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4 Quantum Models of Light

notation |n⟩. These are known as the number or Fock states and are eigenstates of the number operator ̂ = n|n⟩ â † â |n⟩ = n|n⟩

(4.14)

The ground state, also known as the vacuum state, is deﬁned by â |0⟩ = 0

(4.15)

It cannot be lowered any further. The energy of the ground state has already been discussed in Section 4.1.3 and is given by 1∑ ̂ ⟨0|H|0⟩ = ℏ𝜔k (4.16) 2 k where we sum over all frequencies. The operators â and â † are raising and lowering operators on a series of equally spaced eigenstates: √ √ (4.17) â |n⟩ = n|n − 1⟩ and â † |n⟩ = (n + 1)|n + 1⟩ They can be used to generate all possible number states starting with the ground state |0⟩ by a successive application of the creation operator (̂a† )n |n⟩ = √ |0⟩ n!

(4.18)

They are orthogonal and complete, i.e. ∑ ⟨n|n′ ⟩ = 𝛿nn′ and |n⟩⟨n| = Î

(4.19)

n

where Î is the identity operator. The number of photons n is exactly known for a number state; hence the variance of the photon number is zero for a number state: ̂ ̂ 2=0 Vn = ⟨Δn2 ⟩ = ⟨n|n̂ n|n⟩ − ⟨n|n|n⟩

(4.20)

We encountered the single photon number state in Section 3.5 where we found that a consideration of their properties in an interferometer led us to a radically diﬀerent view of light from the classical one. We can illustrate this further by contrasting the average value of the quadrature amplitudes for classical light and light in a number state. Recall that in a classical description of light, an arbitrary quadrature amplitude, X 𝜃 , can be described as a function of the phase angle, 𝜃, according to X 𝜃 = cos[𝜃]X1 + sin[𝜃]X2

(4.21)

We can convert this equation to the correct quantum mechanical form by simply replacing the quadrature variables by quadrature operators. The average value of the quadrature amplitude for light in a particular quantum state is then found by taking the expectation value over the state of interest. However what we ﬁnd for a number state is that ̂ 𝜃 ⟩n = ⟨n|X ̂ 𝜃 |n⟩ = 0 ⟨X

(4.22)

4.2 Quantum States of Light

where the orthogonality of the number states ensures that all terms like ⟨n|̂a|n⟩ and ⟨n|̂a† |n⟩ are zero (see Eq. (4.17)). The results in Eqs. (4.21) and (4.22) could hardly be more diﬀerent. In the classical case the quadrature amplitude oscillates as a function of the quadrature angle, 𝜃, whilst for the number state the average value is always zero, independent of the quadrature angle. It is clear that number states represent quite exotic states of light, quite dissimilar to what is produced directly by a laser. We will return to consider them again later but for now we look for light states with more familiar behaviour, which, in turn, turn out to be those most naturally produced. For now we use the number states as a basis set and ﬁnd more realistic superpositions. However, one particular number state plays a major role in all quantum optical experiments: the vacuum state |0⟩. 4.2.2

Coherent States

The number states do not represent an optical state with a well-deﬁned phase. To describe laser light we require a state that has a more precisely deﬁned phase. This is the coherent state |𝛼⟩, which is the closest quantum approximation to the ﬁeld generated by a laser [6]. Consider, in an idealized way, how a light wave is produced in the classical picture. Initially the energy in the electric and magnetic ﬁelds is zero. Then, by some means, the electric ﬁeld is displaced to some non-zero value, say, |𝛼|. This in turn displaces the magnetic ﬁeld and so on, producing a propagating light ﬁeld. Suppose we follow the same idealized recipe in quantum mechanics. This suggests that we should displace the quantum mechanical vacuum state to some non-zero value. We might hope that this will produce an oscillating ﬁeld. However, because the quantum mechanical vacuum has non-zero energy, we also expect uniquely quantum features. ̂ acting on the vacuum ket has the eﬀect of displacThe displacement operator D ing it by an amount 𝛼, where 𝛼 is a complex number that can be directly associated with the complex amplitude in classical optics (see Chapter 2). We deﬁne ̂ |𝛼⟩ = D(𝛼)|0⟩ The displacement operator is given by ̂ D(𝛼) = exp(𝛼 â † − 𝛼 ∗ â )

(4.23)

It is a unitary operator, i.e. ̂ −1 (𝛼) = D(−𝛼) ̂ ̂ † (𝛼) = D D and has the properties ̂ ̂ † (𝛼)̂aD(𝛼) = â + 𝛼 D † † ̂ ̂ D (𝛼)̂a D(𝛼) = â † + 𝛼 ∗

(4.24)

The family of states |𝛼⟩ is called coherent states, and, as we will now show, they behave in many ways like classical laser light. Due to the unitarity of the displacement operator, they are normalized to unity: ⟨𝛼|𝛼⟩ = 1

(4.25)

99

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4 Quantum Models of Light

Also the coherent states are eigenstates of the annihilation operator â , as can be veriﬁed from ̂ † (𝛼)̂aD(𝛼)|0⟩ ̂ ̂ † (𝛼)̂a|𝛼⟩ = D = (̂a + 𝛼)|0⟩ = 𝛼|0⟩ D ̂ Multiplying both sides with D(a) results in the eigenvalue condition â |𝛼⟩ = 𝛼|𝛼⟩

(4.26)

Using Eq. (4.26) we now see that the expectation values for the quadrature amplî 𝜃 ⟩𝛼 , are identical to the classical ̂ 𝜃 |𝛼⟩ = ⟨X tude operator in a coherent state, ⟨𝛼|X variables, provided we identify the displacement 𝛼 with the classical complex amplitude. That is ̂ 𝜃 ⟩𝛼 = cos[𝜃]X1 + sin[𝜃]X2 ⟨X

(4.27)

Hence we see that the average value of the ﬁeld oscillates as a function of quadrature angle. Similarly the expectation value for the photon number in a coherent state is ⟨̂a† â ⟩𝛼 = 𝛼 ∗ 𝛼

(4.28)

which again is the same as the classical value. Thus we see that the average values of the observables for a coherent state are identical to those for a classical ﬁeld. However, unlike a number state or a classical ﬁeld that contains a deﬁnite amount of energy, a coherent state does not contain a deﬁnite number of photons. This follows from the fact that a coherent state |𝛼⟩ is made up of a superposition of number states: ∞ ∑ |n⟩⟨n|𝛼⟩ (4.29) |𝛼⟩ = n=0

Using Eq. (4.18) and the completeness of the number states, Eq. (4.19), one obtains ) ( 1 𝛼n (4.30) ⟨n|𝛼⟩ = √ exp − |𝛼|2 2 n! which is the contribution of the various number states. Now consider the relationship between two diﬀerent coherent states. Using Eqs. (4.29) and (4.30), we ﬁnd that their scalar product is ) 2 ( 1 | | |⟨𝛼1 |𝛼2 ⟩|2 = | exp − (|𝛼1 |2 + |𝛼2 |2 ) + 𝛼1∗ 𝛼2 | | | 2 2 = exp −|𝛼1 − 𝛼2 | (4.31) which is not zero. That means two diﬀerent coherent states are not orthogonal to each other. However, for two states with signiﬁcantly diﬀerent photon numbers, that is, |𝛼1 − 𝛼2 | > 1, the product is very small. Such states are approximately orthogonal. One consequence is that the coherent states form a basis that is over-complete. But despite this, the coherent states can be used as a basis on which any state |Ψ⟩ can be expanded [7]. In summary, the coherent states have properties similar to those of classical coherent light and in fact represent its closest quantum mechanical

4.2 Quantum States of Light

approximation. Just as under certain conditions laser light can be approximated as idealized classical coherent light, so in situations where a quantum mechanical description is needed, laser light can often be well approximated by a coherent state. We will ﬁnd in Section 5.1 that coherent states are stable in regard to dissipation, which means one coherent state is transformed into another coherent state when attenuated. Consequently, the coherence properties of the light remain similar after attenuation. Coherent states will be used extensively in our modelling of laser experiments. Being a complex number, the range of possible values of 𝛼, and thus the range of possible coherent states, requires a two-dimensional representation. This is only natural since we can distinguish diﬀerent states of light experimentally by two parameters, intensity and phase. But we note that only one of them, the intensity, has a simple quantum mechanical equivalent (i.e. the number operator). A similarly simple quantum mechanical operator for the phase is not available. Phase operators can only be approximated in a rather complex way through a limit of a series of operators [7–9], and their deﬁnition has stimulated a lively debate amongst theoreticians. However, the real and imaginary parts of the annî and X𝟐, ̂ are both hilation operator â , i.e. the quadrature amplitude operators X𝟏 well-behaved operators. This leads to a pair of Hermitian operators that describe the ﬁeld. 4.2.3

Mixed States

So far we have only considered pure ﬁeld states. More generally we can have ﬁelds that are in an incoherent mixture of diﬀerent pure states. This is referred to as ̂ a mixed state. A mixed state is normally represented by a density operator, 𝝆, such that 𝝆̂ = 𝑤1 |𝜎1 ⟩⟨𝜎1 | + 𝑤2 |𝜎2 ⟩⟨𝜎2 | + · · · represents a mixture of the states |𝜎1 ⟩, |𝜎2 ⟩, … with fractional weightings 𝑤1 , 𝑤2 , …, respectively. It is important not to confuse mixtures of states and superpositions√ of states; they are very diﬀerent. Consider the equal superposition state |𝜙⟩ = 1∕ 2(|a⟩ + |b⟩). Written as a density operator, this state is 𝝆̂ p = |𝜙⟩⟨𝜙| 1 1 1 1 = |a⟩⟨a| + |a⟩⟨b| + |b⟩⟨a| + |b⟩⟨b| 2 2 2 2 In contrast an equal mixture of the states a and b is just

(4.32)

1 1 |a⟩⟨a| + |b⟩⟨b| (4.33) 2 2 The coherences between the states that are present for the superposition are absent in the case of the mixture. Expectation values for mixed states are given by 𝝆̂ m =

̂ = Tr{x̂ 𝝆} ̂ ⟨x⟩ where the trace, Tr, is taken over a complete set of eigenstates. An important example of a mixed state in optics is the thermal state, which can be represented

101

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4 Quantum Models of Light

as a mixture of number states by ) ( ) ( 1 G−1 G−1 2 𝝆̂ th = |2⟩⟨2| + · · · |1⟩⟨1| + |0⟩⟨0| + G G G

(4.34)

where G is a parameter that ranges from one for a zero temperature thermal ﬁeld (the vacuum) up to a high temperature thermal ﬁeld for G → ∞.

4.3 Quantum Optical Representations 4.3.1

Quadrature Amplitude Operators

Let us now look in some detail at the quadrature amplitude operators deﬁned by ̂ = â + â † X𝟏

̂ = i(̂a† − â ) and X𝟐

(4.35)

We will sometimes refer to them as the amplitude quadrature and the phase quadrature, respectively, for reasons that will become apparent when we discuss bright ﬁelds (Section 4.5). A variety of nomenclature for the quadrature ̂ X𝟐} ̂ ≡ amplitude operators exist in the literature – some common ones are {X𝟏, + ̂− ̂ ̂ ̂ ̂ p}. ̂ {X , X }; {X, P}; {x, ̂ We can deﬁne eigenstates of these operators via X𝟏|x1⟩ = x1|x1⟩ and similarly ̂ X𝟐|x2⟩ = x2|x2⟩. The eigenvalues xi form an unbounded continuous set. Because ̂ and X𝟐, ̂ it is not possible to of the non-zero commutation relation between X𝟏 ̂ ̂ be in a simultaneous eigenstate of X𝟏 and X𝟐. The resulting uncertainty principle constrains the variances of the operators in any state to ̂ 2⟩ ≥ 1 ̂ 2 ⟩⟨ΔX𝟐 ⟨ΔX𝟏

(4.36)

where the quadrature ﬂuctuation operators are deﬁned: ̂ = X𝟏 ̂ − ⟨ΔX𝟏⟩ ̂ ΔX𝟏

̂ = X𝟐 ̂ − ⟨ΔX𝟐⟩ ̂ and ΔX𝟐

(4.37)

̂ or X𝟐 ̂ are unphysical as inﬁnite uncertainty It follows that exact eigenstates of X𝟏 in an unbounded spectrum implies inﬁnite energy. This is analogous to the situation for position and momentum of a free particle. We saw in the previous section that the expectation values of the quadrature operators in a coherent state were equal to their classical values. It is straightforward to show that the uncertainty product for the quadratures in a coherent state is given by ̂ 2 ⟩𝛼 = 1 ̂ 2 ⟩𝛼 ⟨ΔX𝟐 ⟨ΔX𝟏

(4.38)

which means a coherent state is a minimum uncertainty state. Equally important is the fact that the uncertainty of the coherent state is split equally between the two quadratures. The uncertainty relationship is symmetric: ̂ 2⟩ = 1 ̂ 2 ⟩ = ⟨ΔX𝟐 ⟨ΔX𝟏

(4.39)

Hence the coherent state can be thought of as the best allowed compromise for a simultaneous eigenstate of the two quadratures. The quadrature operators allow us to represent a beam of light graphically. These so-called phasor diagrams are

4.3 Quantum Optical Representations

very popular in quantum optics – ever since their introduction by C. Caves [10]. Any state of light can be represented on a phasor diagram of the operators, that ̂ versus ⟨X𝟏⟩. ̂ is, a plot of ⟨X𝟐⟩ In such a diagram a particular coherent state |𝛼⟩ is ̂ ̂ equivalent to an area centred around the point given by the value (⟨X𝟏⟩, ⟨X𝟐⟩) for 2 ̂ ̂ this state. The size of the area is given by the variances ⟨ΔX𝟏 ⟩ and ⟨ΔX𝟐2 ⟩ and ̂ 2 ⟩, and the area of the is usually represented just by a circle with diameter ⟨ΔX𝟏 circle describes the extent of the uncertainty distribution. This phasor diagram is a qualitative description that shows several important results: 1. For a coherent state the uncertainty area is symmetric. Any measurement of the ﬂuctuations will yield the same result, independent of the projection angle 𝜃 chosen. 2. The uncertainty area is of constant size, independent of the intensity |𝛼|2 of the state. ̂ X𝟐 ̂ 3. Two states are only distinguishable if their average coordinates in the X𝟏, ̂ 2 ⟩. plane are separated by more than ⟨ΔX𝟏 The last result is signiﬁcant. It gives us a practical rule when two states are measurably diﬀerent. The uncertainty area sets a minimum change in amplitude or phase that we have to make before we can distinguish two states. In a classical system such an uncertainty region would be associated with noise, due to technical imperfections. However, here the uncertainty area is a consequence of the quantum rules – not technical imperfections. Consequently, the uncertainty area represents the eﬀect of quantum noise on a measurement, whether it is amplitude, phase, or any other property of the light. Experimental measurements cannot be more precise than the quantum noise. The uncertainty area is two-dimensional. One can imagine the entire parameter space being covered, or tiled, with circular uncertainty areas of uniform size (see Figure 4.2). The number of tiles, which are not all overlapping, approximately determines the number of states that can be distinguished. As a practical example, consider the light in an optical communication system. The information is transmitted by changing either the intensity or phase of the light at one modulation frequency. We assume the intensity is limited. How many distinguishable Figure 4.2 Uncertainty areas can cover the entire parameter space like tiles.

ˆ 2 ΔX2

X2

ˆ 2 ΔX1

X1

103

104

4 Quantum Models of Light

states of light, that is, diﬀerent pieces of information, could we send? The light is represented by a two-dimensional plane with a circle that indicates the largest possible intensity Imax . Inside this circle we can place our tiles. The number of tiles estimates the upper limit to the number of states we can use, that is, the number of diﬀerent pieces of information that a beam of light modulated at one frequency can contain. As described in Section 10.2, in 2018 many more systems are approaching this quantum resolution, and ultimately technology will reach this limit in most cases. 4.3.2

Probability and Quasi-probability Distributions

The uncertainty area is a simple, useful tool that can, under certain circumstances, be used to completely characterize the state. To understand under what conditions this is possible, we need to look at the probability distributions ̂ and X𝟐. ̂ Such measurements can be carried out associated with measuring X𝟏 using the technique of balanced homodyne detection, which will be described in detail in Chapter 8. The probability of obtaining a result in a small interval, 𝛿x, ̂ performed on a system around a particular value x1 from a measurement of X𝟏 in the state |Ψ⟩ is given by PΨ (x1 ) = |⟨x1 |Ψ⟩|2 𝛿x

(4.40)

̂ the probability of obtaining results in a and similarly for a measurement of X𝟐 small interval around x2 is PΨ (x2 ) = |⟨x2 |Ψ⟩|2 𝛿x

(4.41)

In other words |⟨xi |Ψ⟩| is the probability density at the point xi . These distributions can be evaluated for various states using the relationships 2

d ̂ ⟨x1 |X𝟐|Ψ⟩ = −i ⟨x |Ψ⟩, dx1 1

d ̂ ⟨x2 |X𝟏|Ψ⟩ =i ⟨x |Ψ⟩ dx2 2

(4.42)

For example, for a coherent state, we can show ⟨x1 |̂a|𝛼⟩ = 𝛼⟨x1 |𝛼⟩ 1 ̂ + iX𝟐|𝛼⟩ ̂ = ⟨x1 |X𝟏 2( ) d 1 ⟨x1 |𝛼⟩ x1 + = 2 dx1

(4.43)

Solving the relevant diﬀerential equation then gives 2 1 |⟨x1 |𝛼⟩|2 = √ e−(x1 −x1 ) π

(4.44)

where x1 = (𝛼 + 𝛼 ∗ )∕2 is the real part of 𝛼. We see that our x1 results form a Gaussian distribution. The width or variance of the distribution is 1 as expected. Similarly we ﬁnd 2 1 |⟨x2 |𝛼⟩|2 = √ e−(x2 −x2 ) π

(4.45)

4.3 Quantum Optical Representations

for the probability distribution of x2 results. Here x2 = i(𝛼 − 𝛼 ∗ )∕2 is the imaginary part of 𝛼. Again, we ﬁnd a Gaussian distribution of width 1. Because the coherent states are Gaussian, they can be completely characterized by the ﬁrst̂ and X𝟐 ̂ and thus the uncertainty area of the preand second-order moments of X𝟏 vious section. This property leads to an elegant approach based on the covariance matrix, which will be discussed in Section 4.3.4. It turns out that many experimentally accessible optical quantum states have Gaussian quadrature distributions and therefore are quite easily characterized. This is not in general true however. For example, the quadrature distributions for a single photon number state are given by |⟨x1 |1⟩|2 =

2 1 √ 2x2 e−x1 1 π

(4.46)

and similarly 2 1 |⟨x2 |1⟩|2 = √ 2x22 e−x2 π

(4.47)

These distributions are clearly non-Gaussian. A two-dimensional probability distribution for a state |Ψ⟩ is given by the Q-function deﬁned as |⟨𝛼|Ψ⟩|2 π which is positive and bounded by QΨ (𝛼) =

(4.48)

1 π Physically the Q-function represents the probability density of obtaining the ̂ and result 𝛼 = (x1 + ix2 )∕2 from an optimal simultaneous measurement of X𝟏 ̂ ̂ ̂ X𝟐. Because X𝟏 and X𝟐 do not commute, such a measurement is necessarily nonideal. By optimal we mean that the resolution of the measurement is as good as the uncertainty principle allows. Such an optimal measurement can be achieved using the techniques of heterodyne or dual homodyne detection, which will be discussed in Chapter 8. For a coherent state |𝛽⟩, the Q-function is simply QΨ ≤

exp(−|𝛼 − 𝛽|2 ) |⟨𝛼|𝛽⟩|2 = (4.49) π π It is a Gaussian function centred around the point (e(𝛽), m(𝛽)). The width of the Gaussian is two, that is, twice that of what we found when the quadratures were measured individually. The additional noise is a result of the nonideal measurement in which one extra unit of quantum noise is added. Note that this function has a width that is independent of the average photon number of the state. A strong coherent state has the same-sized Q-function as a weak ﬁeld – just displaced further from the origin. The Q-function for a number state is quite diﬀerent to that of a coherent state and is given by Q𝛽 =

Qn =

|𝛼|2n exp(−|𝛼|2 ) |⟨𝛼|n⟩|2 = π πn!

(4.50)

105

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4 Quantum Models of Light

Figure 4.3 Uncertainty area of a number state.

X2

X1

This distribution has the shape of a ring (see Figure 4.3). The maximum probability is a circle around the origin, and the distribution in the radial direction is Gaussian centred at the average photon number with a width that decreases very rapidly, with n! That means for large photon numbers the radial distribution is approaching a delta function. A number state contains no phase information at all. The ring shape does not favour any particular phase. A two-dimensional distribution more closely related to our homodyne distributions is the Wigner function, which can be written as WΨ (𝛼) =

̂ D(−𝛼)|Ψ⟩ ̂ ̂ 2⟨Ψ|D(𝛼) 𝚷 π

(4.51)

̂ is the parity operator. The parity operator measures the symmetry of where 𝚷 the probability amplitude distribution of a state. The number states are parity eigenstates with eigenvalues +1 for n even and −1 for n odd. So the action of the parity operator on an arbitrary superposition of number states is ̂ 0 |0⟩ + c1 |1⟩ + c2 |2⟩ + · · ·) = (c0 |0⟩ − c1 |1⟩ + c2 |2⟩ − · · · ) 𝚷(c This is eﬀectively a π rotation on the phasor diagram. Thus the Wigner function is the overlap between the state, displaced by −𝛼, and itself, also displaced by −𝛼 but then also rotated by 180 degrees. For a coherent state |𝛽⟩, we will obtain 2⟨𝛽 − 𝛼|𝛼 − 𝛽⟩ 2e−2|𝛼−𝛽|2 = (4.52) π π This is a symmetric Gaussian with width 1, centred at the point (e(𝛽), m(𝛽)). Its projections along the X1 and X2 directions correspond with the homodyne probability distributions of Eqs. (4.44) and (4.45). In this way the noise introduced by the quantum ﬂuctuations can be interpreted as producing ‘classical’ noise distributions. However, the Wigner function is a quasi-probability function. Although for Gaussian states it is always positive and generally well behaved, it can become negative for non-Gaussian states, such as the number state, and thus cannot be interpreted as a true probability distribution. For such states the quantum ﬂuctuations do not have a classical’ noise interpretation. Producing and analysing exotic states with negative Wigner functions is technically challenging but is now W𝛽 (𝛼) =

4.3 Quantum Optical Representations

Figure 4.4 The randomness of the quadrature values represented by a weighted random distribution of dots. Each dot represents one particular instantaneous value. The histogram of all ̂ and X𝟐 ̂ dots both in the X𝟏 directions corresponds to the projection of the Wigner function.

ˆ X2

ˆ X1

achieved routinely by several groups. We will look at some examples of negative Wigner functions in Chapter 12. The uncertainty areas that are shown in the phasor diagrams are simpliﬁed representations of the Wigner function. They are contours drawn at the half maximum value of the full two-dimensional Wigner function (see Figures 4.4 and 4.5). This line is the edge of the uncertainty area. It is described by only two values, the size of the minimum and the maximum axes. From an experimental point of view, we use the measured variance, for a certain angle 𝜃, as a value for the uncertainty of this particular quadrature. That means we select one 𝜃, measure the variance of this quadrature, and obtain two points. We can repeat this for many angles, and in a real experiment we would actually do this to check the theory and reduce the experimental errors. All these points together describe the edge of the uncertainty area. Whilst the Wigner function of a coherent state is circularly symmetric, this is not the only option. The width of the distribution in orthogonal directions could be diﬀerent, and it is possible for one of the variances to fall below 1. The only requirement is that the product of the two widths, that is, the product of the variances, satisﬁes the uncertainty relationship. We shall see in Chapter 9 that such states can be generated in optical experiments, which are known as squeezed states. It follows from the special properties of Gaussian functions that for such Figure 4.5 A two-dimensional Gaussian distribution and its contour. This is the origin of the circular uncertainty area.

X2

X1

107

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4 Quantum Models of Light

a state the contour at the half maximum value is always an ellipse. The diameter along the minor and major axes corresponds to the minimum and maximum quantum noise. The various distributions we have looked at can be generalized to mixed states such that PΨ (xi ) = ⟨xi |𝝆̂ 𝚿 |xi ⟩ QΨ (𝛼) =

⟨𝛼|𝝆̂ 𝚿 |𝛼⟩ π

and 2 ̂ ̂ ̂ Tr{D(−𝛼) 𝝆̂ 𝚿 D(𝛼) 𝚷} (4.53) π where in each case 𝝆̂ 𝚿 is the density operator of the state to be characterized. For example, the Wigner function for a thermal state (see Eq. (4.34)) is given by WΨ (𝛼) =

2

2e−2|𝛼| ∕𝜎 (4.54) π which is again a symmetric Gaussian with width 𝜎 = 2G − 1. Finally, let us make a comment on other possible representations. Another obvious representation would generate a delta function for any particular coherent state. A representation with this property is the P-representation. A plot of the P-function would immediately show which coherent state is present. However, there are a number of mathematical problems that arise with the P-function once, for example, squeezed states are considered. Since for a squeezed state the variance of one of the quadratures is smaller than for a coherent state, and since a delta function cannot be made any narrower, the P-function will get negative values and even singularities. However, regularized versions of the P- function have been developed, which avoid some of these mathematical problems, and these can form a useful way of theoretically analysing quantum optical systems. It can actually be shown that all three representations (Q, P, and W ) are only limiting cases of a whole continuum of possible representations. Wth (𝛼) =

4.3.3

Photon Number Distributions

In the previous section we looked at results of measurements of the quadrature amplitudes. We now consider photon number measurements of a single optical mode. These are modelled as measurements of the observable n̂ = â † â . In general the probability distribution, 𝜌 (n), of a state described by the density operator 𝝆̂ is given by its diagonal elements in the photon number basis, i.e. ̂ 𝜌 (n) = ⟨n|𝝆|n⟩

(4.55)

For a photon number state, there is a unique photon number; it will be exactly the same in every measurement, and hence Eq. (4.55) has only one non-zero element. However, for a coherent state, one expects to ﬁnd a spread in the results of many measurements of the photon number. We can use the expansion in Eq. (4.29) to determine the probability 𝛼 (n) of ﬁnding n photons in the coherent state |𝛼⟩: 𝛼 (n) = |⟨n|𝛼⟩|2 =

|𝛼|2n exp(−|𝛼|2 ) n!

(4.56)

4.3 Quantum Optical Representations

0.2 nα = 4

0.1

nα = 1000

nα = 36

0.05

2Δn = 64

P(n)

P(n)

0.15

0.035 0.03 0.025 0.02 0.015 0.01 0.005

0 0

10

20

30

40

50

60 n

950

1000

1050

1100

Figure 4.6 Poissonian distribution of photons for three coherent states: (i) n𝛼 = 4, (ii) n𝛼 = 36, and (iii) n𝛼 = 1000.

This is a Poissonian distribution with a mean value of |𝛼|2 , which represents the average number n𝛼 of photons in this mode. The intensity ﬂuctuates according to this distribution. The noise is not a result of technical imperfections of the experiment. This is a quantum property of the light. Mathematically speaking it is a direct consequence of the quantum mechanical commutation rules. One important property of any Poissonian distribution is that the variance of the distribution is equal to the√mean value. Consequently, the width of the distribution Δn is equal to Δn = n𝛼 . Examples of three such distributions for n𝛼 = 4, n𝛼 = 36, and n𝛼 = 1000 are shown in Figure 4.6. Evaluating the relative uncertainty of√the mean photon number of a coherent state results in Δn∕n = √ n𝛼 ∕n𝛼 = 1∕ n𝛼 , which decreases with n𝛼 . A state with a high photon number is better deﬁned than that of a state with a small number of photons. We can interpret this result in the photon picture: a photodetector will not see a regular stream of photons, but it will see a somewhat randomized stream such that the measurement of the absorption events – for a perfect detector equal to the number of photons incident – has a distribution given by 𝛼 (n). Each event creates a ﬁxed number of electrons, and the stream of electrons leaving the detector forms the photocurrent. The distribution 𝛼 (n) is directly represented by the variation of the photocurrent. The intensity noise properties of the light could be expressed in terms of the distribution function 𝛼 (n). Thinking in the time domain, this means that the photons arrive in unequal time intervals as shown in Figure 4.7d. For large intensities 𝛼 (n) tends to a normal Gaussian function. In contrast, for low intensities, the Poissonian distribution diﬀers markedly from a Gaussian function. A convenient way to classify the photon number distribution, particularly for bright ﬁelds, is via its width relative to Poissonian. In many practical situations light has more ﬂuctuations than a coherent state, and its photon distribution is wider than a Poissonian function. In this case we would call it super-Poissonian. On the other hand, we will ﬁnd that non-linear processes can be used to make the distribution narrower. This we will call sub-Poissonian light. In order to quantify the noise one can deﬁne the Fano factor, which is given by the ratio of the actual width of the distribution to the width of a Poissonian distribution with the same mean photon number. For some arbitrary state 𝜌𝛽 with

109

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4 Quantum Models of Light

δX2

δX2

δX2 α2 δX1

α1

(a)

X

δX1

(d) I(t)

P(n)

I2

[I2]

[I1]

Vnα1

n nα2

nα1

(b)

t (e) I(Ω)

Counts

α2 1

α1 t

(c)

0

Ω (f)

Im(α(Ω)) Re(α(Ω))

(g)

Ω

Figure 4.7 Diﬀerent representations of a coherent state of light, shown for states |𝛼1 ⟩, |𝛼2 ⟩ with diﬀerent intensities. (a) Phase space diagram, (b) photon number distribution, (c) arrival times of photons, (d) Wigner function, (e) time evolution of the intensities, (f ) variance of the intensities, and (g) the noise at diﬀerent frequencies.

4.3 Quantum Optical Representations

mean photon number n𝛽 and distribution V n𝛽 , we deﬁne the Fano factor as f𝛽 =

V n𝛽 n𝛽

(4.57)

which classiﬁes the states in the following way: f f f f

=0 1

number state sub-Poissonian state coherent state super-Poissonian state

Note that the Fano factor is determined only by the variance. In fact the Fano factor is identical to the normalized variance of the photon ﬂux, which, for many practical situations, is equivalent to the variance V 1 of the quadrature amplitude, as we shall see in Section 4.5. In much of this guide, we will use V 1 interchangeably with the Fano factor. Another important way to classify photon number distributions, particularly for low intensity ﬁelds, is the second-order coherence function, g (2) , previously introduced for classical ﬁelds, which we will discuss in detail in Section 4.6.1. For zero time delay, we have the following relationship between the second-order coherence function and the Fano factor: 1 g (2) (0) = f + 1 − (4.58) n 4.3.4

Covariance Matrix

The second-order quadrature moments can be arranged into a 2 × 2 matrix V, referred to as the covariance matrix [11]: ( ) 1 ̂ X𝟐 ̂ + ΔX𝟐Δ ̂ X𝟏⟩ ̂ ̂ 2⟩ ⟨Δ X𝟏Δ ⟨ΔX𝟏 2 V= 1 ̂ (4.59) ̂ + ΔX𝟐Δ ̂ X𝟏⟩ ̂ ̂ 2⟩ ⟨ΔX𝟏ΔX𝟐 ⟨ΔX𝟐 2 where the quadrature ﬂuctuation operators are deﬁned in Eq. (4.37). Similarly, the displacement vector, d, can be deﬁned via the ﬁrst-order moments: ( ) ̂ ⟨X𝟏⟩ d= (4.60) ̂ ⟨X𝟐⟩ For Gaussian states the covariance matrix plus the displacement vector completely characterizes the state. For simplicity, the covariance matrix is often presented in its standard form: ( ) ̂ ′2 ⟩ 0 ⟨ΔX𝟏 ′ V = (4.61) ̂ ′2 ⟩ 0 ⟨ΔX𝟐 ̂ + sin 𝜃 X𝟐 ̂ and X𝟐 ̂ ′ = cos 𝜃 X𝟐 ̂ − sin 𝜃 X𝟏 ̂ are rotated ver̂ ′ = cos 𝜃 X𝟏 where X𝟏 sions of the quadratures with 𝜃 speciﬁcally chosen to make the oﬀ-diagonal

111

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4 Quantum Models of Light

terms vanish. Such a 𝜃 can always be found. Physically this corresponds to rotating the quadratures until they line up with the major and minor axes of the uncertainty ellipse (see Section 4.3.2). In other words, in the standard form, ̂ ′2 ⟩ is minimized, or vice versa. ̂ ′2 ⟩ is maximized and V 2 = ⟨ΔX𝟐 V 1 = ⟨ΔX𝟏 Many useful properties about Gaussian states can easily be extracted from the covariance matrix. This will be particularly important when we consider multimode states and their correlations. A simple single-mode example is that the uncertainty principle can be expressed as DetV ≥ 1. Here DetM is the determinant of the matrix M, reducing to the familiar Heisenberg form where the covariance matrix is in its standard form (Eq. (4.61)). 4.3.4.1 Summary of Diﬀerent Representations of Quantum States and Quantum Noise

We have examined a number of representations of quantum optical states, several based on quadrature amplitudes, plus the photon number distribution. All representations are complete, so the choice of which to use for a particular problem may depend on mathematical tractability, the physics one wishes to highlight, or simply taste. Being quasi-probability distributions, these representations often suggest interpretations in terms of noise. Quantum noise is the most easily observable consequence of the quantization of light. It is a direct result of the use of operators for describing the amplitude of the electromagnetic ﬁeld and of the commutation rules applied to these operators. In other words, the minimum uncertainty principle applied to light leads to the quantum noise that aﬀects many optical measurements. For Gaussian states a noise interpretation is always possible for the quadrature amplitudes; however in general such an interpretation is at odds with photon number measurements. For non-Gaussian states this fails even for quadrature measurements as negative probabilities’ can result. A useful quantum state, sometimes seen as lying on the boundary between classical and quantum states, is the coherent state. It is an idealized state of light representing the output of a perfect laser source. In mathematical terms it is the eigenstate of the annihilation operator. A physical consequence of this is that attenuation changes one coherent state into another coherent state, just with lower energy. The characteristic properties of the coherent state are preserved. In many situations the coherent state can be interpreted as a classical light ﬁeld with the addition of stochastic noise. The coherent states can be used as a basis set, and all types of light can be described as a superposition of these basic states. This expansion can be represented by a probability distribution for measuring particular 𝛼, the Q-function, which includes the intrinsic measurement penalty. The characteristic properties of a coherent state of light can be summarized in Table 4.1. The corresponding graphical representations of quantum noise are shown in Figure 4.7.

4.4 Propagation and Detection of Quantum Optical Fields

Table 4.1 Properties of coherent states. Properties of coherent states

Average complex amplitude

𝛼 = |𝛼|ei𝜙

Average photon number

n𝛼 = |𝛼|2

Quadrature amplitude operators

̂ X𝟐 ̂ corresponding to a complex Operators X𝟏, amplitude 𝛼

Wigner function

Quasi-probability distribution function is given by a two-dimensional Gaussian function of unity width. Contour of the Wigner function is a circle

Fluctuations of the quad. amplitude

Variance of the operator is given by V𝛼 (𝜃) = V 1𝛼 = 1, independent of quadrature and detection frequency. Standard quantum limit (SQL)

Intensity noise

V n𝛼 = n𝛼 , which is equivalent to the shot noise formula. The intensity noise spectrum is ﬂat, independent of detection frequency

Photon number distribution

Poissonian: 𝛼 (n) = |⟨n|𝛼⟩|2 =

Photon statistics

Time sequence of photon arrival is irregular as shown in Figure 4.7c. The statistics of the number of photons arriving in a time interval times follows a Poissonian distribution

|𝛼|2n n!

exp(−|𝛼|2 )

Fano factor

f𝛼 = 1 for all detection frequencies

Classical ﬁeld

Carrier component has an amplitude |𝛼|, with optical phase arg(𝛼). It has completely uncorrelated noise sidebands

4.4 Propagation and Detection of Quantum Optical Fields Up till now we have been implicitly considering a stationary single frequency mode in a cavity and the behaviour of observables of that mode without specifying how they would be measured. Now we wish to address the experimentally relevant situation of propagating ﬁelds and photodetection. We begin with a quite general discussion of the problem whilst still limiting our discussion to that of single-mode ﬁelds. In free space there is an inﬁnite continuum of frequency modes that must all be quantized. The approach described in this section can be a satisfactory approximation in situations where, over the detection bandwidth, the frequency spectrum of the source and the frequency response of the optical elements are both ﬂat. That is, although there are many modes present, they are all doing the same thing’. A single mode can also be constructed from a superposition of frequency modes as we shall describe in the ﬁrst part of this section. The approach of this

113

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4 Quantum Models of Light

section is also appropriate if the optical elements act coherently on a single mode of any kind to map it to another single mode. In the following section we will discuss a more speciﬁc approach in which the multimode eﬀects of non-uniform spectral responses are included. 4.4.1

Quantum Optical Modes in Free Space

A single propagating mode is deﬁned quantum mechanically in an analogous way to the classical approach (see Section 2.2): ̃k dk e−i𝜔t F(k, x)A (4.62) ∫ ̃ k are continuum annihilation operators. They obey the commuThe operators A tation relation ̃ † ] = 𝛿(k′ − k) ̃ k′ , A (4.63) [A â (x, t) =

k

and the mode function is normalized such that ∫

dk|F(k, x)|2 = 1

(4.64)

As a result it is easy to show that [̂a(x, t), â † (x, t)] = 1

(4.65)

as required for Boson creation and annihilation operators. Two such modes, say, ̂ are said to be independent or orthogonal if they commute, i.e. â and b, [̂a(x, t), b̂ † (x, t)] = 0

(4.66)

Notice that if the mode function for the mode b̂ is G(k, x), then this implies ∫

dkF(k, x)G(k, x)∗ = 0

(4.67)

which is identical to the deﬁnition of independent classical modes in Section 2.2.3. Indeed all the properties of classical modes discussed there transfer across to the quantum realm – a feature exploited in many of the experiments we will discuss later in the book. An idealized quantum optical mode that is nonetheless a good approximation in many situations is the plane wave Gaussian pulse: â (z, t) = where

√ ̃𝜔 = A

∫

̃𝜔 d𝜔e−i(𝜔t−kz) F(𝜔)A

−2zo k⃗ 2 ko ⃗ ko A ̃k dke √ 4zo π ∫

(4.68)

(4.69)

and k⃗ = (kx , ky ), k ≡ kz is the direction of propagation and 𝜔 = 2π𝜈 = |k|∕c. Also, ko is the average wave number of the beam. The parameter zo is deﬁned in Section 2.1.5. Note that ̃ † ′ ] = 𝛿(𝜔 − 𝜔′ ) ̃ 𝜔, A (4.70) [A 𝜔

4.4 Propagation and Detection of Quantum Optical Fields

Physically, Eq. (4.68) describes a pulse of light in a transverse Gaussian mode (see Sections 2.1.5 and 2.2.3) on axis and close to its waist. The pulse shape as a function of frequency is given by F(𝜔). The relatively simple form of Eq. (4.68) comes from the fact that the wave fronts of a Gaussian mode are approximately plane waves close to their waist such that the transverse and longitudinal degrees of freedom decouple. It may seem very restrictive to have the requirement that we are close to the waist of the beam; however, lenses can be used to focus a Gaussian beam down to its waist at any spatial location. Typically in experiments it is arranged that optical beams interact with optical elements at their waists. A simple case of a pulse shape that is easy to work with is if the pulse shape is also Gaussian: √ 1 −(𝜔−𝜔o )2 (4.71) F(𝜔) = √ e 2𝑣 𝑣π where 𝜔o = ko ∕c is the average frequency of the beam and 𝑣 is the variance around this mean. With this pulse shape, the following conditions on the parameters for Eq. (4.68) to be a good approximation close to the waist: √ are required √ z, 1∕𝑣 ≪ zo and 𝑣 ≪ 𝜔o . Now consider the quantum mode ̂ t) = b(z, where

√ B̃ 𝜔 =

d𝜔e−i(𝜔t−kz) F(𝜔)B̃ 𝜔

(4.72)

−2zo k⃗ 2 ko ⃗ ko eixo kx A ̃k dke √ 4zo π ∫

(4.73)

∫

Equations (4.68) and (5.47) describe two pulses with identical spectral characteristics co-propagating in parallel Gaussian modes separated by a distance xo . √ ̃ 𝜔 , B̃ † ′ ] = 0 and the two modes will be indepenProvided xo ≫ ko ∕zo , then [A 𝜔 dent. Often when we talk about independent modes, we will be thinking of this type of situation with spatially separated optical beams; however, as discussed in Section 2.2, there are many other degrees of freedom that can be used to create independent modes. In the rest of this section, we will work at the level of the mode operators without being speciﬁc about the type of modes involved. 4.4.2

Propagation in Quantum Optics

The propagation of quantum optical modes through optical elements is modelled ̂ † . Two distinct pictures ̂ where U ̂ −1 = U via unitary transformation operators, U, can be employed: (i) the Schrödinger picture, in which the evolution of the state of the mode |Ψ⟩ → |Ψ′ ⟩ is calculated via ̂ |Ψ′ ⟩ = U|Ψ⟩ and the system observables remain stationary, or (ii) the Heisenberg picture, in which the evolution of a mode observable â → â ′ is calculated via ̂ ̂ † â U (4.74) â ′ = U

115

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4 Quantum Models of Light

and the state of the system is stationary. The two pictures are equivalent in the sense that all expectation values are equal in the two pictures because ̂ ̂ † â U|Ψ⟩ = ⟨Ψ|̂a′ |Ψ⟩ ⟨Ψ′ |̂a|Ψ′ ⟩ = ⟨Ψ|U

(4.75)

However, depending on the particular situation, the two pictures oﬀer diﬀerent levels of computational complexity and diﬀerent physical insights. For propagation through stationary elements (that do not change with time), we can write ̂ = ei Ĥ 𝜏∕ℏ U

(4.76)

̂ is the interaction Hamiltonian characterizing the particular optical elewhere H ment and 𝜏 characterizes the strength of the interaction or the time for which it acts. For the latter case it is easy to show that the equation of motion for an arbitrary Heisenberg operator is then 1 dâ ̂ = [̂a, H] (4.77) dt iℏ The Heisenberg picture most resembles the classical treatment thanks to canonical quantization that requires that the equations of motion of the annihilation operator â are identical to those of the classical complex amplitude 𝛼. Thus in many situations it is straightforward to write down the propagation transformations in the Heisenberg picture, and one is able to retain a considerable amount of classical intuition in the quantum treatment. This is particularly true for Gaussian states as will be discussed in the next section. However, this is not always the case, and in certain situations the higher-order moments of the annihilation and creation operators may have equations of motion quite dissimilar from their classical counterparts. The Schrödinger picture on the other hand gives a uniquely quantum mechanical view of the transformations and as such tends to be most useful when quantum eﬀects dominate, such as in photon counting experiments. In general it is not trivial to convert from a potentially straightforward analysis of an optical circuit in the Heisenberg picture to the corresponding Schrödinger state transformation. An exception is propagation through linear optical elements, that is, passive elements that do not change the photon number in the modes. Clearly the action on the vacuum mode of unitary operators representing such elements will be the identity, that is, ̂ lin |0⟩ = |0⟩ U

(4.78)

Now suppose we write out the state we wish to transform in the number basis: |𝜓⟩ = c0 |0⟩ + c1 |1⟩ + c2 |2⟩ + · · · ) ( 1 †2 † = c0 + c1 â + c2 √ (̂a ) + · · · |0⟩ 2

(4.79)

4.4 Propagation and Detection of Quantum Optical Fields

where we have used Eq. (4.18). Now we ﬁnd ( ) 1 † † 2 ̂ lin c0 + c1 â + c2 √ (̂a ) + · · · U ̂ |0⟩ ̂ lin |𝜓⟩ = U ̂† U U lin lin 2 ( ) 1 ′′† ′′† 2 = c0 + c1 â + c2 √ (̂a ) + · · · |0⟩ 2

(4.80)

̂ aU ̂ † is the inverse Heisenberg picture annihilation operator, where â ′′ = Û obtained by inverting the standard Heisenberg equations of motion. This technique is easily generalized to optical elements that mix more than one mode and, where applicable, is a convenient tool for moving between the two pictures. 4.4.3

Detection in Quantum Optics

The basic detection tool in quantum optics is photodetection. One way or another almost all measurements of optical observables rely on photodetection. For bright ﬁelds photodetection measures the intensity of the ﬁeld; for weak ﬁelds it measures the photon number of the ﬁeld. Although quite distinct technologies must be employed in the bright and weak ﬁeld limits, the process can be modelled in both cases as the measurement of the photon number observable â † â . More sophisticated detection models will be discussed in later chapters. As for propagation, detection in quantum optics is modelled diﬀerently in the Heisenberg and Schrödinger pictures. In the Heisenberg picture photodetection can be viewed as the transformation of the quantum mode â ′ into the classî a represents the photocurrent or, more gen̂ a = â ′† â ′ . The mode N cal mode N erally, the information generated by the detector and is classical in the sense that ̂ †a ] = 0. Nevertheless N ̂ a is an operator and must be so to correctly predict ̂ a, N [N ̂ a can be investigated by calculating expectation all behaviour. The statistics of N values of its various moments: ̂ a ⟩ = ⟨Ψ|N ̂ a |Ψ⟩ ⟨N j

j

(4.81)

where |Ψ⟩ is the initial state of the mode â . Using the technique of homodyne detection (see Section 8.1.4), photodetection can also be used to measure the quadrature observable â ′ ei𝜃 + â ′† e−i𝜃 . This then can be modelled as a transfor̂ 𝜃a = â ′ ei𝜃 + â ′† e−i𝜃 . mation of the quantum mode â ′ into the classical mode X 𝜃 ̂ Once again, although Xa represents a classical mode, it must be modelled as an ̂ a the statistics of the quadrature amplitude can be calculated operator. As for N by taking expectation values over the initial state of mode â . The operators representing the photocurrents can be manipulated as they might be in an experiment. For example, correlation measurements can be modelled by adding, subtracting, or multiplying photocurrent operators representing the measurement of diﬀerent modes.

117

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4 Quantum Models of Light

In the Schrödinger picture photodetection can be viewed as projection onto the eigenstates of â † â , i.e. the number states |n⟩. Thus we can predict the probability that a particular result n will occur via Pn = |⟨n|Ψ′ ⟩|2

(4.82)

Average values, variances, etc. can be calculated from the probability distribution thus generated. Similarly if the experiment is conﬁgured such that a quadrature observable is measured, then the projection is onto the eigenstates of â ei𝜃 + â † e−i𝜃 , |x𝜃 ⟩, and the probabilities are given, as was used in Section 4.3.2, by PΨ (x𝜃 ) = |⟨x𝜃 |Ψ⟩|2 𝛿x

(4.83)

It is important to note that the predictions about the photocurrent generated in the Schrödinger picture are just numbers, unlike the operators generated in the Heisenberg picture. A major conceptual diﬀerence arises between the two approaches when multimode systems are considered. Suppose a system initially describable by a pure state in a single mode interacts with a number of other modes, also initially in pure states, during propagation. In the Heisenberg picture there will now be a number of output modes that are all a function of the input modes. Now suppose we have lost’ all but one of these modes, â ′ . Nevertheless we proceed as before by ̂ a = â ′† â ′ , just as we would with classical modes. This analysing the properties of N scenario looks quite diﬀerent in the Schrödinger picture however. After propagation we are left with a state that describes the global properties of all the modes. If we have lost the other modes, we must describe that explicitly by averaging over all the possible detection results the other modes may have given. This is achieved by tracing over all the lost modes such that 𝝆̂ a = Trc≠a {𝝆̂ ′ }

(4.84)

where 𝜌 is the global state of the system, whilst 𝝆̂ a is the reduced density operator that describes the state of only mode â . The subscript c ≠ a indicates that the trace is over all modes except â . Under certain conditions the trace will not change the state of mode â . The state is then said to be separable as it indicates that the portion describing mode â can be factored out. More generally though the state will be inseparable or entangled, meaning no factorization is possible. Then inevitably 𝝆̂ a will be a mixed state, and ′

Pn = ⟨n|𝝆̂ a |n⟩

(4.85)

will give the probability of a particular detection result n. If the results of measurements on the other modes are known, then we must insert them explicitly so |Ψa ⟩ = {⟨nb |⟨nc |⟨nd | …} |Ψ′ ⟩

(4.86)

where now |Ψa ⟩ is the conditional state of the system given that the results nb , nc , nd , … were obtained for the other modes. 4.4.4

An Example: The Beamsplitter

To illustrate some of the ideas introduced in this section, we consider a simple example: a single photon number state passing through a beamsplitter. A more

4.4 Propagation and Detection of Quantum Optical Fields

general discussion of beamsplitters will be given in Chapter 5. The beamsplitter is assumed to have transmittance 𝜂. The other mode entering the beamsplitter is unoccupied. We label the mode containing the single photon state â and the unoĉ Thus the initial state can be written as |1⟩a |0⟩b . Let us consider the cupied mode b. Heisenberg approach ﬁrst. As described in Section 5.1, the classical equations for mixing two modes with amplitudes 𝛼 and 𝛽 on a beamsplitter can be written as √ √ 𝛼out = 𝜂 𝛼 + 1 − 𝜂 𝛽 √ √ 𝛽out = 1 − 𝜂 𝛼 − 𝜂 𝛽 (4.87) Canonical quantization then allows us to immediately write √ √ â ′ = 𝜂 â + 1 − 𝜂 b̂ √ √ b̂ ′ = 1 − 𝜂 â − 𝜂 b̂

(4.88)

Hence, if we photodetect â , we obtain ′

̂ a = â ′† â ′ N = 𝜂 â † â + (1 − 𝜂)b̂ † b̂ +

√ 𝜂(1 − 𝜂)(̂a† b̂ + b̂ † â )

(4.89)

The average photon number detected is then ̂ a ⟩ = ⟨1|a ⟨0|b â ′† â ′ |0⟩b |1⟩a n = ⟨N ̂ b+ = 𝜂⟨1|a â † â |1⟩a + (1 − 𝜂)⟨0|b b̂ † b|0⟩ √ ̂ b + ⟨0|b b̂ † |0⟩b ⟨1|a â |1⟩a ) 𝜂(1 − 𝜂)(⟨1|a â † |1⟩a ⟨0|b b|0⟩ =𝜂

(4.90)

where we have used the relationships of Eq. (4.17). In a similar way the variance is ̂ 2a ⟩ − ⟨N ̂ a ⟩2 Vn = ⟨N

= 𝜂(1 − 𝜂)⟨1|a â † â |1⟩a ⟨0|b b̂ b̂ † |0⟩b = 𝜂(1 − 𝜂)

(4.91)

The result for the average photon number can be understood from a classical ﬁeld point of view: the intensity is reduced according to the transmission of the beamsplitter; the non-zero variance after some loss at the beamsplitter reﬂects the particle aspect of light. More on this is said in Chapter 5. Now let us consider the same problem in the Schrödinger picture. Using the technique of Eq. (4.80), we ﬁnd the state evolution through the beamsplitter is given by |Ψ′ ⟩ = â ′′ |0⟩a′ |0⟩b′ √ √ = ( 𝜂 â ′ + 1 − 𝜂 b̂ ′ )|0⟩a′ |0⟩b′ √ √ = 𝜂|1⟩a′ |0⟩b′ + 1 − 𝜂|0⟩a′ |1⟩b′

(4.92)

In order to consider only detection of the â mode, we need to trace over the b̂ mode as per Eq. (4.84). Writing 𝝆̂ ′ = |Ψ′ ⟩⟨Ψ′ |, we ﬁnd 𝝆̂ a′ = ⟨0|b′ 𝝆̂ ′ |0⟩b′ + ⟨1|b′ 𝝆̂ ′ |1⟩b′ = 𝜂|1⟩a′ ⟨1|a′ + (1 − 𝜂)|0⟩a′ ⟨0|a′

(4.93)

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4 Quantum Models of Light

Using Eq. (4.85), we ﬁnd that detection of this state gives the probabilities P1 = ⟨1|𝝆̂ a′ |1⟩ = 𝜂 P0 = ⟨0|𝝆̂ a′ |0⟩ = 1 − 𝜂

(4.94)

which leads to the same average value and variance as predicted from the Heisenberg result. Now, instead of losing’ the b̂ mode, let us suppose that we have photodetected it. If the photodetection of the b̂ mode gave the result 0’, then the conditional state of the â mode will be √ (4.95) |Ψ⟩a′ = {⟨0|b′ }|Ψ′ ⟩ = 𝜂|1⟩ √ indicating that a photon will be found in mode â with certainty. The factor of 𝜂 indicates that the probability of this state being prepared √ (i.e. the probability of the measurement of the b̂ mode giving the 0’ result) is ( 𝜂)2 = 𝜂. Similarly, if the measurement of the b̂ mode gave the result 1’, then the conditional state of the â mode would be √ |Ψ⟩a′ = {⟨1|b′ }|Ψ′ ⟩ = 1 − 𝜂|0⟩ indicating that it is certain that no photon will be found in the â mode. How could we predict this latter result in the Heisenberg picture? Well, we found that ̂ one was not found in â and vice versa. whenever a photon appeared in mode b, Another way of saying this is that the two modes are anti-correlated. To test for anti-correlation in an experiment, we might add the photocurrents obtained from the photodetection of the two modes and look for reduced noise (variance). Adding the photocurrents, we would obtain (using Eq. (4.88)) ̂ b = â ′† â ′ + b̂ ′† b̂ ′ ̂a+N N = â † â + b̂ † b̂

(4.96)

which has zero variance indicating perfect anti-correlation as expected. Although the results match in the two pictures, as they must, the calculational techniques and the physical pictures suggested are quite diﬀerent.

4.5 Quantum Transfer Functions As we saw in Section 4.3, in general a complete description of a state of light, including its quantum properties, can be carried out through various probability and quasi-probability distributions [7]. For the description inside an optical cavity, a single function will suﬃce. In contrast, for a freely propagating beam, the situation is more involved [12]. We can expand our beam in a basis of single frequency modes. We will then ﬁnd that we need a whole continuum of distribution functions to describe these modes. These functions can be derived in the Schrödinger picture as the solutions of complex stochastic diﬀerential equations, in many cases master equations [7, 13, 14]. Whilst complete, this approach is often more general than required in practice. Here we present an alternative route based on the Heisenberg picture in which we obtain operator equations that describe how the individual frequency modes transform – the

4.5 Quantum Transfer Functions

quantum transfer functions. We restrict our attention to systems that are linear or can be linearized. Linearization can be done if the modulations and the ﬂuctuations we are concerned with are small compared with the amplitude of the laser beam. This is usually correct for situations with optical powers larger than microwatts. A continuous-wave (CW) laser beam is composed of many independent single frequency modes and so is inherently a multimode situation. In contrast a single pulse from a pulsed laser may be in a single mode described by a coherent superposition of many frequency modes. In either case a complete description of the quantum evolution of the beam is available via the quantum transfer functions. Working in the Heisenberg picture means the operator equations are very similar to the classical equations encountered in Section 2.3.2, and similar interpretations can be made. This approach is not directly applicable to experiments that resolve single photons using photon counting technology. However, in Chapter 12, we will adapt this approach to treat such experiments. 4.5.1

A Linearized Quantum Noise Description

Consider a propagating CW laser beam in a single spatial mode. We will represent ̂ where we use capitals to remind us that the beam by the continuum operator A, this is a continuum operator with units s−1∕2 . We can break this operator up into two contributions: (i) the mean value of the amplitude, given by one complex number 𝛼, and (ii) the ﬂuctuations and modulations, described by the operator ̂ 𝜹A(t), representing small ﬂuctuations around the mean. We thus write ̂ = 𝛼 + 𝜹A(t) ̂ A

(4.97)

Note the similarity here with the classical treatment of small ﬂuctuations, Eq. (2.15), the diﬀerence being that now the ﬂuctuating term is represented as an operator. Note also that Eq. (4.97) is really just a special case of a ﬂuctuation ̂ = 𝛼. We use small 𝛿 here to emphasize that operator (see Section 4.3) where ⟨A⟩ these ﬂuctuation operators are assumed small compared to their expectation ̂ is values. If the beam is detected by a photodetector, then the photocurrent N given by ̂ ̂ † (t)A(t) ̂ =A N ̂ ̂ † (t))(𝛼 + 𝜹A(t)) = (𝛼 + 𝜹A ̂ † (t) + 𝛼 𝜹A(t) ̂ + 𝜹A ̂ † (t) 𝜹A(t) ̂ = 𝛼 2 + 𝛼 𝜹A ̂ = 𝛼 2 + 𝛼 𝜹X𝟏(t)

(4.98)

where we have taken the arbitrary phase of 𝛼 real and linearized by assuming the ﬂuctuations are small and hence neglecting higher than ﬁrst-order terms in the ﬂuctuations. By small’, we mean that ultimately the contribution to the calculated expectation values due to these terms will be negligible. We see that from the quantum point of view, we are eﬀectively measuring the in-phase ̂ is often referred ̂ = 𝜹A ̂ ̂ † (t) + 𝜹A(t). For this reason 𝜹X𝟏 quadrature operator 𝜹X𝟏 to as the amplitude quadrature operator. By introducing another bright beam as a reference, it is also possible to measure the ﬂuctuations in the phase of the

121

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beam (see Chapter 8). Given the linear approximation, these turn out to be ̂ = i(𝜹A ̂ † (t) − 𝜹A(t)), ̂ proportional to the out-of-phase quadrature, 𝜹X𝟐 which is thus often referred to as the phase quadrature. This link is illustrated in ̂ and 𝜹X𝟐 ̂ have continuous eigenvalue spectra. This Figure 4.7a. Recall that 𝜹X𝟏 implies that under such measurement conditions, discrete single-photon events cannot be observed, regardless of the resolution of the detectors. The discussion is more naturally continued in frequency space. The Fourier transform of the operator in Eq. (4.97) is ̃ ̃ A(Ω) = 𝛼 𝛿(Ω) + 𝜹A(Ω)

(4.99)

where 𝛿(Ω) is the Dirac delta function. Recall that 𝛼 is deﬁned in a frame rotating at the mean optical frequency 𝜈o (see Section 2.1). Thus Ω = 2π(𝜈o − 𝜈)

(4.100)

is the frequency diﬀerence from the mean optical frequency. The mean coherent amplitude of the ﬁeld appears as a delta function spike, the carrier, 𝛼 𝛿(Ω), which is only non-zero for Ω = 0, i.e. at the mean optical frequency. We can consider the Fourier transformed ﬂuctuation operator, in some small frequency interval around radio frequency Ω, as approximately describing a single quantum mode at this frequency. Such modes obey the commutation relations ̃ ̃ † (Ω′ )] = 𝛿(Ω − Ω′ ), [A(Ω), A

̃ ̃ ′ )] = 0 [A(Ω), A(Ω

(4.101)

This is a useful and practical picture because experimentally we can perform Fourier analysis on the photocurrent produced by photodetection using a spectrum analyser, discussed in Chapter 8. We can also place information on the laser beams in the form of modulation at various diﬀerent frequencies Ωmod , which is measured in Hz, and each frequency can be thought of as an independent information channel. From Eq. (4.98), we ﬁnd that the operator describing the photocurrent in Fourier space is ̃ out (Ω) ̃ out = 𝛼 2 𝛿(Ω) + 𝛼 𝜹X𝟏 N

(4.102)

Now the ﬁrst term describes the average power, concentrated in a DC spike, and the second term describes the measurement of the amplitude quadrature at frequency Ω. The variance of the photocurrent at frequency Ω is given by 2 ̃ out (Ω)|2 |𝛼⟩ = 𝛼 2 ⟨𝛼||𝜹X𝟏 V Nout (Ω) = 𝛼out out

V 1out (Ω)

(4.103)

We have a direct link between the measured values V Nout (Ω) and the property V 1out (Ω) of the state. In particular we see that ultimately the intensity noise of a bright laser is determined by the quantum states of its frequency sidebands. If the laser mode is subjected to amplitude or phase modulations (either deterministic signals or stochastic noise), these can be incorporated into the description as c-number variables. Thus we can modify Eq. (4.99) to read ̃ ̃ ̃ A(Ω) = 𝛼𝛿(Ω) + 𝜹A(Ω) + 𝛿 S(Ω)

(4.104)

where the ̃ indicates the Fourier transform of a classical variable. If the power ̃ spectra of 𝛿 S(Ω) are much larger than those associated with the quantum part

4.5 Quantum Transfer Functions

̃ (𝜹A(Ω)), then the beam will behave classically. However, once again, if all classical noise is suppressed, we will still be left with ﬂuctuations due to the quantum noise. If we are trying to measure or send information via small modulations of the beam, then the signal to noise associated with this measurement or communication will likewise be determined by the quantum noise. 4.5.2

An Example: The Propagating Coherent State

Let us illustrate the preceding discussion with an idealized and then more realistic example. In Section 4.2.2 we described a coherent state as the displacement of a single cavity mode initially in a vacuum state. We may think of a propagating coherent state in a similar way as a beam with all its power at a single frequency. Notice that this implies a CW laser, as a single frequency corresponds ̃ to a constant output in time. This also means that the quantum state of the 𝛿 A(Ω) sideband modes is the vacuum. From Eq. (4.103) and using Eqs. (4.15) and (4.101), we immediately ﬁnd 2 ̃ out |2 |0⟩ ⟨0||𝜹X𝟏 V Nout (Ω) = 𝛼out ̃ 2 + (𝜹A ̃ † )2 + 𝜹 A ̃ † 𝜹A ̃ + 𝜹A𝜹 ̃ A ̃ † |0⟩ = 𝛼 2 ⟨0|(𝜹A out

2 = 𝛼out

(4.105)

where the absolute square is taken because we are in Fourier space. The laser intensity exhibits Poissonian statistics at all frequencies. In a similar way it follows that the quadrature variances are at the quantum noise limit (QNL) for all angles and all frequencies. These results are as might be expected for a travelling ﬁeld generalization of a coherent state. Of course this is an idealization. Real lasers have a ﬁnite linewidth and may have structure in their ﬂuctuation spectra (see Chapter 6). Nevertheless, for lasers oscillating on single longitudinal and spatial cavity modes, so-called single-mode lasers, at suﬃciently high frequencies, the power in the sidebands will become negligible. At or above such frequencies, the laser will behave like a coherent state, and it is in this sense that lasers are often referred to as approximately producing a coherent state. A typical solid-state laser will be coherent’ at frequencies above about 10 MHz. Another way of saying this is that if we sample the ﬁeld on short timescales, the laser can be thought of as producing a coherent state. On the other hand, as we go to low sideband frequencies (long timescales), eventually the power in the sidebands will become an appreciable fraction of the central peak power and our linearization assumption will break down. 4.5.3

Real Laser Beams

We will ﬁnd in Chapter 6 that a real laser beam contains much more noise than a coherent state. Figure 4.8 shows an example of a typical laser that has an intensity noise spectrum V 1(Ω) well above the QNL V 1(Ω) = 1 of the coherent ̃ state. This ﬁgure shows the link between the values for the quadratures 𝜹X𝟏, ̃ 𝜹X𝟐 and the actual measured intensity noise. In addition such a laser beam will contain modulation signals at certain frequencies. We need a technique to derive

123

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4 Quantum Models of Light

ˆ (Ω) δX1

ˆ (Ω) δX2

V1(Ω)

its 1

r un

e oov

H

0 Ω

Ω

Figure 4.8 The Fourier components of the noise spectrum for a typical real laser. On the right the ﬂuctuations for six diﬀerent detection frequencies Ω are represented by their uncertainty areas. On the left is the corresponding continuous intensity noise spectrum V1(Ω).

such spectra and to describe how the noise and the signals will change whilst the light propagates through the apparatus. 4.5.4

The Transfer of Operators, Signals, and Noise

We are now in a position to describe a complete optical system and to evaluate the propagation of the noise and the signals through the system. Each optical component of the system can be thought of as having several inputs, described ̃ in1 , by their coherent amplitudes 𝛼in1 , 𝛼in2 , … etc., their ﬂuctuation operators 𝜹A ̃ in2 , … , etc., and their states |Φin1 ⟩, |Φin2 ⟩, … , etc. The corresponding spectral 𝜹A variances for each input are ̃ inj (Ω)|2 |Φinj ⟩ V 1inj (Ω) = ⟨Φinj ||X𝟏 ̃ inj (Ω)|2 |Φinj ⟩ V 2inj (Ω) = ⟨Φinj ||X𝟐

(4.106)

where j = 1, 2, … Inside the optical components the modes are combined. The algebraic equations for the coherent amplitudes are solved to give the output coherent amplitude. We usually choose the arbitrary overall phase of the input ﬁelds such that the output coherent amplitude is real. After linearization the equations of motion for the ﬂuctuation operators can be solved in frequency space. The ﬂuctuation operators in the frequency domain for one output mode

4.5 Quantum Transfer Functions

can then be described by equations of the form ̃ in1 (Ω) + c1c (Ω) 𝜹A ̃ † (Ω) ̃ out (Ω) = c1a (Ω) 𝜹A 𝜹A in1 ̃ ̃ † (Ω) + · · · + c2a (Ω) 𝜹Ain2 (Ω) + c2c (Ω) 𝜹A in2

̃ in1 (Ω) + c∗ (−Ω) 𝜹A ̃ † (Ω) ̃ † (Ω) = c∗ (−Ω) 𝜹A 𝜹A 1c 1a out in1 ̃ in2 (Ω) + c∗ (−Ω) 𝜹A ̃ † (Ω) + · · · + c∗ (−Ω) 𝜹A 2c

2a

in2

(4.107)

and similarly for all other output modes. The coeﬃcients ckl (Ω) are complex numbers that describe the frequency response of the component. Note that the response is not necessarily symmetric about the carrier. The cka and ckc are linked via commutation requirements. For linear systems the ckl do not depend on the beam intensities and all ckc = 0. We can also treat the case of propagation through non-linear media, such as 𝜒 (2) and 𝜒 (3) media. This includes non-linear processes such as frequency doubling, parametric ampliﬁcation, the Kerr eﬀect, or non-linear absorption. Equation (4.123) remains linear, but in these cases the coeﬃcients c1a , c1c , c2a , c2c , … are dependent on the average optical power of the beams. In various examples throughout this book, we will show how the coeﬃcients cij can be derived from the physical properties of the component, ̃ out (Ω) for such as reﬂectivities, conversion eﬃciencies, etc. Once the operator 𝜹A the output mode in question has been found, the variances can be determined using Eq. (4.106). Let us consider some general cases: (1) The simplest case is where all the inputs are independent: ̃ inl (Ω)⟩ = 𝛿kl . This will be true if all the input beams represent ̃ ink (Ω)𝜹X𝟏 ⟨𝜹X𝟏 beams from independent sources or diﬀerent vacuum inputs. Secondly, we require that the coeﬃcients obey c∗ij (−Ω) = cij (Ω), i.e. magnitude and phase of the frequency response are symmetric. This will be true if there are no detunings or strong dispersion in the optical system. In this case we obtain the simplest type of transfer function: V 1out (Ω) = |c1a (Ω) + c1c (Ω)|2 V 1in1 (Ω) + |c2a (Ω) + c2c (Ω)|2 V 1in2 (Ω) + · · · V 2out (Ω) = |c1a (Ω) − c1c (Ω)|2 V 2in1 (Ω) + |c2a (Ω) − c2c (Ω)|2 V 2in2 (Ω) + · · ·

(4.108)

Notice that for a linear system (c1c = c2c = · · · = 0), the transfer functions for both quadratures are identical. (2) The coeﬃcients still obey c∗ij (−Ω) = cij (Ω), but we allow for inputs that are not independent. Now there can be correlations between the quadrature variances of diﬀerent inputs. However, we can always regain the form of Eq. (4.108) by breaking down each of the inputs into a function of its own inputs until we end up with all the inputs being independent. (3) The coeﬃcients obey c∗ij (−Ω) = c∗ij (Ω), i.e. the magnitude of the frequency response is symmetric but not the phase. This will occur if the ﬂuctuations

125

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suﬀer a diﬀerent phase rotation to that of the coherent amplitude. In this case we get cross-coupling between the transfer functions of V 1out and V 2out such that now V 1out (Ω) = |c1a (Ω) + c1c (Ω)|2 cos2 (𝜃)V 1in1 (Ω) + |c1a (Ω) − c1c (Ω)|2 sin2 (𝜃)V 2in1 (Ω) + · · · V 2out (Ω) = |c1a (Ω) − c1c (Ω)|2 sin2 (𝜃)V 2in1 (Ω) + |c1a (Ω) + c1c (Ω)|2 cos2 (𝜃)V 1in1 (Ω) + · · ·

(4.109)

where 𝜃 is deﬁned by cij (Ω) = |cij (Ω)|e . We have assumed that V 1 and V 2 are the quadratures of minimum and maximum ﬂuctuations. (4) The coeﬃcients obey c∗ij (−Ω) ≠ c∗ij (Ω). This will occur when the frequency response of the optical components is asymmetric around the carrier frequency, for example, due to detunings. It is not possible to represent the output as a transfer function of quadrature variances in this case. i𝜃

The simplest example of a transfer function describes the inﬂuence of loss terms only and has the form V 1out (Ω) = |c1a |2 V 1in1 (Ω) + |c2a |2 + |c3a |2 + · · ·, where all the inputs apart from V 1in1 (Ω) are vacuum inputs that have the numerical value of 1. Such a transfer function has identical eﬀects on both quadratures. An example of case (2) would be where either a modulation is common or two or more of the input modes have correlated noise. This is the case when the beams originate from one source, such as two beams formed by a beamsplitter, or where two beams are generated by an above threshold optical parametric oscillation (OPO). Examples of case (3) can be as benign as mixing beams on a beamsplitter where the coherent amplitudes of the inputs have diﬀerent phases, or components with strong dispersion where the refractive index changes within a frequency interval of Ωdet , or as complex as single-ended cavities with optical detuning. The simplest example of case (4) is a double-ended optical cavity with detuning. An appropriate name for Eq. (4.108) is quantum noise transfer function. This equation is very similar to the classical transfer functions used by communication engineers. The classical function can be derived using a description of the various inputs as classical waves with modulation at Ω. Noise behaves exactly in the same way as the modulations. Quantum noise is not included. The propagation and interference of these waves are determined, and the result is the size of the modulation at the output. In contrast, Eq. (4.108) includes all the quantum eﬀects of the light, in particular the eﬀects of quantum noise. We can directly see how the signals vary in comparison to the quantum noise and can evaluate the coeﬃcient for the transfer of the signal-to-noise ratio from input to output. This approach has been developed since the 1980s by a number of research groups, very prominently by the groups in France who pioneered squeezing [12]. 4.5.5

Sideband Modes as Quantum States

We will now outline how we can relate the quantum transfer functions to the detailed quantum state of the ﬁeld at diﬀerent frequencies. In particular we will

4.5 Quantum Transfer Functions

consider the example of modulation of a quantum noise limited beam and see how this can be viewed as producing a coherent state at the modulation frequency. In Section 2.1.7 we saw how small modulations of a classical light beam resulted in coherent power being injected symmetrically into upper and lower sidebands around the carrier. In Section 4.5.1 we described classical modulations in the Heisenberg picture as evolution of the quantum ﬂuctuation operator ̃ ̃ ̃ 𝜹A(Ω) → 𝜹A(Ω) + 𝛿 S(Ω)

(4.110)

This Heisenberg evolution is equivalent to the Schrödinger evolution of displacement (see Section 4.2.2), i.e. ̃ ̃ A(Ω))|𝜓⟩ |𝜓⟩Ω → D(𝜹 Ω

(4.111)

Now consider the amplitude quadrature operator in Fourier space (Figure 4.9). It ̃ † (Ω), is is easy to show that in general the Fourier transform of a conjugate, e.g. A † ̃ , via a change in sign related to the conjugate of the Fourier transform, e.g. A(Ω) † ̃ ̃ † (Ω) = A(−Ω) . Using this property of the Fourier transform, we can of Ω, e.g. A observe that ̃ † (Ω) ̃ ̃ 𝜹X𝟏(Ω) = 𝜹A(Ω) + 𝜹A † ̃ ̃ = 𝜹A(Ω) + 𝜹A(−Ω)

(4.112)

Thus we see that a quadrature measurement actually probes both the upper and lower sidebands. Suppose the initial state of the sidebands is a vacuum. Using Eq. (4.111) we ﬁnd that modulation produces the coherent sideband state ̃ ̃ |𝜓 ′ ⟩ = |𝛿 S(Ω)⟩ Ω |𝛿 S(−Ω)⟩−Ω

(4.113)

where, in response to Eq. (4.112), we have written the state explicitly in terms of an upper and lower sideband part. We are ignoring the low frequency contributions from the carrier. For an arbitrary mix of ideal amplitude and phase modulation, we have ̃ 𝛿 S(Ω) = 𝛽(𝛿(Ω + Ωmod ) + 𝛿(Ω − Ωmod ))

(4.114)

The complex number 𝛽 = Ma ∕4 + iMp ∕4, where Ma and Mp are the amplitude and phase modulation depths, respectively, and Ωmod is the modulation X2

E

vL–Ω

vL+Ω

X1 Random distribution in time (a)

vL

(b)

Random components at all frequencies

v

Figure 4.9 Comparison between (a) the phase space diagram and (b) the sideband picture. The quantum properties are encapsulated in noise sidebands at all modulation frequencies above and below the laser frequency 𝜈L .

127

128

4 Quantum Models of Light

frequency (see Section 2.1.7). Averaging over a small range of frequencies around Ωmod , we can write our state as |𝜓 ′ ⟩ = |𝛽⟩Ωmod |𝛽⟩−Ωmod |0⟩Ω≠±Ωmod

(4.115)

A pair of identical coherent states has been created at the upper and lower sideband frequencies. As quadrature measurements probe the sidebands symmetrically (see Eq. (4.112)), the results of quadrature measurements are identical to those for a single frequency coherent state. For example, the expectation value of X1 in the Schrödinger picture is given by † ̃ ̃ ̃ + ⟨𝛽|𝜹A(−Ω ⟨𝜹X𝟏(Ω mod )⟩ = ⟨𝛽|𝜹A(Ωmod )|𝛽⟩Ω mod ) |𝛽⟩−Ω mod

mod

= 𝛽 + 𝛽∗

(4.116)

This is of course the same as the result that would be obtained in the Heisenberg picture. This result generalizes to other states provided the transfer functions remain symmetric about the carrier. Most interesting are states for which the upper and lower sideband modes become entangled as this has no classical analogue. Squeezed states, which are described in Chapter 9, are an example of such states. Using a homodyne detector it is possible to measure the variances VΘ (Ω) at all frequencies Ω. The properties of the light in a small interval around each frequency can then be described by one Wigner function W (Ω). The interval is assumed small enough such that the frequency response is ﬂat. In this way the entire beam can be described as a series of quantum states, one for each small frequency interval. This is a Schrödinger picture view of the travelling ﬁeld, which complements very well the actual technology used in optical communication. Here each of these frequency intervals is used as a separate communication channel. The variance VΘ (Ω) can be determined mathematically by ﬁrst projecting all parts of the Wigner function onto one axis at angle Θ. This produces a series of one-dimensional distribution functions DF(Θ) that describe in the form of a histogram the time-varying photocurrent in a small frequency range. In a second step the variances of these functions are evaluated, which provides information about the width of DF(Θ). In general we would need to evaluate the moments of DF(Θ) to all orders. However, as we have seen, for the restricted class of Gaussian functions, a direct link exists between the Wigner functions and the variances, and a complete description is given by Vmin (Ω), Vmax (Ω), and Θmin (Ω), as shown in Section 4.3.2. It is also possible to obtain the average photon number in the sidebands from the quadrature phase measurements. Consider the sum of the amplitude and phase quadrature variances written in terms of annihilation and creation operators. We have † ̃ † ̃ ̃ ̃ 𝜹A(Ω) + 𝜹A(−Ω)𝜹 A(−Ω) ⟩ V 1(Ω) + V 2(Ω) = 2⟨𝜹A(Ω) † † ̃ ̃ ̃ ̃ 𝜹A(Ω) + 𝜹A(−Ω) 𝜹A(−Ω)⟩ +2 = 2⟨𝜹A(Ω)

(4.117)

Hence we can write 1 † ̃ ̃ 𝛿N(|Ω|) = ⟨𝜹A(|Ω|) 𝜹A(|Ω|)⟩ = (V 1(Ω) + V 2(Ω) − 2) 4

(4.118)

4.5 Quantum Transfer Functions

For a coherent state (of the entire beam), we expect V 1 = V 2 = 1 at all frequencies, and so from Eq. (4.118) there are no photons in the sidebands as discussed in Section 4.5.2. This discussion requires a caveat: we have assumed throughout that the sidebands are symmetric around the carrier, i.e. our transfer function is not of type (4) as described in the previous section. Asymmetric transfer functions can occur. For example, it is possible to produce single-sideband modulation. The state produced in such a situation is |𝜓 ′ ⟩ = |𝛽⟩Ωmod |0⟩Ω≠Ωmod

(4.119)

Such states are not fully characterized by homodyne quadrature measurements. 4.5.6 Another Example: A Coherent State Pulse Through a Frequency Filter So far we have only considered CW or quasi-CW situations where our beam can be approximated as many independent single frequency modes (or mode pairs as in the case of the sidebands). Now we wish to illustrate how the quantum transfer functions can also be used to describe the propagation of pulses through optical elements with non-uniform spectral response. The example we will discuss is a pulse of light in a coherent state transmitted through a frequency ﬁlter. We consider a plane wave Gaussian mode as described by Eq. (4.68). Written in the nomenclature of this section, we have â =

∫

̃ dΩF(Ω)A(Ω)

(4.120)

Suppose we ask: what is the expectation value of an amplitude quadrature measurement that projects onto this mode when the mode is prepared in a coherent state of amplitude 𝛼? Using the techniques of Section 4.4 and Eq. (4.24), we ﬁnd ̂ † (𝛼)(̂a + â † )D ̂ â (𝛼)|0⟩ ⟨𝛼|̂a + â † |𝛼⟩â = ⟨0|D â = ⟨0|̂a + â † + (𝛼 + 𝛼 ∗ )

dΩF(Ω)F(Ω)∗ |0⟩ ∫ = ⟨0|̂a + â † + 𝛼 + 𝛼 ∗ |0⟩ = 𝛼 + 𝛼 ∗

(4.121)

As expected we obtain 2 e 𝛼. The intermediate step of the calculation shown in the second line is included to emphasize that when generalizing from the displacement of a single frequency mode (Eq. (4.24)) to that of a pulse, the overlap between the mode that is displaced and that which is detected appears. In this case the detected mode is identical to the displaced mode, and so the integral evaluates to unity via normalization (Eq. (4.64)). Now suppose that a frequency ﬁlter is placed in front of the detector. The detector mode is now â o =

∫

̃ o (Ω) dΩF(Ω)A

(4.122)

̃ o is determined by the transfer function of the frequency ﬁlter. where A A frequency ﬁlter can be modelled as the transmission through an optical cavity,

129

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4 Quantum Models of Light

as described in Section 5.3. Here we just quote its transfer functions: 𝜅 iΩ ̃ ̃ B(Ω) (4.123) A(Ω) + 𝜅 − iΩ 𝜅 − iΩ ̃ where 𝜅 is the linewidth of the ﬁlter and B(Ω) is a second mode, initially in the vacuum state, coupled in by the ﬁlter. Now the expectation value of the amplitude quadrature becomes ̃ o (Ω) = A

̂ † (𝛼)(̂ao + â †o )D ̂ â (𝛼)|0⟩ ⟨𝛼|̂ao + â †o |𝛼⟩â = ⟨0|D â ) ( 𝜅 𝜅 F(Ω)F(Ω)∗ |0⟩ = ⟨0|(̂ao + â †o ) + dΩ 𝛼 + 𝛼∗ ∫ 𝜅 − iΩ 𝜅 + iΩ ( ) 𝜅2 Ω𝜅 ∗ = dΩ|F(Ω)|2 (𝛼 + 𝛼 ∗ ) 2 − i(𝛼 − 𝛼 ) ∫ 𝜅 + Ω2 𝜅 2 + Ω2 (4.124) In general the ﬁlter produces two eﬀects: a rotation of the quadratures such that a component of the imaginary part of 𝛼 now appears in the expectation value and attenuation by a Lorentzian factor. Notice that if |F(Ω)| = |F(−Ω)|, then the m 𝛼 term goes to zero as it is an odd function of Ω. Hence, if the signal is symmetric and the centre frequency is on resonance with the ﬁlter, there will be no rotation. Restricting to this case we can get some insight √about the attenuation by 2

−(Ω) 1 √ e 2𝑣 (c.f. Eq. 𝑣 π ⟨𝛼|̂ao + â †o |𝛼⟩â ≈ 𝛼 +

considering two limits. Suppose F(Ω) is the Gaussian

(4.68).

𝛼 ∗ , i.e. If 𝑣 ≪ 𝜅, then we can approximate F(Ω) ≈ 𝛿(Ω) and if the frequency spread of the pulse is narrow compared to the linewidth of the ﬁlter, almost all the power of the pulse is transmitted to the detector. On√the other 1 √ and hand, in the opposite limit where 𝑣 ≫ 𝜅, we can approximate F(Ω) ≈ 𝑣 π

⟨𝛼|̂ao + â †o |𝛼⟩â ≈ 0, i.e. if the frequency spread of the pulse is broad compared to the linewidth of the ﬁlter, almost none of the power of the pulse is transmitted to the detector. 4.5.7

Transformation of the Covariance Matrix

Throughout this section we have been considering linear transfer functions arising from linear, or linearized, optical interactions. Such transfer functions always map one set of Gaussian states to another set of Gaussian states and as such are known as Gaussian operations. Thus, if our system is initially in a Gaussian state and it is transformed according to linear transfer functions, then the ﬁnal state will also be Gaussian. This means that in such a situation all the information required to completely describe the output state can be obtained by ﬁnding the transfer functions of the elements of the input displacement vector and covariance matrix (see Section 4.3.4). For transfer function cases (1)–(3), a covariance matrix initially in standard form, Eq. (4.61), will remain in standard form, and it is only necessary to calculate how the variances transform. See also Section 2.1.5. However, for case (4), a covariance matrix initially in standard form will transform to one not in standard form, and hence the transformation of all matrix elements must be calculated.

4.6 Quantum Correlations

4.6 Quantum Correlations A key diﬀerence between classical optical ﬁelds and their quantum counterparts is that the correlations that can exist between spatially separated beams in quantum mechanics can be stronger than is allowed by classical theory. We now look at examples of how this diﬀerence can be quantiﬁed both in the single-photon and linearized domains. 4.6.1

Photon Correlations

In Section 2.3.3 we saw that the intensity correlations between diﬀerent beams could be characterized by the second-order correlation function, g (2) , which was deﬁned by g (2) (𝜏) =

⟨I(t)I(t + 𝜏)⟩ ⟨I(t)2 ⟩

(4.125)

We also saw that for classical ﬁelds g (2) (0) ≥ 1. In contrast we argued in Section 3.6 that g (2) (0) < 1 should be possible for quantum mechanical ﬁelds. We will now treat this question with the full quantum formalism. The calculation that led to the restriction on g (2) in the classical case relied on the fact that ⟨A2 ⟩ ≥ ⟨A⟩2 . This relationship still holds if A is an operator, so how is it that we can have g (2) (0) < 1 for a quantum mechanical ﬁeld? To answer this we must look more carefully at what the actual measurement of g (2) involves. As outlined previously we need to divide the ﬁeld in half at a beamsplitter and then separately measure the intensities of the two halves with a variable time delay, 𝜏, introduced. For a classical ﬁeld the action of the beamsplitter is I(t) → 1∕2I(t). Thus by ﬁnding the average of the product of the intensities of the two beams and normalizing by their individual averages, we arrive at the above expression for g (2) . Now consider the quantum mechanical case. As we have seen in Section 4.4.4 (see Chapter 5 for more details), the correct quantum mechanical expressions for the outputs of a 50: 50 beamsplitter are √ √ â ′ = 0.5 â + 0.5 b̂ √ √ (4.126) b̂ ′ = 0.5 â − 0.5 b̂ where â represents the beam whose intensity we are measuring, whilst b̂ is initially in the vacuum state. If we now form the ratio of the average of the product of the intensities of the two beams normalized to their individual averages, we obtain g (2) (𝜏) =

̂ ̂ + 𝜏))⟩ ⟨(̂a† (t) + b̂ † (t))(̂a(t) + b(t))(̂ a† (t + 𝜏) − b̂ † (t + 𝜏))(̂a(t + 𝜏) − b(t ̂ ̂ + 𝜏))⟩ ⟨(̂a† (t) + b̂ † (t))(̂a(t) + b(t))⟩⟨(̂ a† (t + 𝜏) + b̂ † (t + 𝜏))(̂a(t + 𝜏) − b(t (4.127)

Using the ﬁeld commutation relations and the fact that the b̂ mode is initially in the vacuum state (see Section 4.2.1), this expression can be signiﬁcantly

131

132

4 Quantum Models of Light

simpliﬁed to g (2) (𝜏) =

⟨̂a† (t)̂a† (t + 𝜏)̂a(t)̂a(t + 𝜏)⟩ ⟨̂a† (t)̂a(t)⟩2

(4.128)

where we have also assumed the individual beam averages are not a function of time (the system is ergodic). Equation (4.128) is then the correct quantum mechanical expression for the second-order correlation function. Notice that the ordering of the operators under the averages is such that all creation operators stand to the left whilst all annihilation operators stand to the right. This is referred to as normal ordering and is often a convenient way of expressing results. We can now consider two simple examples. First, suppose the mode â was initially in a coherent state. Using Eq. (4.128) we immediately ﬁnd that for a coherent state, g (2) (0) = 1, as expected for an ideal classical ﬁeld. However, if the mode â is in a number state, then the numerator is immediately zero, as two applications of the annihilation operator to the one photon state produces zero. Thus, for a single photon number state g (2) (0) = 0, in stark contrast to the classical limit. In Chapters 12 and 13, we will see other examples of quantum intensity correlations as functions of diﬀerent degrees of freedom such as polarization. 4.6.2

Quadrature Correlations

We can also consider correlations between the quadrature amplitude ﬂuctuations of spatially separated beams, recorded by separate detectors. These correlations can be quantiﬁed by measuring the quadrature diﬀerence and sum variances deﬁned by 1 ̃ 2 )2 ⟩ ̃ 1 + 𝜹X𝟏 V 1s = ⟨(𝜹X𝟏 2 1 ̃ 2 )2 ⟩ ̃ 1 + 𝜹X𝟐 V 2s = ⟨(𝜹X𝟐 2 for the sum variances of the two quadratures and

(4.129)

1 ̃ 2 )2 ⟩ ̃ 1 − 𝜹X𝟏 V 1d = ⟨(𝜹X𝟏 2 1 ̃ 2 )2 ⟩ ̃ 1 − 𝜹X𝟐 V 2d = ⟨(𝜹X𝟐 (4.130) 2 for the diﬀerence quadrature variances. Here the subscripts label the beams 1 and 2. If the beams are correlated, say, in the amplitude quadrature, we would expect V 1d < V 1s . Conversely, if the beams were anti-correlated, we would ﬁnd V 1d > V 1s . Two coherent beams are found to be uncorrelated, with their noise at the QNL; that is, for two coherent beams, we have V 1d = V 1s = V 2d = V 2s = 1. Beams are said to be quantum correlated if they have sum or diﬀerence quadrature variances that fall below 1. Such correlations cannot be consistently accommodated in a classical theory. In particular, if, for example, V 1d < 1 and V 2s < 1, then the quantum state of the beams is necessarily said to be entangled (see Section 12.3) and no consistent local realistic description of the correlations is possible.

4.6 Quantum Correlations

More generally we might consider the conditional variance of, say, beam 1’s amplitude quadrature, given that we have measured beam 2’s amplitude quadrature. This tells us how accurately we can infer the value of beam 1’s amplitude quadrature given the measurement of beam 2 and is deﬁned as ̃ 1 + g 𝜹X𝟏 ̃ 2 )2 ⟩ V 11|2 = ⟨(𝜹X𝟏

(4.131)

Here g is a gain that is chosen such as to minimize the value of V 11|2 . By calculus we can ﬁnd that the optimum value of g is g=−

̃ 2 𝜹X𝟏 ̃ 1 𝜹X𝟏 ̃ 2 + 𝜹X𝟏 ̃ 1⟩ ⟨𝜹X𝟏 2 V 12

(4.132)

Substituting back into Eq. (4.131), we obtain V 11|2 = V 11 −

̃ 2 𝜹X𝟏 ̃ 2 + 𝜹X𝟏 ̃ 1 ⟩2 ̃ 1 𝜹X𝟏 ⟨𝜹X𝟏 2 V 12

(4.133)

Similarly for the phase quadrature conditional variance, we have V 21|2 = V 21 −

̃ 2 𝜹X𝟐 ̃ 2 + 𝜹X𝟐 ̃ 1 ⟩2 ̃ 1 𝜹X𝟐 ⟨𝜹X𝟐 2 V 22

(4.134)

For coherent states we ﬁnd V 11|2 = V 21|2 = 1, that is, the conditional variance is no better than the single beam variance indicating no correlation. Quantum correlation occurs when a conditional variance falls below 1. A special situation, which can only happen when the beams are entangled, is when both V 11|2 and V 21|2 fall below 1, and this will be discussed in Chapter 13. It is convenient to deﬁne the expectation value of the product of the quadratures of the two beams as the amplitude quadrature correlation coeﬃcient, C1, i.e. 1 ̃ ̃ 2 𝜹X𝟏 ̃ 2 + 𝜹X𝟏 ̃ 1⟩ (4.135) ⟨𝜹X𝟏1 𝜹X𝟏 2 Experimentally we can obtain C1 by measuring the sum and diﬀerence quadrature variances, i.e. 1 C1 = (V 1s − V 1d ) (4.136) 2 In terms of C1 we then have that uncorrelated beams have C1 = 0, correlated beams have C1 > 0, and anti-correlated beams have C1 < 0. Similarly for the phase quadrature correlation coeﬃcient, we have C1 =

C2 = 4.6.3

1 (V 2s − V 2d ) 2

(4.137)

Two-Mode Covariance Matrix

If the state of our two-mode system is Gaussian, then a complete description is provided by the covariance matrix and displacement vector. The covariance matrix of a two-mode system can be written in the following way: ( ) V1 C V1,2 = (4.138) CT V 2

133

134

4 Quantum Models of Light

where Vi is the covariance matrix of the ith mode as deﬁned in Eq. (4.59). The matrix C is deﬁned as ) (1 ̂ 1 ΔX𝟏 ̂ 2 + ΔX𝟏 ̂ 2 ΔX𝟏 ̂ 1 ⟩ 1 ⟨ΔX𝟏 ̂ 1 ΔX𝟐 ̂ 2 + ΔX𝟐 ̂ 2 ΔX𝟏 ̂ 1⟩ ⟨ΔX𝟏 2 2 C= 1 ̂ 2 ΔX𝟐 ̂ 1 + ΔX𝟐 ̂ 1 ΔX𝟏 ̂ 2 ⟩ 1 ⟨ΔX𝟐 ̂ 1 ΔX𝟐 ̂ 2 + ΔX𝟐 ̂ 2 ΔX𝟐 ̂ 1⟩ ⟨ΔX𝟏 2 2 (4.139) and describes the correlations between the modes. The matrix CT is the transpose of C. In standard form the two-mode covariance matrix reduces to

V1,2

⎛ V 11 ⎜ ⎜ 0 =⎜ ⎜ C1 ⎜ ⎝ 0

0 V 21 0 C2

C1 0 V 12 0

0 ⎞ ⎟ C2 ⎟ ⎟ 0 ⎟ ⎟ V 22 ⎠

(4.140)

where the subscripts on the variances indicate the mode and C1 and C2 are deﬁned in the previous section. The displacement vector of the two-mode state is ̃ 1⟩ ⎞ ⎛ ⟨𝜹X𝟏 ⎜ ⎟ ̃ 1⟩ ⎟ ⎜ ⟨𝜹X𝟐 d=⎜ (4.141) ̃ 2 ⟩ ⎟⎟ ⎜ ⟨𝜹X𝟏 ⎜ ̃ ⎟ ⎝ ⟨𝜹X𝟐2 ⟩ ⎠ The covariance matrix approach will turn out to be very useful when analysing quantum information protocols with Gaussian states.

4.7 Summary We have introduced quantum optical theory and have discussed various quantum models of light. As is generally true in quantum mechanics, there are diﬀerent pictures and/or representations that can be used in calculations that seem very diﬀerent but lead to the same solutions. Which one to use can depend on the application. Here we have focused mainly on single photon states and coherent states as examples of the most quantum and most classical states, respectively; however the approaches described will be applied more generally in the coming chapters. 4.7.1

The Photon Number Basis

The photon picture is intuitive and obvious for the explanation of emission and detection processes. However, care has to be taken since photons do not behave like classical particles. Their quantum properties have to be taken into account. The description of state evolution in the Schrödinger picture using the photon number basis is optimized for situations with low photon numbers and photon

4.7 Summary

counting. Technology for generating and detecting single photons and photon coincidences has been developed that matches this representation well. The special quantum properties of particles are directly accessible in this regime, and they are used extensively for quantum applications such as cryptography and quantum information processing, described in Chapter 13. We will ﬁnd elsewhere in this guide, Chapter 9, that the photon number representation is very useful in explaining the properties of non-linear processes of CW laser beams, such as second harmonic generation (SHG) or OPO. Mathematically the discrete form of the number basis can have calculational advantages and is particularly useful for numerical solutions. 4.7.2

Quadrature Representations

We can also expand our quantum optical state in terms of the eigenstates of the quadrature observables in various ways, leading to continuous probability and quasi-probability distributions in phase space. The most popular probability distribution is the Q-function, which can be interpreted as the probability density of measuring the state to be in the coherent state |𝛼⟩. The most popular quasi-probability distribution is the Wigner function whose marginals are the probability densities for measuring particular quadrature outcomes, but which can take on negative values for non-Gaussian states. For the special but ubiquitous case of Gaussian states, the ﬁrst and second quadrature moments are suﬃcient to completely characterize the state. It is useful then to tabulate these moments into a displacement vector and covariance matrix that can be used to calculate various properties of the state. This is a compact representation particularly when one is considering multimode states. 4.7.3

Quantum Operators

A useful quantum optical approach is to calculate how the quantum mode operators evolve in the Heisenberg picture. Starting from some ﬁducial initial state, often the vacuum state, the transformation of the annihilation operators of the various modes involved completely characterizes the ﬁnal quantum state of the system. We have described how propagation and detection are modelled in the Heisenberg and Schrödinger pictures and highlighted the diﬀering physical interpretations associated with each. For many practical applications a simpliﬁed linearized model is fully suﬃcient. In this case we can completely describe the beam by the evolution of the operators ̃ ̃ 𝜹X𝟏(Ω) and 𝜹X𝟐(Ω). This model can be used to derive the noise transfer functions for V 1(Ω) and V 2(Ω), describing how the quantum (and classical) noise propa2 ̃ gates through diﬀerent devices, where V (Ω) = ⟨|𝜹X(Ω)| ⟩. In this way we can analyse complete optical systems from source to detector by evaluating the operators at the various locations. This model of transfer functions will be extremely useful to derive the properties of light beams that have deeper quantum properties, such as squeezed light (Chapter 9), entanglement between the quadratures, and CW quantum processing such as optical teleportation (Chapter 13).

135

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4 Quantum Models of Light

If all the interactions are linear or linearizable and our initial state is the vacuum, then all output states will be Gaussian, and so the transfer function approach can conveniently be used to calculate the covariance matrix elements and their evolution. 4.7.4

The Quantum Noise Limit

We have found that the QNL deﬁnes the quietest light that can be produced with a conventional source, such as thermal light from a lamp, or coherent light from a laser. Quantum noise is the way the quantum nature of light manifests itself in CW laser beams and in individual pulses of laser light. However, there is more to come, and other types of light states are possible. Within the quantum model it is possible to deﬁne light that has reduced ﬂuctuations in one of its quadratures. Such squeezed states of light and their generation are the topic of Chapter 9. We ﬁnd that for all devices that are sensitive enough to detect quantum noise, and these are getting more and more common due to the advances in technology, it is important to understand the quantum properties of the light. They will inﬂuence the performance of the devices. In many cases the quantum eﬀects will set a fundamental limit to the performance, the QNL. We will ﬁnd tricks to avoid these limitations. In other cases the performance will be quite diﬀerent near the QNL compared with the classical regime well above the QNL. Finally we will show applications and devices that are only possible because of the existence of these quantum eﬀects.

References 1 Goldstein, H. (1980). Classical Mechanics. Singapore: Addison-Wesley. 2 Power, E.A. (1964). Introductory Quantum Electrodynamics. Longmans. 3 Vogel, W. and Welsch, D.-G. (1994). Lectures on Quantum Optics. Berlin:

Akademie Verlag. 4 Casimir, H.B.G. and Polder, D. (1948). The inﬂuence of retardation on the

London - van der Waals forces. Phys. Rev. 73: 360. 5 Kitchener, J.A. and Prosser, A.P. (1957). Direct measurement of the

long-range van der Waals forces. Proc. R. Soc. A 242: 403. 6 Glauber, R.J. (1962). Coherent and incoherent states of the radiation ﬁeld.

Phys. Rev. 131: 2766. 7 Walls, D.F. and Milburn, G.J. (1994). Quantum Optics. Springer-Verlag. 8 Pegg, D.T. and Barnett, S.M. (1988). Unitary phase operator in quantum

mechanics. Europhys. Lett. 6: 483. 9 Freyberger, M., Heni, M., and Schleich, W.P. (1995). Two mode quantum

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Further Reading

13 Scully, M.O. and Zubairy, M.S. (1997). Quantum Optics. Cambridge Univer-

sity Press. 14 Gardiner, C.W. and Zoller, P. (2000). Quantum Noise, 2e. Berlin:

Springer-Verlag.

Further Reading Louisell, W.H. (1973). Quantum Statistical Properties of Radiation. New York: Wiley. Mandel, L. and Wolf, E. (1996). Optical Coherence and Quantum Optics. Cambridge University Press. Schleich, W.P. (2001). Quantum Optics in Phase Space. Wiley-VCH. Grynberg, G., Aspect, A., and Fabre, C. (2010). Introduction to Quantum Optics. Cambridge University Press.

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5 Basic Optical Components Real experiments consist of many components – at ﬁrst sight, any quantum optics experiment looks like a bewildering array of lenses, mirrors, etc. ﬁlling an optical table in a more or less random fashion. However, they all serve a purpose, and their individual characteristics are described in this chapter. In quantum optics experiments we consider light as laser beams or modes of radiation. Using optical components, we want to change the parameters of the mode, such as beam size, direction, polarization, or frequency. This means we wish to transform one mode into another mode. This can be achieved by a whole range of optical components, all of which are linear, which means that their output is proportional to their input, independent of the intensity of the beam. One of the simplest optical processes is attenuation by a beamsplitter, and we will see how it aﬀects both the intensity as well as the ﬂuctuations of the light. Whilst the ﬁrst is simple, that is, the intensity is reduced, the latter requires more detailed modelling. The complementary process, gain, is similar but requires much more elaborate equipment and theory, which will be covered in Chapter 6, describing lasers. Combining several mirrors allows us to build interferometers and cavities. Several other optical components are used to manipulate the light: spatial parameters can be controlled with lenses and curved mirrors; polarization can be inﬂuenced with various crystals and surface eﬀects; and frequency can be shifted using electro-optic modulators (EOM). In this chapter we will ﬁrst describe the eﬀect of the various components on a single spatial and temporal mode. The quantum description in Chapter 4 will be used to demonstrate and model the distinctive quantum eﬀects. The concept of linear systems, beamsplitters, interferometers, and resonators is immensely useful and can be used to describe many devices. Any machine that is based on the propagation of waves and modes will contain these components. This could be as simple as the unavoidable losses that take power out of the mode, any coherent splitting and recombination that leads to interference, and any resonators that have tuned transmission functions. Finally, we summarize the many practical ways of building beamsplitters, interferometers, and resonators that have been developed to create devices that use the principles of quantum physics. These go well beyond optics.

A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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5.1 Beamsplitters One of the most common and simplest optical components is a mirror. This is made either of thin layers of metal or multilayer dielectric ﬁlms deposited on glass. A mirror has two output beams for every input beam, one transmitted and the other reﬂected. Thus it is referred to as a beamsplitter. Let us analyse the properties of a beamsplitter. We can use the diﬀerent models described in Chapter 4; they will obviously lead to the same conclusion but will provide diﬀerent interpretations of the eﬀect of a beamsplitter. 5.1.1

Classical Description of a Beamsplitter

Consider a beamsplitter as shown in Figure 5.1 with an incoming beam described classically by the complex amplitude 𝛼in exp(i𝜙in ). There are two waves leaving the beamsplitter: the transmitted beam, 𝛼t exp(i𝜙t ), and the reﬂected beam, 𝛼r exp(i𝜙r ). We describe the properties of the beamsplitter by the intensity reﬂection coeﬃcient 𝜖: |𝛼r |2 = 𝜖|𝛼in |2 |𝛼t |2 = (1 − 𝜖)|𝛼in |2

(5.1)

The special case 𝜖 = 12 describes a balanced 50/50 beamsplitter. However, there is also a second input axis deﬁned such that a wave 𝛼u exp(i𝜙u ) would produce at the output a wave in the same place and propagating in the same direction as 𝛼t and 𝛼r . For a simple beamsplitter only 𝛼in exists and 𝛼u = 0. In other applications two waves 𝛼in and 𝛼u are superimposed and, in this way, are made to interfere. The optical frequencies of all input and output beams are identical. All the frequencies, including all the sidebands, are preserved. Assuming it has no losses, the mirror will preserve energy. This means |𝛼in |2 + |𝛼u |2 = |𝛼r |2 + |𝛼t |2

(5.2)

The intensity response to the other input beam can be written similarly to Eq. (5.1). The response for amplitude transmission and reﬂection is not that simple, since the eﬀect of the beamsplitter on the phase of the light has to be considered. Let us arbitrarily set the phase of the two incoming waves to zero, which means 𝜙in = 𝜙u = 0. The relative phase between the two output beams is of interest: √ √ 𝛼t = (1 − 𝜖) 𝛼in exp(i𝜙in,t ) + 𝜖 𝛼u exp(i𝜙u,t ) √ √ 𝛼r = 𝜖 𝛼in exp(i𝜙in,r ) + (1 − 𝜖) 𝛼u exp(i𝜙u,r ) Φu

αu

ε

αin

Figure 5.1 The classical beamsplitter.

αt

Φt

Φin Φr

αr

5.1 Beamsplitters

There is not suﬃcient information to determine all four relative phase terms 𝜙in , t; 𝜙n , r; 𝜙u,t ; and 𝜙u,r directly. This could be done by solving the electromagnetic boundary conditions explicitly. The solutions, known as Fresnel equations, can be found in textbooks on physical optics. These equations are diﬀerent for metal and dielectric mirrors, and they depend on the polarization of the input wave. However, we can apply energy conservation and ﬁnd a general condition for the four waves: |𝛼in |2 + |𝛼u |2 = |𝛼t |2 + |𝛼r |2 = (1 − 𝜖)|𝛼in |2 + 𝜖|𝛼u |2 + 𝜖|𝛼in |2 + (1 − 𝜖)|𝛼u |2 √ + 𝜖(1 − 𝜖) 𝛼u∗ 𝛼in exp(i(𝜙in,t − 𝜙u,t )) + c.c. √ + 𝜖(1 − 𝜖) 𝛼in 𝛼u∗ exp(i(𝜙in,r − 𝜙u,r )) + c.c.

(5.3)

which leads to exp(i(𝜙in,t − 𝜙u,t )) + exp(i(𝜙in,r − 𝜙u,r )) = 0

(5.4)

The usual choice is to set three of the relative phases to zero: 𝜙in,t = 𝜙in,r = 𝜙u,r = 0, and 𝜙u,t = π, which means √ √ 𝛼r = 𝜖𝛼in + 1 − 𝜖𝛼u √ √ 𝛼t = 1 − 𝜖𝛼in − 𝜖𝛼u (5.5) The normal explanation is that one of the reﬂected waves has a phase shift of 180∘ in respect to all other waves. For a simple dielectric layer, this would be the reﬂection of the layer with the higher refractive index. But this is only one possible choice; various mirrors have diﬀerent properties. All mirrors can be represented by Eq. (5.5). The above analysis had been discussed as early as 1852 by Stokes [1, 2] and leads to a more general matrix format: ( ) ( ) ( )( ) 𝛼r 𝛼in Mr,in Mr,u 𝛼in =M = (5.6) 𝛼t 𝛼u Mt,in Mt,u 𝛼u If there is no loss, the matrix M has to be a unitary transformation, which means the following conditions apply: |Mr,in | = |Mt,u | and |Mt,in | = |Mr,u | |Mr,in |2 + |Mt,in |2 = 1 Mr,in M∗t,in + Mr,u M∗t,u = 0 and our choice of the beamsplitter matrix is ( √ ) √ 𝜖 1 − 𝜖 √ √ 1−𝜖 − 𝜖

(5.7)

We should note that any real beamsplitter will introduce losses due to absorption and internal scattering. This can be described by a loss term (1 − A) such that |𝛼t |2 + |𝛼r |2 = (1 − A)(|𝛼in |2 + |𝛼u |2 )

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In most cases the transmitted and the reﬂected waves are aﬀected by the same amount. For metal mirrors typical values are A = 0.05; for dielectric mirrors, A of 10−2 –10−4 is common, but losses as low as 5 × 10−6 have been achieved. 5.1.1.1

Polarization Properties of Beamsplitters

In some cases we might want a beamsplitter that acts equally on all polarizations. In the other extreme case one could imagine a polarizing beamsplitter (PBS) that reﬂects only one polarization and transmits the orthogonal polarization. Both these cases can be approximately achieved with diﬀerent types of beamsplitters. Another possibility would be a partially polarizing beamsplitter (PPBS) that reﬂects all of one polarization but only partially transmits the orthogonal polarization. In general, the polarization response is not trivial, and the reﬂectivity 𝜖 is polarization dependent. For this purpose we deﬁne two orthogonal polarization states: S and P linear polarization. For P polarization the electric ﬁeld vector is parallel to the surface of the mirror. This is out of the plane of the drawing in Figure 5.1. For S polarization, the electric ﬁeld is perpendicular (or senkrecht in German) to the P polarization. This is in the plane of Figure 5.1. The S polarization contains a component that is perpendicular to the surface of the mirror, and the boundary condition on the mirror surface for the parallel and perpendicular ﬁelds is diﬀerent. Consequently the S and P components of the light are reﬂected diﬀerently, both in amplitude and in phase, and in general 𝜖P and 𝜖S differ from each other. They depend on the way the mirror is manufactured and changes with the input angle and wavelength. For a simple dielectric material, such as uncoated glass, the problem can be solved easily, resulting in the Fresnel reﬂection coeﬃcients [3]. For real multilayer mirrors such calculations are rather involved. One special solution is 𝜖S = 0 for a dielectric surface under Brewster’s angle. Stacks of tilted glass plates, near Brewster’s angle, are used as polarizers. As long as we consider only light linearly polarized in the S or P direction, the polarization is maintained, and the relative phase shift between the S and the P wave is of no concern. For any other light the polarization is modiﬁed by the mirror. The mirror acts like a combination of a non-PBS and a wave plate. In summary, to describe a mirror it is not suﬃcient to state just the reﬂectivity 𝜖; the following properties have to be speciﬁed as well: input angle, wavelength, polarization, and losses. Finally, the following practical hints are useful for the selection of mirrors: there is a choice of materials for mirrors. For non-PBS, that means ideally 𝜖P = 𝜖S . Metal surfaces are very useful, and 𝜖S = 𝜖P = 0.48 can be readily achieved, but the losses are considerable. With dielectric mirrors the diﬀerence between 𝜖P and 𝜖S is usually signiﬁcant, unless they are speciﬁcally designed for the speciﬁc application. Dielectric mirrors should preferably be used with linearly polarized light aligned in the S or P direction. The losses can be small ( 1. The second beamsplitter recombines the beams. Figure 5.5 The components of a classical interferometer.

M1

ε1

α|

αin

M3

M2

M4

α||

∆ϕ

α1

ε4 α2

153

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5 Basic Optical Components

5.2.1

Classical Description of an Interferometer

In the classical wave model, a balanced interferometer with 50/50 beamsplitters can be described by the following evolution of the complex amplitudes: √ 𝛼I (t) = 1∕2 𝛼in (t) √ 𝛼II (t) = 1∕2 𝛼in (t) √ √ 𝛼1 (t) = 1∕2 𝛼I (t) exp(iΔ𝜙) − 1∕2 𝛼II (t) √ √ (5.25) 𝛼2 (t) = 1∕2 𝛼I (t) exp(iΔ𝜙) + 1∕2 𝛼II (t) √ √ where the amplitude reﬂectivity 𝜖 has been set to 1∕2. The calculation can easily be generalized to other values of 𝜖. Here the convention of Eq. (5.7) is used. The minus sign in the third equation represents the 180∘ phase shift imposed by one, but not the other, of the reﬂections. Strictly speaking, this set of equations applies only to plane waves, but it can also be applied to spatial Gaussian beams. The output intensity of the interferometer is given by I1 (Δ𝜙) = Iin cos2 (Δ𝜙∕2) I2 (Δ𝜙) = Iin sin2 (Δ𝜙∕2)

(5.26)

and I1 + I2 = Iin . There are many applications for interferometers: they can be used to measure physical processes that aﬀect the relative phase diﬀerence Δ𝜙. One should think about an interferometer as a device that transforms changes in the optical phase into changes of intensity. Equation (5.26) holds for any interferometer that is based on the interference of two waves – independent on the layout and the details of the apparatus. This includes Michelson interferometers, Lloyd’s mirrors, ﬁbre-optic interferometers, polarimeters, etc. The main task in describing a speciﬁc conﬁguration is to evaluate Δ𝜙; once this is done the intensity distribution can be directly evaluated from Eq. (5.26). Under practical conditions the interference will not be perfect. The overlap of the two wave fronts will not be perfect, or the two waves might not be completely coherent. The spatial or temporal coherence between the two waves, which means the correlation between 𝛼I and 𝛼II , will determine the strength of the interference. In the case of perfect correlation, the result in Eq. (5.26) is valid. In the case of no correlation between 𝛼II and 𝛼I , or if they are incoherent, the resulting intensity will be I1 (Δ𝜙) = (𝜖4 |𝛼II |2 + t4 |𝛼I |2 )hΩ = constant I2 (Δ𝜙) = (t4 |𝛼II |2 + 𝜖4 |𝛼I |2 )hΩ = constant I1 (Δ𝜙) + I2 (Δ𝜙) = Iin

(5.27)

where we are now considering an interferometer that is not necessarily balanced. Thus the visibility, or contrast, of the interference system can be used to measure the degree of coherence. We can measure the visibility VIS by comparing the extrema Imax and Imin of the intensity in one output beam and use the deﬁnition VIS =

Imax − Imin Imax + Imin

(5.28)

5.2 Interferometers

We can then compare the measured value of VIS with the best possible value VISperfect that we would have achieved for completely correlated beams, where 𝛼II and 𝛼I completely interfere: √ 2 III II (𝛼II + 𝛼I )2 − (𝛼II − 𝛼I )2 VISperfect = = (5.29) (𝛼II + 𝛼I )2 + (𝛼II − 𝛼I )2 III + II VISperfect can be determined separately by individually blocking the beams and by recording the intensity of the beams independently. Combining these four measurements of Imax , Imin , III , and II in Eq. (5.29) yields the value =

VIS VISperfect

(5.30)

This is the value for the coherence of the beams, assuming a perfectly aligned interferometer, which is also the value of the correlation between the beams. This measurement cannot produce easily the sign of the correlation since correlated and anti-correlated lights produce fringe systems of identical visibility – only shifted with respect to each other. A knowledge of the relative phase between 𝛼II and 𝛼I at the maximum interference would allow us to determine the sign. Please note that this technique can be used for any situation; the intensity of the interfering beams do not have to be balanced. However, a simple, direct evaluation is only possible for the balanced situation |III | = |II | since in this case VISperfect = 1 and = VIS. In all cases we have = |C(𝛼II , 𝛼I )|. One application of this procedure is to determine the degree of spatial coherence of a beam of light using a double slit experiment. In this particular case 𝛼II and 𝛼I are parts of the same wave but from spatially diﬀerent sections of the beam. The phase diﬀerence Δ𝜙 is a purely geometrical term that is given by Δ𝜙 = d sin(Θ) where d is the separation of the two slits and Θ the angle of detection measured from the axis normal to the plane of the two slits. If the detection occurs at a distance much larger than the slit separation, the fringes are intensity minima and maxima that are parallel and equally spaced. The visibility of these fringes is a quantitative measure of the spatial coherence of the light. Similarly, a measurement of the visibility can be used to ascertain the correlation between two electronic signals, for example, the currents from two photon detectors. By adding and subtracting the signals, or by adding them with a variable time delay, interference signals can be generated. The visibility, and thus the correlation coefﬁcient, can be determined. In Section 7.3.4 this application is discussed in detail. 5.2.2

Quantum Model of the Interferometer

We now consider a quantum model of the interferometer. As in the classical case, we will assume that the instrument is perfectly aligned and that the spatial overlap of the two beams at the recombining beamsplitter is perfect. The eﬀect of spatial and temporal mode mismatch will be discussed in Section 5.2.6 (Figure 5.6). The interferometer is illuminated by modes with the operators â in and â u ; inside we have the operators â I and â II . After recombination we have the two output operators â 1 and â 2 , which are detected individually. As with the classical model, the case of a Mach–Zehnder interferometer with a phase diﬀerence Δ𝜙 is treated

155

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5 Basic Optical Components

Avac Ain

ε1

Figure 5.6 Description of an interferometer in the quantum model.

A|

A||

A1

∆ϕ

ε4 A2

ﬁrst. The results can be generalized to any other interferometer and to applications where Δ𝜙 is a function of time or space. By canonical quantization the internal operators are linked to the input operators via √ √ â I = 1 − 𝜖1 â in − 𝜖1 â u √ √ â II = 𝜖1 â in + 1 − 𝜖1 â u (5.31) The output operators are linked to the internal states via √ √ â 1 = exp(iΔ𝜙) 1 − 𝜖4 â II − 𝜖4 â I √ √ â 2 = exp(iΔ𝜙) 𝜖4 â II + 1 − 𝜖4 â I

(5.32)

For the balanced interferometer, we can set 𝜖1 = 𝜖4 = 12 , and the equations can be combined to give â 1 = exp(iΔ𝜙∕2)(cos(Δ𝜙∕2) â in + i sin(Δ𝜙∕2) â u ) â 2 = exp(iΔ𝜙∕2)(i sin(Δ𝜙∕2) â in + cos(Δ𝜙∕2) â u ) 5.2.3

(5.33)

The Single-Photon Interferometer

The output state of a single-photon incident on a balanced interferometer can be calculated by inverting Eq. (5.33). For the input state |1⟩in |0⟩u = â †in |0⟩in |0⟩u , we obtain the output state cos(Δ𝜙∕2) |1⟩1 |0⟩2 + i sin(Δ𝜙∕2) |0⟩1 |1⟩2

(5.34)

Depending on the phase shift in the interferometer, the photons can be made to exit from one or other output ports with certainty or be in a superposition of both ports. This eﬀect is clearly a wave eﬀect and cannot be understood by thinking of the beamsplitters as random photon partitioners. The behaviour of the photons depends on path length diﬀerences between the arms of the interferometer on the order of fractions of the optical wavelength. 5.2.4

Transfer of Intensity Noise Through the Interferometer

We now consider the intensity noise observed at the interferometer output when a bright beam is incident. In order to determine the photon ﬂux, for example, at

5.2 Interferometers

output A2 , we have to evaluate the operator n̂ 2 = â †2 â 2 . However, for a bright beam we can consider this as a classical part 𝛼 for the average values and the operator 𝛿 â for the ﬂuctuations, e.g. â 2 = 𝛼2 + 𝛿 â 2 . To calculate the average intensities, we only have to consider the terms in 𝛼; all terms with 𝛿 â make no contribution, and we obtain N 1 = N in cos2 (Δ𝜙∕2) N 2 = N in sin2 (Δ𝜙∕2)

(5.35)

Not surprisingly, this is the same result as for the classical calculation. The interferometer is like a variable beamsplitter that distributes the energy between the two output ports. The ratio of the distribution can be adjusted by the diﬀerential phase Δ𝜙. Now we come to the transfer function for the ﬂuctuations. The direct calculation of the ﬂuctuations would require the evaluation of V N1 = ⟨𝛿 â †1 𝛿 â 1 𝛿 â †1 𝛿 â 1 ⟩ − ⟨𝛿 â †1 𝛿 â 1 ⟩2 . This would be very laborious, but for bright beams we can linearize (see ̃ 1 (Ω)|2 ⟩. Consequently, turning Section 4.5) and use V N1 = 𝛼12 V 11 (Ω) = 𝛼12 ⟨|𝜹X1 Eq. (5.33) into a transfer function for the amplitude quadratures, the result is V 11 (Ω) = cos2 (Δ𝜙∕2) V 1in (Ω) + sin2 (Δ𝜙∕2) V 2u (Ω) V 12 (Ω) = sin2 (Δ𝜙∕2) V 1in (Ω) + cos2 (Δ𝜙∕2) V 2u (Ω)

(5.36)

Again, this is not a surprising result. The ﬂuctuations of the two outputs are proportional to the average photon ﬂuxes. The size of the ﬂuctuations reﬂects the division of the ﬂux between the two outputs. However, it is important to note that the ﬂuctuations of the phase quadrature of the unused port enter these calculations, not the phase ﬂuctuations of the input laser beam, which cancel completely. It was ﬁrst pointed out by C.M. Caves [7] that the noise in the output can be interpreted as a beat between the coherent input beam and the vacuum state with V 2u = 1. In this balanced interferometer the phase noise of the input beam cancels; it does not contribute. If we have only one input beam, we can make the assumptions that (i) the second port is not used, but the other input beam is the vacuum mode and V 1u = 1, and (ii) the photon ﬂux of the input beam is dominant (N in ≫ 1). Now we obtain V 11 (Ω) = cos2 (Δ𝜙∕2) V 1in (Ω) + sin2 (Δ𝜙∕2) V 12 (Ω) = sin2 (Δ𝜙∕2) V 1in (Ω) + cos2 (Δ𝜙∕2)

(5.37)

This is identical to the transfer function of a single beam with a beamsplitter of variable reﬂectivity, determined by the phase shift diﬀerence Δ𝜙 in the interferometer. Again, the spectrum of the ﬂuctuations remains unchanged.

5.2.5

Sensitivity Limit of an Interferometer

How should we operate the interferometer to get the best sensitivity? What is the limit in sensitivity? It is necessary to determine the signal that can be created by modulating the phase diﬀerence. For a small change 𝛿𝜙 of the phase

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shift (𝛿𝜙 ≪ π), the change in the optical ﬂux at the detector is equal to the ﬁrst derivative of the output photon ﬂux in regard to the phase multiplied with 𝛿𝜙: 𝛿N2 = 𝛿𝜙

d(N2 ) 𝛿𝜙 1 = 𝛿𝜙2 cos(Δ𝜙∕2) sin(Δ𝜙∕2) Nin ≈ sin(Δ𝜙)Nin d𝜙 2 2 (5.38)

An AC detector measures the signal power psig that is proportional to the square of the change in optical power (see Section 2.1) and includes 𝜏, the inverse of the detection bandwidth: ( )2 𝛿𝜙 (5.39) sin(Δ𝜙) Nin 𝜏 psig = 2 As illustrated in Figure 5.7, the signal is largest when the change in power is most rapid, that is, at Δ𝜙 = π∕2, in the middle between the maximum and minimum of a fringe. The signal can be compared with the variance of the photon number or the quantum noise on the detector: pqnoise = V N2 = sin2 (Δ𝜙∕2) Nin 𝜏 and we can evaluate the SNR: sin2 (Δ𝜙) 𝛿𝜙2 SNR = Nin 𝜏 2 4 sin (Δ𝜙∕2)

(5.40)

(5.41)

which has the largest value when Δ𝜙 = 0, 2π, where the output optical power is at a minimum. In the jargon of the trade, we operate the interferometer at a dark fringe. This is a very diﬀerent operating point than one would use for a classical interferometer with intensity-independent noise. The reason for this result is that whilst both signal and noise disappear at Δ𝜙 = 0, the quantum noise disappears more rapidly than the signal and the ratio of the two reaches a maximum (see the SNR trace (c) in Figure 5.7). Note that this is a somewhat academic result. In practice the perfect dark fringe is not an optimum since there are always some additional intensity-independent noise terms such as electronic noise. The optimum will be at a point where the quantum noise component in the current ﬂuctuations is larger than the additional 20

Linear units (QNL)

158

15

(b) Quantum noise

(c) SNR

10

(a) Signal

5

0 0

π/2

π

3π/2

Phase offset (∆ϕ)

2π

Figure 5.7 Results for an interferometer that is limited by quantum noise. Shown as a function of the detuning Δ𝜙 are (a) the optical power that also corresponds to a signal with ﬁxed modulation depth, (b) the quantum noise, and (c) the signal-to-noise ratio.

5.2 Interferometers

noise. Details are given in Section 10.3. We deﬁne a QNL instrument as a device where the quantum noise is the dominant noise source. The minimum detectable phase change 𝛿𝜙QNL in a QNL interferometer corresponds to an SNR of 1. Near the optimum dark fringe, we can approximate sin2 (Δ𝜙) (Δ𝜙)2 =4 ≈ sin2 (Δ𝜙∕2) (Δ𝜙∕2)2 1 𝛿𝜙QNL = √ Nin 𝜏

(5.42)

for a coherent input state. This is the standard QNL for interferometry. It is the smallest achievable measurement of a change in phase using coherent light. A comment on units is that 𝛿𝜙QNL is measured in rad. Nin is the number of photons travelling from the interferometer to the front face of a detector in one measurement interval. Thus Nin is a number with no dimensions. The value of 𝛿𝜙QNL is not ﬁxed; it depends on the integration time 𝜏. If we spend more time on the measurement, integrate for longer, the value would improve. But the integration time cannot be longer than the period of the signal. As an example the QNL for an interferometer operated with 1 (mW) of light at 𝜆 = 1(μm) and a detection interval of 1 (ms) is 4.4 × 10−7 (rad). This quantum description is not complete. Equation (5.42) shows that the eﬀect of the quantum noise decreases with more optical power. It seems that it would be possible to reach any arbitrary precision; this must be impossible. It is one of the deep truths of quantum mechanics that we need to complement (using an expression coined by Niels Bohr) the concept of quantum noise of the light with a description of the measurement process. We have to consider both the system we investigate and the process of measurement. In straightforward analogy to the Gedanken experiment of Heisenberg’s microscope, there is a back eﬀect of the measurement process onto the object we measure. In an interferometer we investigate the position of the mirrors, which deﬁnes the optical path length of the interferometer. The measurement process is the interference of the two reﬂected modes of light. The reﬂection of a beam of light with power Preﬂ exerts a force on the mirrors: 2Preﬂ c The ﬂuctuations of this force are due to the quantum noise ﬂuctuations in the photon ﬂux, as discussed above, and the amplitude spectral density of the force will be 2 ℏ Preﬂ 2 (Ω) = (5.43) 𝛿FRP c𝜆 which is independent of the detection frequency. Such a stochastic force will produce ﬂuctuations in the mirror position and results in a position noise spectrum [8]: √ ℏPreﬂ 1 1 2 2 𝛿F (Ω) = (5.44) 𝛿x (Ω) = m(2πΩ)2 RP mΩ2 8π3 c𝜆 FRP =

159

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5 Basic Optical Components

where m is the mass of the mirror. This displacement occurs in both arms of the interferometer but with opposite sign. The total eﬀect gives a spectrum of the angular ﬂuctuations: ℏ √ Nin 𝜏 𝛿𝜙RP (Ω) = mΩ2 𝜆2 This noise contribution increases with the input power. It also rises at lower detection frequencies. The total noise will be 𝛿𝜙2QM = 𝛿𝜙2QNL + 𝛿𝜙2RP

(5.45)

and the balance between the two terms depends on detection frequency and the power level chosen. Only at low frequencies and high powers does the radiation pressure dominate. At higher frequencies and lower powers, the system is limited by intensity quantum noise alone (QNL). Very special arrangements, such as high-power lasers and very lightweight mirrors, are required to make the light pressure visible. An example for this balance is shown in Figure 10.10 in Chapter 10. For each frequency there is a minimum noise level, also called the standard quantum limit (SQL). This limit can be reached for an optimum power of Popt = πc𝜆mΩ2 , which is typically quite large. For example, with m = 1 kg, 𝜆 = 1 μm, and Ω = 100 Hz, one obtains Popt = 9.4 MW. At the optimum condition the system reaches its limit in sensitivity as dictated by the quantum physics of the measurement process. The QNL described in Eq. (5.45) cannot be avoided with any measurement process based on normal coherent light, but it can be surpassed by using correlated photon states and quantum non-demolition techniques that are described in Chapters 9–11.

5.2.6

Eﬀect of Mode Mismatch on an Interferometer

So far we have assumed that the modes aI and aII that meet at the second beamsplitter in the interferometer are perfectly mode-matched. In general this will not be true, leading to the visibility of the interferometer being reduced. There are many ways in which mode mismatch can occur, for example, the transverse modes might not perfectly overlap, either because they are displaced from each other or because their spot sizes are unequal; their polarizations might not perfectly match due to some birefringence in one arm; or their longitudinal mode functions might not perfectly overlap. Qualitatively, all these diﬀerent possibilities lead to similar eﬀects. As a particular example we will consider the latter case for a single photon through an interferometer. We consider a single-photon input state as in Section 5.2.3 but now show its mode decomposition explicitly using the techniques of Section 4.4.1: â †in (ti ) |0⟩in |0⟩u =

∫

̃ † |0⟩in |0⟩u d𝜔e−i𝜔ti F(𝜔)A in,𝜔

(5.46)

where ti is some initial time. We model propagation through an interferometer as shown in Figure 5.6 but now allow for an arbitrary time delay, td , in the lower arm.

5.2 Interferometers

Transforming the frequency operators and assuming a balanced interferometer, we obtain the output state |𝜙⟩1,2 =

∫

1 ̃ † + ((e−i𝜔(t+td ) + e−i𝜔t )A ̃ † ) |0⟩1 |0⟩2 d𝜔F(𝜔) ((e−i𝜔(t+td ) − e−i𝜔t )A 1,𝜔 2,𝜔 2 (5.47)

This is a spectrally entangled state. To evaluate the visibility of the interferometer, we need to introduce a particular detection model. We model broadband photon counting of the output modes of the interferometer by the operator n̂ i =

∫

̃† A ̃ d𝜔 A i,𝜔 i,𝜔

(5.48)

̃ represent ̃† A Here it is assumed that the although the individual operators A i,𝜔 i,𝜔 number detection with strong frequency resolution, the output photocurrent of the detector averages over all frequencies. The photon counting rates are then given by 𝜔t 1 d𝜔|F(𝜔)|2 |1 − ei𝜔td |2 = d𝜔|F(𝜔)|2 sin2 d ∫ ∫ 2 2 𝜔 t 1 d𝜔|F(𝜔)|2 |1 + ei𝜔td |2 = d𝜔|F(𝜔)|2 cos2 d ⟨n̂ 2 ⟩ = ∫ ∫ 2 2 ⟨n̂ 1 ⟩ =

(5.49)

We can identify two limits to this expression. If 𝜔 td is suﬃciently small, the sine and cosine terms will be approximately constant over the width of the mode function and so can be taken outside the integrals giving 𝜔o td 𝜔t d𝜔|F(𝜔)|2 = sin2 o d 2 ∫ 2 𝜔 t 𝜔 t 2 o d ⟨n̂ 2 ⟩𝜔tdlim d𝜔|F(𝜔)|2 = cos2 o d →0 = cos 2 ∫ 2

2 ⟨n̂ 1 ⟩𝜔tdlim →0 = sin

(5.50)

where 𝜔o is the central frequency of the single-photon pulse. In this case the time delay is so small that it only imparts a phase shift. The result agrees with those of Section 5.2.3. In this limit there is perfect mode matching and the visibility is unity. Alternatively, if 𝜔td is suﬃciently large, the sine and cosine terms will oscillate very rapidly. As a result, over intervals for which the mode function is approximately constant, both the sine and cosine terms will average to one-half. We obtain 1 1 d𝜔|F(𝜔)|2 = 2∫ 2 1 1 ⟨n̂ 2 ⟩𝜔tdlim d𝜔|F(𝜔)|2 = →∞ = 2∫ 2 ⟨n̂ 1 ⟩𝜔tdlim →∞ =

(5.51)

Now the time delay in the lower arm is so long that there is no overlap at the ﬁnal beamsplitter between pulses travelling along the upper arm and the lower arm. In this limit there is complete mode mismatch, and the visibility is zero. For delay times in between these limits, there will be partial mode match, and the visibility will be between zero and one.

161

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5 Basic Optical Components

From the point of view of the broadband detection model (Eq. (5.48)), it is clear that the visibility is reduced because, although individual frequency components still display interference, the fringes shift their position with frequency and hence are washed out when we average over many frequencies. Interestingly, Eq. (5.48) can also arise from a completely diﬀerent detection model in which our individual detection operators have excellent temporal resolution, but we then integrate the photocurrent over time. Speciﬁcally 1 ̂† A ̂ d𝜏 A i,𝜏 i,𝜏 2π ∫ ′ 1 ̃† ̃ i,𝜔′ d𝜔′ ei𝜔 𝜏 A = d𝜏 d𝜔e−i𝜔𝜏 A i,𝜔 ∫ ∫ 2π ∫

n̂ i =

=

∫

̃† A ̃ d𝜔 A i,𝜔 i,𝜔

(5.52)

̂† A ̂ represent number Now it is assumed that the individual operators A i,𝜏 i,𝜏 detection with strong temporal resolution. The expansion in moving from lines one to two uses the Fourier transform relation between the time and frequency domains. The simpliﬁcation in moving from the second to third line results from recognizing that the integral over the detection time, 𝜏, creates the Dirac delta function 𝛿(𝜔′ − 𝜔). We arrive at the same measurement operator as for the broadband case, and so the visibility also decreases with increasing time delay for this detection model. However, now the reduction of visibility can be attributed to which-path information in the interferometer. In principle the time-resolved detectors can resolve the arrival times of the photons at the two outputs with high precision. When the time delay is very small, the arrival time of the photons reveals no information about which path they took through the interferometer. However, when the time delay is large, photons that take the lower arm will arrive noticeably later than those that take the upper arm, thus revealing the path taken through the interferometer and erasing the interference. Even though we integrate over diﬀerent times and hence do not reveal the arrival times to the experimenter, the in-principle existence of which-path information erases the interference.

5.3 Optical Cavities One component of widespread use in quantum optics experiments is the optical cavity. In the simplest case this consists of two mirrors that are aligned on one optical axis such that they exactly retro-reﬂect the incident light beam. In the more general case, it is a group of mirrors aligned in such a way that the beam of light is reﬂected in a closed path, such that after one round trip, it interferes perfectly with the incident wave. The curvature of the mirrors is chosen such that they match the curvature of the internal wave fronts at all reﬂection points. The optical path length for one round trip through the cavity is the perimeter p, which takes into account all the refractive indices inside the cavity. The light is reﬂected

5.3 Optical Cavities

(a)

(b)

Figure 5.8 Schematic diagrams of linear and ring cavities.

many times around the cavity. Each round trip generates one partial wave; all these waves will interfere (Figure 5.8). For the case that p is an integer multiple of the wavelength of the incident light, a strong ﬁeld will be built up inside the cavity due to constructive interference; the cavity is on resonance. The resonance frequencies satisfy the condition 𝜈cav =

jc ; p

j = integer

(5.53)

Note that for a linear cavity the optical length is p = 2nd12 , where d12 is the physical separation between the two mirrors and n is the refractive index. If either the cavity length or the wavelength of the light is scanned, the cavity will go in and out of resonance. The cavity resonance frequencies are equally spaced, and the diﬀerence between two adjacent resonance frequencies is called the free spectral range (FSR): FSR = c∕p

(5.54)

The magnitude of the enhancement of the ﬁeld and the width of the resonance depend on the reﬂectivities of the mirror and losses inside the cavity. Cavities are used for a number of purposes: the laser itself is a cavity with an active medium inside that provides the gain. The cavity provides the optical feedback required to achieve oscillation, hence the name laser resonator. The cavity determines the spatial properties of the laser beam. In addition, it selects the frequencies of the laser modes. A cavity can be used to investigate the frequency distribution of the incident light. For this purpose, the length of the cavity is scanned, and the transmission is plotted as a function of the length. In this mode of operation, a cavity is called an optical spectrum analyser. On the other hand, a cavity can be used as a very stable reference for the frequency of the light. The detuning of the incident light from the resonance frequency can be measured, and the frequency of the laser light can be controlled via electronic feedback. In this way the laser is locked to the cavity, and the frequency of the laser light is stabilized. For details see Section 8.4. The light reﬂected from the cavity provides optical feedback. This is very undesirable for many laser systems and sometimes even destructive. However, feedback can be used with some lasers, such as diode lasers, to form an external cavity and thereby control the laser frequency. The high optical ﬁeld inside the cavity can be used to enhance non-linear optical eﬀects. Such build-up cavities are used in many of the CW squeezing experiments described in Chapter 9.

163

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5 Basic Optical Components

On resonance, the cavity will set up internal spatial modes. In the case of a laser resonator, this determines the shape of the laser mode and its waist position and waist size. Only certain combinations of mirror curvatures will lead to a stable resonance condition [9]. A passive cavity will only be resonant with that component of the incident ﬁeld that corresponds to its internal spatial mode. All other light will be directly reﬂected. In other words, the incident light has to be mode-matched to the cavity in order to get into the cavity. In return, the transmitted light has a very well-deﬁned, clean spatial mode. For this reason cavities can be used as spatial mode cleaners. There are several applications of cavities for inﬂuencing the ﬂuctuations of the light. The build-up of the internal ﬁeld is not instantaneous. It requires the interference of many waves that have to travel through the entire cavity. A steady state can be achieved after a time tcav . This is also the time it takes for the internal ﬁeld to decay in the absence of an incident light ﬁeld. Thus the cavity acts like an integrator on the ﬂuctuations with a time constant tcav . The high frequency components of the incident light, at frequencies larger than 𝛾 = 1∕tcav , will not enter the cavity; they will be reﬂected. These frequency components will be strongly attenuated in the transmitted light. On resonance, the cavity acts like a low-pass ﬁlter on the intensity ﬂuctuations. Finally, there are interesting applications of a cavity that is close to, but not exactly on, resonance with the input ﬁeld. The phase of the reﬂected amplitude, compared with the phase of the incident light, varies strongly, by about π, when scanned across one cavity linewidth. Thus the sidebands of the light, which have diﬀerent cavity detunings to the carrier frequency, are reﬂected with diﬀerent optical phases. As a consequence, the quadrature of the reﬂected light is rotated in respect to that of the incident light. We will ﬁnd (in Section 9.3) that cavities can be used to analyse the quadratures of light, coherent, or squeezed and as a device to change the squeezing quadrature. 5.3.1

Classical Description of a Linear Cavity

The intensity of the internal, reﬂected, and transmitted light and the relative phase of these beams can be evaluated with a purely classical model. Let us consider a simple cavity with input mirror M1 and output mirror M2 . This could be a standing wave cavity with two mirrors only or a ring cavity where all other mirrors are near-perfect reﬂectors. The cavity can be described by the parameters and the corresponding total cavity properties in Figure 5.9 and Table 5.1. These properties can be derived using the classical equations for multiple interference of waves. Here we will use the standard textbook approach of adding partial waves and combine it with the more elegant approach of solving d12

Figure 5.9 Properties of a linear cavity.

Loss: e–α2d R1 T1

R2 T2

5.3 Optical Cavities

Table 5.1 Parameters and properties of a linear cavity. Parameters of a linear cavity

T1

= intensity transmission of the input mirror

T2

= intensity transmission of the output mirror

R1

= intensity reﬂectivity of the input mirror

R2

= intensity reﬂectivity of the output mirror

p

= optical path length for one round-trip

d12

= distance from input to output mirror

𝛽

= the loss coeﬃcient for internal absorption or scattering

exp(−𝛽p)

= round-trip intensity loss of the cavity

gm

= loss parameter

Pin

= power of the incident beam

Pout

= power of the transmitted beam

Preﬂ

= power of the reﬂected beam

𝛾

= cavity linewidth

Δ

= normalized cavity detuning = (𝜈L − 𝜈cav )∕𝛾

Properties of a linear cavity

FSR

= free spectral range = c∕p

F

= ﬁnesse = FSR∕𝛾

self-consistency equations at the boundaries of the cavity. The amplitude 𝛼in of the incident ﬁeld will be partially transmitted through the √ ﬁrst mirror, and the internal ﬁeld just past the ﬁrst mirror ﬁeld will be 𝛼0 = T1 𝛼in . It will circulate inside the cavity, and during each round trip it will be reduced in amplitude by a factor [9]: √ (5.55) gm = R1 R2 exp(−𝛽p) which includes the amplitude reﬂectivities at the mirrors and the internal losses. The wave will also pick up a phase diﬀerence 𝛿𝜙 = 2π(𝜈L − 𝜈cav )p∕c for every round trip (Figure 5.10). T1 R1 A1 Ein

Pin

T2 R2 A2

E0 e–αp ei δϕ E1 E2 E3 E4

Erefl

Prefl

Ecav

Pcav

Eout

Pout

Figure 5.10 Multiple amplitude interference inside a linear cavity.

165

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5 Basic Optical Components

Let us describe the ﬁeld amplitude 𝛼ca𝑣 inside the cavity as a sum of all circulating terms: 𝛼cav = 𝛼0 + 𝛼1 + 𝛼2 + 𝛼3 + 𝛼4 + · · · =

∞ ∑

𝛼j

j=0

where 𝛼j+1 = g(𝜈)𝛼j and g(𝜈) = gm exp(−i𝛿𝜙(𝜈))

(5.56)

is a complex number that describes the total eﬀect of the cavity during one round trip. The amplitude of 𝛼2 is related to the amplitude of 𝛼1 by the same factor. The total ﬁeld is described by 𝛼0 (5.57) 𝛼cav = 𝛼0 (1 + g(𝜈) + g(𝜈)2 + g(𝜈)3 + g(𝜈)4 + · · · ) = 1 − g(𝜈) Note that this is equivalent to the eﬀect of a closed feedback system that adds a component g(𝜈)𝛼0 inside the feedback loop. The intensity, and thus the power of the light inside the cavity, is built up to |𝛼0 |2 |1 − g(𝜈)|2 |𝛼0 |2 = |1 − gm exp(−i𝜙)|2 |𝛼0 |2 = (1 − gm )2 + 4gm sin2 (𝛿𝜙∕2)

Pcav ∝ |𝛼cav |2 =

(5.58)

Thus we get 1 1 + (2F∕π)2 sin2 (𝛿𝜙∕2) )2 ( 𝛼 with Pmax = 1−g0 and m √ π gm F = Finesse = (1 − gm ) Pcav = Pmax

(5.59)

The ﬁnesse is generally used as a measure for the quality of cavity. It is the number most frequently quoted in experimental publications. We can now link the amplitudes of the various ﬁelds through the following boundary conditions, where we take 𝛼cav as the internal ﬁeld just on the inside of mirror M1 : √ 𝛼0 = T1 𝛼in √ 𝛼out = T2 exp(−𝛽d12 + i𝛿𝜙∕2)𝛼cav √ √ − R1 𝛼in + T1 𝛼reﬂ = (5.60) √ R1 𝛼cav

5.3 Optical Cavities

The second condition takes into account the internal losses on the path d12 from mirror M1 to mirror M2 and the transmission through mirror M2 . The last condition takes into account that the wave at the outside of mirror M1 is out of phase (minus sign) with the wave transmitted through this mirror from the inside. Combining Eq. (5.60) with Eq. (5.57) leads to the link between the transmitted and reﬂected powers with the input power Pout exp(−2𝛽d12 ) = T1 T2 Pin |1 − g(𝜈)|2 Preﬂ |R − (R1 + T1 )g(𝜈)|2 = 1 Pin R1 |1 − g(𝜈)|2 Pcav T1 = Pin |1 − g(𝜈)|2

(5.61)

Consider the special case of a linear cavity with no losses in the mirrors, which means p = 2d12 and R1 + T1 = 1, R2 + T2 = 1. This is commonly discussed in textbooks as the case for an optical spectrum analyser or a Fabry–Perot etalon. In this case we obtain the result Pout T1 T2 |g(𝜈)|2 = Pin R1 R2 |1 − g(𝜈)|2 Preﬂ |R1 − g(𝜈)|2 = Pin R1 |1 − g(𝜈)|2 Pcav T1 = Pin |1 − g(𝜈)|2

(5.62)

The cavity build-up can be dramatic, and the transmission of a cavity can be much larger than the individual transmissions of the mirrors. Note that these equations hold for all detunings; on resonance, simply replace g(𝜈) by gm . For illustration it is worthwhile to consider some special cases: (i) A symmetric, lossless cavity. That means T1 = T2 = T and 𝛽 = 0. The loss parameter for this case is gm = R = (1 − T), and we obtain from Eq. (5.61) Pout ∕Pin = T 2 R∕R(1 − R)2 = 1 and Preﬂ ∕Pin = (1 − R)2 ∕R(1 − R)2 = 0. Such a cavity is perfectly transmitting and has no reﬂected power at all. This clearly illustrates the dramatic eﬀect of the constructive interference. The internal power is Pcav = Pin T∕(1 − R)2 = Pin ∕T, which means that mirrors of transmission T = 0.01, corresponding to R = 0.99, are required for a power build-up by a factor of 100. The ﬁnesse of such a cavity can be approximated as F = π∕T = 300. (ii) A cavity with loss. All cavities will have some losses. An interesting case is a cavity where the input transmission T1 just balances the sum of all losses. That means R1 = gm , which is equivalent to R1 = R2 exp(−𝛽p). For this case we get Preﬂ = 0; there is no reﬂection. The leakage of the intra-cavity power out through the input mirror is equal to the directly reﬂected amplitude. They cancel each other. This case is known as the impedance matched case,

167

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5 Basic Optical Components

in analogy to the matching of electronic transmission lines: Under-coupled:

R1 < R2 exp(−𝛽p)

Impedance matched:

R1 = R2 exp(−𝛽p)

Over-coupled:

R1 > R2 exp(−𝛽p)

The two other cases, under-coupled and over-coupled, lead to some reﬂection, and the transmission is not optimum. Whilst both cases result in similar reﬂected power, the sign of the reﬂected wave changes from the under-coupled to the over-coupled case. However, in most experiments this sign cannot be detected. All these cases are illustrated in Figure 5.11 by the traces (i), (ii), and (iii). 1 (iii) A single ported cavity (T2 = 0). This obviously has no transmission. However, it is still useful. It shows internal power build-up, and all the leakage back out is through the input mirror. The cavity will have a partial power reﬂection. This light will contain information about the internal ﬁeld. The reﬂected beam can be separated and detected by using a linear cavity and an optical isolator or alternatively a ring cavity where the light is reﬂected into a diﬀerent direction. The loss parameter gm describes the reduction in the intensity of the internal ﬁeld on resonance with the cavity after one round trip. The fraction (1 − gm ) leaves the cavity during each round trip. One round trip takes the time p∕c. Thus the energy loss from the cavity is (1 − gm )Pcav c∕p measured in J/s. Since this loss is proportional to the energy stored in the cavity, the power Pcav will decay exponentially in time in the absence of an incident ﬁeld: Pcav (t) = Pcav (0) exp(−t∕tcav ) c 1 = (1 − gm ) tcav p The amplitude of the ﬁeld decays with twice the time constant, and we can deﬁne a coupling rate 𝜅 for the amplitude 𝛼cav (t) = 𝛼cav (t = 0) exp(−t∕2tcav ) = 𝛼cav (t = 0) exp(−𝜅t) with 𝜅 = (c∕p)(1 − gm ). This decay constant is the appropriate term for the temporal description of the cavity response. It can be linked to the cavity linewidth 𝛾. The Fourier transform of the intensity decay gives the spectral response function of the cavity. It results in a function proportional to 1∕(1 + i4πtcav ). The resonance of the spectral response has a full width half maximum (FWHM) of 𝛾 = 1∕(2πtcav ) and is measured in units of Hz. The same result could have been obtained by determining the FWHM of the cavity response function given in Eq. (5.59). 1 Note that the deﬁnition of under-coupled and over-coupled used here is the opposite of the conventional deﬁnition used, for example, in the book Lasers by A.E. Siegman [9].

5.3 Optical Cavities 0.4 π/2

(ii)

Pout

Φout (Rad)

0.3 0.2

Pin

0.1

(i)

0 0.064

(a)

(iii) 0

0

−π/2 0.064

0.064

0

0.064

0

0.064

0 Detuning (∆ν)

0.064

40 π/2

Φrefl (Rad)

30

Pcav 20

Pin

0

10 −π/2

0 0.064

(b)

0

0.064

0.064

1

π

Φcav (Rad)

0.75

Prefl ___ 0.5 Pin

0

0.25

−π 0 0.64

(c)

0

0.64

0.064

Detuning (∆ν)

Figure 5.11 Illustration of the cavity response. On the left side is the power response, and on the right side the phase response. Shown are the three results: (a) transmitted light, (b) the internal ﬁeld, and (c) the reﬂected light. For each situation three diﬀerent cases are shown. (i) The largely under-coupled case (long dashes), (ii) the impedance matched case (solid line), and (iii) the over-coupled case (dotted line).

5.3.2

The Special Case of High Reﬂectivities

It is useful to treat the transmission through the mirrors and the loss inside the cavity in a similar way. They are all processes for the leakage of the ﬁeld out of the cavity. For this purpose one can describe the properties of the mirrors and the losses with coupling constants 𝛿 [9]. These replace the exponential terms that are required for cavities with large losses. We convert the mirror reﬂectivity into

169

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5 Basic Optical Components

a mirror coupling constant via Ri = exp(−𝛿mi )

or 𝛿mi = ln(1∕Ri ) = − ln(Ri )

and deﬁne an equivalent coupling constant for the intra-cavity losses as 𝛿lo = 𝛽p. In this way the loss parameter gm , deﬁned in Eq. (5.55) for a cavity with two partially reﬂecting mirrors, turns into gm = exp(−(𝛿m1 + 𝛿m2 + 𝛿lo )) = exp(−𝛿) An important case in quantum optics is the case of a high ﬁnesse cavity. Here the round-trip losses are small. That means 𝛿mi ≪ 1, 𝛿lo ≪ 1, and thus 𝛿 ≪ 1. This eﬀectively means that the amplitude is changing so little per round trip that spatial eﬀects along the propagation axis can be neglected. We assume a uniform internal amplitude. We also assume that losses inside the mirrors can be neglected, which means (Ri + Ti ) = 1. This leads to the following approximations: gm = exp(−𝛿) ≈ 1 − 𝛿 𝛿mi = − ln(Ri ) ≈ 1 − Ri = Ti (1 − gm )c (𝛿 + 𝛿m2 + 𝛿lo )c 𝛿c 𝜅= ≈ = m1 2p 2p 2p

(5.63)

We can deﬁne the loss rates for the total cavity and for the individual processes as 𝜅 = 𝜅m1 + 𝜅m2 + 𝜅lo

with 𝜅mi =

Ti c 2p

and 𝜅lo =

𝛽c 2

(5.64)

These deﬁnitions are commonly used in describing high ﬁnesse cavities. We ﬁnd them, for example, in laser models (Chapter 6) and many papers on non-linear processes inside cavities. They are very useful for most quantitative models. However, the limitations of this approximation should not be forgotten. There are many experiments where intra-cavity losses as large as several percent are unavoidable. Or we might have a strong non-linear conversion out of the internal mode into another mode. One example would be eﬃcient second harmonic generation (SHG). In these cases the approximations of the high ﬁnesse limit might not hold, and the exact equations should be used.

5.3.3

The Phase Response

The cavity introduces a phase shift between the incoming, the intra-cavity, the reﬂected, and the transmitted ﬁelds. The boundary conditions for the electric ﬁeld, given in Eq. (5.60), can be used to evaluate these phase shifts. Let us deﬁne all phase shifts in respect to the phase of the incoming amplitude. That means 𝜙cav = arg(𝛼cav ) − arg(𝛼in ) = arg(𝛼cav ∕𝛼in ), where the function arg(z) determines the phase angle 𝜙 of the complex number z. Using the results of Eq. (5.60) and

5.3 Optical Cavities

some algebraic manipulation, the following results are obtained: ( ) gm sin(𝛿𝜙) 𝜙cav = arctan 1 − gm cos(𝛿𝜙) ( ) − sin(𝛿𝜙) 𝜙out = arctan g − cos(𝛿𝜙) ) ( m √ −T1 R1 sin(𝛿𝜙) 𝜙reﬂ = arctan √ √ R1 + T1 R1 (1 − gm cos(𝛿𝜙))

(5.65)

A graphical evaluation of these equations for three special cases of under-coupled, over-coupled, and impedance matched is shown in Figure 5.11 (i)–(iii) alongside the results for the transmitted powers. On resonance, which means 𝛿𝜙 = 0, all three phase terms are equal to zero, modulus π. The electric ﬁelds of the incoming, intra-cavity, and transmitted light are all in phase. The reﬂected light is either in phase or out of phase. Around the resonance the phase terms show a dispersive behaviour. The width of the dispersion curve (right-hand side) is comparable with the cavity linewidth 𝛾 (left-hand side). There is a 2π phase change right across the cavity resonance. The intra-cavity ﬁeld and the transmitted ﬁeld have a phase advance over the incoming ﬁeld that increases with detuning. For the reﬂected light the phase can be advanced (+ sign) or retarded (− sign), depending on the parameters of the cavity. If the directly reﬂected light dominates (under-coupled case), the phase diﬀerence is 180∘ on resonance, and the phase is retarded. If the light from the cavity dominates (over-coupled case), the phase is advanced. Note that sidebands of the incoming wave can experience diﬀerent attenuation and phase shifts. For the limiting case where the modulation frequency Ωmod of the sidebands is less than the cavity linewidth (Ωmod ≪ 𝛾), this eﬀect is negligible. In this case the sidebands remain in phase with the cavity. For modulation frequencies much larger than the cavity linewidth (Ωmod ≫ 𝛾), the two sidebands are shifted by +π∕2 and −π∕2, respectively, and again they appear to be in phase with the central component. If, however, the modulation frequency is of the order of the cavity linewidth, the two sidebands at ±Ωmod will experience very diﬀerent phase shifts. This eﬀect can be exploited to measure the detuning of the cavity. The incoming light is modulated, and the transmitted or reﬂected light intensity is detected. At Ωmod we obtain a signal that is the sum of two beat signals between the sidebands and the central laser frequency. Commonly, phase modulation is used with the consequence that on resonance these two beat signals are out of phase and cancel each other. On resonance, there is no detectable intensity modulation on either the incoming, the transmitted, or the reﬂected light. However, for a detuned cavity the diﬀerence in the phase shifts for the two sidebands will prevent this cancellation. A modulation signal at frequency Ωmod appears, and phase modulation is partially turned into intensity modulation. The signal strength can be measured by demodulating the photocurrent, for example, with a mixer driven at Ωmod . The shape of this response as a function of detuning is closely linked to the dispersion curve for the individual phase terms. It can be derived analytically, and details are given in Section 8.4.

171

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5.3.4

Spatial Properties of Cavities

So far we have treated the cavity and the light in the cavity as one-dimensional systems, and we have lost all spatial information. However, not all cavities with the same mirror reﬂectivity and length are the same. The curvature of the mirrors and the shape of the wave front have to be taken into account. There are certain combinations of mirror separation and mirror curvature where a resonance is possible. In this case the wave front, returning after one round trip, is keeping its size and is not expanding signiﬁcantly. A spatial resonance will build up, and as a consequence the curvature of the wave front of the resulting resonant mode will be identical to the curvature of the mirror. As the simplest example let us consider a linear two-mirror cavity. Here the resonator is deﬁned by the separation d12 of the mirrors and the mirror curvatures 1 and 2 . The focal length of the mirrors is given by f1 = 1 ∕2 and f2 = 2 ∕2. In the standard notation we deﬁne the mirror parameters [9] g1 = 1 −

d12 2f1

and g2 = 1 −

d12 2f2

(5.66)

The condition for a stable resonator, which means a resonator that builds up a resonant wave, can be derived [9] and results in the stability criterion 0 ≤ g1 g 2 ≤ 1

(5.67)

Based on this criterion we ﬁnd that for a given set of mirror curvatures (focal lengths), resonance only occurs for a restricted range of cavity lengths. We have to choose the length and the curvatures together. Some examples of stable resonators are: • The (near) concentric cavity: d12 = 2 1 = 2 2 , which means g1 = g2 = −1 and g1 g2 = 1. • The (near) planar cavity: 1 = 2 = ∞, which means g1 = g2 = 1 and g1 g2 = 1. • The confocal cavity: d12 = 1 = 2 , which means g1 = g2 = 0 and g1 g2 = 0. The concentric and planar cavities are right at the limit of the stability criterion. They can be stable, but even minor changes in length or small misalignments tend to aﬀect the performance. These cases are not recommended. It is better to satisfy the stability criterion with some margin. The confocal cavity is in the middle of the stability region and thus very useful as a robust design for a resonator. Cavities can support higher-order TEMi,j spatial modes, and if the cavity is not confocal, this will occur at cavity lengths diﬀerent to the resonance length for the TEM0,0 mode. This feature can be used to devise techniques for locking a cavity to the frequency of a laser beam [10] by analysing the spatial properties of the transmitted and/or reﬂected beam (see Section 8.4). 5.3.4.1

Mode Matching

The cavity has its own internal geometric mode, determined by the curvature of the mirrors. The wave front of the mode of light illuminating the cavity might have a diﬀerent wave front than the internal mode. The incoming wave can only fully interact with the cavity if the incoming wave front and the internal wave front are

5.3 Optical Cavities

matched. This is known as the mode-matched case. The spatial properties of the incoming wave are matched, the intensity distribution (mode size) is identical to the internal mode, and the wave front curvature is identical to the curvature of the mirror, which in turn is the curvature of the internal mode. In this particular case, which is highly desirable but in practice requires extensive alignment work, the description of the cavity and the coupled light is simple, and the one-dimensional model in the previous section is correct. All the light interacts with the cavity. Most theoretical models assume mode matching. What will happen to light incident on a cavity that has a diﬀerent geometric mode? The cavity will support one fundamental mode and a whole series of higher-order spatial modes. In general, these modes will not be resonant simultaneously – diﬀerent detunings, which means small shifts in the mirror separations, will be required for the resonance condition of the higher-order modes. For any given mirror separation, the cavity will only support one mode. The one special case, where all spatial modes are resonant simultaneously, is the confocal cavity. For all other cases the resonators are non-degenerate – the resonances occur at diﬀerent cavity detunings. On resonance, the cavity will select the part of the external ﬁeld that corresponds to the resonant internal mode, an eigenmode of the cavity. In order to calculate the reﬂection and transmission of a cavity, it is useful to describe the external ﬁeld in terms of all the internal eigenmodes of the cavity. These spatial eigenmodes are a useful basis set for a mathematical expansion of the external ﬁeld. Let us classify the spatial modes as Hermite–Gaussian modes (see Section 2.1) with the classiﬁcation TEMmn describing the possible modes. We can calculate the fraction Pmn of the total power that corresponds to the resonant internal mode TEMmn . In most cases we align the cavity to resonate in the fundamental mode TEM00 . Only this light will enter the cavity and contribute to the build-up of the internal power. The remaining power will be reﬂected, assuming that the front mirror has a high reﬂectivity. The equations for the transmitted and reﬂected power (Eq. (5.61)) have to be modiﬁed to include the fraction Pmm . For example, the transmission will then be Pmn × Ptrans ∕Pin , and the reﬂectivity will be (1 − Pmn ) + Pmn × Preﬂ ∕Pin . Obviously, P00 = 1 corresponds to the perfectly mode-matched situation. To achieve P00 > 0.95 requires considerable patience and good-quality optical components. Once aligned to the fundamental mode, both the internal and the transmitted ﬁelds will have the spatial intensity distribution of a Gaussian beam. The cavity selects the Gaussian component from the incident ﬁeld; it acts like a spatial ﬁlter. Such a cavity is therefore also referred to as a spatial mode cleaner. How can the mode-matched condition be achieved, and how can the mode mismatch be quantiﬁed? By detuning the cavity, which means by changing the mirror separation, a small fraction of a wavelength, the cavity can be set on resonance with the higher-order spatial modes. The exact resonance frequencies can be evaluated by taking into account the phase diﬀerences between the various higher-order modes and the fundamental mode, known as Gouy phase shifts. These phase √ shifts depend on the order numbers m and n and can be expanded as cos−1 (± g1 g2 ) where g1 and g2 are the mirror parameters deﬁned in Eq. (5.66). The changes in the resonance frequencies Δ𝜈mn for the modes TEMmn away from

173

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5 Basic Optical Components

the resonance of the fundamental mode are given by [9] √ (n + m + 1)cos−1 (± g1 g2 ) c Δ𝜈mn = π2d12

(5.68)

with the following special cases: =0

for confocal cavity

1 FSR for near planar cavity 2 = FSR for near concentric cavity

Δ𝜈mn =

If the cavity is not confocal, the higher-order spatial modes map out a whole series of resonances as shown in Figure 5.12b. These can be made visible by scanning the cavity length or the frequency of the laser. At each side peak the cavity is in resonance with one of the higher-order spatial modes, and the transmitted light will show the corresponding spatial patterns of this mode. In most practical situations we want to achieve the mode-matched case. That means we want to minimize the higher-order modes, as shown in Figure 5.12. By using lenses to change the wave front curvature and by optimizing the position of the lenses systematically, mode matching can be achieved. In the same way the alignment of the beam to the axis of the cavity is improved. This is largely an iterative process, since in most situations, there are six degrees of freedom that have to be optimized: two degrees for the tilt of the input beam in regard to the cavity, two degrees for a lateral shift, and two degrees for the wave front curvature. Measuring the size of the high-order peaks Pmn is far more sensitive than measuring the reﬂected power Preﬂ . Clever strategies have been devised to detect misalignment and mode matching separately. There have been several approaches how alignment can be automized [11–13], but the problem of automatic mode matching is still largely unresolved. 5.3.4.2

Polarization

Polarization was not included in the above discussion. For a completely isotropic cavity, the polarization does not matter; the description using a single, arbitrary polarization state of light would be correct. However, in most practical cases at least one of the components of the cavity, or inside the cavity, is anisotropic and induces some birefringence or dichroism. In that case one has to ﬁnd the polarization eigenstates of the cavity. Light in these states will keep its polarization whilst P00 Pmn

FSR

(a)

νcav

(b)

νcav

Figure 5.12 The frequency response of a scanning cavity that is longer than confocal to (a) a mode-matched input beam and (b) a mismatched beam. Improved alignment will get us from case (b) to the optimum case (a).

5.3 Optical Cavities

propagating. It will return after one complete round trip with the same polarization. Note that at any individual point inside the cavity, these eigenstates can be quite diﬀerent. There are always two such orthogonal states. Losses, detuning, and thus the ﬁnesse can be diﬀerent for the two states, and the response of the cavity is described by two independent sets of equations, one for each polarization eigenstate. The best-known example is the unidirectional device inside a ring laser that uses polarization and losses to suppress laser oscillation in one of the two counter-propagating directions. 5.3.4.3

Tunable Mirrors

In many practical situations it would be very desirable to be able to tune the reﬂectivity of a mirror in a cavity. This can be achieved by making one of the mirrors a cavity itself or by using a trick called frustrated internal reﬂection. Here the two uncoated surfaces are used with a small gap to form a mirror. The light is aligned such that each surface would totally reﬂect the light. However, when the surfaces have a separation of a wavelength or less, the evanescent ﬁeld, on the outside of the totally reﬂecting surfaces, reaches to the next surface, and light is transmitted. The magnitude of this eﬀect can be controlled by changing the separation, thus a mirror of tunable reﬂectivity can be constructed [3, 14]. It should be noted that such mirrors are polarization dependent. 5.3.5

Equations of Motion for the Cavity Mode

For the rest of our discussion of cavities, we will assume that the technical requirements for obtaining high ﬁnesse operation have been satisﬁed. Under such conditions the equations of motion of a single cavity mode can be obtained in a straightforward way. Conceptually it is easiest to consider the case of a ring cavity as depicted in Figure 5.13. The cavity has two partially transmitting mirrors with transmissions T1 and T2 and internal loss with a transmission of Tl . External ﬁelds 𝛼in1 and 𝛼in2 enter at the ﬁrst and second mirrors, respectively. We calculate the change in the circulating mode amplitude, 𝛼c , incurred in making one round trip of the cavity under the assumption that that change is small. We have √ √ √ 𝛼c (t + 𝜏) = 𝛼c (t)ei𝜙 1 − T1 1 − T2 1 − Tl √ √ √ + 𝛼in1 (t)ei𝜙 T1 1 − T2 + 𝛼in2 (t)ei𝜙 T2 (5.69) where 𝜏 is the cavity round-trip time and 𝜙 is the diﬀerential phase shift acquired by the ﬁelds in that time. If T1 , T2 , 𝜙 ≪ 1, then the change in the ﬁeld in one round trip will be small. As such we expand to ﬁrst order the LHS of Eq. (5.69) in a Taylor Figure 5.13 A cavity described in the quantum model.

Aout1

Ain2 acav

Ain1

κ1

κloss

Aout2

κ2

175

176

5 Basic Optical Components

series and make small T approximations on the RHS. We obtain ) ( ( √ √ ) 1 1 1 𝛼c (t) + 𝜏 𝛼̇ c (t) = 𝛼c (t) 1 − T1 − T2 − Tl + 𝛼in1 T1 + 𝛼in2 T2 ei𝜙 2 2 2 Making a small 𝜙 (modulo 2π) approximation, i.e. assuming we are close to a cavity resonance, and rearranging, we obtain ( ) √ √ i𝜙 − 12 T1 − 12 T2 − 12 Tl √ √ T1 T2 𝜏 𝛼̇ c = 𝜏𝛼c + 𝛼in1 + 𝛼in2 (5.70) 𝜏 𝜏 𝜏 √ Equation (5.70) is normally presented with the following substitutions: 𝛼 = 𝜏𝛼c is the amplitude of the standing mode of the cavity; Δ = 𝜙∕𝜏 is the detuning of the cavity; and 𝜅i = Ti ∕(2𝜏) are the decay rates of the cavity. With these substitutions we obtain the following equation of motion for the cavity mode: √ √ (5.71) 𝛼̇ = 𝛼(iΔ − 𝜅1 − 𝜅2 − 𝜅l ) + 𝛼in1 2𝜅1 + 𝛼in2 2𝜅2 This equation of motion will be the starting point for our quantum description. The ﬁeld exiting from the ﬁrst mirror is √ √ 𝛼out1 = T1 𝛼c − 1 − T1 𝛼in1 √ ≈ 2𝜅1 𝛼 − 𝛼in1 (5.72) and similarly the ﬁeld exiting from the second mirror is √ √ 𝛼out2 = T2 𝛼c − 1 − T2 𝛼in2 √ ≈ 2𝜅2 𝛼 − 𝛼in2

(5.73)

Equations (5.72) and (5.73) are referred to as the boundary conditions for the cavity. 5.3.6

The Quantum Equations of Motion for a Cavity

The fundamental object of the quantum model of light is the cavity mode. The cavity is the experimental equivalent to the abstract concept of a single frequency mode and its subsequent identiﬁcation with the quantum model of a harmonic oscillator. In general, a description of the quantum evolution of a cavity requires the speciﬁcation of the Hamiltonian describing the interactions within the cavity as well as those between the cavity mode and the continuum of ﬁeld modes outside the cavity. Equations of motion for the ﬁeld can be obtained either in the Heisenberg picture (operator Langevin equations) or in the Schrödinger picture (master equation) [15]. Combining this with quantum boundary conditions [16] allows calculations of the properties of ﬁelds leaving the cavity. It was techniques such as these that were used to rigorously establish the behaviour of quantum ﬁelds that we have described in this and the previous chapters. In later chapters we will use such approaches. However, here we proceed via the simpler approach of canonical quantization of the classical equations of motion (Eq. (5.71)). This leads directly to √ √ √ ̂ in1 + 2𝜅2 A ̂ in2 + 2𝜅l A ̂l (5.74) â̇ = −(𝜅 − iΔ)̂a + 2𝜅1 A

5.3 Optical Cavities

̂ j are travelling modes where â is the standing cavity mode, whilst the A and 𝜅 = 𝜅1 + 𝜅2 + 𝜅l is the total decay rate of the cavity. As for the beamsplitter and the interferometer, the essential diﬀerences between the quantum and classical equations are the transformation to operators and the explicit inclusion of an operator representing the vacuum mode introduced by loss. Equation (5.74) is an example of an operator quantum Langevin equation. The boundary condition for the transmitted ﬁeld √ ̂ in2 ̂ out2 = 2𝜅2 â − A (5.75) A and the reﬂected ﬁeld √ ̂ out1 = 2𝜅1 â − A ̂ in1 A

(5.76)

can also be obtained directly from the classical boundary conditions (see Eqs. (5.72) and (5.73)). 5.3.7

The Propagation of Fluctuations Through the Cavity

In order to analyse the eﬀect of a cavity on the noise of an incident propagating mode of light, we consider the transfer functions for the cavity. We use the model described in Chapter 4. We approach the problem step by step: 1. The ﬂuctuations of the input ﬁeld are described as the spectrum of the amplitude variance V 1Ain (Ω). 2. The cavity is described through the equation of motion. The resulting ﬂuctuations are analysed. We ﬁrst consider the amplitude quadrature of the ﬂuctuations. 3. The variances V 1out of the various output ﬁelds are determined. They are given by a linear combination, the transfer function, of the variances of all the input ﬁelds. The variances of the ﬂuctuations of the internal ﬁeld are of academic interest only. They cannot be measured by conventional means. We ﬁnd that, for a cavity on resonance, the output amplitude ﬂuctuations are linked directly to the input amplitude ﬂuctuations. For a cavity with detuning, both quadrature amplitudes of the input contribute to the output amplitude ﬂuctuations. The output noise spectrum can be written as a linear function of the variance of the input mode and of contributions from the various unused input ports, which is the major diﬀerence to the classical model. Whilst the unused ports do not contribute to the spectrum of the output power, they contribute to the noise spectra of the light. In a way, the eﬀect of the cavity on the ﬂuctuations becomes similar to that of an electronic ﬁlter. The response is frequency dependent; it is linear, and it can be described by a transfer function. We consider the system shown in Figure 5.13 and include the ﬂuctuations. As usual each operator is written as a classical complex value for the amplitude and ̂ in1 , etc. The classical ̂ in1 = 𝛼in1 + 𝜹A ̂ A an operator for the ﬂuctuations: â = 𝛼 + 𝜹a, parts obey Eqs. (5.71)–(5.73), whilst the operator parts obey Eqs. (5.74)–(5.76). Considering just the quantum ﬂuctuations, we ﬁnd √ √ √ ̂̇ = −𝜅 𝜹a ̂ + 2𝜅 𝜹A ̂ + 2𝜅 𝜹A ̂ ̂ + 2𝜅 𝜹A (5.77) 𝜹a 1

in1

2

in2

l

l

177

178

5 Basic Optical Components

By adding the complex conjugate, an equation for the amplitude quadrature can be formed: √ √ √ ̂̇ = −𝜅 𝜹X𝟏 ̂ + 2𝜅1 𝜹X𝟏 ̂ in1 + 2𝜅2 𝜹X𝟏 ̂ in2 + 2𝜅l 𝜹X𝟏 ̂ l 𝜹X𝟏 (5.78) The boundary conditions of Eqs. (5.75) and (5.76) are rewritten for the ﬂuctuations √ ̂ out2 = 2𝜅2 𝜹X𝟏 ̂ − 𝜹X𝟏 ̂ in2 𝜹X𝟏 √ ̂ out1 = 2𝜅1 𝜹X𝟏 ̂ − 𝜹X𝟏 ̂ in1 (5.79) 𝜹X𝟏 The next step is to consider the Fourier components of the ﬂuctuation. Using the property of Fourier transforms (df (t)∕dt) = −i2πΩ (f (t)), we can combine Eq. (5.78) with Eq. (5.79) and obtain √ √ ̃ in1 + (2𝜅2 − 𝜅 + i2πΩ) 𝜹X𝟏 ̃ l ̃ in2 + 4𝜅1 𝜅l 𝜹X𝟏 4𝜅1 𝜅2 𝜹X𝟏 ̃ out2 = 𝜹X𝟏 𝜅 − i(2πΩ) (5.80) In order to obtain the spectrum of the ﬂuctuations transmitted through the cavity, we determine the variance. This is the desired set of transfer functions. It shows how the diﬀerent input variances are linked and how the ﬂuctuations propagate through the system. We can repeat the above calculations and obtain for the reﬂected beam V 1out1 (Ω) = {((2𝜅1 − 𝜅)2 + (2πΩ)2 ) V 1in1 (Ω) + 4𝜅2 𝜅1 V 1in2 (Ω) + 4𝜅1 𝜅l V 1l (Ω)} 1 × 2 (5.81) 𝜅 + (2πΩ)2 and for the transmitted beam 4𝜅 𝜅 V 1in1 + ((2𝜅2 − 𝜅)2 + (2πΩ)2 ) V 1in2 (Ω) + 4𝜅2 𝜅l V 1l (Ω) V 1out2 (Ω) = 2 1 𝜅 2 + (2πΩ)2 (5.82) The ﬁrst useful check is to consider the special case where all inputs are QNL, which means they are not either used or illuminated by a coherent state. We ﬁnd V 1out2 (Ω) = V 1out1 (Ω) = 1 As expected, the two output noise spectra for this special case are also QNL. This is an important result that can be applied to all passive systems no matter how complicated the instrument is: if all the input beams are at the QNL, then also all the output states are at the QNL. But note that even when all output variances have the variances V 1out = 1, the origin of this noise can still vary with the frequency since the components of the transfer function itself are frequency dependent. This becomes important when at least one of the input beams has ﬂuctuations above or below the QNL, and either noise or squeezing is propagating through the cavity. In general, the transfer function is frequency dependent.

5.3 Optical Cavities

Consider the special case of a perfect ring cavity, with 𝜅1 = 𝜅2 = 𝜅∕2. Then the set of transfer functions will become 𝜅 2 V 1in1 + (2πΩ)2 V 1in2 𝜅 2 + (2πΩ)2 2 (2πΩ) V 1in1 + 𝜅 2 V 1in2 V 1out1 (Ω) = 𝜅 2 + (2πΩ)2 V 1out2 (Ω) =

We can now examine the low and high frequency performances of the cavity, which are distinctly diﬀerent. For low frequencies, we see that in the limit Ω → 0 the spectra approach V 1out2 (0) = V 1in1 V 1out1 (0) = V 1in2 In other words, the input noise passes directly through to the output, and the noise of the unused port, normally vacuum noise, passes straight through into the reﬂected beam. The cavity is transparent to noise in either direction. Interestingly, at high frequencies, we see that in the limit of Ω → ∞, these spectra approach V 1out2 (∞) = V 1in2

(5.83)

V 1out1 (∞) = V 1in1

(5.84)

This is the opposite situation to that above. The cavity is completely opaque to noise at high frequencies. The input noise is reﬂected into the reﬂected ﬁeld, and the noise of the unused port is reﬂected into the output ﬁeld. This result is true for all cavities, as described by Eqs. (5.82) and (5.81). For high frequencies ((2πΩ)2 ≫ 𝜅 2 ), the noise from the input source will not be able to propagate through the cavity. Only low frequency noise, within the linewidth of the cavity, is transmitted. The cavity acts like an integrator and smoothes out fast ﬂuctuations. However, these ﬂuctuations are not lost; they appear in the reﬂected light. For low frequencies, (2πΩ)2 ≪ 𝜅 2 , the noise is transmitted. For frequencies around (2πΩ) = 𝜅, there is a smooth transition. This response is shown in Figure 5.14 for the two cases of a noisy input (V 1in1 = 2) and a squeezed input (V 1in1 = 0.1). 2 Vrefl = Vout2

Vin = 2

1.5 Vtrans = Vout1 V

Figure 5.14 Noise response of a cavity as a function of the detection frequency. Both the transmitted and the reﬂected noise are shown. Two cases: (i) a cavity with noisy input (V1in1 = 2) and (ii) a cavity with an input noise below the QNL (V1in1 = 0.1).

1 Vtrans = Vout1 Vin = 0.1

0.5 Vrefl = Vout2

0 0

2

4

Ω/∆ν

6

8

179

180

5 Basic Optical Components

In practice, it means that for a laser beam with modulation at Ωmod incident on a cavity, the transmission function for the laser carrier frequency 𝜈L and for the modulation sidebands is not the same (compare Eqs. (5.61) and (5.82)). In particular, for high frequencies the modulation sidebands cannot be transmitted; they are reﬂected. A cavity can be used as a ﬁlter for noise. The cavity is placed after the output coupler of the laser, all high frequency classical noise is rejected, and the output beam is QNL at these frequencies. Please note that all of the above equations have treated the resonant case. The transfer functions for the phase quadrature, V 2, or indeed any arbitrary quadrature angle, are equivalent to those we have calculated for the amplitude quadrature, V 1, for this case. It is straightforward to include detunings from cavity resonance, but this generates signiﬁcantly more complex equations. The main diﬀerence is that in the detuned case both quadrature amplitudes, X1 and X2, have to be taken into account. A detuned cavity couples the ﬂuctuations of the two quadratures – it links amplitude and phase ﬂuctuations. This eﬀect can be exploited for very sensitive measurements of phase ﬂuctuations, for cavity locking (see Section 8.4), and for the rotation of the squeezing axis (see Section 9.3).

5.3.8

Single Photons Through a Cavity

We now look at the eﬀect of the cavity on a single-photon ﬁeld. As for previous elements we will use the techniques of Section 4.4 to generate the state evolution. However, in contrast to the beamsplitter and interferometer, here we need to explicitly work in the frequency picture. Let us consider the case of a resonant, lossless, two-sided cavity with the input and output mirrors of equal reﬂectivity. Using Eqs. (5.74), (5.75), and (5.76) with 𝜅1 = 𝜅2 = 𝜅∕2 and 𝜅l , Δ = 0, we obtain the following operator evolutions through the cavity in frequency space: ̃ ̃ ̃ out2 = 𝜅 Ain1 + i𝜔 Ain2 A 𝜅 − i𝜔

(5.85)

̃ ̃ ̃ out1 = 𝜅 Ain2 + i𝜔 Ain1 A 𝜅 − i𝜔

(5.86)

and

where 𝜔 = 𝜔o − 𝜔cav is the diﬀerence between the optical frequency of the mode and the cavity resonant frequency in radians per second. Let us consider a single-photon input at the ﬁrst input port and vacuum at the other. We can write the input state as ∞

|1⟩f |0⟩ =

∫−∞

̃ † |0⟩in1 |0⟩in2 d𝜔f (𝜔)A in1,𝜔

̃ † is the photon creation operator at frequency 𝜔 in the input mode to where A in1,𝜔 ̃ † |0⟩in1 = |1⟩in1,𝜔 . Also |f (𝜔)|2 d𝜔 is the probability to the ﬁrst port, such that A in1,𝜔 ﬁnd the photon in the small frequency interval d𝜔 around 𝜔.

5.3 Optical Cavities

Using the inverted forms of Eqs. (5.85) and (5.86), we can evolve the state through the cavity giving ̃ U(𝜔)|1⟩ f |0⟩ =

∞

∫−∞

d𝜔f (𝜔)

̃† ̃ † + i𝜔 A 𝜅A out2 out1 𝜅 + i𝜔

|0⟩out1 |0⟩out2

(5.87)

As expected, the eﬀect of the cavity in frequency space is to act like a frequency-dependent beamsplitter, transmitting the low frequency components of the single-photon state but reﬂecting the high frequency components. For example suppose the input photon has a very narrow frequency range centred at 𝜔 = 0. We can write approximately f (𝜔) = 𝛿(𝜔), and so the input state is just |1⟩in1,0 |0⟩in2 , whilst the output state is |0⟩out1 |1⟩out2,0 , that is, the photon is deﬁnitely transmitted through the cavity. On the other hand if the photon has a narrow frequency range centred on some frequency 𝜔 ≫ 𝜅, then the output state will be |1⟩out1,𝜔 |0⟩out2 , that is, the photon will be completely reﬂected by the cavity. For an input state with a very broad frequency distribution, which we can approximate by supposing f (𝜔) = K, where K is a constant, the output state will be in a superposition of the two output modes: ̃ U(𝜔)|1⟩ f |0⟩ = K

∞

∫−∞

d𝜔

̃ † + i𝜔 A ̃† 𝜅A out2 out1 𝜅 + i𝜔

|0⟩out1 |0⟩out2

(5.88)

If we consider the reﬂected ﬁeld to be ‘lost’, then the transmitted ﬁeld will be a mixed state of the vacuum and a single photon state with the Lorentzian frequency distribution 𝜅2 (5.89) + 𝜔2 In this way cavities select speciﬁc components of the radiation ﬁeld, even in the limit of single photons. 𝜅2

5.3.9

Multimode Cavities

In our quantum treatment, so far we have focused on a single high ﬁnesse cavity mode. This occurred because we made several approximations that assumed high reﬂectivity mirrors and low round-trip loss and that we were close to a single cavity resonance, to obtain Eq. (5.71), before applying canonical quantization to obtain Eq. (5.74). The advantage of doing this was it gave a simple equation of motion, useful in many situations, which corresponds to the quantum Langevin equation one obtains from a quantum Hamiltonian approach. However, it is also possible to simply quantize the exact expression (Eq. (5.69)) in order to obtain the more general quantum expression √ √ √ ̂ c (t + 𝜏) = A ̂ c (t)ei𝜙 1 − T1 1 − T2 1 − Tl A √ √ √ ̂ in1 (t)ei𝜙 T1 1 − T2 + A ̂ in2 (t)ei𝜙 T2 +A (5.90) In principle this expression can be used to describe multiple resonances of a cavity of arbitrary ﬁnesse. Here we will still consider a high ﬁnesse cavity but

181

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retain the multimode feature. Taking T1 , T2 ≪ 1 and using the deﬁnitions in Section 5.3.5, we obtain √ √ √ ̂ in1 (t) + 2𝜅2 A ̂ in2 (t) + 2𝜅l A ̂ l (t)) â (t + 𝜏) = ei𝜙 (𝜅 â (t) + 2𝜅1 A (5.91) i𝜏𝜔 ̃ Using the Fourier transform relationship â (t + 𝜏) = a(𝜔)e and the input–output relations (Eqs. (5.75) and (5.76)), we can solve for the output ﬁeld in Fourier space to obtain √ √ ̃ in2 + 4𝜅1 𝜅l A ̃ in1 + (2𝜅2 − 𝜅 + Γ(𝜔)) A ̃l 4𝜅1 𝜅2 A ̃ out2 = (5.92) A 𝜅 − Γ(𝜔)

where 1 − ei(𝜔−Δ)𝜏 (5.93) 𝜏 This now describes a multimode cavity in which cavity resonances explicitly occur whenever (𝜔 − Δ)𝜏 = 2πn (with n as an integer). Γ(𝜔) =

5.3.10

Engineering Beamsplitters, Interferometers, and Resonators

Whilst in Section 5.1 we focussed on one speciﬁc type of beamsplitter, the well-known device made of a plain piece of glass with speciﬁcally prepared surface, many other devices have the same eﬀect. Some of them transfer the power into one or several speciﬁc other modes; this could, for example, be done by a grating that reﬂects into diﬀerent orders. Or we can use surface or evanescent coupling when two modes approach each other by bringing two surfaces close to each other, or we can achieve this inside an optical ﬁbre coupler. We can set up periodic nanostructured materials that couple diﬀerent propagating modes. The list includes various forms of active modulators that produce gratings or other spatial modes that couple into other spatial modes and those that create diﬀerent frequency components. We can combine several beamsplitters into interferometers and cavities and resonators. This creates many technological options for coherent linear devices. This is illustrated in Figure 5.15, which shows the diverse range of optical resonators that are used today. All are using diﬀerent approaches and techniques, ranging from simple optical cavities made out of two mirrors to multi-mirror ring cavities, ﬁbre couplers, and ﬁbre rings, evanescent waves in discs, toroids and spheres, Bragg reﬂectors, planar optical devices, and photonics crystals. The ﬁrst generation of devices shown in the top row of Figure 5.15 was designed as resonators for lasers. One of the main design criteria is to make the resonator stable, which means a large build-up after many round trips is possible and minor mechanical perturbations have only a small eﬀect. A second design goal is to minimize losses such that the components can handle high internal ﬁelds. All of these resonators have several spatial eigenmodes, which are the diﬀerent stable solutions for the resonance. For example a long linear resonator can support the family of Hermite–Gaussian spatial modes, shown in Chapter 2 in Figure 2.2, and these modes can be generated in a laser. For some of these conﬁgurations, in particular the case of the confocal resonator, the spatial modes

Planar mirror

Spherical mirror

Microtoroid

Figure 5.15 Various optical resonator designs.

DBR

Microdisk

Microsphere

Rectangular cavity

Integrated optic ring

DBR

Fibre ring

Ring mirror

Micropillar

Photonic crystal

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exist at the same resonance frequency; they are degenerate and will appear at the same time. However, for most geometries the spatial modes have diﬀerent optical resonance frequencies, the modes are non-degenerate, and the laser either operates on several modes with diﬀerent frequencies or in one spatial mode at a time. There is a considerable knowledge about the engineering of such resonators to make them insensitive against mechanical vibration and also to make their resonance frequencies tunable. A good starting point for the design techniques can be found in the widely used book by A.E. Siegman [9]. The lower part of the diagram shows a variety of resonators that have emerged in the last few years and have been optimized to create resonators that are stable, have low losses, and more recently are designed to be very small and mass produced as building blocks for complex systems. Each technology has its own design limits, but in general they are improving quite dramatically to achieve almost perfect resonators. The ever-increasing quality of these devices is well described in a series of review articles and textbooks [17].

5.4 Other Optical Components 5.4.1

Lenses

We require optical components to change the spatial properties of a laser beam or mode of light. Actually, in most of optics imaging is the main concern. The ability of lenses and mirrors to form and reproduce images is the reason for using these components. The eﬀect of lenses is a change of the wave front curvature and consequently a transformation from one Gaussian input beam to another Gaussian beam with diﬀerent waist position and size. In practice, lenses are used to transform the spatial mode coming from the laser into the mode required in the experiment, to match it to any cavity that exists in the experiment, and ﬁnally, to focus the mode onto the detector (Figure 5.16). The quality of a lens or mirror is determined by the shape of the lens, its surface ﬁnish, and the uniformity of the material. Most imaging lenses are multi-element lenses, and the details of the lens design are of crucial importance to the quality of the image, the spatial resolution that can be achieved. For practical details consult one of the handbooks on optical instruments [18]. For most cases the paraxial transformation will hold, and we can use the results discussed in Section 2.1. It

Lens Gaussian beam

Figure 5.16 Spatial mode transformation by a lens.

Gaussian beam

5.4 Other Optical Components

assumes that the input is in a single spatial mode and that the lens is well represented by a parabolic phase term. Consequently, the output will be in a single spatial mode. Imperfections of the lens will result in deviations from a Gaussian beam, and aspherical aberrations are fairly common. In this case the light has no longer a parabolic wave front; it is distributed over a variety of spatial modes. This will make any interference experiment, including homodyne detection, very diﬃcult. It will reduce the visibility of the interference fringes and therefore make it an ineﬃcient process. The loss of light due to reﬂection on the lens surfaces is a major experimental problem. As an example, for glass lenses, n = 1.5, the intensity reﬂection per surface is typically 4% for every surface. This can quickly add up in a whole experiment. Thus most surfaces have an anti-reﬂection coating that can reduce the reﬂection to less than 0.5%. Scattering losses inside the components are usually even smaller than 0.1% and will only be of concern if the optical component is placed inside a cavity or is extremely sensitive to scattered light. Whilst losses are one concern in quantum experiments, the other is the need to maintain one single mode. Assume we have a mode of light with an interesting intensity noise statistics that is sub-Poissonian. It passes a lens with distortions and turns into a non-Gaussian mode. If we detect all of this light, we still ﬁnd a sub-Poissonian result. If, however, the detector is sensitive to only the Gaussian mode, because it contains a cavity or it is a homodyne detector that requires the interference and beating with an optical local oscillator, the results will be diﬀerent. If the partition between the modes is random, then the other modes will appear like a power loss, and the statistics will be closer to Poissonian. If the partition is somehow deterministic, then additional noise could be introduced. Fortunately all passive components, such as mirrors and lenses, result in a random partition. The other modes can be treated as loss terms. In contrast, active systems such as lasers are diﬀerent and can introduce additional partition noise, as shown in Chapter 6. A problem that exists with high intensity experiments, and in particular pulsed experiments, is thermal lensing. Glasses will absorb some small fraction (typically 0.1 [%/cm]) of the light, and the absorbed power can lead to local heating of the glass that results in local changes of the refractive index. At such high intensities, light becomes non-linear. In this case self-focussing can occur since the intensity-dependent refractive index eﬀectively forms the proﬁle equivalent to a lens, formed by the beam itself. Self-focussing leads to high higher intensities at the core of the beam increasing the eﬀect event more. This results in positive feedback, which can have disastrous eﬀects. Many lenses have been destroyed with holes and channels burned into the glass. With clever design, and the right materials and pulse length, this can be used to create a localized bubble in the glass, which can be seen from the outside and can serve as on voxel for a 3D structure, very popular with artists and tourists. 5.4.2

Holograms and Metasurfaces

The concept of using structured surfaces to alter the spatial wave front and to manipulate the shape of beams of light has a very long tradition. Fresnel lenses,

185

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5 Basic Optical Components

which use circular phase shifts or even absorption, were a very early invention that could take on the task handled by lenses. Early versions, such as those used in light houses, had structures that were much larger than the wavelength of light, frequently several millimetres, and can be understood in the same way as lenses. Later it was possible to make small structures using photographic processes, and the understanding of coherent light with known phase distributions allowed the concept of holograms by D. Gabor [19] even lasers were available and in many versions since. Holograms can be used to record and play back complex waveforms that result, for example, in the reconstruction of 3D images. Fourier optics, which means the spatial frequencies of the light ﬁeld, created a whole new ﬁeld of understanding imaging systems, designing instruments, and shaping light. Holograms have been reﬁned, and it is not possible to create computergenerated hologram using the high-resolution spatial light modulators (SLM) available today. That means we can now design optical components that can be manipulated from the outside. This allows the shaping of light for many purposes, such as displays, microscopy, optical traps, and optical tweezers for atoms and particles. Practical examples are devices for the investigation of biological processes or the functional imaging of brain cells. The external control also allows the use of the modulators, and phase structures inside feedback loops, which allows for intelligent alignment and imaging processes. Recent development in the design and manufacture of structured surfaces allow the direct control of the wave fronts by very thin components. New nanoscale building techniques can create the required three dimensional structures and have resulted in a completely new class of materials and of passive and active components for the manipulation of light. The properties of metamaterials were explored in 1990s, ﬁrst as a curiosity, and initially tested at longer wavelengths, such as microwaves. Now we have optically thin structured materials made from sub-wavelength nanostructures, sometimes referred to as meta-atoms, which allow us to inﬂuence the properties of the light interacting with them. Metasurfaces enable the manipulation of the wave front at the surface. They can be used as optical elements for beam shaping and holography. At the same time, metasurfaces enable the control of light in the intermediate and near-ﬁeld zones. By tuning the phase properties of waves in the intermediate regime, also known as the Fresnel zone, it is possible, for example, to focus beams or engineer-desired point spread functions. At shorter distances, and straight after the surface where the behaviour of light is governed by its near-ﬁeld characteristics, carefully crafted components allow the concentration of the local intensity as required for the non-linear generation of new frequencies, surface-enhanced Raman scattering, or enhancement of the interaction with atoms via the Purcell eﬀect. For an overview of the design and use of metasurfaces, see, for example, S. Keren-Zu et al. [20]. As long as the metasurfaces, and thicker strictures made of metamaterials, behave in a linear way, they can be seen as the equivalent of the bulkier conventional optical components, such as lenses, interferometers, and optical cavities. They will not inﬂuence the photon statistics or add noise. Increasingly these will be the technique of choice for designing optical instruments, for focussing, and for the generation of multiple spatial modes.

5.4 Other Optical Components

5.4.3

Crystals and Polarizers

Glass and crystals vary in two very important points: crystals are non-isotropic, and they can show an electro-optics response and they can show non-linear eﬀects. The anisotropy of the crystals results in properties such as birefringence, optical activity, and dichroism. The details obviously vary with diﬀerent types of crystals and their composition and geometrical structure. It also varies with the alignment of the light beam with the crystal axis. All these properties can be used to control the polarization properties of the light. The idea here is to introduce a relative phase change between two diﬀerent polarization components of the ﬁeld. For example, introduce a π∕2 phase change of one linear polarization and not the other, and thereby turn linearly polarized light, aligned with the axis of the crystal, into circularly polarized light. This would be called a quarter wave plate. The results of any such component can be analysed using the rules of classical optics, in particular the formalism of Jones matrixes [3]. Birefringent crystals can be manufactured into wave plates and polarizers. In this regard quantum optics experiments are no diﬀerent to classical optics experiments. All tricks available to control and inﬂuence the polarization will be used in the various experiments. One particularly useful component is the optical isolator. This device is equivalent to a diode in electronics; it provides a diﬀerent eﬀect in the forward and backward direction. The light propagating in the reverse direction is redirected and does not go back to the source. This avoids interference between the incoming beam and the retro-reﬂected light. This eﬀect is described as optical feedback that can lead to very noisy or even unstable performance of a laser. At the same time an optical isolator will give access to the retro-reﬂected beam, which is important in experiments where all of the light has to be detected. It is also useful for situations where locking of a cavity to a laser is important. In this case the redirected light contains the information about the cavity detuning. Two types of optical isolators are commonly used: the Faraday isolator and wave plates (see Figure 5.17). The Faraday isolator uses a material, usually a special glass or crystal, with a high Verdet constant. This means a large rotation of the plane of polarization for linearly polarized light can be induced by a static magnetic ﬁeld along

Laser

PBS

B Mirror Faraday isolator

(a)

Laser

PBS

4 Mirror Quarter wave plate

(b)

Figure 5.17 Optical isolators using either a Faraday rotator (a) or a quarter wave plate (b).

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the crystal. The isolator is tuned to the wavelength used in the particular experiment by adjusting the magnetic ﬁeld strength to give a 45∘ rotation. The return beam is rotated again by 45∘ in the same direction, which means both beams are linearly polarized at 90∘ to each other and can be separated by a PBS. The second design uses wave plates (see Figure 5.17b). The source generates linearly polarized light. A quarter wave plate is used to turn this into left-hand circularly polarized light (𝜎+ light). Upon reﬂection in the experiment, the returning light is right-hand circularly polarized (𝜎− light), and the wave plate turns this into a counter-propagating beam with orthogonal linear polarization. This light can be separated with a PBS. Crystals can also contribute entirely new eﬀects. They can have a non-linear response that allows us to generate new frequencies, such as twice the frequency (SHG), half the frequency (optical parametric gain), or sums and diﬀerent frequencies (four-wave mixing (4WM)). Crystals can show a Kerr eﬀect, which means an intensity-dependent refractive index, or non-linear absorption. These eﬀects will inﬂuence not only the intensities but also the ﬂuctuations of the light, and consequently they are discussed in Chapter 9 on squeezing experiments. 5.4.4

Optical Fibres and Waveguides

Optical ﬁbres are a convenient component to deliver light from one place to another. The idea is simple: the ﬁbres have a refractive index structure that guides the light. For suﬃciently small core sizes, a few times the optical wavelength, the ﬁeld inside the ﬁbre has one single mode with low loss, and all other modes are highly attenuated over length. In addition, larger diameter ﬁbres show many modes. The mode properties of ﬁbres are complex and are discussed in great detail in the specialized literature [21]. It is possible to design a lens that will couple very eﬃciently a free-space Gaussian laser beam into a single-mode ﬁbre. This requires a lens of short focal length, such as a microscope objective, which has a numerical aperture matched to the ﬁbre geometry. When the beam parameters are correctly chosen, at least 80% of the power can be coupled into the ﬁbre. The losses along the ﬁbre itself are so exceedingly small that the ﬁbre length is normally of no consequence. Fibres can be spliced together with low losses, and eﬃciencies of 95% are feasible. In this way whole ﬁbre networks, including couplers, can be built. In recent years it has also been possible to create other optical waveguides, frequently through direct writing optical circuits inside blocks of glass and other transmitting materials. Materials that are structured on nanometre scale have been used as so-called photonics crystals that are used as waveguides and optical circuits. All of this has extended the options for guiding and distributing light tremendously. However one should note that in communication applications an overall loss of 3 dB or 50% is fully acceptable. This is also the case for some single-photon experiments, where the overall quantum eﬃciency may not be critical. However, in CW applications these devices are not yet suitable since the high losses reduce the precious quantum properties. It should be noted that simple symmetric ﬁbres completely scramble the polarization of the light. Any polarization dependence of the optics after the ﬁbre

5.4 Other Optical Components

can lead to enormous intensity ﬂuctuations introduced by the ﬁbre. There are special non-symmetric polarization-preserving ﬁbres, which have to be carefully aligned. And even these are very sensitive to mechanical movements of the ﬁbre. The output mode shape from a single-mode ﬁbre is very close to a Gaussian beam. As a result ﬁbres can be used as spatial mode cleaners, and they will transform an input beam with a complex wave front into a very nice Gaussian beam. The light that is not matched will be lost by scattering. The higher losses of a ﬁbre mode cleaner are frequently compensated by its simplicity and the fact that, unlike a mode cleaner cavity, it requires no active locking. Modern technology now allows the writing of waveguides inside solid materials, and the advantages of ﬁbres can be extended to more complex structures that can contain splitters and combiners and other components all integrated into one chip. One of the main attractions is the high internal intensity inside the ﬁbre or waveguide, which make it very attractive for non-linear interactions. Some of the ﬁrst and some of the best squeezing experiments have been carried out in ﬁbres (see Section 9.4.4). Apart from producing quantum eﬀects, the non-linearities in waveguides can be exploited for future active control and the operation of quantum logic operations, all ﬁrmly integrated into solid components [22].

5.4.5

Modulators

Modulators are an essential component of most electro-optical systems. They allow us to encode information onto laser beams. The most common application is to modulate a single beam, which means one single optical mode, without noticeably changing the mode but simply adding frequency sidebands that contain the information. The classical description of modulation (see Section 2.1.7) can be transferred into the quantum mechanical description (see Section 4.5.5) by introducing a time-dependent component to the model of the beamsplitter. The properties of the classical transfer equation remain the same. However, a quantum mechanical vacuum operator appears that represents the coupling of vacuum ﬂuctuations back into the beam. This term is responsible for the special quantum noise properties that are so important in any system operating close to the QNL. For example an intensity modulator is essentially a time-dependent beamsplitter, where an external modulation function controls the strength of the beam splitter. In general a time-dependent beamsplitter that mixes two input modes ̂ transforms them such that ̂ and b(t) represented by the operators a(t) √ √ ̂ â o (t) = 𝜂(t)̂a(t) + 1 − 𝜂(t)b(t) √ √ ̂ − 1 − 𝜂(t)̂a(t) (5.94) b̂ o (t) = 𝜂(t)b(t) where 𝜂(t) is the time-dependent transmitivity of the beamsplitter. Transforming into the frequency domain, we obtain ̃ ̃ o (Ω) = 𝜈(Ω) ̃ ̃ ∗ A(Ω) + 𝜇(Ω) ̃ ∗ B(Ω) A ̃ ̃ ̃ ∗ B(Ω) − 𝜇(Ω) ̃ ∗ A(Ω) B̃ o (Ω) = 𝜈(Ω)

(5.95)

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5 Basic Optical Components

√ where 𝜈(Ω) ̃ is the Fourier transform of 𝜂(t) and 𝜇(Ω) ̃ is the Fourier transform √ of 1 − 𝜂(t). The ∗ symbol indicates a convolution, that is, M(Ω) ∗ L(Ω) =

∫

dΩ′ M(Ω′ )L(Ω − Ω′ )

(5.96)

A good model for the modulator is a beamsplitter that takes oﬀ a small fraction of the light 𝜖o and then adds an even smaller modulation term with amplitude 𝜖(t). Hence we can write 𝜂(t) = 1 − 𝜖o + 𝜖(t) cos(2πΩmod t)

(5.97)

where Ωmod is the modulation frequency and 𝜖o > 𝜖(t). If we assume the modulation amplitude is very small such that 1 − 𝜖o > 𝜖o ≫ 𝜖(t), then we can make the approximations √ √ 𝜖(t) 𝜂 ≈ 1 − 𝜖o + √ cos(2πΩmod t) 2 1 − 𝜖o √ √ 𝜖(t) 1 − 𝜂 ≈ 𝜖o − √ cos(2πΩmod t) (5.98) 2 𝜖o and hence we can write √ 1 𝜖(Ω) ̃ ∗ (𝛿(Ω + Ωmod ) + 𝛿(Ω − Ωmod )) 𝜈(Ω) ̃ = 1 − 𝜖o 𝛿(Ω) + √ 4 1 − 𝜖o √ 1 𝜇(Ω) ̃ = 𝜖o 𝛿(Ω) + √ 𝜖(Ω) ̃ ∗ (𝛿(Ω + Ωmod ) + 𝛿(Ω − Ωmod )) (5.99) 4 𝜖o where 𝜖(Ω) ̃ is the Fourier transform of 𝜖(t). Let us assume that the modulation is turned on at some time −T∕2 and turned oﬀ at T∕2 and has the value 𝜖 when on. Hence 𝜖(Ω) ̃ = 𝜖T sinc(πΩT) where sinc(x) = sin(x)∕x. Further we assume that ̃ ̃ the {b(t) mode is initially in the vacuum state and write B(Ω) ≡ V(Ω). As a result, in the expectation values, non-zero modulated terms coming from the vacuum input will only appear to second order or higher in 𝜖 and so can be neglected. Thus, inserting the convolutions, we obtain the following expression for the output mode from the intensity modulator: √ √ ̃ o (Ω) = 1 − 𝜖o A(Ω) ̃ ̃ A + 𝜖o V(Ω) 𝜖T ̃ ′ + Ωmod ) + A(Ω ̃ ′ − Ωmod ))sinc(π(Ω − Ω′ )T) dΩ′ (A(Ω + √ 4 1 − 𝜖o ∫ (5.100) In the limit that the modulation is on for a long time (T ≫ Δtmod ), the sinc function approaches a 𝛿 function, and the sidebands become very sharp (as assumed in Section 4.5.5). In contrast, if the modulation is only on for a short time, as a burst, the sidebands are spread out over a range of frequencies around the sideband frequencies. Similarly the information about a slowly varying time evolution of the modulation will be contained in the spectral components close to the pair of sidebands.

5.4 Other Optical Components

5.4.5.1

Phase and Amplitude Modulators

1

1

FM

n

atio

cal

ti Ver (a)

riz pola

Fast axis

Modulators normally operate across a frequency bandwidth: some are broadband, and others are more resonant. By feeding a signal at a frequency Ωmod into the modulator, we create the speciﬁc sidebands, which contain the information we want to impose. Within the bandwidth complete modulation spectra can be used, for example, in communication systems. The modulation can be either a phase or an amplitude modulation, or even a combination of both, depending on the design of the modulator. An intensity modulator controls only the magnitude of the transmitted beam and should leave all other properties unchanged. A phase modulator introduces a time-dependent phase shift and should leave the overall power constant. However, we should note that many instruments, for example, interferometers, convert phase modulation into amplitude modulation, and this can make it diﬃcult to build a pure phase modulator. Reducing crosstalk between amplitude and phase modulation is a technical challenge (Figure 5.18). Many crystals have signiﬁcant electro-optic coeﬃcients. That means the refractive index can be manipulated with an external electric ﬁeld. The direction of the electric ﬁeld that aﬀects the refractive index along a particular optical axis depends on the type of crystal used. In some cases the transverse eﬀect is large, which means the electric ﬁeld is applied orthogonal to the optical beam. In other crystals a longitudinal ﬁeld, parallel to the optical beam, is required. The refractive index will follow the ﬁeld up to very high frequencies; more than 20 GHz are possible. Thus these devices can be used as EOM. Large changes in refractive index will require electric ﬁelds of the order of 100–1000 V/cm. The limit in speed is normally set by the electric power requirements. Fast response and large degrees of modulation can be achieved by using resonant electronic circuits that provide a large electric ﬁeld in a very narrow frequency range. This eﬀect can be used to make a phase modulator. Either a crystal with isotropic response is chosen, or the axis of linear polarization is carefully aligned

ted dula mo ization s t ron polar se f n Pha ange i PBS ch o N

n

atio

ariz

du mo

m n A latio u d mo

tion

riza

ola ar p

l

u Circ (b)

n

latio

de

Pol

axis

1

plitu

1

AM

Slow

Figure 5.18 Electro-optical modulators for phase or amplitude.

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5 Basic Optical Components

parallel to the crystal axis. For a single modulation frequency Ωmod , the output light will get one set of sidebands, Ωmod and its higher harmonics, as described in Eq. (2.21). In the ideal case phase-modulated light does not produce any modulation of the photocurrent. In practice, considerable care has to be taken to avoid amplitude modulation. FM sidebands are frequently used to probe the detuning of cavities and to lock lasers to cavities. Alternatively, a birefringent crystal can be used and aligned such that the linear input polarization is at 45∘ to the optical axis of the crystal. In this case the two orthogonal components of the linear polarization are aﬀected diﬀerently. The voltage induces a diﬀerence in the change of the refractive index, and the modulator is acting like the AC equivalent of a wave plate. The polarization is rotated synchronously with the applied electric ﬁeld. After transmission through a polarizer, the light has a mixture of amplitude modulation, from the polarization rotation, and phase modulation. This is a common conﬁguration for intensity modulation. Normally a small reduction in intensity is chosen, and the intensity can be increased and decreased around this oﬀset point, as per the example in Eq. (5.97). An entirely diﬀerent eﬀect is used in acousto-optical modulators (AOM). These utilize the fact that an acoustic standing wave can be generated inside crystals through an RF electric ﬁeld. Light transmitted through such a standing wave sees a periodic density grating and is diﬀracted. The diﬀracted beam is shifted in frequency by ΩRF and is also shifted in its output direction, proportional to ΩRF (Figure 5.19). Since the diﬀraction angle is linked to the modulation frequency, we can redirect a beam by simply changing the frequency ΩRF of the RF driving voltage, which is typically in the range of 10–1000 MHz as a function of time. We can change the intensity of the diﬀracted beam by changing the RF power. Since both RF and power can be controlled rapidly, the output beam is thus scanned in one dimension. One common application of an AOM is as fast scanners, such as laser printers and displays. The AOM can be used as an intensity modulator by keeping the RF ﬁxed and modulating the RF intensity at frequencies Ωmod ≪ ΩRF . In quantum optics experiments the main application is the shift in optical frequency. For an ideal AOM the diﬀracted beam has the same mode properties as the input beam, apart from a change in frequency and direction. All the ﬂuctuations of the light are simply transmitted, and when carefully aligned there should be no additional noise. There are many applications, such as tuning to Crystal with internal acoustic standing wave vL

Figure 5.19 Acousto-optical modulator. vL + Ωmod vL

~

Ωmod

Modulation frequency

5.4 Other Optical Components

an atomic or cavity resonance or the probing of atoms with a frequency oﬀset. AO modulators can also be used as isolators. The return beam from the experiment, for example, a cavity, will have twice the frequency shift 2ΩRF imposed by the modulator and is less likely to interfere with the laser. This retroﬂection trick can also be used to produce a beam with high frequency modulation and no directional shift and intensity modulation. 5.4.6

Spatial Light Modulators

SLMs allow the modiﬁcation and control of the optical wave front through many pixels by electronic means. They are a fairly new technology that is steadily improving in quality and are now almost omnipresent through their applications in laser scanners, data projectors, and adaptive optics. They are part of much of the optics that is being developed. SLMs allow completely new applications, such as laser movie projection [23], in the form of spatial intensity modulators, or they can modify the shape of the optical wave front in the form of spatial phase modulators, for example, to produce dynamically controlled optical tweezers and traps [24] or neurostimulators [25]. At the same time they can be used to optimize the shape of optical beams, which means they can be used as an alternative to optical phase plates to create diﬀerent spatial modes, for example, beams with orbital angular momentum, or they can dynamically improve a wave front that has been distorted by the atmosphere in the case of astronomy or by propagating through biological samples or tissue in the case of medical applications. In quantum optics laboratories SLMs are used to optimize mode matching and have been used for quantum logic and entanglement measurements. The main advantage of the SLM is the high speed and the ability to control through suitable software very complex wave fronts and intensity distributions. The commonly quoted characteristics are the number of pixels controlled, now up to several megapixels; the contrast ratio, which is important for intensity modulators and can now reach up to 100 000 or higher; and the dynamic range, in multiples of full wavelength in the case of phase modulators. The other factor is the speed, which is linked to the ability to switch one pixel, typically on the timescale of 1–100 μs. The second aspect of speed is the ability to control so many pixels simultaneously, which means transmission rates of 10 Mbit/s to 1 Gbit/s and is rapidly progressing through the use of dedicated graphics cards and optimized software. Soon speed and resolution will not be a limiting factor in the applications. There are several diﬀerent technologies: the SLMs can be used in transmission and in reﬂection. The transmission mode with very small mirrors, known as digital light projection (DLP) is widely used in display technology such as video and cinema projectors. However, it does not have the characteristics required in quantum optics due to the considerable losses. More useful are phase modulators that work either in transmission mode or deformable mirrors that operate in reﬂective mode. The liquid crystal display (LCD)-type phase modulator is the most common; it combines large pixel numbers with good access speed. Such a phase modulator can be eﬀectively programmed as a hologram and can be used

193

194

5 Basic Optical Components

as variable lens, changing the focal length, compensating for beam distortions, or transforming the mode, for example, from one TEM mode to another TEM mode or from one Gauss Leguerre LG mode to another LG mode, or to create completely diﬀerent type of modes, for example, special modes for optical tweezers. This can be done with reasonable eﬃciency, and in particular when used as a grating, where the transmission into ﬁrst order or higher order is used, the contrast can be very high. For high eﬃciency applications, transformable mirrors perform far better. In particular those mirrors that have a continuous mirror surface can reach eﬃciencies above 90%. These SLMs have a low number of pixels. For reasonable cost one can get about 1000 pixels, which is suﬃcient for a number of mode matching and entanglement applications. The question arises: do SLMs add technical noise that will reduce their usefulness in quantum optics? The LCD-type modulators have considerable phase jitter that is commonly very noticeable due to the strong phase modulations created by the electronic drivers. In many normal display applications, the image on the SLM is refreshed frequently, for example, every millisecond, which means a rapid reorientation of the liquid crystals at that rate. The crystals then tend to oscillate around their equilibrium position, and this produces a strong driving of the phase setting for each pixel, leading to a large phase noise. This eﬀect can be avoided either by using very diﬀerent electronic drivers that have a very fast refresh rate that cuts the oscillation out of the frequency spectrum that is of interest or by using the SLM as a grating. The phase noise will be visible in the zero order beam. However, by observing the ﬁrst-order transmission or reﬂection, the phase noise will appear as translation noise only and thus not be detectable with the normal optical detectors. This is the reason why transmission SLMs have been used so successfully in combination with single-photon detection where optical losses are not so important [26]. For CW applications with squeezed light, the deformable mirror appears to be promising. A systematic test has been performed to show that no noticeable phase noise is present. This experiment shows that the mirror can be used to transform the mode shape of a beam of squeezed light with an eﬃciency of larger than 95% and that no detectable noise is present that would add to the quantum noise and destroy the quantum correlations [27]. That means no fundamental limits have to be considered – the right type of SLM will exist for every experiment. The future generation of SLMs will have further improved speciﬁcations, and this opens up the ﬁeld of quantum optics experiments to electronic mode matching that enhances the reliability of experiment. The controller of the SLM can use learning algorithms to iterate the many pixel settings with the single goal to optimize a chosen global parameter. An example would be to optimize fringe visibility between two interfering beams to get the best possible mode matching. In this way the combination of SLM and intelligent multidimensional controller will tune itself, and it can be an alternative to the manual alignment process performed now. Future students will have at least one SLM in each experiment and will not need to worry about frequent mirror realignment. The SLMs will also allow more complex experiments with multiple beams. They will allow the design

5.4 Other Optical Components

of reconﬁgurable layouts, for example, for quantum logic, and alternative modulation techniques for quantum communication. 5.4.7

Optical Noise Sources

Optical materials can contribute noise. This is equivalent to saying that additional light can be scattered into the mode under investigation. One example is atoms that are excited by one of the laser beams in the experiment and that spontaneously emit at the frequency and into the spatial mode that is detected. In a similar way stimulated processes can add light to the mode. These could be Raman transitions or other non-linear processes, such as 4WM or SHG. We have to be very careful in experiments that contain strong optical ﬁelds; basically any process that can generate light at the frequency of the mode we wish to detect is potentially a source of competing noise that could swamp an eﬀect as small as quantum noise. These types of noise add to the intensity ﬂuctuations in the mode. In addition, the refractive index of the optical material can be aﬀected by ﬂuctuations that will generate noise in the phase quadrature of the transmitted light. For example, a common source of noise in optical ﬁbres is guided acoustic wave Brillouin scattering, abbreviated as GAWBS. This noise occurs in any system where the wave is guided by geometrical structures such as a ﬁbre or planar waveguide. The noise spectrum shows very pronounced maxima, which can be related to the resonances of acoustic waves that are reﬂected by the boundaries of the ﬁbre. It covers such a wide spectrum that the detection of QNL light becomes very diﬃcult. Non-linear crystals can be the source of additional noise [29]. In all experiments a careful characterization of the noise spectrum has to be carried out, before we can assume that quantum noise is actually observed. Chapter 8 describes these techniques in detail. 5.4.8

Non-linear Processes

In summary, all linear components have a very simple eﬀect on the ﬂuctuations: a coherent state of light will remain a coherent state. Even for light that is not a coherent state, the eﬀects are simple; we will ﬁnd that any losses make the ﬂuctuations look more like those of a coherent state. In the limit of large losses, any input light will approach a coherent state (Figure 5.20). Quantum optics would not be much fun if there were only linear processes, but fortunately there is more. Gain combined with saturation allows us to make a laser, the all-important light source (see Chapter 6). One of the main features of non-linear processes is that they can generate additional new frequencies. In particular, SHG and degenerate optical parametric oscillation (OPO) are used to generate beams at twice or half the original frequency. The non-linear Kerr eﬀect transfers changes in the intensity to the refractive index. 4WM links modulation sidebands. The non-linear processes are too numerous to list here explicitly, and instead the extensive literature should be consulted, for example, the textbooks by M.D. Levenson and S.S. Kano [30], and R.W. Boyd [31] as the current reference. For a recent review of non-linear ﬁbre eﬀect, see the review by

195

5 Basic Optical Components

40 Noise power (rel. units)

196

Figure 5.20 An example of optical noise. The noise introduced in a single-mode optical ﬁbre by guided acoustic wave Brillouin scattering (GAWBS) measured with a CW laser. Source: Courtesy of Shelby and Levenson [28].

Hitachi fibre GAWBS

20 10 8 4 2

Quantum noise

1 0

500 MHz

1000

S.M. Hendrickson et al. [22], and for non-linear metamaterials, see the colloquium review by M. Lapine and the team lead by Y.S. Kivshar and coworkers [32]. These non-linear eﬀects allow us to change the wave front of the light, leading to self-focussing or defocussing; they can also turn a single spatial mode, like a TEM modes, into a high-order mode or generate even multimode situations, all as a consequence of the interplay between high local intensities and non-linear materials. Important for quantum optics is the fact that non-linear eﬀects can change the modulation and information carried by the beam. And it can change the statistics of the light and can be used to manipulate quantum noise and thus allows to create light for measurements with better SNR than allowed by the QNL. Non-linear media are essential for the generation of photon pair, sub-Poissonian light, and squeezed and entangled light, and such experiments will be discussed, one by one, in Chapter 9.

References 1 Stokes, G.G. (1852). On the composition and resolution of streams of polar-

ized light from diﬀerent sources. Trans. Cambridge Philos. Soc. 9: 399–416. 2 Mandel, L. and Wolf, E. (1995). Optical Coherence and Quantum Optics.

Cambridge University Press. 3 Born, M. and Wolf, E. (1999). The Principles of Optics. Pergamon Press. 4 Luis, A. and Sanchez-Soto, L.L. (1995). A quantum description of the beam-

splitter. Quantum Semiclass. Opt. 7: 153. 5 Edwards, P.J. (1993). Reduction of optical shot noise from light emitting

diodes. IEEE J. Quantum Electron. 29: 2302. 6 Grenier, C., Hervé, R., Féve, G., and Degiovanni, P. (2011). Electron quantum

optics in quantum Hall edge channels. Mod. Phys. Lett. B 25: 1053–1073. 7 Caves, C.M. (1981). Quantum-mechanical noise in an interferometer. Phys. Rev. D 23: 1693.

References

8 Saulson, P.R. (1994). Fundamentals of Interferometric Gravitational Wave

Detection. World Scientiﬁc. 9 Siegman, A.E. (1986). Lasers. University Science Books. 10 Shaddock, D.A., Buchler, B.C., Bowen, W.P. et al. (2000). Modulation free

11 12

13

14 15 16

17 18 19 20 21 22

23 24 25

26

27 28 29

control of a continuous-wave second-harmonic generator. J. Opt. A: Pure Appl. Opt. 2: 400. Anderson, D.Z. (1984). Alignment of resonant optical cavities. Appl. Opt. 23: 2944. Sampas, N.M. and Anderson, D.Z. (1990). Stabilisation of laser beam alignment to an optical resonator by heterodyne detection of oﬀ-axis modes. Appl. Opt. 29: 394. Morrison, E., Meers, B.J., Robertson, D.I., and Ward, H. (1994). Experimental demonstration of an automatic alignment system for optical interferometers. Appl. Opt. 33: 5041. Schiller, S., Yu, I.I., Fejer, M.M., and Byer, R.L. (1992). A fused silica monolithic total internal reﬂection resonator. Opt. Lett. 17: 378. Gardiner, C.W. and Zoller, P. (2000). Quantum Noise, 2e. Berlin: Springer-Verlag. Collett, M.J. and Gardiner, C.W. (1984). Squeezing of intracavity and traveling-wave light ﬁelds produced in parametric ampliﬁcation. Phys. Rev. A 30: 1386. Saleh, B.E.A. and Teich, M.C. (2007). Fundamentals of Photonics, 2e. Wiley. Bass, M. and Optical Society of America (1995). Handbook of Optics: Fundamentals, Techniques and Design. McGraw-Hill. Gabor, D. (1948). A new microscope principle. Nature 181: 777. Keren-Zur, S., Michali, L., Suchowski, H., and Ellenbogen, T. (2017). Shaping light with nonlinear metasurfaces. Adv. Opt. Photon. 10: 309. Okamoto, K. (2006). Fundamentals of Optical Waveguides. Academic Press. Hendrickson, S.M., Foster, A.C., Camacho, R.M., and Clader, B.D. (2014). Integrated non-linear photonics: emerging applications and ongoing challenges. J. Opt. Soc. A 31: 3193. Hecht, J. (2018). Cutting-edge cinema. Opt. Photon. News 29: 35. Grier, D.J. and Roichman, Y. (2006). Holographic optical trapping. Appl. Opt. 45: 880. Daria, V., Stricker, C., Redman, S., and Bachor, H.-A. (2009). Arbitrary mutisite two-photon excitation in four dimensions. Appl. Phys. Lett. 95: 093701. Dada, A.C., Leach, J., Buller, G.S. et al. (2011). Experimental high-dimensional two-photon entanglement and violations of generalised Bell inequalities. Nat. Phys. 7: 677. https://doi.org/10.1038/nphys1996. Morizur, J.-F., Armstrong, S., Treps, N. et al. (2011). Spatial reshaping of a squeezed state of light. Eur. Phys. J. D 61: 237–239. Shelby, R.M. and Levenson, M.D. (1986). Quantum Optics and Nonlinear Interactions in Optical Fibres. IBM Res. Report. IBM. Kimble, H.J. (1992). Squeezed states of light: an (incomplete) survey of experimental progress and prospects. Phys. Rep. 219: 227.

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30 Levenson, M.D. and Kano, S.S. (1988). Introduction to Nonlinear Laser Spec-

troscopy. Academic Press. 31 Boyd, R.W. (2008). Nonlinear Optics, 3e. Academic Press. 32 Lapine, M., Shadrivov, I.V., and Kivshar, Y.S. (2014). Colloqium; nonlinear

metasurfaces. Rev. Mod. Phys. 86: 1093.

199

6 Lasers and Ampliﬁers 6.1 The Laser Concept The laser is the key component in any quantum optics experiment. It was the invention of the laser (see Figure 6.1) that gave us access to the world of coherent optics, non-linear optics, and the quantum features of light. Without the laser we would be restricted to thermal light – missing out on most modern applications and on almost all the experiments described in this guide. In quantum optics we are concerned with the characteristics of the light generated by a laser. In particular, we will investigate the ﬂuctuations in the intensity of the light. In theoretical terms the laser is the system of choice for the realization of many quantum optics ideas: the laser light itself is close to a pure coherent state. In many cases, the laser can be regarded as the source for a single mode of radiation as described in the theory. If we consider a continuous-wave (CW) laser, the mode will have constant properties, and even pulsed lasers can be considered as constant in the sense that we describe them as a steady stream of pulses – each pulse like any other. Many books have been written about the laser while the technology constantly evolves and makes the devices more powerful and more accessible. One of the most comprehensive introductions remains the masterpiece by A.E. Siegman [1] in which he provides all the technical details required to understand and operate lasers. In simple technical terms, the laser is an optical oscillator (see Figure 6.2) that relies on feedback, much like an electronic circuit. It consists of a narrow band optical ampliﬁer, which generates a well-deﬁned output beam of light and an optical arrangement that feeds part of the light, in phase, back to the input. The ampliﬁcation is provided by a gain medium, atoms or molecules that are excited by an external source of energy, the pump. The realization that such a situation is possible was one of the early results of atomic quantization [2]. The feedback is provided by a cavity surrounding the active medium. The mirrors of the cavity generate a large internal optical ﬁeld, which is ampliﬁed during each round trip. Of course there are also losses inside the cavity, and in order to achieve suﬃcient feedback for oscillation to occur, the gain has to be larger than the losses. Without any input there would be no output, no optical feedback signal, and no laser beam. However, the presence of even a small amount of light, some small optical amplitude within the bandwidth of the resonant ampliﬁer system, A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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6 Lasers and Ampliﬁers

Figure 6.1 The original laser demonstrated by Theodore Maiman in 1960. The ruby crystal with coatings is the gain medium, the ﬂash lamp surrounding it the pump source. The pulses wore out the metal coating of this version of the laser. Source: Courtesy Welling (2015).

Pump Output beam

Feedback Output Amplifier Power (a)

Cavity with gain medium

(b)

Figure 6.2 Comparison between an (a) electronic ampliﬁer/oscillator and (b) a laser.

will initiate the process and lead to a rapid growth of the output. Spontaneous emission from atoms in their excited state can provide such a seed. The amplitude increases until the gain of the ampliﬁer saturates at a level that just balances the losses due to the mirrors and other sources. The output power will stabilize at a steady-state value, and the laser achieves constant CW operation. The conditions for lasing are twofold. Firstly, the ampliﬁer gain must be greater than the loss in the feedback system so that a net gain is achieved in each round trip. All lasers show a threshold, which means they require a minimum pump power Pthr to start steady oscillation and generate an output beam. Secondly, the total phase accumulated in a single round trip must be an integer multiple of 2π to provide phase matching to the feedback signal, i.e. the laser frequency must correspond to a cavity mode. Ignoring small frequency pulling eﬀects, which can be somewhat larger in diode lasers, the laser mode(s) will correspond with the cavity mode(s) closest to the peak of the gain proﬁle. In the simplest case this is a single ﬁxed frequency.

6.1 The Laser Concept

Above threshold the laser produces a steady output beam. The geometrical properties of the output beam are those of the spatial modes of the cavity, in the simplest case the fundamental Gaussian TEM00 beam (see Eq. (2.8)). The combination of these two cases is known as single-mode operation, both transversely (spatially) as well as longitudinally (spectrally). This is the ideal case for the study of the quantum properties of light. The intra-cavity ﬁeld se well described as one quantum mechanical mode of the radiation ﬁeld that extends throughout the cavity. The output beam can be described by a propagating mode or coherent state as described in Section 4.5. As we will see, the present technology gets us extremely close to the realization of such a perfect laser. 6.1.1

Technical Speciﬁcations of a Laser

For CW lasers we can use a wide range of lasing materials, excitation mechanisms, and cavity designs. CW lasers include gas lasers, such as the once widely used He–Ne lasers and the powerful Ar ion and CO2 lasers, to the complex tunable dye lasers. We do not have space here to describe all the technical details and numerous engineering tricks that are used in the design of these lasers. A range of specialized literature is available to provide this information [1, 3]. Nowadays, the majority are the more robust, technically almost simplistic solid-state diode lasers. The trend is to ﬁbre lasers, and various other solid-state systems all pumped by diode lasers [4]. These lasers are starting to dominate the technological applications. In the area of atomic spectroscopy, where tunability and narrow bandwidth is required, diode lasers and Ti/sapphire rapidly replace the traditional dye lasers. For short pulses solid-state lasers have allowed much shorter pulses. For complex experiments and for applications, the compactness and robustness of the diode laser are the main advantage. As a consequence most of the practical examples of laser performance in this book refer to solid-state lasers. The performance of a laser can be described by the following parameters: optical power, spatial mode structure, output wavelength, frequency spectrum, and noise spectrum. The power is quoted as the total optical power Pout of the output beam. Of practical interest is the total eﬃciency of the laser 𝜂total = Pout ∕Ppump , where Ppump is the pump power required to achieve an output power Pout . Well above threshold the power scales linearly, and the slope eﬃciency 𝜂slope = dPout ∕dPpump is constant. This is shown in Figure 6.3. Note that operation near threshold is not reliable. At that point the power is not stable; it tends to be noisy, and Pthr may not be easy to measure. The spatial properties of the laser beams are determined by the properties of the optical resonator. In most cases a stable resonator is used, and the mode analysis in terms of Hermite–Gaussian TEMm,n modes, as discussed in Chapter 2, can be used. There are a few exceptional cases, particularly for the design of high-power lasers where more complicated unstable resonators are used. However, many lasers operate in a single spatial mode, the lowest order TEM0,0 mode, which is particularly simple to describe. In practice the resonator is never a perfect optical cavity, there will always be some spatial distortions in the output beam, and a parameter Mm,n is normally quoted that describes how closely the beam is

201

6 Lasers and Ampliﬁers

Figure 6.3 Optical output power versus pump power. The laser has a threshold above which the output increases linearly with the pump power.

Pout Output power

202

Pthr

Ppump Pump power

approximated by a TEMm,n mode. The perfect match corresponds to Mm,n = 1. Some lasers produce beams that have elliptical rather than circular beam proﬁles. For these lasers it is important to know the aspect ratio of the ellipse and whether the spatial mode is simply a stretched version of a Gaussian beam or a more complex wave front. The output wavelength is determined by the properties of the gain medium and can vary from the UV (excimer laser 169 nm) to the far-infrared NH3 laser (440 μm). Many laser materials have gain for several output wavelengths. For example, the Ar ion laser has several green/blue laser transitions. The desired wavelength can be selected by optimizing the mirror coatings and by using selective intra-cavity components, such as a prism or a grating. Some lasers are tunable, that is, a range of laser wavelengths can be chosen. Some materials with a tuning range have a broad gain proﬁle – prominent examples are dye lasers and the Ti/sapphire laser; in other materials the gain proﬁle is temperature dependent. For example, in a diode laser the wavelength can be tuned by changing the temperature of the device, controlling the current, and using an extended cavity for precise control of the laser frequency. The frequency distribution of the laser output is determined by the cavity properties and the gain proﬁle. In addition, it is important whether the gain proﬁle is an inhomogeneous or homogeneous proﬁle. In the ﬁrst case diﬀerent cavity modes obtain their gain from diﬀerent active atoms, and the various modes can oscillate independently. The result is a series of spectral output modes, separated by the free spectral range of the laser resonator. In the latter case all active atoms in the laser contribute to the laser in the same way. There will be competition between the spectral modes, and normally only one will win and the output spectrum will be dominated by one single spectral mode. Such a single-mode laser still has a frequency bandwidth. The absolute frequency can change with time. The bandwidth is given as the range of frequencies that are present in a set observation time. The bandwidth is a measure of the size of the frequency ﬂuctuations. For example, suppose only frequencies within a 1 MHz interval are measured within one second interval. This information tells us that the laser could be used to resolve a detailed feature of a given spectrum with a resolution of 1 MHz and no less. However, short-term laser bandwidth

6.1 The Laser Concept

and long-term frequency drift can vary widely. The same laser might drift by 100 MHZ in 10 minutes. We would have to carry out the experiment very quickly. A more complete description of the frequency properties is given by the Allan variance, a statistical analysis of the frequency changes for a range of diﬀerent observation times [5]. We will ﬁnd that the frequency ﬂuctuations are commonly a consequence of technical imperfections, such as vibrations and electrical and acoustical noise. The quantum mechanical ﬂuctuations play hardly a role in the frequency noise. The phase of a laser is unconstrained and will drift. This is known as phase diﬀusion. This means that the variance of the phase quadrature (see Chapter 4) will grow unlimited over long time periods. On the other hand for short times, or high frequencies, the phase can be quite stable and will tend to the coherent state level at suﬃciently high frequencies. Finally, there are ﬂuctuations of the intensity of the output beam. These are described as intensity noise spectra, as the relative changes of the laser power, 𝛿Pout ∕Pout , measured with a bandwidth RBW at a frequency Ω. The measured variance of the optical power ﬂuctuations can be directly compared with the modelled variance of the amplitude quadrature V 1las (Ω) (see Chapter 4), which is normalized by the quantum noise associated with a coherent state of the same optical power. In summary, lasers are available that produce output beams, which are very close to the theoretical limit. Their light can be considered as the real-world analogue of a coherent state. But not all lasers do this. Some lasers are not built well enough and produce beams that are distorted or noisy. Some other lasers have noise that is intrinsic to the lasing process. Finally, we will ﬁnd some lasers that actually perform better than a coherent state. 6.1.2

Rate Equations

Although diﬀerent laser systems diﬀer widely in their details, it is possible to model many aspects of laser behaviour from a quite general rate equation approach. The main assumption in such an approach is that coherent atomic eﬀects are negligible. Coherent eﬀects can be neglected in the presence of strong dephasing eﬀects, such as interatomic collisions in a gas or lattice vibrations in a solid. This is an excellent approximation for all but a few lasers, such as far-infrared gas lasers, which we will not consider in this discussion. In a rate equation model, we consider the gain medium to be made up of M atoms. These atoms have internal states of excitation, with energies E1 , E2 , and E3 as shown in Figure 6.4. We assume lasing occurs between the states with energies E3 and E2 . The frequency of the laser light is centred around 𝜈L = (E3 − E2 )∕h. The actual laser frequency will be determined by the cavity resonance, taking into account all the non-linear eﬀects inside the laser resonator. The excitation of the medium is described by the populations M1 , M2 , M3 of the individual states. These populations should really be interpreted as the probabilities J1 , J2 , J3 of ﬁnding any individual atom in one of the three states. The total population is preserved, thus M = M1 + M2 + M3 = M(J1 + J2 + J3 )

(6.1)

203

6 Lasers and Ampliﬁers

Figure 6.4 The semi-classical model of a three-level laser system.

E3 , J3 γ32

G Energy

204

E2 , J2 Γ

γ31 γ21

E1 , J1

A number of processes link the states. Pumping is modelled as a uni-directional transfer of population from state |E1 ⟩ to state |E3 ⟩. Normally any pumping process will also produce de-excitation of the upper pump level at a rate proportional to its population. The uni-directional pumping is to be understood as the limit of pumping to a higher state, which then decays very rapidly to state |E3 ⟩. The population of all excited states decay through spontaneous emission. Stimulated emission and absorption of photons in the laser ﬁeld couples the states |E3 ⟩ and |E2 ⟩. All these processes are described by rates. As an example, the spontaneous emission from state |E2 ⟩ to state |E1 ⟩ results in M2 𝛾21 = MJ2 𝛾21 photons emitted per second, which are leaving the laser in random directions, and a corresponding reduction in the population of state |E2 ⟩ and increase of population of state |E1 ⟩. Similarly the rate of spontaneous emission from state |E3 ⟩ is described by rate constant 𝛾32 . The pumping rate depends on the pump power, the excitation cross section, and the pumping eﬃciency. It is described by one rate M1 Γ. For simplicity, all we neglect spatial eﬀects and consider one spatial mode only. Similarly only one spectral mode, at frequency 𝜈L , is considered. All optical eﬀects are described by the gain and loss of light from this single mode. For a high ﬁnesse cavity, the total loss of the resonator is just the sum of the individual losses (see Section 5.1.7). We write the total loss 𝜅 as the sum of losses due to the output mirror 𝜅m and those due to all other mechanisms 𝜅l . If the mode inside the laser contains nt photons, then the number of photons leaving the cavity per second will be 2𝜅nt . Of these only the fraction 𝜅m ∕𝜅 exits in the output beam; the remainder are dissipated internally. The energy diﬀerence between the atomic states |E3 ⟩ and |E2 ⟩ corresponds to a frequency that is resonant with the mode. Stimulated emission occurs at a rate G32 J3 nt , and stimulated absorption at a rate G32 J2 nt . The rate constant G32 is deﬁned as G32 = 𝜎s 𝜌c∕nr

(6.2)

where 𝜎s is the stimulated emission cross section for the |E3 ⟩ to |E2 ⟩ transition, 𝜌 is the density of lasing atoms in the medium, c is the speed of light, and nr is

6.1 The Laser Concept

the refractive index. The two processes produce an overall change of the photon number of G32 (J3 − J2 )nt (photons per second). The inversion (J3 − J2 ) drives this ampliﬁcation process. At this point we ignore the small contribution to the photon number due to spontaneous emission into the lasing mode. Combining all these individual rates, we can write a system of four coupled rate equations that describe the dynamics of the laser: dJ1 = −ΓJ1 + 𝛾21 J2 dt dJ2 = G32 (J3 − J2 )n + 𝛾32 J3 − 𝛾21 J2 dt dJ3 = −G32 (J3 − J2 )n − 𝛾32 J3 + ΓJ1 dt dn (6.3) = G32 (J3 − J2 )n − 2𝜅n dt where n is the photon number per atom such that nM = nt . We can also write the equation of motion of the cavity ﬁeld (per atom) as d𝛼c (6.4) = G32 (J3 − J2 )𝛼c − 𝜅𝛼c dt where 𝛼c∗ 𝛼c = n. The steady state solution for the photon number per atom, n, can be obtained by setting the time derivatives to zero. We ﬁnd either n = 0 or n=

Γ(𝛾21 − 𝛾32 ) 𝜅 Γ𝛾21 + 𝛾21 𝛾32 + Γ𝛾32 − 2𝜅(2Γ + 𝛾21 ) G32 2𝜅(2Γ + 𝛾21 )

Provided we have 𝛾21 > 𝛾32 , there is a threshold pump rate, Γt , above which the non-zero solution for n will be positive and stable. This represents laser oscillation while the zero solution is unstable. For pump rates below the threshold, the non-zero solution is negative, and it is the zero solution that is stable. That is, the laser does not oscillate. Physically that means the losses in the resonator, per round trip, are larger than the gain so that any light introduced into the cavity is quickly damped away. Generally the lifetime of the lower lasing level will be very short so we can set 𝛾21 ≫ 𝛾32 , Γ. Under this condition the population of the second level can be neglected, and the ground state can be considered undepleted (J2 = 0, J1 = 1). Our equations of motion reduce to dJ3 = −G32 J3 𝛼c∗ 𝛼c − 𝛾32 J3 + Γ dt d𝛼c (6.5) = G32 J3 𝛼c − 𝜅𝛼c dt and we obtain the following simpler solution for the output photon ﬂux per atom of the laser, nout : ( ) 𝜅m 2𝜅 nout = 2𝜅m n = Γ − 𝛾32 (6.6) G32 𝜅

205

206

6 Lasers and Ampliﬁers

The pump threshold is Γt = steady state, 2𝜅 J3 = G32

2𝜅 𝛾 . G32 32

We also have that above threshold, in the (6.7)

and the following alternative form of Eq. (6.6) can be written 𝜅 (6.8) nout = (Γ − 𝛾32 J3 ) m 𝜅 That is, the rate at which laser photons leave the cavity is simply the rate at which atoms are placed in the upper lasing level minus the rate at which photons are lost through spontaneous emission. Threshold occurs when pumping exceeds the loss due to spontaneous emission. The useful output is scaled by the fraction of photons that leave via the cavity mirror and hence depends on the internal losses. The total eﬃciency of the laser, 𝜂total , depends on this fraction, the relative pump power Ppump /Pthr , as well as the pumping eﬃciency. Very close to threshold the performance is obviously very dependent on ﬂuctuations. The laser can switch on and oﬀ, and our simple theoretical model would have to be extended to include such eﬀects. For reliable operation the actual pump rate Γ should be considerably higher than Γt , which means Ppump > Pthr . Above the threshold the output power increases linearly with the pump rate (see Figure 6.3). Hence the slope eﬃciency, 𝜂slope , is constant above threshold, depending only on the internal losses and pump eﬃciency. At even higher pump powers, the output power can level oﬀ. Theoretically this occurs when the pump rate begins to deplete the ground state. In practice this tends to happen at lower pump powers than a simple estimate would suggest, since a number of additional non-linear eﬀects might occur that can have power limiting back eﬀects. For example, laser materials show thermal expansion, due to absorption, which can lead to a non-uniform refractive index and thus thermal lensing. This and other power-induced wave front distortions introduce additional losses in the resonator, which limit the internal power. An alternative possible conﬁguration of the laser is where the lasing transition occurs between state |E2 ⟩ and the ground state, state |E1 ⟩, with a stimulated emission cross section proportional to G21 . Important examples of this type of laser are the solid-state Ruby and Erbium lasers. The analysis proceeds as before. Making the assumption now that state |E3 ⟩ decays very rapidly, we get the following solution for the output photon ﬂux per atom: ) ( 𝜅m 1 2𝜅 (Γ − 𝛾21 ) nout = 2𝜅m n = (6.9) Γ − 𝛾21 − 2 G21 𝜅 The threshold pump rate is given by Γt =

1+

2𝜅 G21

1−

2𝜅 G21

𝛾21

(6.10)

The major diﬀerence between this laser system and the previous one is the threshold behaviours. Whilst in the previous case, the pump threshold could in principle be made arbitrarily small by reducing the ratio G2𝜅 , that is, increasing the 32

6.1 The Laser Concept

cavity ﬁnesse or active atom density, in this case the threshold cannot become less than the spontaneous emission rate between the laser levels, 𝛾21 . Physically this is because we have to at least half empty the ground level before we can create an inversion. 6.1.3

Quantum Model of a Laser

In order to determine the dynamics of the ﬂuctuations of the laser System, a quantum model is required. In the quantum description the laser consists of a laser medium of M atoms with internal states |E1 ⟩, |E2 ⟩, |E3 ⟩, a pump source with pump rate Γ and noise spectrum V 1P (Ω), and an internal optical mode â with an expectation value of 𝛼c . Operators Ĵ1 , Ĵ2 , Ĵ3 describe the collective excitation of the atoms. As in the semi-classical model, given in the previous section, the expectation values of these operators, J1 , J2 , J3 , respectively, give the probability of ﬁnding any one atom in the respective state. Figure 6.5 shows the laser system in this notation. Making the simplifying assumptions that led to Eq. (6.5) (i.e. negligible second-level population and an undepleted ground state), the quantum operator Langevin equations of motion can be written as √ √ √ dĴ3 ̂ p − Γ 𝜹X ̂ 𝛾 + G32 â † â 𝜹X ̂P = −G32 Ĵ3 â † â − 𝛾32 Ĵ3 + Γ + 𝛾32 𝜹X dt √ √ √ dâ G32 ̂ ̂ l + 2𝜅m 𝜹A ̂ m − G32 Ĵ3 𝜹A ̂ †p (6.11) = J3 â − 𝜅 â + 2𝜅l 𝜹A dt 2 where all terms with ‘𝛿’ preﬁxes are vacuum ﬁelds, with zero expectation values. The equations of motion of the expectation values correspond to the semi-classical rate equations (Eq. (6.5)). All the processes are described in a similar way – now by using operators to describe the states of the atoms and the single mode of the light. In addition, the full quantum model has to contain operators describing all possible noise sources. These include the pump ̂ P , and with an intensity noise spectrum noise, characterized by the operator 𝜹X 2 ̂ m , and ̃ V 1P (Ω) = ⟨|𝜹XP | ⟩, the vacuum input through the cavity mirror, 𝜹A ̂ the vacuum input due to other intra-cavity losses, 𝜹Al . The rapid phase decay

|3〉, σ3

Energy

G

V1las |2〉, σ2

Γ

V1vac

V1pump

γ32

γ31 γ21

|1〉, σ1

Figure 6.5 The quantum model of a three-level laser.

V1spont V1dipole V1losses

207

208

6 Lasers and Ampliﬁers

of the atomic coherence due to collisions with atoms or, in the case of solids, lattice-induced disturbances introduces dipole ﬂuctuations that can be charaĉ p + 𝜹A ̂ †p . Finally, noise in the populations due ̂ p = 𝜹A terized by the operator 𝜹X ̂ 𝛾 . The eﬀects of to spontaneous emission has to be included via the operator 𝜹X all these noise sources appear explicitly in the ﬂuctuations of the output operator ̃ las . The derivation of Eq. (6.11) is quite involved and beyond the scope of this A book. Details of the derivation and the identiﬁcation of the noise operators with their various physical origins can be found in Ref. [6]. The steady-state solutions for the probabilities J3 and photon number per atom, n, in the quantum model, obtained when all the time derivatives are set to zero, are identical to those obtained from the semi-classical Eq. (6.5). We proceed by ̂ linearizing around these stable solutions (see Section 4.5.1). Writing â = 𝛼c + 𝜹a and Ĵ3 = J3 + 𝜹Ĵ3 and only keeping terms to ﬁrst order in the vacuum ﬁelds, we obtain the following linearized equations for the quantum ﬂuctuations: √ d𝜹Ĵ3 ̂ a − (𝛾32 + G32 𝛼c2 )𝜹Ĵ3 + 𝛾32 𝜹X ̂𝛾 = −G32 J3 𝛼c 𝜹X dt √ √ ̂ p − Γ 𝜹X ̂P + G32 𝛼c2 𝜹X √ √ √ ̂ G d𝜹a ̂ l + 2𝜅m 𝜹A ̂ m − G32 J3 𝜹A ̂ †p (6.12) = 32 𝛼c 𝜹Ĵ3 + 2𝜅l 𝜹A dt 2 √ where we have taken 𝛼c = n real without loss of generality (see Section 6.1.5). These equations can be solved in Fourier space. The internal mode is coupled to the external √ ﬁeld (see Section 5.3 – resulting in the frequency space ̃ las = 2𝜅m ã − 𝜹A ̃ m , describing the ﬂuctuations of the output ﬁeld. operator, A The result of interest is the noise spectrum V 1las (Ω) of the photon number, or intensity, of this mode. Under our linearization condition this is given by the ̃ A |2 ⟩ where amplitude quadrature spectrum of the output mode: V 1las (Ω) = ⟨|𝜹X † ̃ ̃ ̃ 𝜹XA = 𝜹Alas + 𝜹Alas . We obtain the following solution for the system of transfer functions: ) ( 4𝜅m2 ((2πΩ)2 + 𝛾L2 ) − 8𝜅m 𝜅G32 n𝛾L Vm V 1las (Ω) = 1 + ((2πΩRRO )2 − (2πΩ)2 )2 + (2πΩ)2 𝛾L2 + + + +

2 2𝜅m G32 nΓ

((2πΩRRO )2 − (2πΩ)2 )2 + (2πΩ)2 𝛾L2 2 (2𝜅m G32 n𝛾32 J3 )

((2πΩRRO )2 − (2πΩ)2 )2 + (2πΩ)2 𝛾L2 2𝜅m G32 ((𝛾32 + Γ)2 + (2πΩ)2 ) ((2πΩRRO )2 − (2πΩ)2 )2 + (2πΩ)2 𝛾L2 4𝜅m 𝜅l (𝛾L2 + (2πΩ)2 ) ((2πΩRRO )2 − (2πΩ)2 )2 + (2πΩ)2 𝛾L2

VP (Ω) V𝛾 Vp Vl

(6.13)

where 𝛾L = G32 n + 𝛾32 + Γ, (2π ΩRRO )2 = 2G32 n𝜅, and the solutions for n and J3 are given by Eqs. (6.6) and (6.7). It should be noted that the parameters have to be chosen separately for each laser. The material constants 𝜎s , 𝜌, nr , 𝛾32 , and

6.1 The Laser Concept

𝛾21 are speciﬁc to the type of laser and reasonably well known. In particular the rapid decays from the highly excited states and the lower lasing level for most lasers justify the use of this simpliﬁed three-level model. Other parameters such as the resonator parameters 𝜅l and 𝜅m and the total number of lasing atoms vary with the individual laser. How close to a coherent state with V 1las (Ω) = 1 can this solution be? The general answer is: many lasers approach this limit if they are operated well above threshold. However, there are important diﬀerences for the various types of lasers. We will now discuss a number of typical cases focusing especially on solid-state and diode lasers. 6.1.4 6.1.4.1

Examples of Lasers Classes of Lasers

The ﬁnesse of the laser resonator, determined by the magnitude of 𝜅, is generally chosen to achieve optimum laser eﬃciency. The ﬁnesse is controlled by changing the transmission of the output mirror (𝜅m ). Reducing the mirror transmission lowers the laser threshold and hence acts to increase eﬃciency. However, it also 𝜅 reduces the ratio 𝜅m due to the presence of internal losses (𝜅l ), which decreases the eﬃciency. Thus for a particular laser material, there will be an optimum value of 𝜅. Three classes of lasers with distinctly diﬀerent dynamical behaviours can be identiﬁed on the basis of the relationship between the sizes of 𝜅 and the decay rate of the upper lasing level (𝛾32 ). They are: • Class 1: 𝛾32 ≫ 𝜅 – e.g. dye lasers. • Class 2: 𝛾32 ≈ 𝜅 – e.g. visible gas lasers. • Class 3: 𝛾32 ≪ 𝜅 – e.g. solid-state and semiconductor lasers. We will now examine the predicted behaviour of examples from each of these classes, focusing particularly on the class 3 solid-state and semiconductor lasers. A fourth class of lasers, mentioned earlier, in which the photon decay rate is more rapid than the coherence dephasing rate cannot be described via rate equations and will not be discussed here. 6.1.4.2

Dye Lasers and Argon Ion Lasers

There is a wide range of possible lasers that are used in quantum optics experiments. In particular when the non-linearity is a resonant atomic or molecular medium, it is important to have a laser that can be tuned on or arbitrarily close to the resonance. While even that can be achieved nowadays with diode lasers, using external cavities, the traditional work horse in these situations is the argon ion laser pumped dye laser. In these devices the active medium is a dye solution, made up of large, complex dye molecules that can provide gain continuously over a wide range of optical frequencies. The actual laser frequency is selected with a number of frequency-dependent components inside the laser resonator. Typically a dye laser will have parameters of 𝜅 ≈ 1.5 × 107 1/s and 𝛾32 ≈ 3 × 108 1/s and hence is a class 1 laser. Provided we are operating more than about ﬁve times above threshold, we can approximate the noise spectrum (6.13) by V 1las (Ω) = 1 +

(2𝜅)2 (V 1P − 1) (2𝜅)2 + (2πΩ)2

(6.14)

209

210

6 Lasers and Ampliﬁers

This is a Lorentzian spectrum with a half width equal to the cavity decay rate. At low frequencies the spectrum will be dominated by the noise of the pump laser. At high frequencies (2πΩ ≫ 2𝜅), the spectrum will be quantum noise limited (QNL). In order to obtain QNL operation from a dye laser at frequencies in the tens of MHz, a QNL pump laser is needed. A dye laser is pumped by an Ar ion laser, providing large optical powers (more than 10 W) at wavelengths in the blue end of the visible spectrum. Typically an argon laser will have 𝜅 ≈ 1.0 × 106 1/s and 𝛾32 ≈ 2 × 106 1/s and hence is a class 2 laser. Suﬃciently far above threshold a class 2 laser has a noise spectrum very similar to that of a class 1 laser (see Eq. (6.14)). Due to the slower cavity decay rate and hence narrower pump (electric discharge) noise region, we would expect the noise spectrum of the argon laser to be QNL at frequencies of tens of MHz. However, due to the small gain that the Ar plasma provides, these lasers have long cavities and thus in the presence of inhomogeneous broadening frequently run on several longitudinal modes. The output spectrum of such a pump laser would hence be dominated by strong spectral features with a spacing of typically 100 MHz. This noise from an Ar ion laser will appear on the intensity noise spectrum of the dye laser. It is possible to suppress the higher-order modes using intra-cavity etalons but only with a large power penalty. For quantum optics experiments, which require a laser beam that is as close as possible to the quantum limit, the noise properties of the Ar ion and dye lasers have posed considerable problems. Most experiments were carried out at speciﬁc detection frequencies, in between the large noise spikes, where the noise spectrum happened to be QNL. In recent years the progress in the development of high-power, low noise diode lasers and diode laser pumped solid-state lasers has been so remarkable that now dye lasers are avoided as much as possible. 6.1.4.3

The CW Nd:YAG Laser

We will now discuss the most popular solid-state laser, the Nd:YAG laser. Gain is provided by excited neodymium ions in a yttrium aluminium garnet glass substrate. Traditionally the neodymium ions were excited by a ﬂash lamp, but it is becoming more common these days to use diode laser arrays as the pump source. In a miniature monolithic ring conﬁguration, these lasers can be made to perform very close to the ideal single-mode laser described by our rate equation (6.3). Thus a careful quantitative comparison between theory and experimental results is warranted here. Typically solid-state lasers have 𝜅 ≈ 107 − 108 1/s depending on their size and 𝛾32 ≈ 102 − 103 1/s and hence are class 3 lasers. For a class 3 laser, all the terms in the noise spectrum (Eq. (6.13)) remain important even when operating up to 10 times above threshold and the spectrum is quite complex. It contains contributions from the pump noise V 1P (Ω), spontaneous emission, dipole ﬂuctuations, and vacuum ﬂuctuations. The mixture of these contributions varies with the detection frequency. It is instructive to plot these various contributions separately. This is done in Figure 6.6a, where the various contributions are described in the ﬁgure caption. There are some very clear features, and the spectrum is dominated by a resonant feature, called the resonant relaxation oscillation that occurs at the frequency √ 2π ΩRRO = 2G23 n𝜅 (6.15)

Noise power relative to QNL (dB)

6.1 The Laser Concept

80 60 (v)

40 (iv) 20

(vi)

(i)

QNL

0

0.01

0.1

(a)

Noise power relative to QNL (dB)

(ii)

(iii) –20 0.001

1

10

100

500

10

30

Frequency (MHz)

75 (i) 55

(ii)

35

15

(iv)

(iii) QNL

–5 0.001 (b)

0.01

0.1

1

Frequency (MHz)

Figure 6.6 Noise spectrum of a free-running Nd:YAG laser. (a) The diﬀerent contributions to Eq. (6.3) are drawn separately. Trace (i) is the contribution of the vacuum noise, (ii) is the eﬀect of the pump noise, (iii) is the noise due to spontaneous emission, (iv) is the noise associated with the dipole ﬂuctuations, (v) is the noise due to intra-cavity losses, and (vi) is the total noise for a QNL pump. Part (b) shows experimental results for such a Nd:YAG laser. In this realistic case (i) the pump noise level was about 40 dB above the QNL. For comparison the predicted noise performance is shown in (ii). Trace (iii) was obtained with a pump noise of 20 dB. Trace (iv) is the corresponding theoretical prediction. Source: Adapted from Harb et al. [7].

In this particular case ΩRRO = 400 kHz. This oscillation is due to the interaction of the atomic state inversion (J3 − J2 ) ≈ J3 with the excitation of the internal optical mode. In physical terms, the energy in the laser is moving back and forth between the laser medium and the cavity mode, resulting in a strong modulation of the output power. This oscillation is a prominent feature and corresponds to an underdamped oscillation that is continually excited by the noise sources within the laser [8]. Note that this oscillation does not occur in class 1 and 2 lasers because for them 𝛾L ≫ ΩRRO , making the oscillation overdamped, as can

211

6 Lasers and Ampliﬁers

be seen from the denominator in Eq. (6.13). An alternative expression for ΩRRO in terms of the pump rate is √ √ Γ − Γt (6.16) 2π ΩRRO = 2𝜅𝛾32 Γt That means the harder the laser is pumped, increasing Γ, the higher is the frequency of the relaxation oscillation and the less pronounced it is. This can be seen by comparing the traces in Figure 6.7 that are plotted for the same laser but with diﬀerent pump powers. In some situations ﬁtting the measured frequencies ΩRRO for a range of pump powers is a more reliable way to determine Γt than an attempt to measure the power at which the laser turns on. Above ΩRRO the laser has a long, drawn-out noise tail. For very high frequencies the laser approaches the quantum noise limit. The origin of the quantum noise can be traced back to the various vacuum inputs in the model. It should be noted that the noise of the pump source has no inﬂuence on the laser noise for frequencies Ω ≫ ΩRRO . A noisy pump will not aﬀect the laser at such high frequencies. However, it also means that in this frequency regime the laser output power cannot be controlled by modulating the pump power. This is certainly a disadvantage for communication systems, since pump power modulation is normally the simplest and cheapest modulation scheme. At low frequencies, Ω ≪ ΩRRO , the noise spectrum of the laser is dominated by the noise of the pump source. There is a strong transfer from the pump to the laser output. By carefully choosing the noise properties of the pump, the laser output can be quietened down. Figure 6.6b shows two such cases. The ﬁrst case, traces (i) and (ii), is realistic for a laser with large pump noise. Here V 1p (Ω) is a factor 10 000, or 40 dB, above the standard quantum limit. The second case, –20

129

170

205 266

–40 Power (dBm)

318 360 413

78 kHz

–60 455 –80

1

0.1

–100 0.01

212

Frequency (MHz)

Figure 6.7 Experimental noise trace for a Nd:YAG laser with various pump powers. The relaxation oscillation frequency ΩR is stated for each case in units of kHz. It increases with pump power as stated in Eq. (6.16). Source: Adapted from Harb et al. [7].

6.1 The Laser Concept

traces (iii) and (iv), was measured with a quieter pump, V 1p (Ω) = 20 dB. Note that even for the case of a quantum-limited pump (V 1p (Ω) = 0 dB), the resonant relaxation oscillation would survive. Only for the impossible case of a completely noise-free pump source (V 1P (Ω) = 0) and no cavity losses or dipole ﬂuctuations would the relaxation oscillation completely disappear. 6.1.4.4

Diode Lasers

Perhaps the simplest and most eﬃcient lasers are semiconductor or diode lasers. Diode lasers use the band gap in doped semiconductor materials to produce optical gain directly from an electrical current. Under suitable conditions diode lasers can be described by our rate equation (6.3) as class 3 lasers. Major diﬀerences with solid-state lasers arise from their small size and high gain. Typically a diode laser will have 𝜅 ≈ 1011 1/s and will be run 5–10 times above threshold. Under these conditions the RRO will be pushed out to frequencies of tens of GHz and will be quite small. Most detection schemes are only sensitive up to frequencies of a few GHz; thus for all practical purposes the spectrum can be written as 𝜅 (6.17) V 1las (Ω) = m [1 + 𝜁 (V 1P (Ω) − 1)] 𝜅 where it is understood that 2πΩ ≪ 𝜅 and 𝜁 is the pump eﬃciency. Moreover, unlike the situation of the dye laser where the pump source is a potentially very noisy Ar ion laser, the pump source of the diode laser is an electric current, which can be made very quiet. Indeed we can have V 1las (Ω) = 1 if the pump current is given by V 1P (Ω) = 1. This means that for a quiet Poissonian noise-limited pump current, a diode laser has a QNL output. Equation (6.17) also means the laser output responds directly to modulation of the pump source. This is important for communication applications where we want to use the current driving the diode laser as the input of modulation. In fact, there is no fundamental reason why we cannot have V 1P (Ω) < 1, that is, a sub-Poissonian current. Unlike light, electric currents can easily be made sub-Poissonian. This would imply V 1las (Ω) < 1; the laser output would be sub-Poissonian. Diode lasers have successfully been made to produce squeezed light using this method, and technical details are discussed in Chapter 9. From Eq. (6.17) we see that, at least in theory, the squeezing is limited only by the internal losses and the pump eﬃciency 𝜁 . 6.1.4.5

Limits of the Single-Mode Approximation in Diode Lasers

From the previous discussion of diode lasers, it would appear that, provided sufﬁcient care was taken to suppress noise in the pump current, these lasers would always run at or below the QNL. For a number of reasons, in practice this is not the case. In particular, multimode behaviour can lead to increased noise. Normally one expects single-mode operation from a homogeneously broadened medium, such as that of a diode laser, due to gain competition. However, this is not the whole story; if there is strong spatial dependence in the laser mode, some regions of the gain medium can become gain depleted, whilst others do not, allowing modes with diﬀerent spatial dependence to access the undepleted gain and get above the lasing threshold. Competition between these modes can lead to elevated noise levels. An example is the high-power diode array, consisting

213

214

6 Lasers and Ampliﬁers

of a strip of diode gain medium sharing a common cavity. The cavity can support a number of transverse modes, many of which lase simultaneously due to spatial hole burning, thus introducing excess noise. Further, the far-ﬁeld spatial patterns of the diﬀerent modes are diﬀerent, such that diﬀerent spatial regions of the beam can have diﬀerent noise properties. Even in a purely homogeneously broadened medium, spontaneous emission into the lasing mode, which is neglected in our rate equation (6.3), can allow for the existence of weakly lasing longitudinal side modes [9]. The homogeneity of the medium means that the production of a photon in the main mode implies no photon is concurrently emitted in the side modes and vice versa. Hence there is a strong anti-correlation between the side modes and the main mode. If just the main mode is observed, it is found to be very noisy due to the deletion of photons to the side modes [10]. However, the anti-correlation means that the noise tends to cancel when all the light is observed. Unfortunately, diﬀerences in the losses experienced by the main mode and the side modes can lead to a small amount of excess noise, which does not cancel. This can be suﬃcient to destroy squeezing. This problem has been solved using injection locking or external cavities to suppress the side modes. In this way reliable squeezing has been obtained – the details of these experiments are given in Chapter 9. In summary we ﬁnd that lasers themselves are interesting but in many cases complex light sources. Only the single-mode laser operating well above threshold will be QNL at all detection frequencies. For all other cases we have to explicitly derive the intensity noise spectrum or measure it directly. 6.1.5

Laser Phase Noise

In Section 6.1.2 we solved for the steady-state values of the intra-cavity photon number, n, and the upper laser level population, J3 . We also introduced the intra-cavity complex amplitude 𝛼c . Notice however that it is not possible to solve for the phase of 𝛼c . Physically this indicates that the laser phase is metastable and will tend to drift in time. This eﬀect is known as phase diﬀusion. When we linearized the quantum equations of motion to obtain the equations of motion for the ﬂuctuation operators (Eq. (6.12)), we took the phase of 𝛼c to be constant and real. This is of no consequence for the intensity spectrum because ̂ are phase insensitive. However, we need to be ̂ † A) intensity measurements (A more careful if considering phase-sensitive measurements. We can calculate the phase ﬂuctuation spectrum of the laser output from ̃ las |2 ⟩ (6.18) V 2las (Ω) = ⟨|𝜹X𝟐 ̃ las = i(𝜹A ̃ las − 𝜹A ̃ † ). The phase ﬂuctuation spectrum can be where as usual 𝜹X𝟐 las obtained by making a phase-sensitive measurement of the ﬁeld with respect to a stable phase reference, which means using homodyne detection described in Section 8.1.4. Our assumption of constant classical phase (𝛼c real) will only be valid for the phase spectrum if the ﬂuctuations around that constant phase are small. The solution is given by V 2las (Ω) =

(2πΩ)2 + 2(2𝜅)2 (2πΩ)2

(6.19)

6.2 Ampliﬁcation of Optical Signals

At very high frequencies the phase spectrum tends to the quantum noise level, V 2las (Ω) = 1. As we move to lower and lower frequencies, the phase noise becomes larger and larger, eventually diverging at zero frequency. The divergence of the phase noise spectrum at zero frequency indicates that our linearization breaks down for suﬃciently small Fourier frequencies. On long enough timescales, or low enough frequencies, the phase drifts signiﬁcantly, thus invalidating the linear approximation. This is the signature of laser phase diﬀusion. In the absence of technical noise, this phase diﬀusion determines the laser linewidth. In practice phase diﬀusion does not present a major problem as generally the same laser is used as a phase reference as is used to produce the signal. The phase drift is then common to both modes and so is not observed. Thus provided we stay within the coherence length of our laser (see Section 2.3.3 and work at frequencies for which the laser’s intensity noise is at the QNL, a laser beam can represent an excellent realization of an ideal coherent state. 6.1.6

Pulsed Lasers

The focus in this section is primarily CW lasers since they are very common in quantum optics experiments, in precision metrology. The steady optical power is necessary to avoid unwanted non-linear eﬀects and the related laser noise. They are also easy to explain and comprehend. However, the ﬁrst laser demonstrated was pulse, and much of the research work and many applications use pulsed lasers, ranging from nanosecond (ns) down to femtosecond (fs) pulses for commercial systems and attosecond (as) pulse lengths for the highness optical intensities required in research applications. This advance of laser technology uses non-linear processes and the elegant concept of beam pulse compression and expansion, as ﬁrst shown in the late 1980s [11]. These techniques have revolutionized the use of lasers and were honoured by the award of the Nobel prize to Donna Strickland and Gerard Mourou in 2018. In all of these systems, the optical ampliﬁcation is similar, and the optical resonator shares some commonalities. Most of the pulsed lasers have a high repetition frequency Ωrep , frequently, around 100 MHz for femtosecond lasers, and thus a typical applications uses many of these pulses to detect the desired eﬀect. Each pulse has a high peak power, which is useful to drive non-linear eﬀects, and this is key reason to use them in applications such as two- and three-photon imaging, materials processing, ionization, etc. The signal-to-noise ratio however of a measurement is normally measured at signal frequencies that are much lower than the repetition frequency Ωsig < Ωrep , and the inﬂuence of the noise is at low frequencies as well and basically identical to the case of a CW laser. Clear examples of this for the use of pulsed lasers in quantum optics are shown in Section 9.5.

6.2 Ampliﬁcation of Optical Signals It is possible to use an optical medium to amplify the power of a laser beam. For example, we can imagine an input power of 1 mW and an output power

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of 1 W, which means a power gain G of a factor of 1000. A medium that can achieve this purpose is called an active optical medium. Normally it is just a laser medium without optical feedback from a cavity. Such a device would obviously be very useful for practical applications where high power is required. Some of the biggest and most expensive optical systems are the Nd:glass ampliﬁers that have been built to generate powerful laser pulses for fusion studies using inertial conﬁnement [12]. Another application is in optical communications where the beam is attenuated over distance, for example, by losses inside an optical ﬁbre. In this case it is necessary to amplify not only the power but also the modulation sidebands, which contain the information that is communicated. In a ﬁbre system an in-line ampliﬁer, such as an erbium-doped ﬁbre ampliﬁer, is a very useful and attractive device. The gain bandwidth should extend to the largest modulation sideband that is used. Typically this is a few GHz, which is not diﬃcult to achieve. The next question is: how does an optical ampliﬁer aﬀect both the signal and the noise? Does the signal-to-noise ratio vary? It might be easiest to understand the properties of optical ampliﬁers by ﬁrst describing the simpler and more obvious case of an electronic ampliﬁer. Consider a circuit with a linear ampliﬁer that has a power gain Gel (Ω). The gain is frequency dependent; usually Gel (Ω) drops to less than unity at high frequencies. The input into the ampliﬁer is a current with modulation at Ωm . In linear systems we can consider each Fourier component, each modulation frequency, separately and independently of all the others. Ideally the input and output are related by √ 𝛿iout (Ω) = Gel (Ω)𝛿iin (Ω) The variances of modulations, or ﬂuctuations, of the input and the output are linked by Var(iout ) = Gel

Var(iin )

(6.20)

This ampliﬁer aﬀects both the signal and the noise in the same way. In principle, the signal-to-noise ratio will be maintained. In practice a small additional noise component will be added by the ampliﬁer. However there is no fundamental limit to how small this added noise can be. As long as the added noise is much smaller than the noise ﬂoor of the input, the signal-to-noise ratio is maintained. The performance of this system could be improved further if the ampliﬁer is given a reference signal, which is at the same frequency and phase locked to the modulation signal. In this case a signiﬁcant part of the noise, which has random phase, can be suppressed by this apparatus, normally called a lock-in ampliﬁer. The performance of an optical ampliﬁer is quite diﬀerent [13–15]. The input and the output modes can be described in the Heisenberg picture by the operators â in and â out . Suppose our input state is a coherent state, |𝛼⟩. We expect the coherent amplitude to be ampliﬁed such that √ 𝛼out = ⟨̂aout ⟩𝛼 = G 𝛼in This suggests the following operator description: √ â out = G â in

6.2 Ampliﬁcation of Optical Signals

However, this violates the commutation rule [̂aout , â †out ] = G [̂ain , â †in ] = G ≠ 1 This problem is similar to the quantum theory of damping, where a careful consideration of the role of quantum noise has to be made. The vacuum state entering through any unused port has to be taken into account. See the description of a beamsplitter in Section 5.1 for details. In a similar way we have to include the vacuum state in this ampliﬁcation process. The correct description is [16] √ √ â out = G â in + G − 1 v̂ † (6.21) where v̂ is an operator describing an input mode originally in the vacuum state that is mixed in by the ampliﬁer. It is completely independent of â in . The result is that the light, after ampliﬁcation, is considerably noisier than before. The ampliﬁer is increasing not only the power but also the noise level. The variance of the intensity of the ampliﬁed light is V 1out = G V 1in + (G − 1) V 10

(6.22)

Here V 10 = 1 is at the QNL. For large gains (G ≫ 1), the second term results in an increase of an extra 3 dB compared to the classical case (Eq. (6.20)). As shown in Figure 6.8, both the average amplitude and the ﬂuctuations increase. To see that this is indeed the eﬀect of ampliﬁcation by a laser medium, consider again the quantum mechanical equations of motion for the laser (Eq. (6.11)). Suppose that we are below threshold such that 𝜅 > G32 J3 ∕2 but that instead of ̂ in , is injected. Further let us assume vacuum entering the output mirror, a ﬁeld, A that we are suﬃciently below threshold, that there is very little depletion of the upper lasing level, and that we can therefore replace the operator Ĵ3 with its expectation value J3 . We obtain the following equation of motion: √ √ √ G dâ ̂ l + 2𝜅m A ̂ †p ̂ in − G32 J3 𝜹A (6.23) = 32 J3 â − 𝜅 â + 2𝜅l 𝜹A dt 2 Equation (6.23) is now linear and can be solved exactly in Fourier space. Neglecting internal losses (𝜅l = 0) leads to the following solution for the output from the ampliﬁer: √ 2𝜅G32 J3 𝜅 + G32 J3 ∕2 + i2πΩ ̃ in − ̃ †p ̃ out = (6.24) A 𝜹A A 𝜅 − G32 J3 ∕2 + i2πΩ 𝜅 − G32 J3 ∕2 + i2πΩ Figure 6.8 Eﬀect of optical gain on a quantum state.

X2

X1

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The spectrum for either quadrature is then V iout (Ω) =

(𝜅 + G32 J3 ∕2)2 + (2πΩ)2 2𝜅G32 J3 V iin + (𝜅 − G32 J3 ∕2)2 + (2πΩ)2 (𝜅 − G32 J3 ∕2)2 + (2πΩ)2

which is of the same form as Eq. (6.22) with G(Ω) =

(𝜅 + G32 J3 ∕2)2 + (2πΩ)2 (𝜅 − G32 J3 ∕2)2 + (2πΩ)2

We see that in this case the additional quantum noise is introduced via the dipole ﬂuctuations of the gain medium. The 3 dB penalty is imposed by the optical ampliﬁer when it acts equally on all quadratures. Such an ampliﬁer is called phase insensitive. This penalty can be avoided by using an ampliﬁer that is phase sensitive, as will be discussed in the next section.

6.3 Parametric Ampliﬁers and Oscillators Phase-insensitive ampliﬁcation is of limited utility when dealing with quantum ﬁelds because of the excess noise it introduces. We would really like to be able to amplify our quantum-limited light beam without adding additional noise. Let us consider the fundamental commutation problem again. If we write the annihilation operator in terms of the amplitude and phase quadratures, we see the problem clearly: 1 ̂ ̂ + iX𝟐) (6.25) â = (X𝟏 2 If both quadratures are ampliﬁed equally, then â is just multiplied by some gain factor, and via the arguments of the last section, it is clear that an additional ﬁeld, inevitably introducing noise, must be mixed in by the ampliﬁer. However, consider √ the situation in which the amplitude quadrature is ampliﬁed by some amount, K, but the phase quadrature is de-ampliﬁed by the same amount. The output operator would now look like ( ) i ̂ 1 √ ̂ K X𝟏 + √ X𝟐 (6.26) â out = 2 K It is straightforward to conﬁrm that the commutation relations still hold in this case and that this is a legitimate transformation and no additional noise operators are required. Thus, provided that the product of the gains of the two quadratures is unity, noiseless ampliﬁcation of one of the quadratures can be achieved. This is a very important quantum optical interaction. In analogy with the linear ampliﬁer, we can also write Eq. (6.26) in the form √ √ (6.27) â out = G â in + G − 1 â †in where the gain is G=

(K + 1)2 4K

6.3 Parametric Ampliﬁers and Oscillators

For historical reasons a phase-sensitive ampliﬁer is referred to as a parametric ampliﬁer. 6.3.1

The Second-Order Non-linearity

Classically the response of an atomic medium to an incident optical beam can be described by the induced dipole polarization P(E) caused by the beam’s electric ﬁeld E: P(E) = 𝜒E + 𝜒 (2) E2 + 𝜒 (3) E3 + · · ·

(6.28)

A non-linear optical material is one in which, at the available ﬁeld strength, higher than linear terms in the expansion become signiﬁcant. With the invention of the laser, incident ﬁeld strengths became suﬃciently high that many materials could be made to show non-linear behaviour. We will ﬁnd that phase-sensitive ampliﬁcation and other interesting quantum optical phenomena can be produced by materials exhibiting a second-order 𝜒 (2) non-linearity. The key feature of the second-order non-linearity is its ability to couple a fundamental ﬁeld (oscillating at 𝜈) to a harmonic ﬁeld (oscillating at 2𝜈). The Hamiltonian, or energy function, for such an interaction can be written as H = iℏ𝜒 (2) (𝛽 ∗ 𝛼 2 − 𝛼 ∗ 2 𝛽) where 𝛼 is the fundamental whilst 𝛽 is the harmonic ﬁeld. Canonical quantization then leads to the quantum mechanical Hamiltonian ̂ ̂ = iℏ𝜒 (2) (b̂ † â 2 − â †2 b) H This Hamiltonian has an elegant interpretation in terms of photons. The ﬁrst term represents the annihilation of two photons at the fundamental frequency and the creation of one at the harmonic, whilst the second represents the reverse process of a single harmonic photon being annihilated and two fundamental photons being created. The Heisenberg equations of motion (see Section 4.4) are then given by 1 ̂ ̂ H] = 𝜒 (2) â 2 b̂̇ = [b, iℏ â̇ = −2 𝜒 (2) â † b̂

(6.29)

Notice that because the process is energy conserving (and coherent), i.e. two photons of energy h𝜈 equals one of energy h 2𝜈, then no additional vacuum modes are required to be coupled into the ﬁelds. In order that either the up-conversion (fundamental to harmonic) or the down-conversion (harmonic to fundamental) process takes place with some reasonable eﬃciency, we require not only energy conservation but also that momentum is conserved for the converted beam in some particular direction(s). In this way the emission from many dipoles can coherently add to produce a single output beam. This condition is known as phase matching. Let us consider the situation shown schematically in Figure 6.9. A non-linear crystal is placed inside a pair of cavities, one resonant at the fundamental and the other resonant at the harmonic, which are oriented such that there is good phase

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M1 Ain

vA

M2 ζ

αA

Detunings

αB

Aout

κA κB

Δa

Δb

vB =

1 v 2 A

κℓa κℓb

κa = total loss at vA

κb = total loss at vB

Figure 6.9 A parametric medium inside a cavity. The various input, output, and intra-cavity modes are labelled. This system has a threshold and can operate below threshold as a phase-sensitive ampliﬁer of the vacuum modes and above threshold as an oscillator. The subscript A refers to the fundamental frequency, and the subscript B refers to the subharmonic frequency.

matching between the two spatial modes deﬁned by the cavities. By combining the cavity equations of motion derived in Section 5.3.6 with those of Eq. (6.29), we obtain the equations of motion for the internal cavity modes as √ √ (6.30) b̂̇ = 𝜒 (2) â 2 − 𝜅b b̂ + 2 𝜅B B̂ in + 2𝜅lb 𝜹B̂ l â̇ = −2 𝜒 (2) â † b̂ − 𝜅a â +

√ √ ̂l ̂ in + 2𝜅la 𝜹A 2𝜅A A

(6.31)

where 𝜅A and 𝜅B are the loss rates of the input/output mirrors for the â and b̂ modes, respectively. Similarly 𝜅la and 𝜅lb are the respective internal loss rates for the two modes and 𝜅a = 𝜅A + 𝜅la , 𝜅b = 𝜅B + 𝜅lb . We also have the respective input ̂ in and B̂ in and the respective vacuum modes that couple in due to the modes A ̂ l and 𝜹B̂ l . Equations (6.30) and (6.31) can describe a large range internal losses 𝜹A of important quantum optical experiments. 6.3.2

Parametric Ampliﬁcation

Let us now consider parametric ampliﬁcation. We will assume that the harmonic ﬁeld is an intense ﬁeld, which is virtually undepleted by its interaction with the non-linear crystal, whilst the fundamental ﬁeld is ‘small’. We expect down-conversion under these conditions with energy being transferred from the harmonic to the fundamental ﬁeld. We shall also assume that 𝜅a ≪ 𝜅b such that the dynamics of the harmonic mode can be neglected. Under these conditions we can replace the operator b̂ with the c-number 𝛽. We are then left with just a single, linear equation for the â mode: √ √ ̂l ̂ in + 2𝜅la 𝜹A (6.32) â̇ = −𝜒 â † − 𝜅a â + 2𝜅A A where

√ 𝜒 = 2𝜒 (2) 𝛽 = 2𝜒 (2)

2 𝛽 𝜅b in

(6.33)

6.3 Parametric Ampliﬁers and Oscillators

and 𝛽in = ⟨B̂ in ⟩. The parameter 𝜒 is intensity dependent. Equation (6.32) can be solved in Fourier space to give the output mode at the fundamental frequency ̃ out = A

𝜒 2 + 𝜅a2 + (2πΩ)2 2𝜒𝜅a ̃ in + ̃† A A 2 2 (𝜅a + i2πΩ) − 𝜒 (𝜅a + i2πΩ)2 − 𝜒 2 in

(6.34)

Here we have neglected internal losses to the fundamental mode (𝜅la = 0) for simplicity. Equation (6.34) is formally equivalent to Eq. (6.27). The variance of the amplitude quadrature is given by V 1out (Ω) =

(𝜅a2 + 2𝜒𝜅a + 𝜒 2 + (2πΩ)2 )2 𝜒 4 + (𝜅a2 + (2πΩ)2 )2 − 2𝜒 2 (𝜅a2 − (2πΩ)2 )

V 1in (Ω)

(6.35)

where V 1in is the amplitude quadrature variance of the input at the fundamental. The output is an ampliﬁed version of the input amplitude quadrature with no added noise as hoped. This can be seen more clearly at zero frequency where the expression reduces to V 1out (0) =

(𝜅a + 𝜒)2 V 1in (0) (𝜅a − 𝜒)2

(6.36)

As the harmonic intensity is increased 𝜒 becomes larger, the denominator of Eq. (6.36) becomes smaller, and hence the ampliﬁcation increases. On the other hand the phase quadrature (at zero frequency) is given by V 2out (0) =

(𝜅a − 𝜒)2 V 2in (0) (𝜅a + 𝜒)2

(6.37)

which suﬀers increasing de-ampliﬁcation as 𝜒 increases. Notice that if V 2in is at the QNL, then V 2out will be below the QNL, which means V 2out < 1. This is termed squeezing and is discussed in detail in Chapter 9. 6.3.3

Optical Parametric Oscillator

Examining Eq. (6.36) we note that the spectrum diverges as the value of 𝜒 approaches that of 𝜅a . This indicates that an approximation is failing, in this case that of negligible pump depletion. As the ampliﬁcation of the harmonic ﬁeld becomes larger and larger, eventually it starts to signiﬁcantly deplete the harmonic. To see what happens in this case, we must go back to Eqs. (6.30) and (6.31) and linearize. We continue to consider the experimental situation where the cavity for the harmonic mode is either only weakly resonant or even non-existent, that is, we eﬀectively have 𝜅a ≪ 𝜅b . As a result the evolution of the mode b̂ occurs on a much faster timescale than that of the â mode, and we can adiabatically eliminate it by setting its time derivative to zero. We also continue to assume that 𝜅lb ≪ 𝜅b . That means the losses for the mode b̂ can be neglected and Eqs. (6.30) and (6.31) reduce to √ (2)2 √ √ 2 ̂ †̂ ̂l ̂ in + 2𝜅la 𝜹A ̂ † â 2 − 2𝜒 (2) ̂ + 2𝜅A A ̂̇ = − 2𝜒 𝜹a 𝛿a 𝜹a Bin − 𝜅a 𝜹a 𝜅b 𝜅b

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To proceed we ﬁrst consider the semi-classical steady-state solution of this equation. By taking expectation values, and making the approximation that they may be factored, we obtain √ 2 √ 2 𝜒 (2) ∗ 2 2 ∗ 𝛼̇ = − 𝛼 𝛼 − 2𝜒 (2) 𝛼 𝛽in − 𝜅a 𝛼 + 2𝜅A 𝛼in (6.38) 𝜅b 𝜅b The steady-state solution is obtained by setting the time derivative to zero. If we assume that the ﬁrst term on the RHS of Eq. (6.38) is small and can be neglected, then we obtain the parametric ampliﬁcation considered in the previous section. Instead we now retain that term but consider the situation in which there is no input ﬁeld at the fundamental (𝛼in = 0). The equation that needs to be solved is √ 2 2𝜒 (2) ∗ 2 2 ∗ (2) − 𝛼 𝛼 − 2𝜒 𝛼 𝛽in − 𝜅a 𝛼 (6.39) 𝜅b 𝜅b Equation (6.39) has two solutions: either 𝛼 = 0 or 𝛼 ≠ 0. We recognize this as threshold behaviour similar to that of the laser. At a suﬃciently high value of the harmonic pump ﬁeld, the system will start to oscillate and produce coherent output at the fundamental. Under these conditions the device is called an optical parametric oscillator. If 𝛼 ≠ 0 we have √ 2𝜅b 𝜅 𝜅 ∗ 𝛼 𝛼 = − (2) 𝛽in e−i2𝜙𝛼 − a (2)b 2 (6.40) 𝜒 2𝜒 where 𝜙𝛼 is the phase of 𝛼. To be a valid stable solution, 𝛼 ∗ 𝛼 must be real and positive. This is only possible if 𝜙𝛽 − 2𝜙𝛼 = π. For simplicity we take 𝛽in real and negative (𝜙𝛽 = π); then 𝛼 will be real, and the solution for the output photon number is 𝜅 𝜅 nout = 2𝜅a 𝛼 2 = a(2)b2 (𝜒 − 𝜅a ) (6.41) 𝜒 where we have used the deﬁnition of 𝜒 from Eq. (6.33). When 𝜒 < 𝜅a the 𝛼 = 0 solution will be stable; when 𝜒 > 𝜅a then Eq. (6.41) gives the photon ﬂux of the output ﬁeld at the fundamental. Notice that the threshold condition is the same as the point of divergence of the noise spectrum in the previous section. 6.3.3.1

Noise Spectrum of the Parametric Oscillator

We can now investigate the ﬂuctuations of the above threshold fundamental output by linearizing around the semi-classical solution. By linearizing Eq. (6.34) we obtain the following equation of motion for the ﬂuctuations: √ (2)2 2 ̂ † ̇ = − 2 𝜒 𝛼 2 (𝜹a ̂ † + 2𝜹a) ̂ − 2𝜒 (2) ̂𝛿a (𝜹a 𝛽in + 𝛼 𝜹B̂ in ) 𝜅b 𝜅b √ √ ̂ in + 2𝜅la 𝜹A ̂l ̂ + 2𝜅A 𝜹A (6.42) −𝜅a 𝜹a This equation can be solved in Fourier space to give the quadrature ﬂuctuations of the output ﬁeld: √ 2 2𝜅a (𝜒 − 𝜅a ) 4𝜅a − 2𝜒 − i2πΩ ̃ ̃ ̃ Bin 𝜹X𝟏out = 𝜹X𝟏Ain + 𝜹X𝟏 2𝜒 − 2𝜅a + i2πΩ 2𝜒 − 2𝜅a + i2πΩ √ 2 2𝜅a (𝜒 − 𝜅a ) 2𝜅a − 2𝜒 − i2πΩ ̃ out = ̃ Ain + ̃ Bin 𝜹X𝟐 (6.43) 𝜹X𝟐 𝜹X𝟐 2𝜒 + i2πΩ 2𝜒 + i2πΩ

6.3 Parametric Ampliﬁers and Oscillators

where we have neglected internal losses of the fundamental for simplicity ̃ Ain and 𝜹X𝟐 ̃ Ain are the amplitude and phase quadrature ﬂuctuations and 𝜹X𝟏 entering at the fundamental input/output mirrors, respectively. Notice also that non-negligible pump depletion now couples ﬂuctuations from the pump ̃ Bin and 𝜹X𝟐 ̃ Bin , into the fundamental ﬁeld. The quadrature harmonic ﬁeld, 𝜹X𝟏 variances at zero frequency are V 1(0) = 1 +

𝜅a2 (𝜒 − 𝜅a )2

V 2(0) = 1 −

𝜅a2 𝜒2

(6.44)

where we have assumed that the harmonic pump ﬂuctuations are at the QNL. The coupling of the harmonic ﬂuctuations means that the ampliﬁcation of the amplitude quadrature is no longer noiseless above threshold. Strong noise reduction (or squeezing), that is, V 2 ≪ 1, is still present on the phase quadrature close to threshold. However, it becomes negligible far above threshold where the output tends towards a coherent state.

6.3.4

Pair Production

Finally let us return to the parametric ampliﬁer operating far below threshold for another important application of the 𝜒2 interaction. Far below threshold, with no input at the fundamental, we may form the following physical picture: the harmonic beam is mostly transmitted without change through the non-linear medium; however every now and then a harmonic photon is converted into a pair of fundamental photons. This picture suggests that if we counted photons at the fundamental output under these conditions, we should only ever see them arrive as pairs. To test this proposition, we consider the Schrödinger evolution of the fundamental far below threshold where its initial state is the vacuum. The evolution of the fundamental mode is not passive (energy conserving) as energy is being added; however it is unitary (reversible) as no noise is being added. This means we cannot directly use the techniques described in Chapter 4 but we can use a modiﬁed version. Let us work in the Fourier domain and use the fact that we can ̃A ̃ † |0⟩ that is clearly still the vacwrite the vacuum state in the following way: A uum state as the action of creation and then annihilation nullify each other. This motivates us to write the output state as ̃A ̃ † |0⟩ ̃ ̃A |𝜙⟩ = U|0⟩ =U ̃ †U ̃U ̃A ̃ ̃A ̃ † )(U ̃ † )(U|0⟩) = (U ̃ ′A ̃ ′† |𝜙⟩ =A

(6.45)

Equation (6.45) shows that the output state is the eigenstate of the operator ̃ ′A ̃ ′† with eigenvalue ‘one’. Here the dashes indicate reverse time HeisenỸ = A berg evolution (see Section 4.4). Solving this eigenvalue equation leads to (at worst) a second-order recurrence relation for linear or linearized systems.

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Inverting Eq. (6.34) and its corresponding conjugate equation, we obtain † ̃ ̃ ̃ ̃ ̃ − pq∗ A(Ω) Y(Ω) = |p|2 A(Ω) A(Ω) A(−Ω) ∗ ̃ †̃ † 2̃ ̃ −p qA(−Ω) A(Ω) + |q| A(−Ω)† A(−Ω)

(6.46)

where 𝜒 2 + 𝜅a2 + (2πΩ)2 (𝜅a + i2πΩ)2 − 𝜒 2 2𝜒𝜅a q= (𝜅a + i2πΩ)2 − 𝜒 2

p=

(6.47)

Note the presence of creation and annihilation operators for both upper and lower sidebands in the result (see Section 4.5.5). Solving the eigenvalue equation under the assumption that 𝜒 ≪ 𝜅a leads to the following output state for a particular frequency component: |𝜙⟩Ω = |0⟩ +

2𝜒𝜅a 𝜅a2

+ (2πΩ)2

|1⟩Ω |1⟩−Ω

(6.48)

As expected, most of the time ‘vacuum in’ results in ‘vacuum out’. However, occasionally a pair of photons is produced: one in the upper sideband and one in the lower sideband, as required for energy conservation. Given that the input state is a superposition of all frequencies, we may write the total output state as |𝜙⟩T = |0⟩ +

∫

dΩ

2𝜒𝜅a 𝜅a2

+ (2πΩ)2

|1⟩Ω |1⟩−Ω

(6.49)

Operating in this conﬁguration the system is often referred to as a down-converter or photon pair generator. The technical speciﬁcations have been improved steadily via a variety of new non-linear media, engineering of the non-linear crystals and novel optical arrangements. The main emphasis is on improving the brightness of the source, measured as the number of pairs per second, the suppression of individual photons and other noise sources, and the quality of the optical modes of the pairs. Optical down-converters and related systems are presently the workhorses of single-photon experiments, as described in Chapters 12 and 13. The quality of the entanglement that has been measured is a direct indication of the quality of the source of photon pairs.

6.4 Measurement-Based Ampliﬁers Ampliﬁcation of quantum states can also be implemented via techniques that employ partial measurement of the state followed by feedforward of the measurement results. We will refer to such devices as measurement-based ampliﬁers. These can be divided into two distinct classes: (i) deterministic and (ii) heralded. Deterministic measurement-based ampliﬁers implement ampliﬁcation techniques like those discussed in the previous section. However, they avoid the need for in-line, non-linear crystals or other gain media. This can be quite an advantage as coupling into and out of non-linear media can be ineﬃcient and imperfections in the media can lead to unwanted technical noise. In contrast,

6.4 Measurement-Based Ampliﬁers

heralded measurement-based ampliﬁers can beat the quantum limits discussed in the previous section. This is allowed because these devices are probabilistic in their operation. Nevertheless successful operation is heralded by particular measurement results. Hence such devices can be useful for certain applications.

6.4.1

Deterministic Measurement-Based Ampliﬁers

We consider ﬁrst a noiseless, phase-sensitive ampliﬁer. The idea here is to amplify one particular quadrature whilst maintaining the signal-to-noise ratio of any signals on that quadrature (i.e. not adding additional noise). Initially we are not interested in what happens to the other quadrature. A brute force measurement-based technique is the complete detection and re-emission of the light using high eﬃciency detectors and LEDs [17, 18]. The disadvantage here is that no phase reference or coherence exists between the input and output beams. An alternative, which does not require any non-linear process whilst its output still retains optical coherence, was developed and demonstrated by Lam et al. using electro-optic feedforward [19]. This scheme is based on partial detection of the light with a beamsplitter and detector (see Figure 6.10A). The light reﬂected from the beamsplitter is detected, and the photocurrent is ampliﬁed and fed forward to the transmitted beam via an electro-optic modulator (EOM). The electronic gain and phase can be chosen such that the intensity modulation signals carried by the input light are ampliﬁed, whilst the vacuum ﬂuctuations that enter through the empty port of the beamsplitter are cancelled, thus making the ampliﬁcation noiseless. Since not all of the input light is destroyed, the output is still coherent with the input beam. Here a half-wave plate controlled the transmittivity, 𝜀1 , of the polarizing beamsplitter. On the in-loop beam, a high eﬃciency (𝜀2 = 0.92 ± 0.02) photodetector was used. The photocurrent is then passed through multiple stages of RF ampliﬁcation and ﬁltering to ensure suﬃcient gain. An amplitude modulator is formed by using the EOM in conjunction with a polarizer, and ﬁnally the ﬂuctuation spectrum of the output is measured using detector Dout and a spectrum analyser. This system can be modelled with the linearized operator approach as used for the electro-optic noise eater (see Section 8.3). We obtain the following transfer functions for the spectrum of the output beam normalized to the QNL: √ |√ |2 Vout (Ω) = 𝜀3 | 𝜀1 + 𝜆 (1 − 𝜀1 )𝜀2 | Vin (Ω) | | √ |√ |2 + 𝜀3 | (1 − 𝜀1 ) − 𝜆 𝜀1 𝜀2 | V1 | | |2 | √ + 𝜀3 |𝜆 (1 − 𝜀2 )| V2 | | + (1 − 𝜀3 )V3 (6.50) where the electronic gain 𝜆(𝜔) is in general a complex number. Vin (Ω) is the amplitude noise spectrum of the input ﬁeld. The vacuum noise spectra due to the beamsplitter V1 , the in-loop detector eﬃciency V2 , and the out-of-loop losses V3 are shown separately to emphasize their origins. All vacuum inputs are QNL, i.e.

225

6 Lasers and Ampliﬁers

ε2

Dil

Filter

Amplifier Dout

PBS Signal

ε1 λ/2

EOM AM

δv1

(A)

ε3

–73 Input signal

2.8 dB (i)

–78 Measured noise power (dBm)

226

(ii)

–83 –75 Output signal

(c)

2.6 dB

–80 (a)

–85 (b)

–90 5 (B)

10

15

20

25

Frequency, f (MHz)

Figure 6.10 (A) Schematic of the feedforward experiment. Dil , in-loop detector; Dout , out-of-loop detector; PBS, polarizing beamsplitter; 𝜆∕2, half-wave plate; AM, amplitude modulator. (B) Top: noise spectra of the squeezed input beam. Bottom: noise spectra of the output beam. (a) Direct detection of the input light after lossy (86%) transmission. (b) Output noise spectrum without feedforward (eﬀectively 98.6% loss). (c) With optimum feedforward gain the signal and noise are ampliﬁed with little loss of SNR. Source: After Lam et al. [19].

V1 = V2 = V3 = 1. Due to the opposite signs accompanying the feedback parameter 𝜆 in Eq. (6.50), it is possible to amplify the input noise (ﬁrst term) while cancelling the vacuum noise from the feedforward beamsplitter (second term). The third and fourth terms of Eq. (6.50) unavoidable experimental √ represent √ losses. In particular if we choose 𝜆 = 1 − 𝜀1 ∕ 𝜀1 𝜀2 , the vacuum ﬂuctuations from the beamsplitter, V1 , are exactly cancelled. Then, under the optimum condition of unit eﬃciency detection and negligible out-of-loop losses (𝜀2 = 𝜀3 = 1),

6.4 Measurement-Based Ampliﬁers

we ﬁnd Vout (Ω) =

1 V (Ω) 𝜀1 in

(6.51)

That is, the ﬂuctuations are noiselessly ampliﬁed by the inverse of the beamsplitter transmittivity. Hence the system ideally can retain the SNR for a signal gain of G = 1∕𝜀1 . This system was used to demonstrate low noise ampliﬁcation of amplitude squeezed light (see Chapter 9). The second harmonic output from a singly resonant frequency doubling crystal (Section 9.4.3) provided the input beam. The top half of Figure (6.10B) shows the input noise spectra. Trace (i) shows the QNL, and trace (ii) is the noise spectrum of the input light. Regions where (ii) is below (i) are amplitude squeezed. The maximum measured squeezing of 1.6 dB is observed in the region of 8–10 MHz on a 26 mW beam. A small input modulation signal is introduced at 10 MHz that has SNRin = 1.10 ± 0.03. Other features of the spectra include the residual 17.5 MHz locking signals of the frequency doubling system (see Section 9.4.3 and the low-frequency roll-oﬀ of the photodetector, introduced to avoid saturation due to the relaxation oscillation of the laser. The bottom half of Figure 6.10B shows the noise spectra obtained from the single output detector. The input beam is made to experience 86% downstream loss. As trace (a) shows, the SNR is greatly reduced by the attenuation. Signal ampliﬁcation can be performed by ﬁrst introducing a beamsplitter of reﬂectivity 90%. This further attenuates the output beam to 𝜀tot = 𝜀1 𝜀3 = 0.014. With no feedforward gain, as trace (b) shows, the modulation signal is completely lost, and the trace itself is QNL to within 0.1 dB over most of the spectrum. Finally, by choosing the optimum feedforward gain, trace (c) shows the ampliﬁed input signal with SNRout = 0.82 ± 0.03 and G = 9.3 ±0.2 dB. Note that both (b) and (c) are of the same intensity; hence the output signal is now signiﬁcantly above the QNL. This means the ampliﬁed output is far more robust to losses than the input. The results are best described by the transfer coeﬃcient Ts = SNRout ∕SNRin . The spectra in Figure 6.10B corresponds to a value of Ts = 0.75 ± 0.02, in good agreement with the theoretical result of Ts = 0.77. This is to be compared with the performance of an ideal phase-insensitive ampliﬁer, with similarly squeezed light, of Ts ≈ 0.4. Signal ampliﬁcation via feedforward is a versatile technique that can be adapted to various diﬀerent applications. For example, by replacing the homodyne detector in the feedforward loop with a dual homodyne detection set-up, one can implement phase-insensitive linear ampliﬁcation. In particular, now the light reﬂected oﬀ the beamsplitter is split again on a 50 : 50 beamsplitter and sent to two homodyne detectors, one of which measures the amplitude quadrature and the other the phase quadrature. The signals are fed forward to amplitude and phase modulators, respectively, beam. By choosing the √ in the transmitted √ feedforward gains to both be 𝜆 = 1 − 𝜀1 ∕ 𝜀1 𝜀2 , one can obtain (in the unit eﬃciency limit) the input/output relation √ â out =

1 â + 𝜖1 in

√

1 − 1 v̂ † 𝜖1

(6.52)

227

228

6 Lasers and Ampliﬁers

Given the substitution G = 𝜖1 , Eq. (6.52) becomes formally equivalent to Eq. 1 (6.21). Hence quantum-limited phase-insensitive ampliﬁcation can be implemented in this way without needing a laser gain medium (and avoiding various technical noise sources associated with gain media). The vacuum mode, v̂ , leading to the quantum ampliﬁcation noise, is coupled in via the dual homodyne beamsplitter in this case. An experimental demonstration of this technique was carried out by Josse et al. at the University of Erlangen [20]. Coherent state sideband states at 14.3 MHz on a 1064 nm Nd:YAG laser were ampliﬁed using this method, achieving transfer coeﬃcients within about 10% of the theoretical limit (see Eq. (6.22)) and in good agreement with the theory when ﬁnite detection eﬃciency is included. For many applications we are only interested in the ampliﬁcation of the quantum state carried on the sidebands (see Section 4.5.5), i.e. the signal. However, if we are considering an optical beam with a bright carrier, an unavoidable consequence of this signal ampliﬁcation technique is the reduction in intensity of the carrier beam. It turns out that injection locking can be used to amplify the intensity of the output beam without aﬀecting the signal or noise (see Section 8.5). When the signal frequency is much larger than the linewidth of the laser cavity, the output spectrum of the injection-locked ﬁeld is the same as the input spectrum (Vout = Vin ). Because the injection-locked output is of a higher optical intensity, we can thus regain the optical intensity lost by the feedforward ampliﬁcation scheme. This was demonstrated by E. H. Huntington et al. [21] using the output of the feedforward set-up to injection lock a Nd:YAG nonplanar ring oscillator. A transfer coeﬃcient of Ts = 0.88 ± 0.05 with signal gain of G = 9.3 ± 0.2 dB was obtained for a range of output powers from 14 to 350 mW. Hence with this set-up it is possible to independently vary the amount of signal and carrier ampliﬁcation with minimal loss of signal-to-noise ratio. 6.4.2

Heralded Measurement-Based Ampliﬁers

So far we have seen how measurement-based ampliﬁers can mimic the performance of quantum-limited ampliﬁers but without the technical diﬃculties associated with non-linear optical media. We now consider a situation in which the measurement result obtained heralds whether the ampliﬁcation has been successful or not. Because such an ampliﬁer is nondeterministic, i.e. only certain results herald success, it is able to exceed the quantum limits that apply to deterministic ampliﬁers. Consider a device that performs the transformation |𝛼⟩⟨𝛼| → 𝜌(𝛼) = P|g𝛼⟩⟨g𝛼| + (1 − P)|0⟩⟨0|

(6.53)

where g is a real number obeying |g| > 1, |𝛼⟩ is an optical coherent state with complex amplitude 𝛼, and P is a probability. The RHS of Eq. (6.53) is a mixed state, but we assume a heralding signal identiﬁes which term in the output density operator has been produced by any particular run of the device: the ﬁrst term being produced by a successful run and the second term by an unsuccessful run. Thus with probability P the coherent amplitude of the input is ampliﬁed

6.4 Measurement-Based Ampliﬁers

whilst keeping the heralded state a pure coherent state. The quantum limits discussed in the previous sections would clearly be violated if this occurred with P = 1. What probability of success is allowed? The linearity of quantum mechanics requires that the distinguishability of two quantum states cannot be increased by any transformation. This condition is equivalent to requiring that the magnitude of their overlap does not decrease under any transformation. Consider the input states |0⟩ and |𝛼⟩. We require |⟨0|𝛼⟩|2 ≤ ⟨0|𝜌(𝛼)|0⟩ = P|⟨0|g𝛼⟩|2 + 1 − P. ′ 2 Inserting the values of the overlaps using |⟨0|𝛼 ′ ⟩|2 = e−|𝛼 | and taking the limit of small 𝛼, we ﬁnd 1 (6.54) P≤ 2 g We conclude that provided P is bounded according to Eq. (6.54), then the transformation of Eq. (6.53) is physically allowed for small coherent states. The transformation of Eq. (6.53) was discussed by Ralph and Lund who referred to it as heralded noiseless linear ampliﬁcation (NLA) [22]. They proposed a method for realising NLA based on single-photon sources, single-photon detection, and linear optics. The simplest NLA circuit is shown schematically in Figure 6.11a. The optical mode to be ampliﬁed is interacted with a single-photon ancilla as shown. This interaction is a generalization of the quantum scissors introduced by D. Pegg et al. [23]. The interaction is successful if one and only one photon is counted at one or other of the indicated ports. |ψ〉 |0〉

|0〉

Ns p l i t t e r

Tˆ |ψ〉

Ns p l i t t e r

A A

A

〈0|

〈0|

(a)

|in〉 A

=

a″

a′

〈1|〈0| or 〈0|〈1|

a′″

|0〉

|out〉

η |1〉 (b)

Figure 6.11 Schematic of the noiseless linear ampliﬁer. (a) The interaction labelled ‘A’ interacts a single-photon ancilla with an input beam as shown. The upper beamsplitter is 50 : 50, whilst the lower beamsplitter has transmission 𝜂 as shown. The interaction succeeds if a single count is recorded at a′ and no count at a′′ , or vice versa. (b) The N-splitter is an array of beamsplitters that evenly divides the input beam. The second N-splitter coherently recombines the beams and is considered to have succeeded if no light exits through the other ports, as determined by photon counters. Each arm contains the interaction shown in (a).

229

230

6 Lasers and Ampliﬁers

Consider the eﬀect of this device on a small input coherent state, i.e. |𝜓⟩ = |𝛼⟩, where we assume 𝛼 ≪ 1. The action of the generalized quantum scissor is to truncate the coherent state to ﬁrst order and simultaneously amplify it. Speciﬁcally, detection of a single photon at output port a′ and zero photons at output port a′′ , or detection of a single photon at output port a′′ and zero photons at output port a′ , produces the transformation ) √ √ ( 2 𝜂 1−𝜂 † − |𝛼|2 |𝛼⟩a′ → e (6.55) â 𝛼 |0⟩a′′′ 1± 2 𝜂 where the plus sign corresponds to the former case and the minus sign to the latter case. If the latter case occurs, the phase ﬂip can be corrected by feeding forward to a phase shifter. Equation (6.55) is approximately equivalent to 1

(1−g 2 )|𝛼|2

(6.56) |𝛼⟩ → 𝜂 2 e− 2 |g 𝛼⟩ √ where g = (1 − 𝜂)∕𝜂, provided that g𝛼 ≪ 1 holds. Thus for 𝜂 < 1∕2 we have g > 1, and hence the device in Figure 6.11a achieves NLA as per Eq. (6.53). The probability of success is given by the norm of the state P = 𝜂e−(1−g

2

)|𝛼|2

≈

1 g2 + 1

(6.57)

in agreement with the condition of Eq. (6.54). This device can be scaled up such that it can amplify larger coherent states as depicted in Figure 6.11b; however the complexity of the set-up increases rapidly, and the probability of success decreases rapidly [22]. The basic operation of the NLA in the low photon number regime was ﬁrst demonstrated by G.Y. Xiang et al. [24], F. Ferreyrol et al. [25], and A. Zavatta et al. [26]. The ﬁrst two publications implemented the quantum scissor approach described above. However, Zavatta et al. used a diﬀerent approach based on single-photon subtraction and addition that was able to noiselessly amplify somewhat larger photon number states with high ﬁdelity. The NLA has been proposed as a tool for improving various quantum information protocols, and such demonstrations are discussed in Chapter 13.

6.5 Summary In this chapter we have examined the theory of laser oscillators, the conditions under which real lasers behave like the simpliﬁed theory, and the characteristics of the laser light under those conditions. We conclude that well-built lasers can produce output beams that behave approximately like coherent states. We call these QNL lasers or beams. We considered the theoretical limits to ampliﬁcation of quantum limited light and discussed the theory of the non-linear 𝜒2 interaction. We found a general description of this process and considered the speciﬁc examples of parametric ampliﬁcation, parametric oscillation, and pair production. We also discussed measurement-based quantum ampliﬁers based on electro-optic feedforward and heralded devices based on photon counting. The latter type could circumvent

References

the quantum limits to ampliﬁcation because they are nondeterministic. Experiments based on examples discussed in this chapter and many variations on these themes will feature strongly in the following experiments in quantum optics.

References 1 2 3 4 5

6 7

8 9

10 11 12

13 14 15 16 17

18

19

Siegman, A.E. (1986). Lasers. University Science Books, and many reprints. Einstein, A. (1917). Zur Quantentheorie der Strahlung. Phys. Z. 18: 121. Saleh, B.E.A. and Teich, M.C. (2007). Fundamentals of Photonics. Wiley. Koechner, W. (1999). Solid State Laser Engineering, 5e. Springer-Verlag. Allan, D.W. (1987). Time and frequency (time-domain) characterization, estimation, and prediction of precision clocks and oscillators. ICEE Trans. Ultrason. Ferroelectr. Freq. Control 6: 647. Ralph, T.C. (2003). Squeezing from lasers. In: Quantum Squeezing (ed. P. Drummond and Z. Ficek), 141. Springer-Verlag. Harb, C.C., Ralph, T.C., Huntington, E.H. et al. (1997). Intensity-noise dependence of Nd:YAG lasers on their diode-laser pump source. J. Opt. Soc. Am. B 14: 2936. Yariv, A. (1989). Quantum Electronics. Wiley. Inoue, S., Ohzu, H., Machida, S., and Yamamoto, Y. (1992). Quantum Correlations between longitudinal-mode intensities in a multi-mode squeezed semi-conductor laser. Phys. Rev. A 46: 2757. Marin, F., Bramati, A., Giacobino, E. et al. (1995). Squeezing and inter-mode correlations in laser diodes. Phys. Rev. Lett. 75: 4606. Strickland, D. and Mourou, G. (1985). Compression of ampliﬁed chirped optical pulses. Opt. Commun. 56: 219. Di Nicola, J.M., Bond, T., Bowers, M. et al. (2018). The national ignition facility: laser performance status and performance quad results at elevated energy. Nucl. Fusion 59, 3. Haus, H.A. and Mullen, J.A. (1962). Quantum noise in linear ampliﬁers. Phys. Rev. 128: 2407. Caves, C. (1982). Quantum limits on noise in linear ampliﬁer. Phys. Rev. D 26: 1817. Haus, H.A. (2000). Electromagnetic Noise and Quantum Optical Measurements. Springer-Verlag. Vogel, W. and Welsch, D.-G. (1994). Lectures on quantum optics. Berlin: Akademie Verlag. Goobar, E., Karlsson, A., and Björk, G. (1993). Experimental realization of a semiconductor photon number ampliﬁer and a quantum optical tap. Phys. Rev. Lett. 71: 2002. Roch, J.-F., Poizat, J.-Ph., and Grangier, P. (1993). Sub-shot-noise manipulation of light using semiconductor emitters and receivers. Phys. Rev. Lett. 71: 2006. Lam, P.K., Ralph, T.C., Huntington, E.H., and Bachor, H.-A. (1997). Noiseless signal ampliﬁcation using positive electro-optic feedforward. Phys. Rev. Lett. 79: 1471.

231

232

6 Lasers and Ampliﬁers

20 Josse, V., Sabuncu, M., Cerf, N.J. et al. (2006). Universal optical ampliﬁcation

without nonlinearity. Phys. Rev. Lett. 96: 163602. 21 Huntington, E.H., Lam, P.K., Ralph, T.C. et al. (1998). Independent noiseless

signal and power ampliﬁcation. Opt. Lett. 23: 540. 22 Ralph, T.C. and Lund, A.P. (2009). Nondeterministic noiseless linear ampliﬁ-

cation of quantum systems. AIP Conf. Proc. 1110: 155. 23 Pegg, D.T., Phillips, L.S., and Barnett, S.M. (1998). Optical state truncation by

projection synthesis. Phys. Rev. Lett. 81: 1604. 24 Xiang, G.Y., Ralph, T.C., Lund, A.P. et al. (2010). Noiseless linear ampliﬁcation

and distillation of entanglement. Nat. Photon. 4: 316. 25 Ferreyrol, F., Barbieri, M., Blandino, R. et al. (2010). Implementation of a non-

deterministic optical noiseless ampliﬁer. Phys. Rev. Lett. 104: 123603. 26 Zavatta, A., Fiurášek, J., and Bellini, M. (2011). A high-ﬁdelity noiseless ampli-

ﬁer for quantum light states. Nat. Photon. 5: 52.

233

7 Photon Generation and Detection What are photons? In the simplest language, they are what make a photon counter click. A practical working deﬁnition of a photon links directly with the process of detection. Photons are not directly detected in the laboratory: they are nearly always converted to electrical signals, and it is these events that are detected and analysed. In this way we can recover the information that is encoded on the light and is carried by the photons. Light propagates as a high frequency electromagnetic wave, and, as with all such ﬁelds, there are two methods of detection: ﬁeld detection, where both amplitude and phase information can be measured, and intensity detection, which yields no phase information. In the radio, microwave, and mid-infrared (mid-IR) regions of the spectrum, ﬁeld detection can be done directly. In the near-infrared (near-IR), visible, and ultraviolet (UV) regions, ﬁeld detection is achieved indirectly via homodyne or heterodyne techniques: the light ﬁeld is interfered with a reference ﬁeld of known amplitude and phase of either the same frequency (homodyne) or shifted frequency (heterodyne), and the resultant ﬁeld is detected with intensity detectors, as described in Section 8.1.4. Thus in the optical region of the spectrum, intensity detectors are used for both ﬁeld and intensity detection. Broadly speaking there are two diﬀerent ways of recording light intensity: we can detect individual photons or we can measure currents generated by a stream of photons. The time response of the photodetector and the magnitude of the photon ﬂux determine which mode of operation is suitable. Both technologies exist in parallel and are used for diﬀerent purposes.

Figure 7.1 The origin of quantum noise. A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

234

7 Photon Generation and Detection

Table 7.1 Comparison of diﬀerent detection modes, the type of detection, the output results, the processing required, and the applications made possible with this type of detection. Type

Detection

Result

CW weak

Fast, single photon

|0⟩, |1⟩, …

Many samples Pulsed

Fast, synchronized

weak

Single photons

|1⟩ on demand

Processing

Application

Store

Q-info, Q-logic

each event

QKD

Store

Q-info, Q-logic

each event

Many samples CW strong

Slow

Modulation

Cont. homodyne

Sensors, anologue +

averages

Noise variance

Record + FFT

dig. communication

Wigner function

Tomography

QKD

Pulsed

Slow

Modulation

Cont. homodyne

Sensors, analogue +

strong

averages

Noise variance

Record + FFT

dig. communication

Δtmeas ≫ Δtrep

Wigner function

Tomography

Hybrid

Fast, CW, and

Negative

Cont. homodyne

single photon

Wigner function

and post-selection

Many samples

Cat states

Tomography

Q-info, Q-logic

Table 7.1 provides an overview of the types of detection that are required for the diﬀerent applications. All of these are now possible with the diﬀerent photodetectors and processing electronics that are described in this chapter (Figure 7.1). If the photon ﬂux is low enough and the detector is fast enough, with a recording time Δtmeas shorter than the time intervals between the photons, we can record the individual events, resulting in a sequence of numbers, typically 0 for no photon and 1 for at least one photon in each of the counting intervals. This would be the most fundamental way of sending information using this elementary quantum state of light. In addition photons can carry more complex information that is encoded in the quantum state using additional degrees of freedom such as polarization, momentum, or frequency. In these cases we use photons in beams that have several independent modes, for example, a beam with orthogonal polarizations, with multiple spatial modes or optical frequencies. Photons are ideal for these types of applications; since they travel at the speed of light (ﬂying qubits), they interact only weakly with the environment, even over long distances, and photons can be manipulated very eﬃciently and with linear and non-linear optics. By directing the modes to separate detectors, we can record the state of the qubit. Many quantum communication protocols make use and demand exactly one single-photon travelling in a mode, or channel, and more than one photon recorded in one click would compromise the results. For this reason increasing eﬀort is made to both create single photons on demand and to have detectors that can alert to the presence of additional photons that could compromise the security of the communication. Ideally, we would have detectors

Photon Generation and Detection

that can distinguish the number of photons in every counting interval and could then work with multiphoton states. As an alternative to photon counting, we can integrate the individual events and continuously record a photocurrent i(t) that varies in time. We average during the measurement interval Δtmeas over a large number of photons and analyse the optical power P(t) that contains information in the form of a slowly varying modulation of the light, up to a maximum modulation frequency Ωmod < 1∕2Δtmeas . There will always be some fast ﬂuctuations, or noise, and the more accurate the information measured in the photocurrent will get, the more photons have been detected. We can characterize the statistics of the ﬂuctuations by recording and analysing large sets of these short measurement intervals. Depending on the nature of the light source, the photons arrive with quite diﬀerent temporal distributions: for example, thermal (a natural light source), Poissonian (a coherent light source such as a laser), or regularly spaced in time (a Fock state or single-photon source). The information of the recorded power in a particular beam of light is frequently not dependent on the underlying statistics of the light source. Indeed, many quantum optics phenomena based on self-interference are insensitive to the photon arrival statistics with the general caveat that the light is suﬃciently attenuated such that the chance of a multiple photon count in any given measurement interval is very low. Examples include quantum key distribution (QKD) and ‘interaction-free’ measurements, which are discussed in Chapters 12 and 13. In principle, photons are detected as discrete events, and the experiment is about registering the place and time of arrival of these events. In arguably the ﬁrst quantum optics experiment, G.I. Taylor attenuated a thermal light source down to the single-photon level, passed the light through a double slit, and recorded the photon arrival positions as an image on photographic ﬁlm, as discussed in Chapter 3. It was the race to develop television that led to the development of the photomultiplier in the 1930s, which allowed the multiplication of the charge created by a single photon into a useful electronic pulse that could be made audible as a click or as a dot on a screen [1, 2]. In the 1960s the Geiger-mode avalanche photodiode (APD) made solid-state optical single-photon detection possible [3]. Many more innovations, some in the last decade, have extended our detector technology. We can now combine speciﬁcally designed light sources with the corresponding detectors to cater for the various applications. This includes imaging, sensing with strong laser beams or a few photons, and reading optical memories. It could be classical communication with either analogue or digital signals or quantum information applications such as secure communication with QKD, or it could be quantum logic, quantum memories, or potentially quantum computation. For each case there are now sources and detectors available. Many sophisticated applications involve the interference of several modes, which are all individually in a single photon state. These include non-classical interference, teleportation, and quantum computation as they are discussed in Chapter 13 on single-photon experiments and quantum information. In these experiments it is critical that the various modes are indistinguishable. That is, they have identical distributions in energy, which means photon number and frequency, momentum (spatial modes), and time. In general the modes must

235

236

7 Photon Generation and Detection

also have states of deﬁnite photon number, i.e. Fock states. In the case of pulsed modes, this requires pulses that are identical from pulse to pulse. Finally we want to know when such a pulse arrives, providing us with a source for photons on demand, which would be the ultimate quantum light source [4].

7.1 Photon Sources In practical terms, it is not easy to build a source that meets all these requirements. Approaches tried to date have been numerous and include single atoms [5], colour centres [6], molecules [7], atomic ensembles [8], quantum wells, and quantum dots [9]. We can describe and classify photon sources by a number of critical parameters and compare them with the requirements for the ideal single-photon source. Are they deterministic or probabilistic? Probabilistic sources, such as spontaneous down-converters, produce pairs of photons that are entangled but appear at random times. They can be used as heralded sources, where one photon is used to determine the time when the other photon can be observed. Many of the more recent approaches hold the promise of a deterministic photon source where the photons arrive at predetermined times or can be triggered from the outside [4], allowing the use of a range of advanced quantum information protocols. To date in 2018 we are approaching the goal of building such an idealized photon gun [10, 11]. At what temperature do they operate? Some materials work well close to room temperature (300 K), and some require precise temperature control at an elevated temperature in order to optimize the non-linear processes through phase matching. Others, in particular those designed for photon number resolution, work with ultra-cold atomic samples; individual atoms or ions trapped in a vacuum might require cryogenic conditions. In which wavelength region can they be used, and are they broadband or restricted to an atomic transition? Whilst the non-linear crystalline materials have broadband characteristics and work in the visible regime, some require near-IR wavelengths or near-UV wavelengths. Some sources are tunable and can be used in systems that include optical materials with speciﬁc spectral properties, in particular atomic samples or speciﬁc organic materials such as dyes or even biological samples. What is the eﬃciency of extracting the photon from the source and sending it into the detector, including any spectral ﬁltering that would be necessary for a typical quantum information application? This property determines the choice of protocol to be used. A low eﬃciency makes the apparatus slow. Whilst we can only process a small subset of the pulses, the results otherwise do not change. In deterministic sources we know that we will have only one photon at a time, but we do not know exactly when it works. The ideal photon on-demand source has close to 100% eﬃciency. Do they produce a good single spatial mode? This is an important ability because if the probability of ﬁnding the single photon is spread across incoherently related modes, the photon state will be mixed. In many situations, we want

7.1 Photon Sources

to overlap the diﬀerent modes at the output to create a quantum interference at the output of the apparatus. The interference will be maximized if each source operates in a single spatial and frequency mode. One measure of the quality of the single-photon source is the g (2) function. Ideally the value should be zero, which indicates that no more than one photon is contained in each of the outputs from the source and the apparatus is perfect. Values as low as 4 × 10−4 have been achieved, which is extremely useful in practical terms and almost closes potential loopholes for quantum demonstrations. A faint laser would be the simplest approach. It allows to direct all the light on to the detector and can create a single mode. However, photon arrival times are random and thus g (2) = 1. Many sophisticated sources are now available, as the review written by Eisaman et al. shows very clearly [12]. Table 7.2 lists the most common and advanced sources and presents their typical speciﬁcation. However, note that g (2) says nothing about the eﬃciency or modal purity of the source. The most common technology used in quantum optics experiments to date is still spontaneous parametric down-conversion (spontaneous PDC) and is probabilistic. PDC sources are bright with high photon count rates, can produce clean optical beams with well-deﬁned frequency and spatial characteristics [13–16], and can produce tunable entangled modes [17, 18]. In a spontaneous PDC, high frequency photons are passed through a non-linear crystal, where they are converted down into two lower frequency daughter photons, born within 10s of Table 7.2 Comparison of diﬀerent photon sources and their speciﬁcations.

Source

Type

Temperature (K)

Wavelength

Eﬃciency

Mode

g(2)

Faint laser

Prob

300

vis-IR

1

Single

1

PDC bulk

Prob

300

vis-IR

0.6–0.8

Multi

0.0014

PDC pp

Prob

350 ctr

vis-IR

0.85

Multi

—

PDC waveguide pp

Prob

350 ctr

vis-IR

0.07

Single

0.0007

FWM in ﬁbre

Prob

300

vis-IR

0.02

Single

0.02

PDC multiplexed

Prob/Det

300

vis-IR

0.1

Single

0.08

Q-dot (GaN)

Det

300

340–370 nm

10 dark counts per second with moderate Peltier cooling), and have a high spectral bandwidth (detecting light from 185 to 900 nm). Unfortunately, they are not very eﬃcient at turning photons into electrons: 𝜂 ranges from fractions of a percent at high wavelengths (600–900 nm) to a few percent at low wavelengths (185–300 nm) to tens of percent near the peak wavelengths. Currently the best commercial PMT achieves 𝜂 ∼ 40% at 550 nm. A photon counter with higher eﬃciencies than the PMT is the APD used in a regime where it operates as single-photon avalanche diode (SPAD). An APD works via the internal photoelectric eﬀect, where light incident on a semiconductor surface promotes an electron from the valence to the conduction band,

241

7 Photon Generation and Detection

Electrons

Energy

242

Ebg

hν

Figure 7.2 Semiconductor and band gap.

Photon

Holes

as illustrated in Figure 7.2. Of course, this requires a material with a band gap smaller than the photon energy, h𝜈 – a diﬃcult task in certain regions of the spectrum. As with a PMT, a single electron is typically liberated by a single photon, so SPADs are also run in Geiger mode to amplify the single electrons to macroscopic current pulses. Commercially available APDs are often packaged with an ampliﬁer, a discriminator, and a transistor–transistor logic (TTL) output driver, which turns the current pulses into pulses with well-deﬁned height and rise time, so-called TTL voltage pulses (deﬁned as Logic 1 = 2–5 V and Logic 0 = 0–0.8 V), which make them very user-friendly. These detectors are also reasonably fast, with maximum counting rates of ∼10 × 106 photons/s, linear to ∼ 106 photons/s, and suﬀer moderate dark noise (>100 dark counts per second). The wavelength range is smaller than PMTs but decidedly more eﬃcient, depending on the way they are designed. Examples of the possible peak eﬃciencies are for silicon SPADs are 50% at 450 nm, 70% at 600 nm, and 50% at 830 nm and for InGaAs SPADs, up to 33% at 1060 nm. An even more eﬃcient device is the solid-state photomultiplier (SSPM). This is a semiconductor material doped with neutral atoms where the energy levels of the dopant atoms are very close to the conduction band of the semiconductor (e.g. Si doped with As has a 54 meV gap). Brieﬂy, at an operating temperature of 6 ∼ 7 K, the electrons are frozen onto the impurity atoms, as they lack enough thermal energy to be excited into the conduction band. When an incoming photon is absorbed, it generates an electron–hole pair: the electron is accelerated to the anode, impact ionizing other impurities as atoms on the way and so generating an electron cascade within the device. Since little energy is needed to photoexcite an impurity atom so close to the conduction band, SSPMs are sensitive well into the infrared (IR): from 30 μm down to the visible, except for wavelengths in the silicon absorption band, which unfortunately includes the common telecom wavelengths. The broad IR sensitivity means that SSPMs must be carefully screened so that only light at the desired wavelength is coupled to the detector – or else ambient IR will quickly lead to detector saturation. So screened, and corrected for optical losses, the quantum eﬃciency of an SSPM in the visible has been measured to be 96±3% [17]. In an SSPM, photoabsorption and gain occur throughout the bulk of the detector: in a variant known as the visible light photon counter (VLPC), the absorption and gain regions are separated. This reduces

7.2 Photon Detection

the sensitivity to 1–30 μm radiation to ∼2% whilst maintaining the visible spectral response. At 694 nm, uncorrected quantum eﬃciencies of 88±5% have been measured [34], corresponding to corrected eﬃciencies of 95±5%. Unlike an APD or a PMT, the SSPM and VLPC can distinguish between one and several photons hitting the detector at the same time: an SSPM has been shown to distinguish two photons with 47% quantum eﬃciency [35]. Apart from requiring cryogenic operation, typically at 7 K, the other disadvantages to these systems are their cost and relatively high dark noise, typically 2 × 104 dark counts per second. They also have a remarkable wide wavelength range and thus require shielding from IR photons. 7.2.1.3

Photo-thermal Detectors

The superconducting transition-edge sensor (TES) detector operates as a bolometer where the energy of the absorbed photon is detected as a rise in temperature. In order to achieve the sensitivity required to detect a single photon, the heat capacity of the absorber has to be very small, and the sensor must create a large response to a very small temperature change. This is achieved by using a thin layer of superconducting material, deposited onto an insulating material. The device is operated at a temperature between superconducting and normal resistances, such that a minimal change in temperature yields a large change in resistance. The device is kept at the operating temperature by electrothermal control, which also produces the output current. The current sensitivity can be enhanced by using an array of superconducting quantum interference devices (SQIDs). Single-photon sensitivity was ﬁrst demonstrated in the visible and IR [36]. To date TES detectors have recorded the highest eﬃciency so far [37, 38] and a very low dark count rate. However, they suﬀer so far from several drawbacks: the need to operate at about 100 mK and, until recently, a slow response of typically 100 ns, resulting in low maximum count rates of typically 100 kHz. Major technical advances can be expected soon. There are several further approaches that are being followed. They include heat-to-voltage conversion with digital readout (QVD), superconducting nanowire single-photon detectors, quantum dot ﬁeld-eﬀect transistor-based detectors, superconducting tunnel junction (STJ)-based detectors, up-conversion single-photon detectors, and the use of atomic vapours with lambda three-level systems as reviewed by Eisaman et al. [12]. Even quantum non-demolition sensing of photon number using materials with giant Kerr non-linearities [39] is being pursued. Research into all these devices is very active, and it seems rather likely that the quality of detectors, their applicability, and price will improve steadily, and thus we should assume that in the future, almost perfect single number receiving detection with small pixel size and imaging capability will be achievable and practical (Table 7.3). 7.2.1.4

Multipixel and Imaging Devices

Spatial images can be obtained photoelectrically, although typically the response time is orders of magnitude slower than that possible with a single detection element device like an APD. There are several technologies for achieving

243

244

7 Photon Generation and Detection

Table 7.3 Comparison and typical speciﬁcations of diﬀerent single-photon detector technologies: PMT, SPAD, SSPM, VLPC, and TES. Detector

Eﬃciency (%)

Temperature (K)

Dark counts (1/s)

Max counts (1/s)

PNR

PMT

40 at 500 nm

300

100

10 × 106

Some

SPAD

76 at 600 nm

300

200

16 × 106

None

SSPM

76 at 700 nm

50

7 000

10 × 106

Full

VLPC

88 at 694 nm

7

20 000

10

Some

TES

95 at 1556 nm

0.1

3

0.1

Full

Source: From Eisaman et al. [12].

this, perhaps the most common being the photon counting charge-coupled device (CCD). A traditional conﬁguration is the intensiﬁed charge-coupled device (ICCD), where an image intensiﬁer (e.g. a photoelectric photocathode, followed by an array of multichannel plates for electronic ampliﬁcation, followed by a phosphor screen) is imaged, via free space or ﬁbre bundle, to a low noise CCD camera. More recent devices dispense with the image intensiﬁer, utilizing high voltage to produce an on-chip electron cascade, and thus gain. Current CCDs have a quantum eﬃciency in excess of 92% (with resolutions up to 1024 × 1024 pixels and an associated readout rate of 30 frames per second) and will doubtlessly improve. Figure 7.3 shows the output of a spontaneous down- conversion photon source imaged with an image- intensiﬁed photon counting CCD. The photoabsorbing surface in most photoelectric devices is a metal or semiconductor: high refractive index materials that intrinsically reﬂect a significant proportion of incident light (e.g. see Table 7.4). The quantum eﬃciency of photoelectric detectors can be substantially improved via the application of anti-reﬂection (AR) coatings that allow more light to be coupled into the bulk of the material, where the intrinsic quantum eﬃciency may be quite high, e.g. 𝜂 = 99% for Si [41]. In commercial single element devices, the AR coatings are relatively simple (e.g. a single layer of TiO2 ), and the improvement in quantum eﬃciency is modest. In commercial CCDs the situation is somewhat better: mid-band visible AR coatings are available that improve the eﬃciency from

Figure 7.3 Images acquired with a photon counting CCD camera: output of a type II spontaneous down-converter. Source: After [40].

7.2 Photon Detection

Table 7.4 Properties of photodiode materials. Si

Ge

InGaAs

Band-gap Ebg [eV]

1.11

0.66

≈ 1.0

Cut-oﬀ wavelength [nm]

1150

1880

≈ 1300

Peak detection wavelength 𝜆peak [nm]

800

1600

≈ 1000

Material quantum eﬃciency

0.99

0.88

0.98

𝜆min and 𝜆max for 𝜂mat = 0.5

600–900

900–1600

≈ 900–1200

Refractive index n at 𝜆peak

3.5

4.0

3.7

Reﬂectivity at normal incidence (%) Brewster’s angle (∘ )

31

36

33

74

75

76

𝜂mat at peak wavelength [nm]

∼61% to ∼92%. It is worth remembering that for a speciﬁc wavelength application, a custom coating can lead to a signiﬁcant improvement, up to the intrinsic eﬃciency of the device. 7.2.2

Recording Electrical Signals from Individual Photons

The improvements of electronics and computing in the last decade have resulted in a shift of approach to photodetection. The availability of very fast detectors and digitizing boards, large processing capacities, and data storage has led to a paradigm shift in the way experiments are designed and data are analysed. The technique that most people now consider is to store the entire output from the photodetectors directly and to process these digitally, either in real time or in post-processing. This has many advantages: it allows the implementation of a much wider range of processes; it has capabilities for both single-photon experiments and continuous-wave experiments; the data are stored and can be analysed again and again; and the processing tools can be reﬁned after the data were taken. This allows the experimentalist to concentrate on the experiment for the purpose of getting the best data and separately, at other times, on the analysis. In practice this is very conducive to daytime experimentation and automated repeats of the experiments. But it is still followed by long evenings, in front of computer displays. The ﬁrst and very common approach is the direct counting of photons within regular time intervals. The arrival of one or more photons will result in an electric pulse or click. Only PNR detectors will allow us to classify the pulses according to the number of photons. We store such a pulse above a preset threshold as binary 1, whilst we ignore all small events below the threshold and record them as 0, meaning no photon. The processing is now a simple matter of storing the string of 1s and 0s, which can then be used to plot histograms and analyse trends as a function of various parameters. The average value can be associated with the average photon number ⟨N⟩ in this mode. The next level will be to look for correlations of the output of two photodetectors. The most direct way is to set the system

245

246

7 Photon Generation and Detection

such that it is only triggered by coincidences, that is, when both detectors have produced a pulse above the threshold and thus show a 1. There are many important experiments, such as entanglement, tests of Bell’s inequality, etc., which rely directly on coincidence counts. The results can be used to determine a two-mode correlation function g (2) quantitatively. The alternative to coincidence counting is to store the binary ﬁles and evaluate the sequences during or after the recording. This would yield the same results for coincidences but allows other types of protocols. We can consider conditional recording via post-selection where the pulse in one channel is recorded only when a pulse was present in the other channel. This is used in many protocols where one channel is designed to produce photons that herald a particular event in another channel. It has been used to demonstrate logic gates and describe their properties. Or it can be used to evaluate the conditional variance between the two channels. By recording the sequences of events from several detectors, more complex situations have been investigated, for example, the properties of a system with more than two modes. This could be systems that have entanglement between several modes, or degrees of freedom, or even systems where the modes are linked together to form a logic system. That means a large number of correlation functions between diﬀerent modes can be measured, the result leading to a full analysis of the covariance matrix (see Chapter 4). All of this is possible with the straightforward recording of binary sequences, typically with a sampling time of nanosecond, resulting in datasets with a few megabits. Analysis in these experiments is relatively straightforward, since the important parameter is the rate of photons detected at a given output port, measured over some integration time. If ﬁlm is used, the integration time is set by the length of the exposure. For photodetectors, the integration time is set by the counting time, which in turn is set by the electronics, including capacitors as low-pass ﬁlters, and the frequency of the trigger pulses. The electric current is digitized, e.g. TTL pulses, converted into binary values and counted with a commercial pulse counter; otherwise it can be digitized using a commercial discriminator, fast oscilloscope, or similar device and then counted. In some protocols, such as quantum cryptography, it is important that the photons arrive in a well-deﬁned time period. Even in these cases, any source, thermal, coherent, or Fock state source, can be used, as long as some combination of the source, the optical path, or the photodetector detection time is appropriately modulated. Such pulses will have regular time intervals. Alternatively, photon time-of-arrival information is a useful diagnostic in these experiments and may be obtained by using a time-to-amplitude converter (TAC) to measure the diﬀerence between the expected time, as provided by the modulator, and the actual time, as provided by the photodetector. The modulator provides a periodic start signal, for a measurement, which is ended by a photodetection event if one occurs. The time diﬀerence between the start and stop pulses then sets the amplitude of a regular pulse. The amplitude distribution of these pulses can be measured with a multichannel analyser (MCA), providing information on the distribution of photon arrivals.

7.3 Generating, Detecting, and Analysing Photocurrents

7.3 Generating, Detecting, and Analysing Photocurrents 7.3.1

Properties of Photocurrents

The technology used with CW laser beams is quite diﬀerent from that used with single photons, as described in Section 7.2. Here we analyse the full-time history of the photocurrent i(t), and the results we obtain for a laser beam are a value for the average optical power, which is obtained with a power meter, and in parallel a spectrum of the variance of the optical power, such as shown in Figure 7.9, which is obtained with an electronic spectrum analyser (ESA). For CW detection the electronic signal from the photodetector is integrated over many detection events for a time interval Δt, and this generates a current i(t), which is proportional to the power P(t), measured in Watt, of the light beam on the detector or, in other words, the rate at which photons reach the detector: i(t) =

ne (t)e P(t) e 𝜂det = Δt h𝜈

(7.1)

where ne (t) is the number of electrons released by the detector, e is the charge of a single electron, and 𝜂det is the eﬃciency of the conversion process (see for comparison Section 4.4.3). We will be interested in the average value of the current i = ⟨i(t)⟩. This value is proportional to the average optical power Popt ; thus a device that measures i is generally referred to as a power meter. Each such apparatus will have to be calibrated, and care should be taken that the response is linear, since the device will saturate if the optical power is too high.

7.3.1.1

Beat Measurements

The photodetector provides not only an average current i, the DC current, but also a whole spectrum of time-varying signals, the AC. Any AC component at detection frequency Ω can be traced back to a beat between two components of the optical spectrum with a separation Ω. These beat signals are very useful to identify the mode structure of the light source. Multimode laser operation will lead to strong beat signals, which can be detected and processed with high accuracy and sensitivity using conventional electronic RF equipment. The measurement of beat signals between two lasers, for example, will establish the frequency diﬀerence and the convolution of the linewidth of both lasers. Such observations are extremely useful for the establishment of optical frequency standards and also the measurement of laser linewidth, as discussed in Section 8.1.5. In addition, the AC signal also contains information about the noise and the signal carried by the laser beam, since the ﬂuctuations and modulation of the light intensity corresponds directly to the size of the amplitude sidebands. The presence of these sidebands leads to beat signals, which are similar but much weaker and noisier than the beat signals from two lasers.

247

248

7 Photon Generation and Detection

7.3.1.2

Intensity Noise and the Shot Noise Level

For the quantitative measurement of the ﬂuctuations, we are interested in the variance Δi(t)2 of the photocurrent. This is linked to the variance of the number of single-photon detection events via ( ) ne (t)e 2 Δ(ne (t))2 e2 2 Δi(t) = Δ = (7.2) Δt Δt 2 If we consider the generation of all the photoelectrons as independent processes, we can apply the statistics for independent particles. The counting statistics for the electrons should be a Poissonian distribution with variance Δ(ne (t))2 = ne = iΔt∕e. For large numbers of electrons in each counting interval, this should be a Gaussian distribution, resulting in ie Δt We will be investigating the spectrum of the current ﬂuctuations. In this case the detection interval Δt is related to the bandwidth B of the apparatus via B = 1∕2Δt, which leads to Δi(t)2 =

Δi(t)2 = 2 i e B

(7.3)

assuming that i does not vary during the measurement. This is the well-known shot noise formula in electronics that also applies to photodetection. It should be noted that Eq. (7.3) can be derived for any stream of electrons that are generated independently. However, this is a rather special case for a current, and generally a current can be noisier or quieter. There are many practical situations, such as the use of electronic current controllers, which can generate currents with less noise than the shot noise limit. It is important to note that shot noise is not a property of the detector. The noise reﬂects the ﬂuctuations of the light intensity, and we can use detectors to probe the statistics of the light ﬂuctuations. This will be correct as long as the quantum eﬃciency of the detector is high. For 𝜂det = 1 the statistics of the photons and electrons will be identical, the current ﬂuctuations are directly linked to those of the light, and we can investigate the statistical properties of the light by studying the current ﬂuctuations. It should be noted that after the detection of the light, any quantum properties of the ﬂuctuations no longer exist. Current noise can be treated classically, as already discussed in Section 5.1. For any practical purpose there is no quantum limit for the noise of a current. However, at the same time the current ﬂuctuations are a reliable measure of the quantum properties of the light. This distinction is important in the context of measurement theory, the generation of non-classical light (Chapter 9), and the applications of electro-optic feedback described in Section 8.3. How closely the current noise reﬂects the light ﬂuctuations depends on the eﬃciency 𝜂det . A quantum eﬃciency much less than unity corresponds to a random selection of a fraction 𝜂det of the light. This is fully equivalent to having a perfect detector behind a beamsplitter, which selects a small fraction of the photons. This results in a variance of the current that is closer to the shot noise limit than to the variance of the light intensity. To describe this quantitatively, it is useful to

7.3 Generating, Detecting, and Analysing Photocurrents

consider normalized variances for individual Fourier components of the current ﬂuctuations: ) ( Δi(t)2 Vi (Ω) = (7.4) 2ieB The corresponding statistics of the light is described by the normalized spectrum V 1out (Ω) of the photon ﬂux, as was shown in Chapter 4. The eﬀect of the detection eﬃciency 𝜂det is given by Vi (Ω) = 1 + 𝜂det (V 1out (Ω) − 1) This is identical to treating the ineﬃcient detector as a perfect detector that sees only the light reﬂected by a beamsplitter with reﬂectivity 𝜖 = 𝜂det as calculated in Section 5.1. The measured spectrum Vi (Ω) varies with the fraction of the light detected. For a very ineﬃcient detector, the normalized variance of the current approaches the shot noise limit Vi (Ω) = 1. 7.3.1.3

Quantum Eﬃciency

The detection eﬃciency 𝜂det contains all processes that reduce the conversion from light into a current. In practice, there are a number of such processes, as shown in Figure 7.4: only a fraction 𝜂r of the light reaches the photodetector material, and the rest is reﬂected or scattered at the surface of the detector. The detector absorbs a fraction 𝜂a of the light, and the rest is scattered or transmitted through the detector. The absorption scales exponentially with the thickness of the material and depends on the doping of the material. The material has a quantum eﬃciency 𝜂mat , which depends strongly on the wavelength of the light. This eﬃciency can be very close to unity as long as the wavelength of the light is close to but shorter than the cut-oﬀ wavelength set by the band-gap energy of the material. Finally, there is a chance for the electron–hole pair to combine before it can leave the detector. This recombination is more likely at the surface of the material than inside. At shorter wavelengths the light tends to be absorbed more strongly; it does not penetrate as deeply. Thus more detection processes occur close to a

N(t)

(1 – ηcom) recombination

(1 – ηr) reflection

ηdet =

N(t) i(t)

(1 – ηa) transmission

ηmat conversion

i(t)

Figure 7.4 The diﬀerent loss mechanisms present in a photodetector.

249

7 Photon Generation and Detection

surface, and the losses due to recombination are larger. Only the fraction 𝜂com will actually contribute to the photocurrent. All these losses are independent and can be combined to the total detector eﬃciency 𝜂det : 𝜂det = 𝜂mat × 𝜂a × 𝜂r × 𝜂com The eﬃciency 𝜂det is generally independent of the intensity and the detection frequency Ω. However, it has been noted that 𝜂com will get worse at higher intensities and that some photodetectors get slower at higher intensities when either the detector material is getting hot or there are insuﬃcient fast charge carriers to drive the AC. The eﬃciency 𝜂det has to be determined for the parameters of the light used in the speciﬁc experiment, in particular its wavelength, total power, and focusing geometry. Techniques for increasing 𝜂det in order to approach the ideal case of 𝜂det = 1 are very important in all experiments with non-classical light. 7.3.1.4

Photodetector Materials

Photodetection relies on the interaction of light with materials, which results in the absorption of photons and the generation of electron–hole pairs. These pairs change the conductance of the material, and they contribute to the photocurrent. The material for the photodetector has to be selected for a given frequency of the light. The energy of the photons h𝜈 has to be matched to the band-gap separation Ebg of the material. Semiconducting solids, such as germanium (Ge), silicon (Si), or indium–gallium–arsenide (InGaAs) are well suited for this application. These materials work in the visible and near IR. The properties of these materials are listed in Table 7.4, and the wavelength dependence of the eﬃciency is shown in Figure 7.5. Note that the properties of primary semiconductors (Si, Ge) are ﬁxed but that the properties of tertiary materials (such as InGaAs) depend on the relative composition, which means the concentration of the individual elements. The material quantum eﬃciency 𝜂mat of a very pure and selected material can be very high, up to 𝜂mat = 0.98. This is the limit of the eﬃciency of the detector and can be achieved in practice provided the sample of material used is thick enough to absorb all the light and reﬂection losses are avoided. Since most photodetector 1 Si

Ge

InGaAs η

250

0.5 InSb InGaAsP

Ag–ZnS 0 0.1

0.2

0.4

0.6 1 2 Wavelength λ (μm)

4

Figure 7.5 Quantum eﬃciencies of various photodiode materials.

6

7.3 Generating, Detecting, and Analysing Photocurrents

materials have very high refractive indices, their reﬂectivity is also very high; for example, for Si with a refractive index of 3.5 at the peak detection wavelength of 800 nm, the reﬂectivity under normal incidence is 31% of the intensity. To this the reﬂection loss at the cover glass (typically 8% from both surfaces of the glass) has to be added. In order to increase the quantum eﬃciency, it is common to remove the cover glass and to either tilt the detector closer to Brewster’s angle, which requires large detectors, or use an AR coating on the detector. In addition, the light reﬂected at the surface can be retro-reﬂected on to the same detector. Using these techniques typical quantum eﬃciencies that were achieved by 2005 were 65% at 553 nm (EG&G FND 100 Si photodiode) and 95% at 1064 nm (Epitax 500 InGaAs photodiode). Similarly high eﬃciencies around 1600 nm have also been achieved, which is important due to the interest in optical communication applications and the optimal performance of silicon ﬁbres in this wavelength range. In most cases one would expect technology to simply improve. However, there have been situations where certain optimized components are no longer produced, for example, the Epitax diodes, and it can be costly and time consuming to ﬁnd replacements. Alternatively, several detectors can be cascaded, each subsequent detector receiving the light reﬂected by the previous detector, and all the individual photocurrents are combined. Here electronic delays of the diﬀerent currents have to be carefully avoided in order to keep the signals synchronized with an accuracy better than the inverse of the detection frequency. Using cascaded detectors eﬃciencies close to 100% have been reported for detection frequencies of up to 100 kHz using Si detectors [42] and up to 10 MHz using InGaAs detectors. 7.3.2 7.3.2.1

Generating Photocurrents Photodiodes and Detector Circuit

The photodetection process leads to a change in the conductivity of the semiconductor material. A simple photoresistor could be used to produce a photocurrent. However, the most common conﬁguration is a PIN photodiode, which is a composite structure made out of P and N semiconductors with an intermediate layer. In this device an external voltage, the bias voltage, Ubias , creates an electrical ﬁeld in the region between the P layer and the N layer. Here electron–hole pairs can be separated, and they contribute to the photocurrent. There are three modes of operation for a photodiode. The diode can be operated with an open circuit. This will produce a voltage proportional to the photon ﬂux; no current will ﬂow. Alternatively, a low resistance is switched across the diode, and the circuit is eﬀectively a short circuit. In this conﬁguration a DC current is generated and the photodiode operates like a solar cell. Both these modes have slow responses. It is advantageous to apply a voltage, the reverse bias voltage Ubias , to the detector that accelerates the charge carriers. In this mode the photodetector is used like a diode to generate a voltage signal across an external resistor. The diode current is proportional to the photon ﬂux N(t). For a suﬃciently large bias voltage, the photocurrent is proportional to N(t). A schematic circuit and a typical response curve are shown in Figure 7.6. It is common to

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7 Photon Generation and Detection

i N(t)

DC Ubias

–Ubias

R i(t)

U

N(t) AC –Ubias / R

Figure 7.6 Circuit of a photodiode with reverse bias voltage Ubias and the typical diagram that shows the photocurrent i(t) as a function of the bias voltage for various values of the photon ﬂux N(t).

use the highest possible bias voltage (Ubias = 5–80 V depending on the diode) since this will increase the drift velocity of the charge carriers and thereby the speed of the detector. The photocurrent contains both a DC component and an AC component that can be directed to diﬀerent receivers using simple capacitors or more complicated electronic networks. The AC components contain all the information about modulation and the ﬂuctuations. 7.3.2.2

Ampliﬁers and Electronic Noise

The typical AC noise component is of the order of 1 part in 107 of the DC current. It requires ampliﬁcation. This can be achieved by various types of ampliﬁers, as described in [43]. The most obvious solution is to terminate the photodiode into a 50 Ω resistor and to use a commercial RF ampliﬁer. Such an ampliﬁer can be obtained with wide bandwidth (1–200 MHz), large gain (10–60 dB), and low noise performance. Please note that the gain is normally stated as the power ampliﬁcation in dB, where dB is 10 log Pout ∕Pin (see Eq. (7.9)). The ampliﬁer will increase both signal and noise at the input. Figure 7.7 shows the combination of noise and signal that is generated. The diﬀerent noise contributions are: • Dark noise of the detector Δi(t)2dark . Note that this is not the shot noise; it is only the noise due to false detection events. For most of our measurements with optical powers of a few milliwatt, this can be neglected. • Thermal, or Johnson, noise of the circuit given by Δi(t)2th ≈ 4kB TB∕R where T is the temperature, B is the bandwidth, and R is the impedance of the circuit. • Ampliﬁer noise Δi(t)2amp ≈ 2eGBF. Here G is the voltage gain and B the bandwidth. F is the excess noise factor, a composite of diﬀerent contributions inside the ampliﬁer. The excess noise factor is normally quoted in the technical speciﬁcations. Note that a large bandwidth detector contributes more noise than a narrowband ampliﬁer. All these noise contributions are independent; thus their variances can be added. Note that the noise Δi(t)2optical created by the light is the actual information

7.3 Generating, Detecting, and Analysing Photocurrents

Light

Photodiode

Gain

Output Thermal noise Gain noise

Noise

(a)

Dark noise Quantum noise

Signal

(b)

Modulation and classical noise

Figure 7.7 Signal and noise contributions in a photodetector. Noise terms (a) and signals (b) propagate diﬀerently through the stages of the detectors circuit.

we wish to detect. It corresponds to the signal in a communication application, and in order to achieve a clear detection of the signal, the total noise has to be less than the optical noise: Δi(t)2el = Δi(t)2dark + Δi(t)2amp + Δi(t)2th ≪ Δi(t)2optical

(7.5)

One of the largest contributions is normally the thermal noise. Consider the following example: for a DC current of 1 [mA], equivalent to 1 mW of power at 𝜆 = 1000 nm, the thermal noise at room temperature for an impedance 2 2 of 50 Ω and a detection bandwidth of 100 kHz is Δi(t)2thermal ∕i = 4kB TB∕Ri = 4 ∗ 1.38 × 10−23 ∗ 300 × 105 ∕(50 × 10−6 ) = 3.3 × 10−11 . The relative noise that corresponds to the shot noise for this bandwidth and average power is 2 Δi(t)2 ∕i = 2eB∕i = 2 ∗ 1.6 × 10−19 ∗ 105 ∕10−3 = 3.2 × 10−11 . This means that for the given conditions both quantum noise and electrical noise produce comparable relative ﬂuctuation Δi(t) of about 5 × 10−6 . In this example, the thermal noise is as large as the shot noise at this optical power, and such a detector could only show quantum noise at much higher powers. The importance of this so-called front end of the circuit cannot be overestimated; it will determine the noise characteristics of the detector circuit. Several technical tricks exist to optimize, which is summarized, for example, by P.C.D. Hobbs [44]. Most custom-made photodetectors contain a so-called transimpedance ampliﬁer, a circuit that converts the AC photocurrent into an AC voltage, which is fast and still has a high impedance. In such a circuit the thermal noise will be much lower, about 10−8 at 1 mA DC current, and we can clearly resolve quantum noise. State-of-the-art experiments provide a combined electronic noise of typically −10 dB and in some cases −30 dB below the quantum noise limit (QNL). This is the reason why photodetectors for quantum optics experiments are frequently custom made. For illustration, an example of a complete circuit diagram of such an ampliﬁer is shown in Figure 7.8. It shows the photodiode,

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7 Photon Generation and Detection

3

C1 22u

C2 100n

C3 100n

D1 ETX 500

7

I2 10u

2 – 3 +

5

1N4007 D2

+15

~4PF R1 2K 6

IC1 R2 CLC420 200R

U2 79L05 3 GND 2 IN –5V 1

–15

1N4007 D3

I3 10u C8 22u

C6 100n

C4 100n C10

GND 2

7

U1 78L05 Vin +5V

3 + 2 –

8

1

IC2 CLC430 6

R5

Vout

51R

4

I1 10u

4

254

R3 750R R4 300R

C5 100n

C7 100n

C9 100n

Figure 7.8 Example of an actual circuit diagram of a photodetector. For other examples, see M.B. Gray et al. [43].

the transimpedance ampliﬁer (IC1), the driver ampliﬁer (IC2), and all the ﬁlter components required for a practical low noise circuit. Whilst the development of electronics has continued constantly, the design of photodetectors still follows the same pattern. A good reference in 2017 is still the article by P.C.D. Hobbs [44], which summarizes the concept of the electronic design and the references given in the textbook by the same author [45]. 7.3.2.3

Detector Saturation

In many cases the light has strong modulations at speciﬁc detection frequencies. This could be due to laser noise, such as beats with other laser modes, or due to the sidebands introduced for the purpose of locking the laser to resonators, as discussed in Section 8.4. We expect the photodetector to handle both the very small ﬂuctuations and the modulation without distortions, and consequently we require a very large dynamic range for some applications as large as 60 dB. If the dynamic range of the ampliﬁers is not chosen large enough, the circuit will be saturated. This frequently leads to a decrease of all AC components, even those at other than the modulation frequency that originally caused the saturation. This saturation can be confusing and can mimic false results. It is particularly worrisome if several ampliﬁers are used, a ﬁrst ampliﬁcation stage with wide bandwidth and a later stage with narrow bandwidth. In this situation the ﬁrst stage can saturate from a modulation at frequencies outside the bandwidth of the ﬁnal stage. Thus the modulation is not visible at the output, and the noise level decreases when the signal is turned on – mimicking a noise suppression. In more than one case, this malfunctioning of the detector has fooled researchers and was taken as the demonstration of some quantum optical eﬀect. As a precaution one should always record spectra for the entire bandwidth of the detector, and any

7.3 Generating, Detecting, and Analysing Photocurrents

ampliﬁer and low-pass ﬁlter should best be placed directly after the photodiode. In addition, the scaling of the noise traces with the input optical power should be checked, as outlined in Section 8.1.1, to verify that no saturation exists. 7.3.3

Recording of Photocurrents

In most situations experiments are conducted by recording the photocurrents directly via fast analogue-to-digital (A/D) converters storing the entire information with a large bandwidth and analysing the physical properties, such as histograms and correlation functions. Historically, this is a form of post-processing that was typical for the analysis of single-photon events. However, due to the vastly increased speed of the processors, this can also be used now for all types of data analysis of continuous photon currents. For example, the modulation spectrum of the photocurrent can be determined with a numerical fast Fourier transform (FFT) providing similar results as the direct analogue ESA that were used in the beginning of the quantum experiments. Provided that the speed of recording, the dynamic range of the AD converter, and the size of the memory provides no limitation, we can assume that the Fourier transform will produce a true spectrum. In 2017, A/D converters with 18–20 bit conversion and sample speeds of 100 megasamples/s are available. In addition the size of the memory is now almost no issue, say, a few terabits, and the size is only noticeable in the time it takes to process a trace. However, there are limitations in the digital data recording that have to be considered and tested thoroughly to avoid misinterpretations: the main constraints are digitization speed or sampling frequency (samples per second), the dynamic range (bit per sample), storage capacity (total number of bits), and sensitivity to aliasing, which means the interference between the ﬁxed sample frequency and higher harmonics present in the input, leading to spurious low frequency eﬀects. For a simple practical example, we might consider a technical test where we record a communication channel that transmits signals with a carrier frequency of 100 MHz and modulation sidebands of around 100 kHz. We want to record an experiment for about 10 seconds. And we want to see a signal-to-noise ratio of 40 db. All this seems quite reasonable. Using the Nyquist theorem as a guide, we would set the sampling frequency to about 200 megasamples/s. To achieve a dynamic range, or contrast, of 10 000, which is necessary in order to see the ﬂuctuations and signal in reasonable detail, we would have to use a 16-bit A/D converter. This would generate 2 × 108 ∗ 65 536 ∗ 10 = 1.31 × 1014 = 130 terabits. That is, in excess of 2017 technology and such a brute force approach would simply create an unmanageable amount of data. However, the real information content is actually much smaller. We actually have only a maximum of 1 million possible signal values per second, we can average out the noise to a slowly varying value with an accuracy of one part in 10 000 of the biggest signal. So with little distortion a 12-bit A/D converter would be suﬃcient for simple communication applications. After mixing down the signal from 100 MHz and recording at a speed of 0.1 megasamples/s, only about 106 ∗ 4000 ∗ 10 = 4 gigabits of recording would be generated. This volume of data is comparable with the recoding of a

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high-deﬁnition video signal for that length of time and would be suﬃcient for most applications. In comparison, this could have been an experiment recorded with analogue instrumentation, such as an ESA that creates a printout of a spectrum centred at 100 MHz and a width of 200 kHz with a 50 dB vertical scale with a resolution of 0.5. Whilst such a spectrum contains only an average of the information over the entire 10 seconds scan, it would give us excellent information about the signal-to-noise ratio, and any potential problems or non-linearities within the system bandwidth would be recognized. The printout contains only a hundred values between 0 and 120 db, but this represents the complete spectrum. In order to describe the technical speciﬁcations of the apparatus, such a simple graph or even just a few numbers are suﬃcient in most cases. This example shows that the recording of an arbitrary signal with very ﬁne detail requires extraordinary number of samples. This not only exceeds our storage capacity, but it also makes processing practically impossible, and it is only necessary if we have no predictor of the signal that we wish to ﬁnd. That means that we should tailor our system to exactly what we need; there is no universal solution and just digitizing everything is not eﬃcient. In most technical applications we have a speciﬁc goal in mind. Let us consider two extreme examples. We can store and show with a smartphone or tablet any imaginable sequence of video images. These could potentially change completely from frame to frame, say, 30 times per second. However, in most cases the change from frame to frame might be small. In order to make this practical, the information has to be cleverly compressed, for example, in one of the current standards as MP4. This allows us to store only a fraction of the total information contained in all the pixels at all times. Clever algorithms project the images unto mathematical base functions and select a small amount of data that represents the video. The spatial resolution can be selected. The information can be stored and can be analysed quite readily. This technique requires online compression and storage of a medium number of data points. At the other end of the range of applications is the investigation of something unknown, for example, of gravitational waves (GW) detected as the correlation between the outputs of several optical interferometers placed at diﬀerent locations. Since late 2015 this has been dramatically demonstrated by the global LIGO collaboration. The GW have frequencies from megahertz to kilohertz. The detectors have good sensitivity above about 10 Hz and are being sampled at 40 kilosample/s continuously for 24 hours per day for months on end, since we do not know when the next event will happen and what it will look like. This leads to many terabits of data. Such large datasets have been recorded by LIGO since 2011 and have been constantly analysed by many groups around the world. In quantum optics experiments we typically lie between these extremes, and we have to design our experiments carefully to make use of the existing digital recording and processing technology. We can now record 100 megasamples/s with 16-bit resolution. In 2017 we can handle 1–100 gigabit ﬁles quite readily. However, we should also consider any analogue steps as part of the recording process, such as mixing the RF signal before recording, including low-pass ﬁlters or band ﬁlter to isolate the frequencies we are interested in and to protect

7.3 Generating, Detecting, and Analysing Photocurrents

us from aliasing. Thus the optimum system will be a combination of analogue preprocessing and digital recording and post-processing. 7.3.4 7.3.4.1

Spectral Analysis of Photocurrents Digital Fourier Transform

The time spectrum of the photon current contains a wealth of information that describes the properties of the apparatus. It is thus a widely used test procedure and the key to demonstrating many optical techniques. Figure 7.9 shows a typical spectrum and all the settings of the analyser being used. The most direct approach for creating a spectrum from digitized data is to use an FFT. This commonly available software routine is frequently directly built into the data acquisition system, for example, LabVIEW . The obvious requirements are as follows: a suﬃciently large number of samples, according to the Nyquist limit; the dynamical range, which is limited by the sampling resolution; and protection against interference with other periodic events, for example, at higher frequencies. The latter is an unavoidable consequence of recording with strictly periodic samples and can only be avoided by optimizing the recording electronics. Since FFT deals with stored data, the central RF, resolution bandwidth (RBW), and the averaging, or in traditional terms, video bandwidth (VBW) can all be selected after the recording. The recording should satisfy the following

Figure 7.9 A typical intensity spectrum as displayed by an electronic spectrum analyser. The modulation is about 8.9 MHz. The smaller signal at about 5.2 MHz is a radio interference from another source, aﬀecting both the signal and the dark trace. Here the settings of the instrument are shown together with the data.

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requirements: a sampling frequency must be at least twice the central RF; the number of samples should be at least larger than the ratio of RF to RBW; and the VBW is set by the inverse of the recording length. For slow long-term recordings, the choice is to either process all frequencies simultaneously and display long-term average values or alternatively analyse every section of the spectrum independently one at a time, which provides less averaging but allows to catch transient phenomena. It is worth noting that some of the commercially available ESA are actually a combination of A/D converter, storage, and FFT, rather than the full analogue version of using RF mixers. In these devices all the sample settings are automatically adjusted by selecting the required frequency and bandwidth settings. These machines are limited in their speciﬁcations but avoid the complication of having to individually optimize the digitization and sampling settings. 7.3.4.2

Analogue Fourier Transform

An ESA is used in some experiments to record the Fourier spectrum of the photocurrent. A typical spectrum displayed by such an instrument is shown in Figure 7.9. The principle function of an ESA is to determine and display the power spectrum of the electrical input signal. We consider in detail the transfer function from the spectrum of the input signal to the display. It is important that we understand the many options and settings of the analyser in order to obtain reliable data that can be interpreted as the properties of the light beam. A state-of-the-art analyser is a very powerful instrument with many options, menus, and knobs. An ESA contains a combination of analogue and digital electronics that produce a string of data that will be recorded alongside its actual settings. These machines are designed for the recording of signals, such as radio signals. In this section we will discuss the technical details of the ESA, such as the various bandwidth settings, sensitivity scales, sweep speeds, etc. and the tricks for using the analyser for the recording of noise and signals. Finally, we will discuss alternatives such as electronic mixers, which are useful for more advanced experiments such as tomography. 7.3.4.3

From Optical Sidebands to the Current Spectrum

We are interested in the sidebands of the optical ﬁeld that contain all the information about the ﬂuctuations and the statistics as well as all the information that is transmitted with the light in the form of modulation. Typical sideband frequencies range from kilohertz to gigahertz. These frequency intervals are too small for ordinary spectroscopic instruments, such as grating spectrometers or optical spectrum analysers (OSA), and are just about accessible with optical cavities. However, the most reliable and best separation of the sidebands is by the Fourier analysis of the photocurrents. The information is transferred from the optical domain (𝜈 ≈ 1015 Hz) into the domain of electronic instruments, which means detection frequencies Ω < 10 GHz. For this purpose several light beams are combined on a photodetector. The resulting beat signals contain information about the quadrature of the light. These techniques, homodyne and heterodyne detection, are described in

7.3 Generating, Detecting, and Analysing Photocurrents

Section 8.1.4. In all cases the information is contained in the temporal development of the current i(t) generated by the photodetector. The purpose of an ESA is to measure and display the spectral composition pi (Ω) of the electrical power of an electrical signal i(t). Mathematically this corresponds to a determination of the power spectrum, or the square of the Fourier transform, of the input voltage. The instrument measures the power pi (Ω) of the oscillating ﬁeld in a spectral interval around the detection frequency Ω. The spectral response of the analyser is given by the function res(Ω), which has a width given by the RBW. For a ﬁxed input resistance R, typically 50 Ω, the measured power is ( )2 pi (Ω) = time average res(Ω) i(Ω) R dΩ (7.6) ∫ where i(Ω) is the Fourier transform of the current i(t). We are measuring the square of the AC photocurrent integrated over the detection interval. The ESA is not sensitive to the phase of the electrical input signal. It will display only positive frequencies; there is no diﬀerence in the result of a measurement of signals with positive or negative frequencies, of sine or cosine modulations, of the current. 7.3.4.4

The Operation of an Electronic Spectrum Analyser

Technically, the ESA consists of a mixer that selects a certain frequency; an envelope detector, in most cases a logarithmic detector; and circuitry for averaging the signal, as shown in Figure 7.10. In most modern analysers the signal is digitized and displayed with a wide choice of scales and other options. In the following we analyse the transfer function between the electronic input signal and the display. The ESA is built around the concept of electronic heterodyne detection. A reference signal with ﬁxed amplitude and variable frequency, the local oscillator, is generated internally. The frequency Ωdet of the local oscillator determines the frequency under detection. The input signal is ampliﬁed and mixed with the local oscillator. The output from the mixer is ﬁltered by a narrow band pass ﬁlter with bandwidth RBW. The result of the measurement, at a given Ω, can still ﬂuctuate in time, and we can describe the output of the ESA by a distribution function with the mean value 𝜇disp , and it is this value from which we infer the spectral density pi (Ω) of the photocurrent. In our optical experiments we make relative measurements, which means we are not interested in the absolute values of 𝜇disp ,

Mixer Voltagein

Filter RBW filter

Envelope detector

Log amplifier

Filter VBW filter

Local oscillator

Display

Scale dB power

Figure 7.10 Block diagram of a typical spectrum analyser.

AD sample

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but in relative values, such as the ratio of the 𝜇disp to a calibrated value that represents quantum noise. The output signal is integrated using a low-pass ﬁlter with a cut-oﬀ frequency known as the VBW. The VBW is the inverse of the time constant Δtave with which the measured signals is averaged, Δtave = 1∕VBW. In the limit of small VBW, large integration times, the output is one single, steady value 𝜇disp (Ω) for each detection frequency. In many practical situations we want to see the spectrum rapidly, and a lower value for the VBW is chosen; this does not vary 𝜇disp (Ω) but produces a more noisy trace. Modern RF spectrum analysers cover a wide range of input frequencies, from about 1 kHz to 100 GHz. They can achieve this wide span through a sophisticated combination of several internal mixing processes. Nevertheless, the idea outlined above of a narrow band scanning ﬁlter followed by an integrating power detector is appropriate. The instrument covers a total frequency interval, or span, SPA, and scans it with a set scan speed, measured in Hz/s. The instrument requires at least a time of 1/RBW for each measurement; after that it moves on by a frequency interval RBW. Consequently the maximum scan speed is RBW2 . For typical values of RBW of 104 Hz, the fastest possible speed is 108 Hz/s = 100 MHz/s. Faster scans would lose detailed information. The highest possible VBW is equal to the RBW, but usually the VBW is chosen lower. In particular for noise measurements, it is necessary to select a low VBW in order to obtain a well-deﬁned result. Frequently, measurements with ﬁxed detection frequency Ωdet are carried out. In this case the scan represents a time history of the power of the single Fourier components. 7.3.4.5

Detecting Signal and Noise Independently

It is not possible to distinguish between a pure noise source and a mixture of signal and noise by just looking at a trace recorded at one frequency. In this case we could not say if we have a laser beam with quantum noise alone, a laser beam with quantum and classical noise, or laser beam with noise and modulation. We need additional information. For example, a scan of frequencies would reveal a peak if the signal is narrowband and the noise is broadband. Or a measurement of the orthogonal quadrature, using a homodyne detector, would show the characteristics of a signal and of noise or squeezing and anti-squeezing. We can isolate signal and noise, as measured by the traces on the spectrum, and we can determine the SNR as SNR =

2 2 𝜎(signal+ − 𝜎noise noise) 2 𝜎noise

(7.7)

What would happen if we had more than one noise source and more than one signal present? As long as the noise terms have Gaussian distributions with 2 2 2 widths 𝜎n1 and 𝜎n2 , we can combine them into one value 𝜎noise = 𝜎n1 + 𝜎n2 . In a similar way we can combine two signals S1 and S2 with a random phase relationship. However, if the two signals have a ﬁxed phase relationship, the amplitudes 2 will have to be added coherently, before the variance 𝜎signal is formed. In either

7.3 Generating, Detecting, and Analysing Photocurrents

case, one combined signal intensity Ssignal can be found. Consequently, the calculation in Eq. (7.7) for one noise and one signal term is generally suﬃcient for all cases that we will encounter. 2 for a given noise source by decreasing Note that it is possible to reduce 𝜎noise the RBW, eﬀectively detecting less noise and the same amount of signal. This will improve the balance in Eq. (7.7) and allow smaller signals to be measured. However, this will make the detection process slower. We can also determine the measured value better by averaging the ﬂuctuations for a suﬃciently long time, which means using a low VBW. In the limit of VBW → 0, or extremely long integration times, the response of the analyser to a noise input with constant RMS will be a constant reading. The entire noise distribution would now be reduced to one piece of information. In most practical situations the integration time will be limited to about 0.1 s (VBW = 10 Hz) since systematic drifts inﬂuence the performance of the experiment and do not allow excessively long scans. When the noise is white, which means the ﬂuctuations are independent of the frequency, the spectrum will be a ﬂat horizontal line on the display. 7.3.4.6

The Decibel Scale

Some examples of actual measurements of laser intensity noise are given in Chapter 8. The display of the spectrum analyser is normally logarithmic. The units used are logarithmic power units, called dBm. The conversion of Watt into dBm is given by ( ) P[W] (7.8) pi [dBm] = 10 log 1[mW] The unit dBm is an absolute unit of power, for example, the RF power of 1 mW corresponds to 0 dBm, 0.001 mW = 1 μW corresponds to −30 dBm, etc. Whilst these numbers appear to be rather small, it should be noted that they refer to the power of the AC components only, not the optical power of the beam that generated the photocurrent. It would require a strong modulation of the intensity of the beam to get a signiﬁcant fraction of the energy into the sidebands. For example, a beam with average optical power Popt and single frequency harmonic modulation of the intensity by 1% has both a positive and a negative sideband at the modulation frequency, ±Ωmod . Each contains 1% of the amplitude of the optical ﬁeld, which corresponds to i(Ωmod ) = 0.02 of i and pi (Ωmod ) = 4 × 10−4 Popt . In most situations the relative change of the signal, or the noise, is of interest, and a relative unit of power, the decibel or dB, is used. It describes the ratio of two power levels. On a logarithmic scale this corresponds to a diﬀerence ΔP(Ω) given by ( ) P1 (Ω) (7.9) ΔP1,2 (Ω)[dB] = 10 log P2 (Ω) An expression such as ‘the signal was ampliﬁed by 3 dB’ means that the power of the signal increased by a factor of 103∕10 = 1.995 ≈ 2. Note √ that for such an increase the actual AC current would grow by only a factor 2. Similarly 6 dB corresponds to a factor 4 increase in power. The ratios in Eq. (7.9) are independent of the units; we record them from the display as ratios between values of 𝜇disp , and

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we think of them as the ratio of two noise powers P and interpret them as the ratio of quadrature variances V 1. All results of noise increase or noise suppression are quoted in dB, which gives directly the factor of change in the quadrature variance V 1 of the light. 7.3.4.7

Adding Electronic AC Signals

In many situations more than one signal or one noise component are present at the same detection frequency. Only the total power of all the contributions will be displayed. It is important to be able to determine the contributions from the individual terms, and we need rules for the adding of several AC. These rules depend on the nature and the correlation of the individual currents. Clearly, there is a distinction between currents that are coming from the same source or from different independent sources. First consider the extreme case of two currents, i1 (Ω) and i2 (Ω), which are generated by the same source. They are perfectly correlated, and their cross-correlation coeﬃcient as deﬁned in Chapter 2 is equal to unity: C(i1 , i2 ) = 1. In practical terms that means that the two signals have identical time histories. For single harmonic functions, this means that they have the same frequency and are locked in phase to each other. For these signals the combined power is simply the sum of the amplitudes: Pi1 +i2 (Ω) =

∫

res(Ω)[i1 (Ω′ ) + i2 (Ω′ )]2 R dΩ′

(7.10)

If the ratio of the amplitudes of the two currents is given by the number c = i2 (Ω)∕i1 (Ω), the measured power will change by a factor Pi1 +i2 = (1 + c)2 (7.11) Pi1 or in dB by ΔPi1 +i2 [dB] = 10 log((1 + c)2 )

(7.12)

For the case of two identical signals, i1 (Ω) = i2 (Ω) or c = 1, the power increase is 6 dB. The other extreme case is that of two signals that have a random phase diﬀerence to each other, which means they are uncorrelated. In this case the two signals add in quadrature, which means ΔPi1 +i2 [dB] = 10 log(1 + c2 )

(7.13)

and for the case of two signals with equal amplitude, the power increase is 3 dB. The same technique can be applied to noise terms, which are only diﬀerent from signals insofar as the power of the noise at detection frequency Ω varies in time whilst the power of a signal remains ﬁxed. In particular, signal and noise terms are always uncorrelated to each other. Two noise terms are uncorrelated when they come from diﬀerent sources. They are correlated when they can be traced back to the same source. Correspondingly, these noise signals are added in the respective way, using either Eq. (7.12) or Eq. (7.13). In the ﬁrst case, uncorrelated terms, the variances of the two noise terms are added in quadrature. Note that this procedure is strictly speaking only correct if both noise signals have the same statistical distribution functions, e.g. Gaussian distributions, which we can assume for many practical situations.

7.4 Imaging with Photons

7.4 Imaging with Photons All of the images shown in Figure 2.14 can be described by classical optics. At the same time, they use low intensities and thus photon ﬂux, and the limiting eﬀects of photon statistics are clearly visible. The ﬁrst step is to consider the eﬀects of photons in the generation of light in the light source. This could be due to the various processes that generate the light, such as thermal processes for an incandescent light, excitation of atoms in a plasma in a discharge lamp, excitation of atoms in a semiconductor that operates as a LED, or ampliﬁcation of light in a laser resonator. In each of these situations, the clearest and most elegant description of the process is on the basis of photons. There are well-described probabilities for an atom to emit a photon, and the wavelength of the light is determined by the energy of each photon via Ephoton = hc∕𝜆, and this is controlled by the energy level structure of the atoms or the semiconductor being employed. The ampliﬁcation of light inside a laser resonator is best described in terms of stimulated emission. Starting with the object that is being illuminated and imaged, the major processes are the transmission and absorption of the light from the illumination, where some of the photons are absorbed and the ﬂux is reduced in proportion. In addition photons can be generated in the object plane with new propagation directions through elastic scattering. These photons have the same energy per photon, which means they carry the same wavelength information. There could also be inelastic scattering, or ﬂuorescence, where the emitted photon has less energy than the photon required for the excitation of the atom involved in the process. This leads to new wavelengths being created in the object. There can be time delays between the excitation and ﬂuorescence, which can be used through precise time gating to distinguish between the photons from the illumination and the sample and enhance contrast and sensitivity. The improvement in detectors and gating has allowed imaging and spectroscopy to reach the level of detecting single molecules. This in turn has led to a major expansion of imaging techniques in biological and medical applications [46]. Another option is the use of multiphoton excitation where the presence of several low energy photons is required to created one high energy photon. For example, Figure 2.14d was created using 2-photon excitation of a dye in the brain sample. This has the advantage that the excitation has a much larger probability in the focus of a laser beam since the probability scales with the square to the local photon ﬂux. Therefore superior spatial resolution is achieved in the x, y plane and to some degree in the z direction. It is possible to reach with high precision individual or many points simultaneously for photostimulation and to image light from multiple spots. Higher-order three-photon excitation is possible. Such non-linear processes, which use multiphoton processes, are also at the core of so-called super-resolution techniques, including stimulated emission depletion (STED) microscopy proposed by Stefan Hell in 1994 and demonstrated in 1998 [47, 48], ﬂuorescence photoactivation localization microscopy (FPALM), and other techniques in microscopy that bypass the conventional diﬀraction limits. For a summary of the techniques and some of the wide impact they have had, see, for example, [49].

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As we have seen in Chapter 4, the time-averaged photon ﬂux of the quantized ﬁeld is proportional to the intensity of the classical ﬁeld, N(X, Y ) ∝ I(X, Y ), and the losses, redistribution of light, and change in ﬂux can be described by one standard model. The quantum eﬃciency 𝜂 is the one single parameter that is used to model the change in photon ﬂux. This includes all the ﬁner details of Fourier optics and any of the imaging techniques. As a consequence we can easily translate the intensity distribution to determine the photon ﬂux captured by each of the pixels of the photon detector. This applies independently to each pixel. For all these linear processes, the photon statistics will be either a Poissonian distribution or approach this distribution for lower and lower values of 𝜂. Consequently, we have a universal description of the noise in each part of the imaging system and for each pixel. Moreover, as with multiple beamsplitters, there are no correlations between the ﬂuctuations in each of the pixels; the photon counting noise in diﬀerent pixels is independent and only scales with the photon ﬂux. It is easy to visualize an image that has been taken in very dim light by just a few photons. It is grainy and noisy and is obviously not of the best quality. With modern technology, and cooled CCDs, it is now quite common to work with just a few photons/pixel and time interval. More photons, more intensity, would obviously help until we have a beautiful image at high intensities. Many situations where this is not possible come to mind, in night viewing, astronomy, etc. The human eye is close to the limit of being able to detect individual photons. Whether this is possible is still a topic of active research, and it seems that this is possible for some situations and some people but not for all, as discussed in Section 3.1. Detectors have steadily improved, with improvements to the sensitivity and reduction of other noise sources, as listed in Section 7.2. Thus it is now more common to image with a few photons per pixel and to be photon counting and shot noise limited. This means imaging has in many cases reached the limits set by the quantization of the light. Examples are given in Figure 2.14 where parts (a) and (d) were made using specialized CMOS detectors. In addition, there has been rapid progress with the development of arrays of single-photon detectors based on SPADs as used in part (d) where both fast timing and high sensitivity are important. All these types of single-photon detectors are now employed in more and more low light imaging situations in astronomical, medical, biological, and engineering applications.

References 1 Iams, H.E. and Salzberg, B. (1935). The secondary emission phototube. Proc.

IRE 23: 55–64. 2 Kubetsky, L.A. (1937). Multiple ampliﬁer. Proc. Inst. Radio Eng. 254: 421. 3 McIntyre, R.J. (1961). Theory of Microsplasma instability in silicon. J. Appl.

Phys. 32: 983. 4 Lounis, B. and Orrit, M. (2005). Single-photon sources. Rep. Prog. Phys. 68:

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23 Collins, M.J., Xiong, C., Rey, I.H. et al. (2013). Integrated spatial multiplexing

24 25 26 27 28

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35 36

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40 41 42

of heralded single-photon sources. Nat. Commun.. 4. https://doi.org/10.1038/ ncomms3582. Kim, J., Benson, O., Kan, H., and Yamamoto, Y. (1999). A single-photon turnstile device. Nature 397: 500. McKeever, J., Boca, A., Boozer, A.D. et al. (2004). Deterministic generation of single photons from one atom trapped in a cavity. Science 303: 1992. Riebe, M., Monz, T., Kim, K. et al. (2008). Deterministic entanglement swapping with an ion-trap quantum computer. Nat. Phys. 4: 839. Home, J.P., Hanneke, D., Jost, J.D. et al. (2009). Complete methods set for scalable ion trap quantum information processing. Science 325: 1227. Zhang, J., Pagano, G., Hess, P.W. et al. (2017). Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator. Nature 551: 601. Brouri, R., Beveratos, A., Poizat, J.-Ph., and Grangier, P. (2000). Photon antibunching in the ﬂuorescence of individual color centres in diamond. Opt. Lett. 25: 1294. Beveratos, A., Kuhn, S., Brouri, R. et al. (2002). Room temperature stable single-photon source. Eur. Phys. J. D 18: 191. James, D.F.V. and Kwiat, P.G. (2002). Atomic vapor-based high eﬃciency optical detectors with photon number resolution. Phys. Rev. Lett. 89: 183601. Taylor, G.I. (1909). Interference fringes with feeble light. Proc. Cambridge Philos. Soc. 15: 114. Kwiat, P.G., Mattle, K., Weinfurter, H. et al. (1995). New high-intensity source of polarization-entangled photons. Phys. Rev. Lett. 75: 4337. Takeuchi, S., Kim, J., Yamamoto, Y., and Hogue, H.H. (1999). Development of a high-quantum-eﬃciency single-photon counting system. Appl. Phys. Lett. 74: 1063. Kim, J., Takeuchi, S., Yamamoto, Y., and Hogue, H.H. (1999). Multiphoton detection using visible light photon counter. Appl. Phys. Lett. 74: 902. Cabrera, B., Clarke, R.M., Colling, P. et al. (1998). Detection of single infrared, optical, and ultraviolet photons using superconducting transition edge sensors. Appl. Phys. Lett. 73: 735. Lita, A.E., Miller, A.J., and Nam, S.W. (2008). Counting near-infrared single-photons with 95% eﬃciency. Opt. Express 16: 3032. Fukuda, D., Fujii, G., Numata, T. et al. (2009). Ti/Au TES to discriminate single photons. Metrologica 46: S288. Munro, W.J., Nemoto, K., Beausoleil, R.G., and Spiller, T.P. (2005). High-eﬃciency quantum-nondemoltion single-photon-number resolving detector. Phys. Rev. A 71: 033819. Takeuchi, S. (2001). Beamlike twin-photon generation by use of type II parametric downconversion. Opt. Lett. 26: 843. Zalewski, E.P. and Geist, J. (1980). Silicon photodiode absolute spectral response self calibration. Appl. Opt. 19: 1214. Gardner, J.M. (1994). Transmission trap detectors. Appl. Opt. 33: 5914.

Further Reading

43 Gray, M.B., Shaddock, D.A., Harb, C.C., and Bachor, H.-A. (1998). Photode-

44 45 46 47

48 49

tector designs for low-noise, broadband and high power applications. Rev. Sci. Instrum. 69: 3755. Hobbs, P.C.D. (2001). Photodiode front needs: the real story. Opt. Photon. News, OSA, April. 42. Hobbs, P.C.D. (2001). Building Electro-Optical Systems: Making it All Work. Wiley. Michalet, x., Siegmund, O.H.W., Vallerga, J.V. et al. (2007). Detectors for single-molecule ﬂuorescence imaging and spectroscopy. J. Mod. Opt. 54: 239. Hell, S.W. and Wichmann, J. (1994). Breaking the diﬀraction resolution limit by stimulated emission: stimulated emission depletion microscopy. Opt. Lett. 19: 780. Hell, S.W. (2009). Microscopy and its focal switch. Nat. Methods 6: 24. Chi, K.R. (2009). Super-resolution microscopy: breaking the limits. Nat. Methods 6: 15. https://doi.org/10.1038/nmeth.f.234.

Further Reading Rieke, G. (2002). Detection of Light: From the Ultraviolet to the Submillimeter. Cambridge University Press. Saleh, B.E.A. and Teich, M.C. (2007). Fundamentals of Photonics, 2e. Wiley.

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8 Quantum Noise: Basic Measurements and Techniques We now turn our attention to the measurement of quantum noise – that is the fundamental noise we observe on the photocurrent when we detect optical beams with a large coherent amplitude (|𝛼|2 ≫ 1) using photodetectors. Several theoretical models have already been described in Chapters 2 and 4, and the experimental components have been discussed in Chapters 5 and 7. Based on this let us build up, step by step, the experiments for measuring and manipulating quantum noise.

8.1 Detection and Calibration of Quantum Noise We will concentrate on CW laser beams and the noise associated with the intensity, amplitude, and phase of the light. We have to devise ways of interpreting the electrical noise spectra. How do we distinguish quantum noise from ordinary technical noise? How can we can calibrate our measurement? Three diﬀerent techniques are commonly used: direct detection (Section 8.1.1), balanced detection (Section 8.1.2), and homodyne detection (Section 8.1.4). The choice depends on the properties of laser beam. 8.1.1

Direct Detection and Calibration

The obvious way of detecting quantum noise is to directly detect the light with a single photodetector and to analyse the resulting photocurrent. As long as we can assume that the noise is Gaussian, we will have the full information in the form of the variance of the noise, and we can use an electronic spectrum analyser or we can perform a spectral decomposition of the digitized data using a Fourier transform. This is the same arrangement as one would use to measure an intensity modulation of the laser beam, and both of these eﬀects will be present in the resulting spectrum. In most applications we will be interested in the quantum noise as the ultimately limiting eﬀect in the signal-to-noise ratio (SNR) (Figure 8.1). Direct detection gives us an electrical noise power pi (Ω), which for a perfect laser mode and detection system should be entirely due to the quantum noise of the light. There are obviously some technical pitfalls that need to be avoided, and we need a reliable procedure that convinces us, and anybody else, that we A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

8 Quantum Noise: Basic Measurements and Techniques

V1las(Ω) Laser

Photodetector efficiency η

Amplifier gain gel Noise pel

Spectrum Analyser Vi (Ω) pi (Ω)

Figure 8.1 Schematic diagram of direct detection.

are really measuring quantum noise. The noise power pi (Ω) displayed by the spectrum analyser is a measure of ﬂuctuations of the photocurrent, the ampliﬁcation, and the noise contributions of the electronics. For a quantum noise limited source, we obtain the value pi,QNL (Ω). All measurements will be normalized to this value, resulting in the normalized variance Vi (Ω) =

pi (Ω) pi,QNL (Ω)

(8.1)

Once the recorded noise traces have been calibrated, it is obvious that a new unit should be used. A creative suggestion was to label the standard quantum noise limit (QNL) as 1 Hoover, in recognition of the role of the vacuum noise [1]. All noise diagrams in this guide with linear scales, or a logarithmic scale in decibels (dB) deﬁned on page 261, use this unit. We need to know the total quantum eﬃciency of the detection system to link the variance Vi (Ω) to the ﬂuctuations of the light V 1(Ω) (Eq. (4.57)). All values for the noise power are long-term averages, as shown on the display depicted in Figure 8.2. In this case a quantum noise limited light source with modulation at about 9 MHz illuminates the detector. The left-hand side traces show the display for a spectrum from 6 to 10 MHz with constant resolution bandwidth (RBW) of 100 kHz and variable video bandwidths (VBWs) of 30, 300, and 3000 Hz. Note that the average noise remains constant whilst the actual time ﬂuctuations of the noise increase with VBW. The signal remains unchanged, and the width of the signal is given by RBW. The right-hand side traces show the results for the RBW –86 Noise power (dbm)

270

–90

–94

–98

VBW: 10 Hz RBW: 100 kHz

VBW: 100 Hz RBW: 100 kHz

VBW: 300 Hz RBW: 100 kHz

VBW: 300 Hz RBW: 300 kHz

VBW: 300 Hz RBW: 1 MHz

Figure 8.2 Experimental noise and signal spectra for diﬀerent combinations of resolution and video bandwidth. These are four separate experimental traces, with settings as shown in Figure 7.9. The increases in the noise ﬂoor with larger resolution bandwidth are clearly visible.

8.1 Detection and Calibration of Quantum Noise

of 300 kHz. The average noise level increases with RBW. In contrast, the signal height remains the same, since the signal is much larger than the noise and the signal width increases with RBW. Clearly we gain by selecting a larger RBW and get further away from the electronic noise. However, there are limitations, in particular if we have not a ﬂat but a structured noise spectrum, and there are a number of technical complications: 1. Electronic noise. The photodetector generates independent electronic noise, pel (Ω), which adds to the optical noise spectrum. We can avoid this complication by designing special detectors with low electronic noise (Section 7.3.2) and by independently measuring the dark noise of the detector. This is usually done by simply blocking the detector. If the dark noise is signiﬁcant, we can correct the measured signal by subtracting the dark noise after the measurement. It can be assumed that the electronic noise and the optical noise are independent and are added in quadrature. It should be noted that some exotic ampliﬁers have the unpleasant property that the noise depends on the actual photon current. In these cases a normal dark current measurement is not sufﬁcient, and several measurements at various photocurrents, or intensities, are required to investigate the scaling of the electronic noise. The performance of a given detector is commonly described by the noise equivalent power (NEP), which is the optical power on the detector at which quantum noise and electronic noise are equal (pi,QNL (Ω) = pel (Ω)). That means the recombined trace is 3 dB above the dark noise trace. Values of NEP = 1 mW are typical. Values as low as 10 μW have been achieved. 2. AC response. Most detector circuits will have a frequency-dependent gain, given by gel (Ω). For all detectors there will be a high frequency roll-oﬀ, depending on the type of photodiode and ampliﬁer used. Frequently, there is also some electronic mismatch between the components leading to a sinusoidal frequency response curve. This is more a nuisance than a complication. It is taken into account by normalizing the traces (Eq. (8.1)), and gel (Ω) does not aﬀect Vi (Ω) directly. 3. Eﬃciencies. The detector eﬃciency is not perfect. A low eﬃciency has the same eﬀect as a beamsplitter that diverts only a fraction of the light into the detector. For measurements of the quantum noise alone, this is not crucial. For example, we can measure the QNL even from only a fraction of the beam and compensate with more electronic gain. However, knowledge of the eﬃciency is crucial to calculate the SNR of a signal or the degree of squeezing in the light, which means diﬀerences to the QNL. These eﬀects disappear with high quantum eﬃciency. Note that electronic gain and eﬃciency play diﬀerent roles. The golden rule for any squeezing experiment is that all of the light should be detected and that quantum eﬃciencies have to be close to unity. 4. Power saturation. Detectors can only handle a limited amount of optical power. The actual limit varies from mW for small fast photodiodes up to W for large detectors. Once the limit in power is exceeded, the detector can actually be damaged due to heating processes. The absorbed light will heat the diode material, and, more importantly, the bias current will lead to further ohmic heating. Even below this damage limit, the performance can become

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non-linear. The number of available carriers inside the detection material is limited, and the response drops if the local intensity is too high. Thus the detector has to be uniformly illuminated even at intensities below the recommended power limit. 5. Saturation of the ampliﬁers. Several cases have been reported where a detector showed a peculiar response and the reason was the presence of a strong modulation signal that saturated the entire circuit. Saturation at one radio frequency (RF) can aﬀect the performance of the detector at many other frequencies. Care should be taken that no such strong signal is present by recording large bandwidth spectra up to the roll-oﬀ frequency of the detector. A particular pitfall is to use a series of ampliﬁers where the bandwidth of the earlier stages is broader than that of later stages, for example, by using a low-pass ﬁlter at the output of an ampliﬁer. In this case the large high frequency signal would not be visible in the output, but it is saturating the ampliﬁers ahead of the ﬁlter. Saturation can change the noise spectrum over a wide bandwidth, and it can even lower the noise level, and in some experiments saturated ampliﬁers have generated noise traces that mimicked, and were mistaken for, squeezing. After all precautions have been taken, we can trust the detector. But in order to measure the QNL, we still need to calibrate it. One obvious technique is to calculate the expected noise power pi,QNL (Ω) from the formula for shot noise (Eq. (3.9)), and using the appropriate constants, we get pi,QNL (Ω) (dBm) = −138.0 + 10 log(i × RBW)(dBm) + pel (Ω)(dBm) + gel (Ω)(dB) where RBW is measured in Hz and i in A and an input resistance of 50 Ω is assumed. pi,QNL (Ω) can be determined once the average photocurrent iave is known. This in turn should be done best by a direct measurement of the DC from the photodiode. It also requires a knowledge of the detection bandwidth, which is normally the RBW of the spectrum analyser. The AC gain of the various amplifying stages has to be included. Finally, the actual response characteristics of the spectrum analyser to noise, rather than to a signal, have to be taken into account (Section 7.3.4), which could require corrections as large as 2 dB. After all these measurements and calculations, one could expect to get a value for the noise level with an accuracy of typically 20% (1 dB). This is good enough to test the apparatus for inconsistencies but insuﬃcient to calibrate the traces accurately. A useful check is to investigate the scaling of the detected electrical noise power with the optical power, as shown in Figure 8.3. Quantum noise, electronic noise, and classical photon noise, or modulations, all scale diﬀerently. Electronic or dark noise is usually independent of the optical power Popt . The noise power 0.5 . The noise due to classical optical due to quantum noise is proportional to Popt noise is proportional to Popt . Figure 8.3 shows four cases of diﬀerent mixtures of noise and their scaling. A reliable measurement of the quantum noise should only be attempted for situations that show a linear dependence of the measured noise level on Popt , in this case near Popt = 1.

8.1 Detection and Calibration of Quantum Noise

Log noise power, Pi (Ω)

100 (d) 10 (c) 1 (b) (a) 0.1 0.1

1 10 Log optical power, Popt

100

Figure 8.3 Scaling of diﬀerent types of noise for a beam of light with ﬁxed classical relative intensity noise (RIN = constant) and variable optical power Popt . The noise terms scale diﬀerently with Popt : (a) electronic noise, (b) quantum noise, (c) classical noise, and (d) total noise as seen in an experiment. For convenience all terms are normalized by the quantum noise present at Popt = 1.

8.1.1.1

White Light Calibration

An alternative is to calibrate the detector with a light source that we know is quantum noise limited. For example, we could use a thermal white light source, which has Poissonian photon statistics (Section 2.3). In principle all that is required is to illuminate the detector with white light that generates the same DC photocurrent and to record the RF noise spectrum. However, care has to be taken: the wavelength should be ﬁltered to the same wavelength (about ±20–50 nm) as used in the actual experiment since the response of photodiodes is known to be wavelength dependent. The absorption depth depends on the wavelength. In addition, the illumination pattern has to be similar since localized illumination can lead to charge carrier depletion. Finally, the diode, particularly the very small InGaAs diodes, can easily be geometrically overﬁlled. Once a large fraction of the light misses the photosensitive area, it can heat up the entire chip and distort the results. From experience, large photodetectors can be calibrated using white light to about ±0.1 dB, whilst small InGaAs detectors can be calibrated to an accuracy of about ±0.4 dB. 8.1.2

Balanced Detection

An alternative technique for calibration is to use a balanced detection system. Here the incoming light is divided into two beams of equal optical power, which are detected by two matched photodetectors (Figure 8.4). The two photocurrents are either added or subtracted. Two noise spectra are created by the electronic spectrum analyser and are recorded as the sum and diﬀerence spectrum. This technique can distinguish between optical quantum noise and classical optical noise. The beamsplitter sends classical noise onto both detectors – the classical optical noise in both beams, and thus the photocurrents are correlated to each other. That means they can be subtracted and cancelled. When added, the full

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8 Quantum Noise: Basic Measurements and Techniques

Vvac (Ω)

η

Vlas (Ω)

η

+

– gel (Ω) pel (Ω)

Vi (Ω)

Figure 8.4 Balanced detector.

noise power is recovered. For quantum noise the eﬀect of the beamsplitter is diﬀerent – the two resulting photocurrents are not correlated. That means the two channels add in quadrature for both the sum and the diﬀerence. In the quantum noise limited case, the two spectra are identical. This argument is based purely on the correlation properties of the sidebands containing the information about the noise, as described in Section 4.5. A more formal derivation of these results is given in Appendix B, which describes the balanced detector in terms of the propagation of coherent states through a beamsplitter. Electronic noise from the two detectors is obviously not correlated and adds in quadrature. It cannot easily be distinguished from quantum noise. Consequently, the apparatus can suppress all classical technical noise on this mode of light and leaves only the optical quantum and the electronic noise. The description is p+ (Ω) = gel (Ω) pi (Ω) Vi (Ω) + pel (Ω) p− (Ω) = gel (Ω) pi (Ω) + pel (Ω) Vi (Ω) =

p+ (Ω) − pel (Ω) p− (Ω) − pel (Ω)

(8.2)

In the case of sub-Poissonian intensity noise, the plus trace displays the noise suppression, and the minus trace shows quantum noise only. The technical difﬁculties with this detection scheme is the balancing of the detectors. The two photodetectors have to have the same electronic RF response. We have eﬀectively built an electronic interferometer, and both the gain and the phase have to be matched at all RF that are used. This can be extraordinarily time consuming since the response frequently depends not only on the notional values of the components used but also on the actual physical layout of the circuit. Obviously cable lengths and delay times in the circuits have to be matched to achieve a proper balance. Passive splitters and combiners are used as adders and subtractors at high frequencies (≥ 50 MHz), whilst active circuits are useful for low and medium frequencies. The optical balancing can be tricky since it ideally requires a precise 50/50 beamsplitter. This can be achieved by careful adjustment since the splitting ratio for most dielectric beamsplitters depends on the input angle. Alternatively, a polarizing beamsplitter (PBS) can be used. For a linearly polarized output beam, the ratio can be continuously adjusted by rotating the polarization direction ahead of the beamsplitter with a half-wave plate.

8.1 Detection and Calibration of Quantum Noise

In all these arrangements optical losses have to be avoided if squeezing is to be detected since they would eﬀectively decrease the quantum eﬃciency. It is tempting to compensate a mismatch in the electronic gain with an adjustment in the optical power. But again, this would decrease the quantum eﬃciency. Thus many of the common tricks used in detecting ordinary coherent light are not allowed for squeezed light. 8.1.3

Detection of Intensity Modulation and SNR

The size of a modulation can be measured directly. After all, electronic spectrum analysers were designed for this purpose. The height of the spectral peak is proportional to the power of the Fourier component. The relative intensity noise is linked to the measured power by pi (Ω) = (i RIN(Ω))2 × RBW 1

RIN(Ω) = 10 2 pi (Ω) dBm

(8.3)

For example, a beam that generates an average photocurrent of 1 mA with a modulation of one part in 104 (RIN = 10−4 ) produces a noise peak of −10 dBm when measured with a RBW of 100 kHz. For large modulations, the photocurrent SNR, as deﬁned in Eq. (7.7), can be measured directly as the diﬀerence, in dB, between the signal peak and the noise ﬂoor. For small modulations we have to take into account that the signal peak contains both the signal and the noise and have to calculate the value SNR = (signal peak − noise)/noise. For very small SNR (< 2), the exact response of the spectrum analyser has to be considered (Appendix E). Electronic gain will not aﬀect the SNR. Finally, the total quantum eﬃciency, including any form of attenuation of the light, has to be taken into account when using Eq. (7.7) to evaluate the optical SNR from the measurements of the photon current. 8.1.4

Homodyne Detection

The direct detection schemes described in Section 8.1 are used to measure the intensity as well as the intensity noise of a laser beam. For a single-mode ﬁeld, the combination of two detectors with a beamsplitter can be used to evaluate the contribution of classical modulation and quantum noise. However, the direct detection cannot diﬀerentiate between the quadratures, eﬀectively only measuring the in-phase (or amplitude) quadrature. To measure diﬀerent quadrature angles, we need a reference beam, commonly called the local oscillator. This beam has to be phase-locked to the input; otherwise it cannot provide a phase reference to distinguish between the quadratures. If the local oscillator has the same frequency as the input, we have a homodyne detector. Alternatively, if the local oscillator is frequency shifted, the system is known as a heterodyne detector. 8.1.4.1

The Homodyne Detector for Classical Waves

For a homodyne detector we can assume that the local oscillator is created by the same source as the input beam we want to measure. In practice the link

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Unknown beam αin Laser

M1 M4 Local oscillator αlo

D1

ϕlo Phase shifter

D2

_

i_(t)

Figure 8.5 Homodyne detector.

could be much more indirect. The entire system can be represented by one large interferometer (Figure 8.5). The input beam 𝛼in and the local oscillator beam 𝛼lo are combined by the mirror M4 that has a balanced 50/50 reﬂectivity. The whole apparatus has to be aligned in the same way and with the same precision as a conventional interferometer. The two detectors D1 and D2 will receive the interference signal. Firstly, let us classically analyse the performance of the system. The two beams can be represented by the amplitudes 𝛼in (t) = 𝛼in + 𝛿X1in (t) + i𝛿X2in (t) 𝛼lo (t) = [𝛼lo + 𝛿X1lo (t) + i𝛿X2lo (t)] ei𝜙lo

(8.4)

i𝜙lo

Here e represents the relative phase diﬀerence between the input amplitude and the local oscillator amplitude. It can be controlled by changing the length of the one arm of the interferometer designated as the local oscillator. In practice, this involves the ﬁne control of a mirror position by a piezo or the rotation of a glass plate in the local oscillator beam. The local oscillator should dominate the output signal. We consider the case that the local oscillator beam is far more 2 . In this case we can assume that all the intense than the input beam: 𝛼lo2 ≫ 𝛼in intensity is from the local oscillator and that the intensities on both detectors are the same: 1 |𝛼D1 |2 = |𝛼D2 |2 = |𝛼lo |2 (8.5) 2 All terms proportional to 𝛼in can be dropped in comparison with those proportional to 𝛼lo . The signal at the two detectors can be described using the standard convention for the phase shift in a single beamsplitter (Eq. (5.1)), including a 180∘ phase shift for one of the reﬂected beams (Figure 8.6): √ √ 𝛼D1 = 1∕2 𝛼lo (t) + 1∕2 𝛼in (t) √ √ (8.6) 𝛼D2 = 1∕2 𝛼lo (t) − 1∕2 𝛼in (t) The intensities and the photocurrents from the two detectors are proportional to |𝛼D1 |2 and |𝛼D2 |2 , respectively, ∗ |𝛼D1 |2 = 1∕2 [ |𝛼lo (t)|2 + 𝛼lo (t)𝛼in (t) + 𝛼in (t)𝛼lo∗ (t) + |𝛼in (t)|2 ]

8.1 Detection and Calibration of Quantum Noise

Figure 8.6 Projection through the uncertainty area.

δX2

θ = ϕlo δX1

which can be approximated as |𝛼D1 |2 = 1∕2 [|𝛼lo |2 + 2𝛼lo 𝛿X1lo (t) + 2𝛼lo (𝛿X1in (t) cos(𝜙lo ) + i𝛿X2in (t) sin(𝜙lo )) ]

(8.7)

∗ compared with |𝛼lo |2 and 𝛼lo 𝛿X ∗ compared by neglecting terms such as 𝛼lo 𝛼in ∗ with 𝛼in 𝛿X . In addition all higher-order terms in 𝛿X have been left out. Similarly, the result for the other detector can be found. We can now deﬁne the diﬀerence current i− (t) from the two detectors:

i− (t) ≈ 2𝛼lo (𝛿X1in (t) cos(𝜙lo ) + i𝛿X2in (t) sin(𝜙lo ) )

(8.8)

This is a remarkable result. The current i− (t) scales with the amplitude of the local oscillator, but the ﬂuctuations, or noise, of the local oscillator are completely suppressed. Its noise level could be well above the QNL, and the result would not be aﬀected. On the other hand, the power of the input beam has no inﬂuence, as long as it is small compared to the power of the local oscillator. This is an extremely useful and somewhat magic device. The diﬀerence current i− (t) contains only higher frequency, or AC, terms. Obviously the DC cancel each other. In the experiment we could use a splitter/combiner or a diﬀerential AC ampliﬁer to generate this ﬂuctuating current. Finally, the variance of this current is evaluated. We should note that in this calculation all cross terms of the type 𝛿X1𝛿X2 cancel out in the averaging process. We obtain Δi2− ≈ 4𝛼lo2 (𝛿X12in cos2 (𝜙lo ) + 𝛿X22in sin2 (𝜙lo ))

(8.9)

This means the measured variance is a weighted combination of the variance of the ﬂuctuations in the two quadratures. In particular for the phase 𝜙lo = 0, the variance in the quadrature amplitude is measured, whilst for 𝜙lo = 90 = 𝜋∕2 the variance in the quadrature phase is measured. An alternative interpretation of Eq. (8.9) is that the homodyne detector measures the variance of the ﬂuctuating state under a variable projection angle 𝜃 = 𝜙lo . By scanning the local oscillator phase, the projection angle can be selected. The sideband interpretation says that the diﬀerent types of modulation diﬀer by the relative phase 𝜃 between the two beats between the two sidebands and the carrier. The homodyne detector selects one such phase. The modulation with the other phase (𝜃 + 90) is not detected at all. The homodyne detector ﬁlters out speciﬁc quadrature of the modulation sideband. With the ﬁxed phase shift 𝜙lo for the local oscillator, the detector will have best sensitivity for one particular type of modulation. For example, for 𝜙lo = 0, only amplitude modulation is detected, and

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for 𝜙lo = 90, only phase modulation is detected. The sensitivity of the detector to this quadrature changes with cos2 (𝜙lo ). The homodyne detector is very popular in radio applications and is a key technology in our present communication systems. Signals are detected in phase with a local oscillator, in the case of radio waves an electronic oscillator inside the receiver, which can be controlled in frequency very well, for example, using quartz crystal oscillators, or could even be locked to the incoming RF (known as phase-locked loop (PLL)). Radio homodyne and all the later heterodyne detection techniques have allowed us to place diﬀerent carriers very close to each other in the radio spectrum and to detect very weak signals. This increases dramatically the carrying capacity compared to a simple intensity-coded system and allowed applications such as the mobile phone, satellite communications, etc. Quantum mechanical calculations are required to check the validity of the above classical results for the quantum properties of the light. The balanced homodyne detector is a special case of a Mach–Zehnder interferometer (Section 5.2) with a 50/50 output mirror. The two incident ﬁelds on the beamsplitter M4 can be described by the operators ̂ in (t) + i 𝜹X𝟐 ̂ in (t) â in (t) = 𝛼in + 𝜹X𝟏 ̂ lo (t) + i 𝜹X𝟐 ̂ lo (t)] ei𝜙lo â lo (t) = [𝛼lo + 𝜹X𝟏

(8.10)

where we are using the nomenclature of Section 4.5.1. In complete analogy to the classical derivation, we can now derive the equivalent to Eq. (8.9) but now for the full quantum operators: ) ( ̂ 2 ⟩sin2 (𝜙lo ) ̂ 2 ⟩cos2 (𝜙lo ) + ⟨𝜹X𝟐 Δi2− ≈ 4𝛼lo2 ⟨𝜹X𝟏 in in = 4𝛼lo2 (V 1in cos2 (𝜙lo ) + V 2in sin2 (𝜙lo ))

(8.11)

The derivation of Eq. (8.11) holds for the approximation that 𝛼lo ≫ 𝛼in . In this simple interferometer we have a beautiful device for the measurement of the variance for a single quadrature, including all quantum properties. We can select the quadrature by simply varying the phase angle of the local oscillator. For a coherent state, where all quadratures are identical, this technique has no particular advantage over direct detection. However, once the variances of the quadratures change, as in a squeezed state, the homodyne detector is the essential measuring system. It was ﬁrst proposed by H.P. Yuen and J.H. Shapiro [2] and D.F. Walls and P. Zoller [3] and has since been a key component in all experiments with quadrature squeezed states, as discussed in Chapter 9. Finally, a few practical hints are given: It should be noted that the homodyne detector can be operated even when the intensity of the input is small. As long as |𝛼in |2 is smaller than the intensity |𝛼lo |2 of the local oscillator, all the results given above are correct. Thus the detector can be used to measure squeezed vacuum states as well as bright squeezed light. If |𝛼in |2 is comparable with |𝛼lo |2 , the output intensity will change periodically with 𝜙lo , and the variance will be a mixture of variances from both orthogonal quadratures. Balancing the homodyne detector is the most diﬃcult part of this experiment. Both the amplitude and the wave front curvature have to be matched for both outputs. This generally means that

8.1 Detection and Calibration of Quantum Noise

both the local oscillator and the input beam should travel comparable distances and should experience similar optical components such as lenses and mirrors. In squeezing experiments we cannot aﬀord any losses; otherwise some vacuum noise would be added, and the noise suppression would deteriorate. That means some of the tricks quite common with normal classical interferometers cannot be used here. For example, we have to detect all of the beam, and using an aperture is not allowed. We cannot balance the two beamsplitter outputs by attenuating one of the beams. It is also important that all the electrical components are balanced and they should have the same gain and phase diﬀerence at all detection frequencies used. The quality of the alignment can be judged by measuring the fringe visibility VIS of the system (deﬁnition in Eq. (5.28)). This can be done in some experiments by turning up the power of the input beam to be the same as that of the local oscillator. As long as some input intensity is available, the normalized visibility = VIS∕VISperfect can be measured, where VIS is the directly measured value and VISperfect is the visibility predicted from the measurement of the intensities. For = 1 the homodyne detector records all the correlations accurately. The measurement of the extrema is reduced when < 1, and the correlations between the input and the local oscillator beams appear to be smaller than they really are. The quality of the homodyne detector is described by an overall eﬃciency: 𝜂hd = 𝜂det 2

(8.12)

where 𝜂det is the normal eﬃciency of the photodetector. In most situations 𝜂det is smaller than 2 . Values as good as 2 = 0.98 have been achieved. These details will be discussed in the description of the individual experiments. The homodyne detector is the most common technique to measure the noise and signal properties of both quadratures. It is more complex and requires more patience than direct detection, but it provides the full information. 8.1.5

Heterodyne Detection

Whilst in homodyne detection we are mixing two beams with the same optical frequency on one detector, with a beamsplitter we can also mix beams with diﬀerent optical frequencies 𝜈1 and 𝜈2 . This is known as heterodyne detection. As long as the diﬀerence frequency ΩHT = 𝜈1 − 𝜈2 is within the detection range of the photodetector, and in practice that is limited typically to about 1–2 GHz, we will record a strong, narrow signal at ΩHT , which is surrounded by beat signals created by all the sidebands of either laser beam beating with the respective carriers of the other beam. Note that a spectrum analyser cannot distinguish between positive and negative beat frequencies and these two will be folded together (Figure 8.7). Heterodyne detection is frequently used to determine the linewidth Δ𝜈2 of an unknown laser. The spectrum around the detection frequency ΩHT = 𝜈1 − 𝜈2 contains information about both beams. The lineshape of the heterodyne signal is a convolution of the lineshapes of the two lasers. If one of the laser beams has a narrow linewidth compared with the other, e.g. Δ𝜈1 ≪ Δ𝜈2 , the width ΔΩHT of

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8 Quantum Noise: Basic Measurements and Techniques

Spectrum analyser Laser 1

Laser 2 Mode matching optics

Figure 8.7 Example of heterodyne detection of the bandwidth of two lasers.

the heterodyne signal corresponds to the broader linewidth, in this case Δ𝜈2 . The exact link between ΔΩHT and Δ𝜈 depends on the lineshape. For example, if both lasers have Lorentzian lineshapes, which are quite common, and Δ𝜈1 ≈ Δ𝜈2 , the link is ΩHT = Δ𝜈1 + Δ𝜈2 . Furthermore, the heterodyne spectrum will show any additional laser modes in the input beams. These will appear as strong, narrow spectral features around ΩHT . In heterodyne detection we are measuring signals at high detection frequencies, and outside the prevailing technical noise at low frequencies, the heterodyne technique is very sensitive in the amplitude quadrature. Ideally it is only limited by the QNL of the stronger of the two input beams, and in this case the detection is at the QNL, or commonly called shot noise limited. This technique has been used to detect with very high sensitivity the ﬂuorescence spectrum of individual ions in traps or individual atoms and to determine such properties as the ion trap frequency (as low as hundreds of Hz) and the excitation temperature. In this regime heterodyne detection competes with single-photon detection in sensitivity. 8.1.5.1

Measuring Other Properties

The detection schemes discussed so far concentrate on the measurement of the intensity and the quadratures of the laser light. However, there are other properties of the light, in particular the position and the polarization of the light, that are worth a consideration. Many applications rely on the modulation and detection of these properties, as can be seen from Figure 8.8. Detector (c) measures the beam position by comparing the diﬀerence in the intensity falling onto the two halves of a split detector. This could also be done in both transverse directions using a quadrant detector. Detector (d) uses a PBS to analyse the degree of polarization, in this case horizontal and vertical linear polarization. Other states of polarization can be determined by including a wave plate ahead of the PBS. This comparison shows that in all cases we are using two photodetectors and we are processing the diﬀerence of the photocurrents. In all these cases we will ﬁnd that for a coherent and single-mode beam, the perfect laser beam, each detector produces a photocurrent that includes quantum noise and that simply adds to the quantum noise from the other detector. This means we have a quantum noise for the detection of all these properties: intensity, quadrature position, and

8.2 Intensity Noise

D1 Laser

Laser D2

D1

+/–

(a)

Phase shifter

(b)

D2

_

D1

Laser (c)

Laser (d)

PBS

D2

Figure 8.8 Comparison of detection schemes: (a) balanced detector for V1(Ω), (b) homodyne detector for the quadratures, (c) split detector for the position of the beam, and (d) polarization detector.

polarization. The last one is normalized diﬀerently, and the uncertainty can be expressed in terms of the Stokes parameters that have a cyclic commutation relationship [4, 5], but is still a QNL. We will ﬁnd in Chapters 9 and 10 that for all these properties we can generate non-classical states of light with reduced noise below the QNL.

8.2 Intensity Noise 8.2.1

Laser Noise

Lasers are the light sources in all quantum optics experiments, but they are far from ideal; most lasers have intensity noise well above the QNL as discussed in Chapter 6, page 210. With careful preparation some lasers can be QNL in restricted parts of the spectrum. Good examples are the diode-pumped monolithic Nd:YAG laser, which is one of the quietest and most reliable lasers available (https://www.coherent.com/lasers/main/mephisto-lasers) [6, 7]. These lasers are QNL for frequencies well above 5 MHz, as shown in Figure 6.6, and thus are rather useful light sources for quantum optics experiments. Diode lasers themselves can be excellent low noise light sources at frequencies up to several hundred MHz, as discussed on page 213. In particular the low-power ( 0, the noise is increased. This is not correct. The suppression depends on the value |1 − h(Ω)|2 . This is best explained with a Nyquist diagram (Figure 8.10b). Equation (8.16) means that the feedback loop decreases the variance at all those frequencies where |1 − h(Ω)| > 1, that is, where h(Ω) is outside a circle of radius 1 drawn around the point (1, 0) on the Nyquist diagram. The reduction in the variance is proportional to the square of the distance between h(Ω) and (1, 0). If the line that traces h(Ω) is inside the circle, the variance is actually increased, not reduced. If h(Ω) gets close to (1, 0), the resulting noise is extremely large. The stability of the systems can be described by one simple rule: if the line that traces h(Ω) in the Nyquist diagram encloses the point (1, 0), the system is unstable. It will oscillate madly. This includes the case where h(Ω) = 1. All other systems, where h(Ω) does not enclose (1, 0), are stable. But note that even in a stable system, there can still be frequencies where the noise is increased, as demonstrated in the example shown in Figure 8.11. Finally, we ﬁnd that the variance of the modulation of the beam leaving the classical system is scaled by the transmission 1 − 𝜖 of the beamsplitter and the quantum eﬃciency 𝜂 of the detector, in the same way as the intensity of the output beam, resulting in the transfer Figure 8.11 The action of the noise eater on a large classical signal using the feedback gain shown in the previous diagram. Input noise is V1las (Ω) = 1000, very large compared to the QNL. The resulting output V1out (Ω) remains well above the QNL of V1out (Ω) = 1.

10 000 Vin = 1000 1000 Vout

284

100

10

1

10

100

1000 Ω

10 000

100 000

8.3 The Intensity Noise Eater

function V 1out (Ω) = 𝜂2 (1 − 𝜖) V 1il (Ω) = 𝜂2 (1 − 𝜖)

V 1las (Ω) |1 − h(Ω)|2

(8.17)

where 𝜂2 is the quantum eﬃciency of the output detector. To illustrate Equation (8.17) consider the output noise spectrum shown in Figure 8.11, which is calculated for a large constant laser noise V 1las (Ω) = 1000, very much larger than the QNL. The feedback system has the gain h(Ω) shown in Figure 8.10. We see the eﬀect of a feedback ampliﬁer with a maximum gain at 300 kHz and unity gain points at 6 and 1000 kHz. The gain drops below 1 at frequencies below 6 kHz and above 1000 kHz. This feedback loop has an internal delay time of about 4 μs. The phase response is determined by the somewhat resonant nature of this circuit. In the Nyquist diagram this corresponds to a line that starts with straight negative feedback until h(Ω) changes sign at about 400 kHz. However, the noise is reduced even at higher frequencies up to 1000 kHz since the noise reduction is proportional to the square of the distance to the point (1, 0). The complete spectrum of the output variance is shown in Figure 8.11. The noise is suppressed from very low frequencies to 1000 kHz. The maximum gain results in the best noise suppression at 300 kHz. But when the line h(Ω) enters the circle around (1, 0), the noise increases. A clear hump is visible from 1000 kHz to 40 MHz. This feedback loop functions well at medium frequencies with high gain. But at higher frequencies the phase has increased too much, and excess noise is created. This result is not atypical for a realistic feedback system. In practice one has to work very hard to ﬁnd a design that reduces the gain quickly enough and simultaneously avoids too much electronic phase change. 8.3.2

Quantum Noise Control

The behaviour of the feedback system near the standard quantum limit (QNL) is quite diﬀerent. In this case we have to include the vacuum ﬂuctuations from the unused input port of the beamsplitter. We can also no longer assume that the variance inside and outside the feedback loop will remain the same. Figure 8.12 shows the apparatus including all vacuum ﬁelds. The input into Modulator

Avac

η2

V1las (Ω) Avac

Laser PID h(Ω)

η1

Avac V1out (Ω)

V1il (Ω)

Figure 8.12 Schematic diagram of a noise eater with all vacuum terms.

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8 Quantum Noise: Basic Measurements and Techniques

̃ las (Ω). We distinguish between the in-loop the noise eater is described by 𝜹A ̃ il in front of the beamsplitter, the ﬁeld in front of the in-loop detector ﬁeld 𝜹A ̃ f , the ﬁeld that is leaving the apparatus 𝜹A ̃ o and the two electrical signals, one 𝜹A ̃ el that is fed back to the modulator, and 𝜹A ̃ out from inside the feedback loop 𝜹A the detector that measures the beam leaving the apparatus. We recall that we are using nomenclature introduced in Section 4.5.1 whereby capitals and tildes indicate continuum operators in the frequency domain with commutators: ̃ ̃ † (Ω′ )] = 𝜹(Ω − Ω′ ). [𝜹A(Ω), 𝜹A It is now necessary to deﬁne a quantum description of the feedback. This problem was addressed as early as 1980 and was approached in various ways. The most obvious and direct approach was pioneered by Wiseman et al. [9]. The main feature is that the current signal at time t is not only related to the instantaneous value of the photon ﬂux Nil but is the convolution with a response function that takes the history of the signal into account and simulates the gain of the feedback loop. For frequencies that are smaller than the inverse of the round-trip time of the feedback loop (Ω < 1∕tloop ), the eﬀect is similar to the eﬀect in a classical feedback system, and we can describe the action of the modulator simply by √ ̃ il = 𝜹A ̃ las + g(Ω) 𝜹A ̃ el 𝜹A (8.18) which describes the action of the intensity modulator that combines the input ﬁeld and the electrical signal. The very small absorption of this device, which also corresponds to a vacuum input, has been ignored. The gain of the in-loop electrical ampliﬁer is given by the value g. The various ﬁelds are linked by √ √ ̃ il + 1 − 𝜖 𝜹A ̃ vac ̃ f = 𝜖𝜹A 𝜹A √ √ ̃ el = 𝜂𝜹A ̃ f − 1 − 𝜂 𝜹A ̃ vac 𝜹A √ √ ̃ o = 1 − 𝜖𝜹A ̃ il + 𝜖 𝜹A ̃ vac 𝜹A √ √ ̃ o − 1 − 𝜂 𝜹A ̃ vac ̃ out = 𝜂𝜹A 𝜹A (8.19) where we have assumed that the two detectors have the same quantum eﬃciency 𝜂 and recall that 𝜖 is the reﬂectivity of the feedback beamsplitter. We can now ̃ out , ̃ el and 𝜹A combine the equations in (8.19) to provide explicit expressions for 𝜹A ̃ ̃ convert these into the quadrature operators 𝜹X𝟏el and 𝜹X𝟏out , and use our standard recipe to calculate the noise spectra V 1el (Ω) for the current inside the loop and V 1out (Ω) for the current from the external detector. The details of the calculation are given in Appendix D, and the resulting transfer functions are V 1el (Ω) =

𝜂𝜖(V 1las (Ω) − 1) + 1 |1 − h(Ω)|2

(8.20)

and V 1out (Ω) = 1 +

2 1 − 𝜖 𝜂𝜖(V 1las (Ω) − 1) + |h(Ω)| 𝜖 |1 − h(Ω)|2

(8.21)

√ where h(Ω) = 𝜂𝜖 g(Ω) is the open-loop round-trip gain, which includes both the electronic gain g(Ω) and the optical attenuation. Note that our model

8.3 The Intensity Noise Eater

of the modulator requires the intensity attenuation to be combined with the voltage gain. These results show that the properties of the light inside the controller and the beam leaving the controller are entirely diﬀerent. Inside (Eq. (8.20)), the modulation and the noise are strongly suppressed, and for a large gain the variance V 1el (Ω) can be well below the QNL. The noise eater works well inside, which is possible, since the laser beam inside is a conditional ﬁeld. It only exists inside the feedback system, and the conditions for a freely propagating beam do not apply. The standard commutation rules do not apply and allow this violation of the normal quantum noise rule. Such light is sometimes referred to as squashed light [10]. For high gain, |h(Ω)| > 1, and a low level of laser noise, V 1las (Ω) ≈ 1, the variance V 1el (Ω) of the control current can be substantially less than the QNL. This was observed by A.V. Masalov et al. [11, 12] and others. It was initially concluded that the light inside the loop was sub-Poissonian and could be used for measurements with a better SNR than the QNL would allow. However, it was overlooked that any signal, for example, a modulated absorption, would be controlled by the feedback loop in a similar way as the classical noise. A detailed analysis has shown that even with sub-Poissonian control currents, it is not possible to improve the SNR. Any apparent advantage of the low noise of the control current cannot be used [13]. The properties of the freely propagating output beam (Eq. (8.21)) are quite different. This beam obeys the standard commutation rules and conditions for the quantum noise, and the spectrum V 1out (Ω) in most cases is above the QNL. We ﬁnd that this light has properties that are quite diﬀerent from the predictions of the classical model (see Eq. (8.17)) and contain unique quantum features. In many situations the eﬀect of the noise eater is actually counterproductive, and the machine increases the noise level. This is one of the striking examples where the properties of an optical instrument with small signals, near the QNL, are considerably diﬀerent to the conventional classical predictions, and standard engineering solutions will actually fail. In the area of laser noise control, this lesson had to be learned, and quantum optics rules are important for the technical progress. It is worthwhile to consider the following special cases: 1. For very low gain (h(Ω) → 0), this result shows simply the eﬀect of attenuation by the combination of beamsplitter transmission (1 − 𝜖) and quantum eﬃciency 𝜂 with V 1out (Ω) = 1 + (1 − 𝜖)𝜂 (V 1las (Ω) − 1). This is identical to a simple beamsplitter. Note that the eﬀect of the quantum noise contribution is included. 2. For a QNL input laser (V 1las (Ω) = 1), the feedback loop creates extra noise |h(Ω)|2 that increases with the gain h(Ω) with V 1out (Ω) = 1 + 1−𝜖 . This case 𝜖 |1−h(Ω)|2 is interesting for engineering applications since it means that in systems with very small signals, close to the QNL, the real feedback system shows a behaviour that is quite diﬀerent from the classical approximation. The higher the gain and the smaller the fraction 𝜖 of the light that is used for control, the more excess noise is generated. The physical explanation is that the vacuum noise at the beamsplitter is added and ampliﬁed. For example, if only

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8 Quantum Noise: Basic Measurements and Techniques

10 8 Vlas = 10 6 V1out

288

Vlas = 5 4 Vlas = 1 2 0

0

–1

–2

–3

–4

–5

Round-trip gain, h

Figure 8.13 The eﬀect of the feedback system with 𝜖 = 0.1, for diﬀerent modulations V1las = 10, 5, 1, as a function of the open-loop gain h. At small gains the larger signals are suppressed, and at larger gains (h = −5) all signals approach the same value, in this case V1out = 7.3.

10% of the light is used (𝜖 = 0.1) and the open-loop gain is h = −5 (negative feedback), the output variance will be 1 + ((1 − 0.1)∕0.1) 52 ∕(1 + 5)2 = 7.25. It could be signiﬁcantly larger if the feedback phase is not optimum. In order to avoid this noise penalty completely, all the light would have to be detected (𝜖 = 1), which is clearly impractical. One can minimize the excess noise by detecting most of the light, leaving only a small intensity at the output. In . the limit of very large gain, we obtain the limiting value V 1out (Ω) = 1 + 1−𝜖 𝜖 This value is independent of the gain, a kind of saturation eﬀect producing a ﬁxed noise level, as shown in Figure 8.13. ), the system shows the classical proper3. For very large signals (V 1out (Ω) > h(Ω) 𝜂𝜖 ties with the noise suppression given by V 1out (Ω) = no longer play any role.

𝜂𝜖V 1las . |1−h(Ω)|2

Quantum eﬀects

To illustrate this quantum eﬀect, the performance of the noise eater previously described in Figure 8.10b has been evaluated for a wide range of input laser noise levels. They range from V 1las = 1000 to V 1las = 1. All traces in Figure 8.14 are normalized to the QNL. At the highest frequencies where the gain is much less than one, the noise spectrum is unaﬀected. The top trace, with the large noise level, V 1las = 1000, clearly shows the same response as the classical feedback loop. Biggest suppression is achieved close to the maximum gain 300 kHz and some excess noise at higher frequencies, 2000 kHz. However, the lowest trace for a QNL laser clearly demonstrates the noise penalty. In this particular case the feedback system produces about 8 dB of excess noise. At laser noise levels of about 10 times the QNL, the feedback loop has very little eﬀect indeed since in this case noise suppression and noise penalty balance each other. One explanation for the noise penalty is the fact that the beamsplitter acts as a random selector of photons, not a deterministic junction (see the discussion in Section 5.1). The noise measured by the in-loop detector is a random selection

8.3 The Intensity Noise Eater

104 Vin = 1000

1000

300 100

Vout

100

30 10

10

1

3 1

0.1

1

10

100

1000

104

105

Ω

Figure 8.14 The eﬀect of a noise eater on a laser. Various input noise levels (V1las = 1, 3, 10, … , 1000) are shown. At low noise levels this device creates rather than eats noise.

of the laser noise that is not correlated to the ﬂuctuations transmitted to the output. This small fraction of the noise is detected, converted into a current, ampliﬁed, and added by the modulator to the remaining ﬂuctuations of the light from the laser. Obviously all this has to happen within a time less than 1∕Ω. Unlike in a classical feedback circuit, the noise that was detected cannot cancel the remaining noise since there is no correlation between them. Instead, they are added in quadrature, leading to the excess noise. The more gain we apply, the worse it gets. This phenomenon has been observed and described by a number of researchers. One particularly instructive description was given by Mertz and Heidmann who used a Monte Carlo technique to describe the properties of the beamsplitter [14]. 8.3.2.1

Practical Consequences

The consequence of this performance at the QNL is that we have to reduce the gain of the feedback loop to much less than unity at all detection frequencies where the laser should be quantum noise limited. Most lasers are QNL at large frequencies. Consequently, the task is to develop special electronic ampliﬁers that combine high gain at low frequencies, where classical noise dominates, with low gain (|h| < 10−2 ) at the frequencies where we need QNL performance. The trick to achieve this is a very steep roll-oﬀ in the gain without too much phase lag being incorporated [15], and this requires specialized electronic designs. There are diﬀerent options for intensity control. As discussed so far, an external intensity modulator can be added to the laser. It is also possible to control the pump power to the laser by feedback. In the case of diode lasers, the injection current can be directly controlled, which is a very elegant approach. The performance of this system can be predicted by incorporating the laser noise transfer function for pump noise into the description of the feedback loop. The solution is not trivial. The practical diﬃculty is that most current control units are not

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suitable for this since their external inputs include ﬁlters designed to avoid noise transients. This produces a phase lag that is far too large for a good feedback performance. In many cases it is better to leave the current source untouched and to add a small control current. This was demonstrated successfully by C.C. Harb et al. [16] for a diode laser-pumped CW Nd:YAG ring laser. The relaxation oscillation was suppressed by more than 40 dB, and the noise at low frequencies was largely eliminated. Similar circuits are available in commercial lasers [17]. In conclusion, a feedback intensity controller, or noise eater, is extremely useful to reduce technical noise in the intensity of a laser. With careful design the QNL can be achieved. However, it cannot be used to either generate an output beam that is sub-Poissonian (squeezed) or measure any eﬀect with an SNR better than the standard quantum limit that can already be reached with a coherent state. A complementary design, electro-optic feedforward, has quite diﬀerent properties that can be utilized for noiseless ampliﬁcation (Chapter 10) or quantum information processing (Chapter 13).

8.4 Frequency Stabilization and Locking of Cavities In many experiments it is very important to operate lasers and cavities that are locked to each other, which means they are tracking each other in the frequency domain. Active feedback techniques are required to achieve this aim. The length of either the cavity or the laser resonator is constantly adjusted to keep both of them in resonance at one frequency. Applications for such a system are build-up cavities for non-linear processes, mode cleaner cavities, and the stabilization of the laser frequency. In the latter case, a very quiet reference cavity is chosen, which is protected as much as possible from external inﬂuences, such as mechanical vibrations and temperature ﬂuctuations. The laser is locked to this reference and will have a much narrower linewidth than a free-running laser. The performance of lasers can best be described using the concept of the Allan variance and quoting the timescale on which the smallest variance is observed [18]. Frequency stabilization has been used since the very early days of laser design. Many of the milestones in this technology have been achieved by J. Hall and his colleagues in a continuous series of technical developments since the 1970s. Two current examples illustrate the degree of improvement that can be achieved. A free-running dye laser has a linewidth of many MHz, measured over an interval of 1 second. Using active control techniques, the linewidth can be reduced to about 1 kHz. A free-running diode laser-pumped CW Nd:YAG laser has already a narrow linewidth, 10 kHz measured over 100 seconds, due to its simple mechanical structure. Using active control techniques, linewidths of less than 1 Hz measured in 100 seconds can be achieved. And this is not a fundamental limit [19]. Many tricks are used in the design of the special reference cavities, such as low expansion materials, special mounting systems, and cryogenic control. Extraordinary precision has been achieved through painstaking engineering and some ingenious design. Here only the principles of the locking technique

8.4 Frequency Stabilization and Locking of Cavities

will be outlined, as an introduction to the techniques commonly used. In most CW quantum experiments, at least one cavity locking system is required. Frequently several are used simultaneously. In most cases the locking system can be described by the schematic block diagram shown in Figure 8.15a. The aim of Modulator PBS

Piezo element

Faraday isolator

Reference cavity

Laser

RF signal generator

~ Error signal Mixer

(a)

PBS

Piezo element

Faraday isolator

Reference cavity

Laser

Error signal (b)

Transmitted power (V)

2 TEM00

1.5

TEM10

TEM20

TEM30

(i)

1 0.5

PDH error signal (V)

0 0.05 (ii) 0

Tilt error signal (V)

–0.05 1 0.5

–0.5 –1

(c)

(ii)

0

0

1

2

3 4 5 Scanning time (a.u.)

6

7

Figure 8.15 Frequency locking system. (a) Schematic layout for PDH locking and phase response of the cavity. (b) Layout for tilt locking. (c) Experimental error signals as demonstrated by D.A. Shadock et al. (i) Transmitted power, (ii) error signal generated via PDH locking, and (iii) error signal generated via tilt locking technique. Source: Courtesy of D.A. Shadock et al. [20].

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the device is to dynamically control the detuning Δ𝜈, the diﬀerence between the cavity and the laser frequency, and to maintain resonance (Δ𝜈 = 0). The light from the laser is mode-matched, using a lens system, into the reference cavity. An error signal is generated, which is a measure of the detuning Δ𝜈. This signal is processed using an ampliﬁer of suitable gain, bandwidth, and phase response. Finally, an actuator is used, such as a piezo crystal mounted on a mirror, to control either the laser frequency or the cavity frequency. Let us discuss the most important features of these components. The cavity parameters are chosen to produce a suitable control range. Typically the control range is as wide as the cavity linewidth. For lasers with large frequency excursions, such as dye lasers, a low ﬁnesse cavity is required. For narrowband lasers, such as CW Nd:YAG lasers, a high ﬁnesse (up to 20 000 or more) cavity is chosen to provide a very steep frequency response. The cavities can be confocal, which reduces the diﬃculties of mode matching, or can have a conventional stable resonator conﬁguration.

8.4.1

Pound–Drever–Hall Locking

The error signal has to be responsive to very small changes in Δ𝜈. We require a technique that measures the detuning accurately and with low noise. The most popular technique is to measure the response of the cavity to frequency modulation sidebands. This technique was ﬁrst used by Pound for microwave systems and later reﬁned by several scientists for the optical domain. It is frequently known as the Pound–Drever–Hall (PDH) technique. In this case a phase modulator, driven by a frequency Ωpm , is placed between the laser and the cavity. The light reﬂected from the cavity is detected. Since optical feedback has to be avoided, an optical isolator (Section 5.4) is normally placed between the cavity and the laser, with the reﬂected beam directly accessible. The reﬂected light has modulation sidebands at frequency Ωpm . These are detected using a mixer and, as local oscillator, a part of the modulator driving signal. The amplitude of the resulting low frequency signal provides a suitable error signal. If the cavity is on resonance with the laser beam, which means Δ𝜈 = 0, the reﬂected light will be purely phase modulated. After the mixing process, no signal is available since the two sidebands balance each other exactly. The error signal for Δ𝜈 = 0 is zero. However, once the cavity is detuned, both PM sidebands will experience a diﬀerent phase shift, and the two beat signals will have a phase diﬀerence Δ𝜙out (Δ𝜈) with respect to each other. They will no longer cancel, and there will be some AM, a detectable error signal after the mixer. The shape of the error signal, as a function of cavity detuning, is easily explained for the case that the modulation frequency Ωpm is much larger than the cavity linewidth Γcav . The relative phase of the two beat signals is determined by the phase diﬀerence between the central component 𝜙c and the two sidebands 𝜙− , 𝜙+ . The three components have a distinctively diﬀerent phase, as discussed in Section 5.3. A typical phase response is shown in Figure 8.15c(ii). The corresponding power response, in this case for an impedance-matched cavity, is given in Figure 8.15c(i). We can arbitrarily deﬁne 𝜙(0) = 0 at resonance. That means

8.4 Frequency Stabilization and Locking of Cavities

for large modulation frequencies, we obtain 𝜙(−Ωpm ) = 𝜙(+Ωpm ) = 0 The phase diﬀerence at the output is Δ𝜙out (Δ𝜈) = (𝜙+ − 𝜙c ) − (𝜙c − 𝜙− ) = 𝜙(Δ𝜈 + Ωpm ) − 𝜙(Δ𝜈) − (𝜙(Δ𝜈) − 𝜙(Δ𝜈 − Ωpm ))

(8.22)

This output phase diﬀerence is shown in Figure 8.15c(ii). On resonance both terms are equal, 𝜙(Δ𝜈) = 0, and the two beat signals keep their original phase, Δ𝜙out (0) = Δ𝜙in (0). The modulation remains pure phase modulation. The error signal is zero. For any other detuning we have to evaluate the phase response explicitly. It crosses zero on resonance, Δ𝜈 = 0, and rises steeply on either side. Close to resonance it has a linear slope, which can be measured in rad/MHZ. The steeper the slope, the higher the gain in the feedback loop. At a detuning less than one cavity linewidth, the phase diﬀerence peaks and then rolls oﬀ gently. At such large detunings the feedback system still has the correct sign. However, it has a small gain and the system is only slowly pushed backed to the resonance. The regime of constant feedback sign is called the capture range. As long as the error signal stays within the capture range, the detuning will be reduced and will eventually arrive at resonance; outside it the system moves away from resonance. The capture range of this technique is large, as large as the modulation frequency. A diﬀerent interpretation of the result given in Eq. (8.22) is to say that the vector describing the reﬂected beam is rotated in the phasor diagram in regard to that of the input beam. However, the demodulated signal after the mixer corresponds to a measurement along the quadrature amplitude axis. Phase modulation of the input beam is orthogonal to this axis. The reﬂected light, with a rotated phasor diagram, now has some component of modulation in this direction. Phase modulation is partially converted to amplitude modulation; an error signal appears. The rotation angle 𝜃(Δ𝜈) is the half phase diﬀerence Δ𝜙out (Δ𝜈), and the maximum rotation is ±𝜋∕2. This rotation obviously applies to all types of modulations and also to ﬂuctuations. Error signals can be generated using AM or mixtures of AM and PM. The rotation of the quadrature is also used as a self-homodyning detector for quadrature squeezed light (Section 9.3). 8.4.2

Tilt Locking

An alternative technique for the generation of the error signal is tilt locking. This is based on the interference of spatial modes reﬂected by the cavity. The input ﬁeld is slightly misaligned with respect to the cavity, resulting in a TEM0,0 and other higher-order modes TEMi,j inside the cavity, which are all resonant at diﬀerent detunings Δi,j as a result of the Gouy phase shift (Sections 5.3.3 and 2.1.2). To ﬁrst order, the misaligned beam can be approximated by the sum of the TEM0,0 and TEM0,1 modes that have a phase shift Φ0,0 0,1 between them. Assuming the cavity is not confocal and that TEM0,0 is near resonance, the mode TEM0,1 will be far oﬀ resonance and will be fully reﬂected, like the sidebands in the PDH scheme. The mode TEM0,1 has a phase shift of 𝜋 between the two spatial components. The

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8 Quantum Noise: Basic Measurements and Techniques

mode TEM0,0 will be partially reﬂected, and we now have to combine the amplitudes of the two modes and detect the resulting beam with a split detector that is aligned to resolve the two parts of the TEM0,1 ﬁeld. The interference is critically dependent on Φ0,0 0,1 , which in turn depends on the detuning Δ0,0 . On resonance, Δ0,0 = 0, the two modes are exactly Φ0,0 0,1 = 𝜋∕2 out of phase, and the resulting amplitude on both sides of the split detector has equal magnitude. For a detuning to one side, Δ0,0 < 0, and the phase diﬀerence Φ0,0 0,1 is larger than 𝜋∕2, and this results in a reduction of the combined amplitude on one side of the split detector and an increase on the other side, as shown in Figure 8.15b. The amplitude 𝛼0,0 is rotated in the phasor diagram with respect to the amplitude 𝛼0,1 . The result is, for argument’s sake, a positive signal. For the opposite detuning, Δ0,0 > 0, the roles are reversed, and we will obtain a negative signal. Altogether we generate the error signal as displayed in Figure 8.15c(iii). We see that both techniques are equivalent and produce similar error signals. Tilt locking requires less equipment but is more sensitive to mechanical instabilities. PDH locking is the more commonly employed technique. A third alternative, after Hänsch and Couillaud, is based on the response of the cavity to two input polarizations. Provided the cavity has a higher ﬁnesse for one polarization than the other, the reﬂected light, when analysed in the appropriate polarization direction, will again produce an error signal with the required dispersion shape. Which particular technique should be chosen depends on the speciﬁc technical requirements. 8.4.3

The PID Controller

The other component is the feedback ampliﬁer. It processes the error signal. The gain should be as large as possible, and the bandwidth should span as far as possible to control slow and fast frequency excursions. However, there are severe limitations set by the requirements for the phase of the feedback gain. As already discussed in Section 8.3, a reliable rule for stable operation is that the gain rolls oﬀ with high frequencies and the phase shift at the upper unity gain point should be less than 𝜋. A more precise deﬁnition can be obtained using a Nyquist diagram (Figure 8.10). As a consequence the delay times have to be kept short, and any resonances, mechanical or electronic, have to be avoided. The gain is usually controlled by a PID ampliﬁer, which means a combination of proportional ampliﬁer (middle frequencies), integrator (low frequencies), and diﬀerentiator (high frequencies). All these components contribute to the electronic round-trip phase and have to be carefully optimized. Unfortunately, most electromechanical actuators have resonances in the acoustic range kHz (Figure 8.16), severely limiting the bandwidth of the ampliﬁer that has to reach unity gain at frequencies well below the resonance. One option is to split the control into several independent branches, for example, to use a piezo with large range (length changes of several optical wavelengths) at low frequencies to compensate for a drift of the frequencies and to use electro-optic components, such as a phase modulator inside the cavity, which has a smaller range but no resonance, for the control of the high frequency excursions. Extensive research

8.4 Frequency Stabilization and Locking of Cavities

kPZT, γPZT

Universe

(a)

Counter weight

PZT

kc,γc Universe

Counter weight

mm

mc xc(t)

x=0 Amplitude response

(b)

xm(t)

Phase response

20

0 –0.5 –1 –1.5

10 0 (i)

Amplitude (dB)

(iii)

ϕ control–π

10 0

Stability

Stability region

–10 region 20

0

–π ϕ control –0.5

10 0 Stability –10 region 20

–1 –1.5

Stability region

(iv)

–π 0

ol

ϕ contr

10 0

Stability

–10 region 10

0 –0.5 –1 –1.5

Stability region

20

30

40

50

10

Phase (rad/π)

(ii)

–10 20

20

30

40

–0.5 –1 –1.5 50

Frequency (kHz)

Figure 8.16 (a) Mirror mounted on a PZT modelled as a mass-and-spring system. The counterweight is the mass on which the PZT is pushing. This model includes the option of preloading. (b) Amplitude and phase response of a PZT (i) with small counterweight, (ii) with large counterweight, (iii) with large counterweight and low tension preloading, and (iv) with large counterweight and high tension preloading. Source: From W. Bowen [21].

has been carried out into these control systems, for example, for the applications in frequency standards and precision interferometry. 8.4.4

How to Mount a Mirror

Where the control is done using a lead zirconate titanate (PZT)-mounted mirror, the bandwidth of the feedback control system will be severely limited. The mirror, PZT, and mirror mount eﬀectively form a system with a series of mechanical resonances Ωres,j , from the lowest j = 1 to higher frequencies. Since this system is inside the feedback loop, we can expect stable operation only at frequencies well below the lowest resonance Ωres,1 . Otherwise the phase shift induced by the resonance will interfere with feedback control. The resonance frequency depends on the mass of the mirror, the mass of the mirror mount or counterweight, and the way the PZT and the mirror are connected. The ﬁrst improvement is to use the smallest possible mirror substrate and a PZT glued with a hard

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compound onto a massive backplate as the counterweight. This type of solution is found in many commercial systems, for example, many laser systems with an extended optical resonator. The next step is to modify the mount by preloading it, which means we contain the mirror and PZT in a rigid cage and exert a pressure force onto the mirror. In practical terms the cage is pushing via a rubber O-ring onto the mirror. The result is that the eﬀective spring constants are increased and that the resonance frequencies are increased. This trick has been used by many laboratories that specialize in laser frequency control, and many diﬀerent mechanical systems have been developed. One example was built and analysed by W. Bowen et al. [21] where the counterweight and the preloading tension can be systematically varied. They mounted the mirror as part of a cavity and recorded the cavity locking signal with an electronic network analyser that can record the amplitude and phase response. The results are shown in Figure 8.16b(ii). We can see that the resonances in the spectrum move towards higher frequencies. The shaded area indicates the useful locking range, and this is extended by more and more preloading. The preloading force is limited to values that cause no damage to the mirror. 8.4.5

The Extremes of Mirror Suspension: GW Detectors

This is only one of many examples of optimizing a mirror mount. Many special engineering solutions are being employed, ranging from very small and passive systems to large, complex active systems. The most extreme case is the mounting of the beamsplitter and end mirror in a gravitational-wave (GW) detector as outlined in Section 10.3. Here the mirrors are mounted on a most elaborate and optimized seismic suspension that includes both active and passive components located inside a vacuum system to suppress the vibrations in the environment by many orders of magnitude. The overall concept is to suspend the mirrors on a sequence of pendula, each with diﬀerent frequencies. The combined transfer function will show a large attenuation at all frequencies. The ground vibrations are the largest frequencies due to many seismic and man-made eﬀects from mHz to hundreds of Hz. The multi-pendula suspension in a GW detector has a very sharp cut-oﬀ and is typically used above10 Hz. Whilst very complex both mechanically and in regard to the feedback systems, such suspension systems can be extremely reliable and have been used in continuous long-term operation of GW detectors.

8.5 Injection Locking For a number of reasons, high-power lasers tend to have a much wider linewidth and not have a well-deﬁned output mode. The mechanical design is simple, allowing more mechanical vibrations; the control of the pump source is not very sophisticated, and the electric current driving the pump source could be noisy or more than one pump source may be used. On the other hand, very high-quality lasers of low power but with excellent frequency and intensity noise

8.5 Injection Locking

quality and perfect spatial mode can be built. The trick is to combine the high output power of one laser with the high beam quality of the other. This can be achieved through the technique of injection locking. The basic idea is to seed the high-power laser, referred to as the slave laser, with some of the output from the well-controlled, high-quality laser, the master laser, and force the high-power laser to take on most of the mode qualities of the master laser. The slave laser will change its output frequency and phase to that of the master, and it will remain locked to this frequency. As a consequence the laser linewidth of the slave will be reduced to that of the master. If the slave laser was already operating in a single spatial mode, no changes would occur, and the spatial mode is determined by the geometry of the resonator of the slave laser. However, if it was operating in several modes, one mode will get an advantage, through the injection of light, and the laser can change over to this speciﬁc spatial mode. The output power of the slave laser is basically unaﬀected by the injection locking. Injection locking is the optical equivalent of the locking of two oscillators, an eﬀect that was known for a long time for mechanical systems. It was reported that in 1865 Christiaan Huygens, whilst conﬁned to his bed by illness, observed that the pendula of two clocks in his room moved in synchrony if the clocks were hung close to each other, but became free running when hung apart. He eventually traced the coupling to mechanical vibrations transmitted through the wall from one clock to the other. This observation was later conﬁrmed using electrical oscillators and has developed into a widespread technique in electronic signal processing and microwave technology. Thus it was well known and established before the laser was invented. Optical injection locking will occur simply when the output beam of the master laser is matched into the resonator of the slave laser, both in the spatial waveform and by having the same polarization state. In addition, the optical frequency 𝜈m of the master laser has to be within the locking range 𝜈s ± Δl of the slave laser. Inside this range locking will happen completely. Outside this range no locking will occur at all, and the slave laser will operate as a free-running laser. The change from one state to another is abrupt. In practice this means that the slave laser will track the master laser as long as their independent free-running frequencies are within the locking range, |𝜈s − 𝜈m | = Δms < Δl . It is actually not easy to detect where the slave laser is within the locking range since the free-running frequency cannot be observed. The only indication would be a phase diﬀerence between the injection locked output beam and the master beam. To maintain locking, it might be necessary to add a control system that suppresses frequency drifts of the slave to keep the detuning Δms within the locking range. This can be achieved by a much simpler control system than would be necessary for a full frequency stabilization of the slave laser (Figure 8.17). An expression for the locking range can be derived by comparing the the power Ps of the free-running slave laser with the power of the master laser after ampliﬁcation inside the slave resonator. This later value is given by the product of the amplitude gain |G(𝜈)|2 and the power Pm injected from the master into the slave laser Pm . The gain |G(𝜈)|2 can be estimated as (2𝜅)2 ∕(2𝜋(𝜈 − 𝜈s ))2 where 𝜅 is the amplitude decay rate of the slave resonator that, in many small coupling cases, can be estimated as 𝜅 = (1 − R)∕t, R the reﬂectivity of the output coupler, and t

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V1pm

Slave pump

Master pump V1master

V1slave

Master laser Slave laser

Figure 8.17 Schematic layout for injection locking.

the round-trip time. Setting the two values equal leads to the locking condition 2𝜅 √ |G(𝜈)|2 Pm ≥ Ps or Δl ≤ Pm ∕Ps (8.23) 2𝜋 As a practical example, if the slave laser has a power of 10 W, the laser resonator has a round-trip length L of 1 m, the cavity loss rate is 10% per round trip, and the √ master laser has a power of 0.1 W, the locking range will be Δl = 2(1 − R)c∕2𝜋L Pm ∕Ps = 0.5 MHz. The ﬁrst observation of optical injection locking was carried out in 1966 by Stover and Steier who used two He–Ne lasers for this purpose [22]. They controlled the frequencies of the two lasers and were able to conﬁrm the predicted locking range. The technique is now widespread, and it is used to control the properties of the light emitted by high-power lasers, to improve the mode quality, and to build powerful tunable lasers by locking a low cost powerful diode laser to well-controlled low-power diode lasers. On the other hand, frequency locking can also be an undesirable process. It prevents the measurement of very small frequency diﬀerences. For example, in laser gyroscopes, which are used in navigational applications, a beat signal between the two counter-propagating modes in a ring laser is measured. The modes experience diﬀerent cavity lengths, if the ring laser rotates in an inertial frame. They will beat and the frequency of the beat is a measure of the rate of rotation. These frequencies fbeat are very small, and they are given by 2𝜋fbeat = Ωrot A∕p where A is the area and p the perimeter of the gyroscope. Ωrot is the angular frequency of the rotation. For the Earth’s rotation the value, at the pole, is 2𝜋 in 24 hours or 75 × 10−6 rad/s. For a 1 m2 ring gyroscope located at the pole, the beat frequency is about 20 Hz. Locking of the two modes is very likely, but it obviously has to be avoided; otherwise the beat signal is zero, and it would appear that the Earth stood still. Extraordinarily small scatter losses are suﬃcient to provide enough light to cause locking. As a consequence the demand for laser gyroscopes has been the motivation for most of the recent improvements in low-loss optics and the advances in the technology of mirror manufacture. Quantum optics experiments have gained from these improvements. It is interesting to ask what the noise properties of the injection-locked laser system could be. Does the master laser dominate the system completely? Is it possible to build in this way a high-power quantum noise limited laser? This question was addressed by several researchers in the regime of classical ﬂuctuations [23]. More recently, complete laser models have been developed, and using the

References

65 (iii) Power (dB above QNL)

55 45 35

(ii)

25 15 (i)

5 QNL –5 0.01

0.1

1

10

Frequency (MHz)

Figure 8.18 The spectral intensity noise of an injection-locked laser. (i) A polynomial ﬁt to the experimental data. (ii) Experimental data. (iii) Excess noise due to ampliﬁcation of the master laser’s phase noise by the injection-locked slave laser. Source: From C.C. Harb et al. [25].

linearized quantum theory, it is possible to derive complete noise spectra for the injection locked systems. In these models the laser is literally described as an oscillator with two distinct internal modes, at the frequency of the master and the slave laser, and the combined system is solved within one laser model, similar to that described in Chapter 6 [24]. The outcome of these models and of experiments [25] is that the characteristic of the laser intensity noise varies with the detection frequency. There are clearly distinguishable regions of frequencies that show diﬀerent properties. A practical example is the locking of two Nd:YAG lasers. One region is centred roughly around the relaxation oscillation frequency of the slave laser (Figure 8.18). Here the slave laser operates eﬀectively like an ampliﬁer for the noise of the master laser. Note that if the master laser is quantum noise limited, this means output is considerably noisier than a QNL laser with the same power as the slave laser. At very high frequencies the output can reach the quantum limit, and it maintains the properties of the slave laser. At very low frequencies the dominant source of noise is the pump source of the slave laser. Diﬀerent strategies have to be followed to optimize the system at the diﬀerent frequencies.

References 1 Levenson, M.D., Shelby, R.M., and Perlmutter, S.H. (1985). Squeezing of clas-

sical noise by nondegenerate four-wave mixing in an optical ﬁber. Opt. Lett. 10: 514. 2 Yuen, H.P. and Shapiro, J.H. (1980). Optical communication with two-photon coherent states. I. Quantum-state propagation and quantum-noise reduction. IEEE Trans. Inf. Theory IT-26: 78.

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3 Walls, D.F. and Zoller, P. (1981). Reduced quantum ﬂuctuations in resonance

ﬂuorescence. Phys. Rev. Lett. 47: 709. 4 Robson, B.A. (1974). The Theory of Polarisation Phenomena. Oxford: Claren-

don. 5 Schnabel, R., Bowen, W.P., Treps, N. et al. (2003). Stokes operator-squeezed

continuous-variable polarization states. Phys. Rev. A 67: 012316. 6 Kane, T.J. and Byer, R.L. (1985). Monolithic, unidirectional single-mode

Nd:YAG ring laser. Opt. Lett. 10: 65. 7 Köchner, W. (1996). Solid-State Laser Engineering, 4e. Springer-Verlag. 8 Dorf, R.C. (1992). Modern Control Systems, 6e. Addison Wesly. 9 Wiseman, H.M., Taubman, M.S., and Bachor, H.-A. (1995).

10 11

12 13 14 15

16

17

18

19

20

21 22

Feedback-enhanced squeezing in second-harmonic generation. Phys. Rev. A 51: 3227. Wiseman, H.M. (1999). Squashed states of light: theory and applications to quantum spectroscopy. J. Opt. B: Quantum Semiclassical Opt. 1: 459. Masalov, A.V., Putilin, A.A., and Vasilyev, M.V. (1994). Sub-Poissonian light and photocurrent shot-noise suppression in a closed optoelectronic loop. J. Mod. Opt. 41: 1941. Troshin, A.S. (1991). Photon statistics in a negative-feedback optical system. Opt. Spektrosk. 70: 389. Taubman, M.S., Wiseman, H.M., McClelland, D.E., and Bachor, H.-A. (1995). Eﬀects of intensity feedback on quantum noise. J. Opt. Soc. Am. B 12: 1792. Mertz, J. and Heidmann, A. (1993). Photon noise reduction by controlled deletion techniques. J. Opt. Soc. Am. B 10: 745. Robertson, N.A., Hoggan, S., Mangan, J.B., and Hough, J. (1986). Intensity stabilization of an Argon laser using electro-optic modulator: performance and limitation. Appl. Phys. B 39: 149. Harb, C.C., Gray, M.B., Bachor, H.-A. et al. (1994). Suppression of the intensity noise in a diode pumped neodymium YAG nonplanar ring laser. IEEE J. Quantum Electron. 30: 2907. Kane, T.J. (1990). Intensity noise in a diode pumped single frequency Nd:YAG laser and its control by electronic feedback. IEEE Photon. Technol. Lett. 2: 244. Allan, D.W. (1987). Time and frequency (time-domain) characterization, estimation, and prediction of precision clocks and oscillators. ICEE Trans. Ultrason. Ferroelectr. Freq. Control 6: 647. Day, T., Gustafson, E.K., and Byer, R.L. (1992). Sub-Hertz relative frequency stabilisation of two diode laser pumped Nd:YAG lasers locked to a Fabry-Perot interferometer. Opt. Lett. 17: 1204. Shaddock, D.A., Gray, M.B., and McClelland, D.E. (1999). Frequency locking a laser to an optical cavity by use of spatial mode interference. Opt. Lett. 24: 1499. Bowen, W. (2003). Experiments towards a quantum information network with squeezed light and entanglement. PhD thesis. Australian National University. Stover, H.L. and Staier, W.H. (1966). Locking of laser oscillators by light injection. Appl. Phys. Lett. 8: 91.

References

23 Farinas, A.D., Gustafson, E.K., and Byer, R.L. (1995). Frequency and intensity

noise in an injection locked, solid state laser. J. Opt. Soc. Am. B 12: 328. 24 Ralph, T.C., Harb, C.C., and Bachor, H.-A. (1996). Intensity noise of

injection-locked lasers: quantum theory using linearized input-output method. Phys. Rev. A 54: 4359. 25 Harb, C.C., Ralph, T.C., Huntington, E.H. et al. (1996). Intensity-noise properties of injection locked lasers. Phys. Rev. A 54: 4370.

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9 Squeezed Light 9.1 The Concept of Squeezing So far we have worked with coherent states of light and the limits imposed by quantum noise that aﬀect all quadratures equally. However, it is possible to generate light that has less noise in one selected quadrature than the quantum noise limit (QNL) dictates. Light with this property is called squeezed light. This form of light is clearly the ﬁrst milestone in quantum optics with many photons. Squeezing technology made many applications of quantum optics possible and lets us create many improvements unique to quantum optics. In optical measurements and metrology, we can achieve a better signal-to-noise ratio than is possible with coherent light, and this is now state of the art in interferometry. Squeezed light is the tool for entanglement of laser beams and for techniques such as teleportation that play a role in communication and signal processing. Even quantum logic might be feasible. Combined with optimized detection techniques, squeezed light can be the source of speciﬁc eﬀects, such as Schrödinger’s cat states of light. In this chapter, we start by describing the eﬀect of non-linear optical processes on the photon statistics and the ﬂuctuations of the light. This is ﬁrst done in a classical approach – since it is possible to see some of the key results immediately. We use two examples that are intuitive and instructive. Next we introduce the concept of squeezed states of light rigorously. This will provide the mathematical tools to analyse the experiments. Finally, we discuss the ﬁrst experiments that have been performed since 1985 to generate squeezed light and describe the state of the art in 2018. 9.1.1

Tools for Squeezing: Two Simple Examples

The properties of quantum noise have already been introduced in detail in Chapter 4, and now we wish to reduce the ﬂuctuations of the light. We cannot suppress the ﬂuctuations altogether that would violate the uncertainty principle. But we can rearrange them between the quadratures. We can build experiments, and protocols, where we measure only one quadrature at a time and only the noise in this one quadrature contributes to the ﬁnal result. Consequently, it will be suﬃcient to reduce the ﬂuctuations in this one quadrature alone. The challenge is: What are the tools that would reduce the noise in one quadrature? How can we generate states of light that have asymmetric quadrature noise A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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9 Squeezed Light

Figure 9.1 Tools for squeezing. X2

X1

distributions that are squeezed in one quadrature? We will see in this section that the trick is to start with a normal coherent state and to introduce correlations ̂ and X𝟐 ̂ through non-linear between the ﬂuctuations of the two quadratures X𝟏 interactions (Figure 9.1). 9.1.1.1

The Kerr Eﬀect

To illustrate how correlations can reduce the quantum noise, the case of the non-linear Kerr eﬀect is instructive [1]. Consider a beam of light travelling through a medium, which has an intensity-dependent refractive index. The beam can be described as an optical ﬁeld Ein with both amplitude and phase ﬂuctuations: 𝛼(t) = 𝛼in + 𝛿X1in (t) + 𝛿X2in (t)

(9.1)

This could be a QNL laser beam, in which case 𝛿X1in (t) and 𝛿X2in (t) represent the quadratures of the quantum noise. The non-linear refractive index n(𝛼) can be described by n(𝛼) = n0 (1 + n2 𝛼 2 )

(9.2)

As a consequence ﬂuctuations in the intensity will modulate the refractive index. The refractive index, in turn, will modulate the phase of the transmitted light. For a medium of length L and light at the wavelength 𝜆, the output phase is given by 2πL n 2πL = 𝜙in (t) + (n0 + n2 (𝛼in + 𝛿X1in (t))2 ) 𝜆 𝜆 For small ﬂuctuations this can be approximated by 𝜙out (t) = 𝜙in (t) +

𝜙out (t) = 𝜙Kerr +

4πL n (𝛼 𝛿X1in (t) 𝜆 2 in

(9.3)

(9.4)

9.1 The Concept of Squeezing

ϕin – ϕout αout

(a)

αin

(b)

αin

Figure 9.2 Eﬀect of a Kerr medium on the electric ﬁeld vector: (a) input ﬁeld and (b) output ﬁeld. Note that the ﬁeld illustrated here has a large amplitude, compared to the ﬂuctuations. Thus the origin of the diagram is far away and not shown. Consequently any changes in the phase are orthogonal to changes in the amplitude. The main result is that the change in the phase ﬂuctuations depends on the intensity proportional to 𝛼 2 .

where 𝜙Kerr is the time averaged phase added by the medium. This is illustrated in Figure 9.2. There is no change to the intensity of the light: 2 2 = 𝛼in = 𝛼2 𝛼out

and 𝛿X1out (t) = 𝛿X1in (t)

(9.5)

At the output of the medium, the phase ﬂuctuations are those of the input light plus a component that is linked to the input intensity ﬂuctuations. Using the relationship between the quadratures 𝛼𝛿𝜙 = 𝛿X2, which is valid for small angles, this can be written in terms of the quadrature amplitudes: 4πLn0 n2 2 𝛼 𝛿X1in (t) 𝜆 = 𝛿X2in (t) + 2rKerr 𝛿X1in (t) 2πLn0 n2 𝛼 2 (9.6) with rKerr = 𝜆 where rKerr is a parameter that includes the non-linearity and the average amplitude 𝛼 of the input ﬁeld. Higher-order terms in 𝛿X1 have been neglected since they are exceedingly small, and only the terms linear in 𝛿X1 have been kept. We see from Eq. (9.6) that the non-linear medium couples the ﬂuctuations in amplitude and phase quadratures. The amplitude 𝛼 drives the non-linear refractive index. This in turn controls the quadrature phase X2. At the output of the medium, the ﬂuctuations are no longer independent. The two quadratures are now correlated. What is the eﬀect of this process on the statistics of the ﬂuctuations? On the one hand, we have made the light noisier by adding ﬂuctuations. On the other hand, we now have correlated ﬂuctuations. We can estimate the consequences by plotting the eﬀect not just for one point but for a whole distribution of classical amplitudes. This is done in Figure 9.3. The input and the output ﬁelds are represented by density plots, already discussed in Chapter 4 and shown in Figure 2.13. The axes of the plot for Kerr eﬀect (Figure 9.3A(a)) are the input quadratures X1in and X2in . The mean value is given by the point (𝛼in , 0). The ﬂuctuating amplitude is represented by a group of dots, wherein each dot gives one instantaneous value of (𝛿X1in (t), 𝛿X2in (t)). The density of the dots indicates the probability of ﬁnding a speciﬁc value of the quadrature amplitude ﬂuctuation, as introduced in 𝛿X2out (t) = 𝛿X2in (t) +

305

9 Squeezed Light

Output E2

Input δX2

δY1

δY2 (a)

(a) δX1in δX1

(b)

δX1 δX2

(b)

Histogram

δY1 Histogram

306

δY2

(A)

(B)

Figure 9.3 Simulation of the eﬀect of a Kerr medium on the ﬂuctuations of an electromagnetic ﬁeld. Left-hand side (A) is the input ﬁeld and right-hand side (B) the output ﬁeld. The top part (a) shows the probability distribution of the ﬁeld, and the bottom part (b) shows the histograms of the samples above.

Section 4.1. This simulation uses a random distribution, with two-dimensional (2D) Gaussian weighting, as described in Appendix A. The probability of measuring a certain value X1(t) is given by a histogram of the projection of all dots onto one of the axes. Two such histograms, for both the X1 and X2 coordinates, are shown in Figure 9.3A(b). Both of them are Gaussian curves with the same width. Figure 9.3B shows the distribution at the output of a non-linear Kerr medium for a parameter rKerr = 0.375. Note that the total phase shift 𝜙out has been ignored. The axes are again the quadrature amplitudes X1out and X2out . The distribution is no longer circularly symmetric. The area covered by the distribution is actually larger since we have added ﬂuctuations ((Figure 9.3B(a)). But note that the distribution is narrower in one special direction. The histogram of this distribution, taken for a projection axis Y 1 rotated by the angle 𝜃s to the X1 axis (Figure 9.3B(b)) is narrower than the histogram for the input state (Figure 9.3A(b)). In this particular case the full width at half maximum (FWHM), or standard deviation, is reduced from 1 to 0.79. The histogram for the projection Y 2 (Figure 9.3B(b)), orthogonal to Y 1, is wider, by a factor 1.23, than at the input. The width of the distribution in the squeezing direction is given by √ 2 2 VY 1 = 1 − 2 rKerr 1 − rKerr + 2 rKerr (9.7)

9.1 The Concept of Squeezing

well below the QNL. This result is derived on page 343. This concept can also be used for a QNL laser beam or coherent state. The dot diagram represents the ̂ and X𝟐. ̂ The variance of the rotated measured probabilities of the observables X𝟏 ̂ ̂ quadrature Y𝟏, rotated by 𝜃s with respect to X𝟏, is smaller than that for a coherent state. Just by introducing this correlation between the quadratures, we have created light with noise reduced below the QNL in one speciﬁc quadrature. Note that in this example the uncertainty area has been increased. The output is no longer a minimum uncertainty state, but this is not of great importance for squeezing. ̂ The details of the We are only interested in measurements of the quadrature Y𝟏. size of the noise suppression and of the squeezing angle in a real experiment are described in Section 9.4.4. The Kerr eﬀect acts on all ﬂuctuations, independent of their frequency. The model can be applied to each individual Fourier component of the ﬂuctuations. For each detection frequency the process correlates the quadratures of the modes. The correlations are introduced separately for each Fourier component. In this way, the Kerr eﬀect correlates the amplitude and the phase quadrature of the light, and this applies to modulations as well as the quantum noise. 9.1.1.2

Four-Wave Mixing

A second example of a squeezing tool is four-wave mixing (4WM). This is a well-known non-linear process that occurs in many resonant media, such as atoms. Imagine three laser beams that are interacting with the medium and are tuned close to resonance with one absorption line, as illustrated in Figure 9.4a. This process is best described in the frequency domain. Two of the laser beams, the pump beams, have the optical frequency 𝜈L . They enter from diﬀerent angles. The third beam is at the frequency 𝜈L + Ω. The medium responds with the generation of a new beam at 𝜈L − Ω. This process is energetically allowed; the energy comes from the abundance of photons at 𝜈L . The strength or probability, of this process depends on three factors: (i) a non-linear coeﬃcient 𝜒 (3) that varies from material to material, (ii) the density of the sample, and (iii) a condition that describes how well the momentum of the photons is preserved. The latter is also known as phase matching and determines the geometry of the input and output beams [1]. Conventionally, this process is used as a technique to produce a new beam that emerges at a diﬀerent frequency or direction. This new beam can be used to investigate the properties of the non-linear medium or as a source of light at the new wavelength. This process also works if we have only one single input beam that has a single modulation sideband at 𝜈L + Ω. In this case all frequency components are co-linear, and 4WM produces a new sideband at 𝜈L − Ω, which is phase-locked to the original sideband. The eﬃciency of the process is given by the relative size of the new sideband. It can be quantiﬁed by the ratio C between the amplitude of the new and the original sideband. This applies to any sideband. Now consider the eﬀect of 4WM on an amplitude-modulated laser beam. That means we have pairs of sidebands, each pair in phase with each other, i.e. their respective beat signals are in phase and constructively interfere. Each sideband is partially transferred to the other sideband respectively by 4WM (Section 2.1.7). This will increase the sidebands and thus the modulation by a factor (1 + C).

307

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9 Squeezed Light

4WM vL

Non-linear medium

vL – Ω

vL + Ω

C

vL

vL – Ω (b)

(a)

δX1in

vL

vL + Ω

δX1in vL

vL More AM Less FM

δX1out (c)

δX1out vL

(d)

vL

Figure 9.4 The eﬀect of a 4WM medium. (a) The geometrical layout: a new beam is generated. (b) The eﬀect of 4WM in the frequency domain is a transfer from one sideband to another. (c) As a consequence amplitude modulation (AM) is increased. (d) Phase and frequency modulation (FM) is reduced.

The medium provides gain for amplitude modulation, as shown in Figure 9.4c. In contrast, phase modulation consists of two sidebands 180∘ out of phase. Again, 4WM creates additional components at the opposite sideband frequency. The transfer from one sideband to the other will decrease the phase modulation by a factor (1 − C), which means 4WM reduces phase modulation (Figure 9.4d). In conclusion, the non-linear process of 4WM provides a selective gain to one quadrature and reduces the other. 4WM is an example of a phase-sensitive ampliﬁer (Section 6.2), which has diﬀerent gain for each quadrature. This process applies both to modulation and to noise sidebands. Any amplitude modulation noise is ampliﬁed; any phase modulation noise is reduced. Consequently, a beam that initially has equal amounts of both types of noise will turn into light with enhanced amplitude modulation and reduced phase modulation noise. This classical argument shows us that an input beam that is in a coherent state will turn into a state with suppressed phase noise. This result is conﬁrmed by a full quantum treatment [2]. It is useful to consider the 4WM process in terms of a phasor diagram, much like we have done for the Kerr eﬀect. Again, we will consider one Fourier component of the ﬂuctuations at a time. Any periodic excursion from the long-term average value corresponds, in the phasor diagram, to an oscillation at the modulation frequency Ω. It can be a mixture of amplitude and phase modulation. This can be drawn, as shown in Figure 9.4c, as an additional vector, the sum of the vectors 𝛿X1(Ω) and 𝛿X2(Ω). The 4WM process will increase 𝛿X1(Ω) by a factor (1 + C) and reduce 𝛿X2(Ω) by a factor (1 − C) (Figure 9.4d). This is called phase-sensitive gain. We can test the eﬀect of 4WM on a truly noisy state, which

9.1 The Concept of Squeezing

Input

Output

δX2

δY2

δY1 (a)

(a)

δX1 δX2

(b)

Histogram

(b)

Histogram

δX1

δY2

δY1 (A)

(B)

Figure 9.5 Simulation of the eﬀect of a 4WM medium on the ﬂuctuations of an electromagnetic ﬁeld. Part (A) shows the input state. The distribution is circularly symmetric. Part (B) shows the distribution after the 4WM medium. It is stretched. The top part (a) shows the probability distribution of the ﬁeld, and the bottom part (b) shows the histograms of the samples above.

is represented by a 2D random Gaussian distribution in time. The transformation we apply point by point is 𝛿X1out (t) = (1 + C) 𝛿X1in (t) and 𝛿X2out (t) = (1 − C) 𝛿X2in (t)

(9.8)

Figure 9.5 shows an example of both input and output distributions, Figure 9.5A andB, respectively, based on the random sample from Figure 9.3 and a value of C = 0.3. Clearly the output state has increased amplitude ﬂuctuations and reduced phase ﬂuctuations. The size of the uncertainty area is preserved. This phasor diagram represents a QNL input; clearly the output has phase ﬂuctuations below the QNL. It is worth noting that the arguments used here are exactly the same as those applied in phase conjugation. The only diﬀerence is that there the spatial phase contributions are reduced by the generation of a reﬂected wave with opposite phase whilst in 4WM we consider temporal ﬂuctuations. These two examples, Kerr and 4WM, are only illustrations of a concept. They show that correlations introduced by non-linear processes can be used to generate a new type of light with smaller ﬂuctuations, in one speciﬁc quadrature, as compared with the input light. The above classical examples can be used as a guide for a complete quantum model of this new type of light. In Chapter 4, it was shown that the quadrature probability distribution for a quantum state can be

309

310

9 Squeezed Light

described by the Wigner function. The coherent light has a Gaussian distribution of the ﬂuctuations around the time averaged value, and this will not change for the strength of non-linearities that are available. In particular, for a coherent state, the Wigner function is a 2D Gaussian function with circular symmetry. The new state has a Wigner function that is compressed in one direction and expanded in the other. Certainly squeezed state is a ﬁtting description. The ﬁrst step for a complete description of this light is a formal and rigorous deﬁnition. 9.1.2

Properties of Squeezed States

With squeezed states we mean states that have the following property: the distribution function of the quadratures is not symmetric, that is, the uncertainty area is no longer a circle, and the width of the distribution function in one particular direction Y 1 is narrower than the QNL. The direction Y 1 can be at any angle with respect to the quadratures X1 and X2. Measurements that are designed to be sensitive only to Y 1 and not the orthogonal quadrature Y 2 contain noise less than the QNL. For this reason we call it non-classical light, meaning that it cannot be described by the normal theory of coherent light, given in Chapter 4, which prescribes a minimum uncertainty area that is independent of the quadratures. Quantum mechanics and non-linear optics have to be combined to avoid the symmetry and to go below the standard uncertainty limit (Figure 9.6). X2

X2 δX2

δX1

δX2

X1

Coherent state

(a)

δX1

(b)

X2

Minimum uncertainty state X2

δX2

δX1

(c)

X1

δX2

X1

Squeezed and excess noise

δX1

(d)

X1

Noisy and not squeezed

Figure 9.6 Phasor diagram comparing squeezed and other states. (a) Coherent state. (b) Minimum uncertainty squeezed state that is narrower than the coherent state in one direction. (c) Squeezed state with excess noise. (d) An asymmetric, noisy, but not squeezed state – it is described by an ellipse, but no projection is narrower than the coherent state.

9.1 The Concept of Squeezing

δX2

δX2

(a)

δX1

(b)

δX1

Figure 9.7 Wigner function of (a) coherent state and (b) squeezed state.

The uncertainty area for a squeezed state is generally drawn as an ellipse. Why does it have this particular shape? The Wigner function for any coherent state is a Gaussian distribution in every direction. The contour line for this state is a circle. We found that this corresponds to ﬂuctuations in the measurements of the variable such that V (Ω, 0) = V (Ω, 90) = 1 for all detection frequencies. It was already shown by W. Pauli [3] that any minimum uncertainty state, even squeezed, will have 2D Gaussian distribution functions. The only requirement is that the widths of these functions satisfy the uncertainty principle V (Ω, 𝜃) V (Ω, 𝜃 + 90) = 1 for any direction 𝜃. One of the two variances can be less than 1. The contour line for such a 2D Gaussian function is always an ellipse. An ellipse can be described by exactly two parameters: the direction of the narrowest axis Y 1 and the ellipticity. Note that the area inside the contour can never be smaller than the area inside the uncertainty circle of a coherent state (Figure 9.7). There are states with excess noise, that is, states with V (Ω, 𝜃) V (Ω, 𝜃 + 90) > 1 for some or even all frequencies. Such states need not be described by 2D Gaussian with elliptical contours. However, we ﬁnd that most realistic optical systems can be described by theoretical models that are linearized in the ﬂuctuations (Section 4.5). These systems can have non-linear properties that are essential for the generation of squeezed light, but its description requires only ﬁrst-order terms in regard to the ﬂuctuations. All such models result in Gaussian distribution functions – and the contours are ellipses. It would be of great interest to generate light with more exotic contours, with parts of the Wigner functions that have negative values and strong non-classical properties – but this has so far only been achieved in the single photon counting domain. Further details are given in Section 9.7. For the remainder we shall restrict ourselves to squeezed states that are described by elliptical contours. Squeezed states require a 2D description. The quadrature amplitudes Y 1 and Y 2 provide the most useful coordinate system – the theoretical model is

311

312

9 Squeezed Light

X2

X2

δY2 θs = 0

X1

θs =

δX1

(a)

X1

π 2

δX1

(b)

X2

X2

δY2

δY1

δY2 δY1

θs

X1 (c)

X1 (d)

Figure 9.8 Diﬀerent types of squeezed states. (a) Amplitude. (b) Phase squeezed state. (c) Quadrature squeezed state. (d) Vacuum quadrature squeezed state.

best derived using these parameters. Direct intensity measurements will only determine the intensity ﬂuctuations. Whilst this determines the width of the variance V 1(Ω) of the amplitude, measured along the X1 axis, it does not measure the quadrature ﬂuctuations in any other direction. We require experiments that can measure rotated quadratures, in particular V (Ω, 𝜃s ) and V (Ω, 𝜃s + 90). This is usually achieved by using a homodyne detector (Section 8.1.4). There are diﬀerent types of squeezed states, as shown in Figure 9.8. Each squeezed state ellipse has a narrow axis (Y 1) and, at right angles, the axis with the widest distribution (Y 2). If the narrow axis is aligned with the amplitude axis, i.e. (Y 1) parallel to (X1), we call the light amplitude squeezed. If it is aligned with the quadrature phase axis, i.e. (Y 1) parallel to (X2), it is called phase squeezed light. In general, (Y 1) could be rotated at some arbitrary angle 𝜃s relative to the axis X1. This is called quadrature squeezed light. 9.1.2.1

What Are the Uses of These Various Types of Squeezed Light?

Amplitude squeezed light is useful in absorption or modulation measurements. It can be detected simply by direct detection. Due to the reduced amplitude noise, the signal-to-noise ratio for amplitude modulation is improved. The ﬂuctuations in the other quadrature, the phase quadrature, are above the QNL. There could even be excess noise in the phase quadrature. None of this would aﬀect the direct intensity measurement. Amplitude squeezed light is obviously related to light with sub-Poissonian counting statistics. It has a Fano factor of less than unity. The detailed connection between squeezing and photon statistics has to be determined in each individual case. It depends on the actual photon probability distribution that could be very diﬀerent to a

9.1 The Concept of Squeezing

Poissonian distribution. However, for the generally observed moderate degree of squeezing, the link is simple: F𝛼 = V 1𝛼 . This light is frequently called bright squeezed light. The generation of bright squeezed light using second harmonic generators (SHGs) [4–7] or diode lasers is now a reliable experimental technique and is described in Section 9.4.3. Quadrature phase squeezed light can be used to improve the signal-to-noise ratio of optical measurements. Care has to be taken that only the phase quadrature is measured. This can be achieved in interferometric arrangements where amplitude ﬂuctuations are cancelled out and phase modulations are turned into modulation of the intensity at the output. In such an arrangement only the noise of the light in the phase quadrature will contribute to the ﬁnal signal-to-noise ratio. Finally, there is a peculiar state, centred at the origin of the phasor diagram, that has a squeezing ellipse. This state has an axis of reduced ﬂuctuations. But for each ﬂuctuation there is an equally likely ﬂuctuation in the opposite direction. This state is given the confusing name squeezed vacuum state. We will see that it is closely related to the coherent vacuum state |0⟩. An attempt to detect squeezed vacuum state directly would not result in any measurable intensity. To be precise, this state contains a rather small number of photon pairs (Eq. (9.13)), and the number of such pairs increases the more the state is squeezed. However, this eﬀect is far too small to be detected directly with photodiode technology. Only with photon counting could such small intensities be seen. In any other regard this state has normal properties: it is a beam with a well-deﬁned mode, direction, size, frequency, and polarization. However, once this beam is aligned with a homodyne detector and beats with a strong local oscillator beam, we can see its properties: the quantum ﬂuctuations vary with the detection phase. The ﬂuctuations corresponding to the Y 1 quadrature are smaller than those of the Y 2 quadrature. This squeezed vacuum state can have signiﬁcant practical applications, as shown in Section 5.2. All interferometers, and other instruments based on beamsplitters, contain input ports that are not used. That means there are locations in the experiment that could be used as the input for a real laser beam that would propagate right through the instrument, probably with some attenuation, and ﬁnally reach the detector. Such an input port has the properties of a real mode of light. In practice this port is not illuminated, and it can be ignored in classical calculations. However, in a quantum model of such an instrument, we cannot ignore this port, as shown in Chapter 5, and we have to represent it with a vacuum mode, the coherent state |0⟩. These vacuum input modes are the origin of the quantum noise in the output, as was shown in Section 5.1. Consequently the performance of the instrument can be improved by illuminating the unused input port by a squeezed vacuum mode. The squeezing quadrature Y 1 has to be aligned such that it is now the driver of the quantum ﬂuctuations that we measure in our apparatus. The result will be reduced quantum noise and a better signal-to-noise ratio. This was proposed by C.M. Caves [8] for the case of interferometers and has been demonstrated in an impressive landmark experiment by M. Xiao et al. [9] who achieved an improvement of the signal-to-noise ratio of 3 dB. The details and limitations of this technique will be discussed in the Chapter 10.

313

314

9 Squeezed Light

9.2 Quantum Model of Squeezed States In Chapter 4 we described pure states of light with Gaussian quadrature statistics as coherent states. We now want to expand this to include a much larger class of states, the squeezed states. They have to be based on a wider deﬁnition and can be derived from the lowest energy state, the vacuum state. 9.2.1

The Formal Deﬁnition of a Squeezed State

Squeezed states are represented by |𝛼, 𝜉⟩

and 𝜉 = rs exp(i2𝜃s )

(9.9)

where |𝛼|2 is the intensity of the state, 𝜃 the orientation of the squeezing axis, and rs the degree of squeezing. The squeezed states can be generated from the vacuum state: ̂ ̂ S(𝜉)|0⟩ |𝛼, 𝜉⟩ = D(𝛼) ̂ where the squeezing operator S(𝜉) is deﬁned as ) ( 1 1 ̂ S(𝜉) = exp 𝜉 ∗ â 2 − 𝜉 â †2 2 2 The properties of the squeeze operator are

(9.10)

(9.11)

̂ Ŝ † (𝜉) = Ŝ −1 (𝜉) = S(−𝜉) ̂ = â cosh(rs ) − â † exp(−2i𝜃s ) sinh(rs ) Ŝ † (𝜉)̂aS(𝜉) ̂ = â † cosh(rs ) − â exp(2i𝜃s ) sinh(rs ) Ŝ † (𝜉)̂a† S(𝜉)

(9.12)

̂ in Eq. (9.10) is possible. This results ̂ Note that the reverse order of D(𝛼) and S(𝜉) in the so-called two-photon correlated state, which was introduced by H.P. Yuen even before the name squeezed state was coined [10]. However, this distinction has little relevance for experiments. It makes no detectable diﬀerence for any realistic state. The formal description allows us to derive the noise properties of a squeezed state exactly. The squeezed states have the following expectation values for the annihilation and creation operators: ⟨𝛼, 𝜉|̂a|𝛼, 𝜉⟩ = 𝛼 ⟨𝛼, 𝜉|̂a2 |𝛼, 𝜉⟩ = 𝛼 2 − cosh(rs ) sinh(rs ) exp(2i𝜃s ) ⟨𝛼, 𝜉|̂a† â |𝛼, 𝜉⟩ = |𝛼|2 + sinh2 (rs )

(9.13)

The last equation means that the intensity of a squeezed state is slightly larger than that of the coherent state with the same value |𝛼|. However, this is a minute eﬀect for any laser beam with detectable intensity. Using these properties we can ̂ derive the variances of the generalized quadrature X(𝜃) = exp(−i𝜃)̂a + exp(i𝜃)̂a† , the quadrature that would be measured at the rotation angle 𝜃. This variance is given by ̂ Var(X(𝜃)) = cosh(2 rs ) − sinh(2 rs ) cos(2 (𝜃 − 𝜃s ))

(9.14)

9.2 Quantum Model of Squeezed States

10

V(θ) (dB)

5

0

–5

–10

(a)

π/2

0

π

3π/2

2π

1

0.75 >7 [5, 7] [3, 5] [1, 3] [−1, 1] [−3, −1] [−5, −3] [−7, −3]

θs X1

n δX1(Ω)

δX2(Ω)

V1(Ω) 1

Ω

0

δX2

δX1

Ω S(Ω)

V(θ)

θ

1

1

0 θs

–Ω

0

Ω

Figure 9.15 Summary of the various graphical representations of a squeezed state.

of losses is easy to predict since all passive losses behave in the same way and can be simulated by beamsplitters. Only non-linear processes are excluded from this model. In terms of the sideband picture, the eﬀect of the beamsplitter is easy to describe. The squeezed light corresponds to light with partially correlated sidebands 𝛿E+Ω and 𝛿E−Ω . For a beamsplitter with reﬂection 𝜖, transmission √ 𝜂 = (1 − 𝜖), the correlated part of the noise sidebands is reduced by 𝜂, in the same way as modulation sidebands are reduced (Section 2.1), whilst simultaneously the uncorrelated part is increased to keep the magnitude of the quantum noise sidebands constant. This is equivalent to including the quantum noise sidebands of the vacuum state from the empty port of the beamsplitter. The combined sidebands 𝛿E+Ω,out , 𝛿E−Ω,out at the output are less correlated than those at the input, and the degree of squeezing is reduced (Figure 9.16a). In terms of the phasor diagrams, the beamsplitter replaces the squeezed Wigner function with one that is less squeezed (Figure 9.16b), which means the contour ellipse

9.3 Detecting Squeezed Light

X2

Partially αin correlated noise

δY2 Ω

δY1 θs

V(θ) θ

1

X1 θs X2

αout Less correlated noise

V(θ) δY2

Ω (a)

δY1 θs

θ

1

X1 (b)

(c)

θs

Figure 9.16 The eﬀect of a beamsplitter on squeezed light, input at the top and output at the bottom. Three representations are used: (a) correlated sidebands, (b) phasor diagrams, and (c) quadrature variances. The top row is representative of the light at the input, and the bottom row at the output of the beamsplitter.

is not as narrow. The corresponding variance V (𝜃) at the output is to the QNL (Figure 9.16c). Loss transforms the contour from an elongated ellipse towards a circle, as shown in Figure 9.17. This might not be obvious if one considers that the ﬂuctuations of the input of the beamsplitter simply add. Just adding the two Wigner functions would suggest a rather more complicated result, as indicated in Figure 9.17b, for the case of adding two squeezed states with diﬀerent squeezing angle 𝜃s . However, all features of the Wigner function have to be taken into account, including the phase terms – and the answer is actually simple: the result is another ellipse with the new orientation 𝜃s and smaller ellipticity. It has to be an ellipse since we know that any solution of the linearized model will result in a 2D Gaussian quasi-probability distribution (Section 9.2). Thus the only parameter that can vary is the variance VY 1 of the squeezed state after the beamsplitter. If the input state is a squeezed state with the general parameters (𝛼, rs , 𝜃s ), the output state, after the beamsplitter, is (𝛼 ′ , rs′ , 𝜃s′ ). We know that the intensity is reduced such that |𝛼 ′ |2 = 𝜂|𝛼|2 . The beamsplitter has a second input port for vacuum ﬂuctuations and is described in the standard way: √ √ ̃ in + 1 − 𝜂 𝜹X ̃ vac ̃ out = 𝜂 𝜹X(𝜃) (9.21) 𝜹X(𝜃) The calculation can be carried out independently for each angle 𝜃. As a result, the minimum will remain at 𝜃s since |0⟩ has no angular dependence. Thus the squeezing angle is unchanged, 𝜃s′ = 𝜃s . We have to evaluate the variances and this is a linear operation √ that leads to the sum of two contributions. The second term simply results in 1 − 𝜂 since any contribution from the vacuum state results in 2 ̃ ⟩ = 1. As a result we obtain Vvac = ⟨|𝜹X(𝜃)| VY 1out = 𝜂VY 1in + (1 − 𝜂) or (1 − VY 1out ) = 𝜂(1 − VY 1in )

(9.22)

327

328

9 Squeezed Light

(a)

No

Yes (b)

Figure 9.17 Wigner functions cannot be added directly; the phase terms must be taken into account. (a) Attenuation of a squeezed state by a beamsplitter results in a state with reduced squeezing. (b) The combination of two squeezed states with diﬀerent 𝜃s values results in a normal squeezed state, not a Wigner function with four contributions.

This means the noise suppression (1 − VY 1out ) is reduced linearly with loss 𝜂. The noise suppression is aﬀected in the same way as the intensity. Note that the noise suppression is commonly quoted on a logarithmic scale in dB for the normalized variance and this value is not reduced linearly. The eﬀect of 𝜂 is shown in Figure 9.18 where the output variance VY 1out is plotted in dB versus the input variance VY 1in . The eﬀect of losses is more severe on strongly squeezed light. A loss of 50%, (𝜂 = 0.5) transforms VY 1in = −3 dB into −1.3 dB, whilst for the much better squeezing of VY 1in = −10 dB, the output is very similar, namely, −2 dB. Strong squeezing is easily lost in a system with limited eﬃciencies. The minimum variance of a squeezed state is VY 1𝛼,𝜉 = exp(−2rs ). One could formally try to link the squeezing parameter rs′ to rs . The link is exp(−2rs′ ) = exp(−2rs )𝜂 + (𝜂 − 1). This is not a simple relationship, and the squeezing parameter is not practical for this type of calculation. Since the conversion of the squeezing due to losses is constantly required in the evaluation

9.3 Detecting Squeezed Light

Output log (VY1(α′,ξ′)) (dB)

0

2

0.5 0.6

4

0.7 0.8

6

0.9 η = 1.0

8

10

0

2

4

6

8

10

Input log (VY1(α, ξ)) (dB)

Figure 9.18 The quantitative eﬀect of diﬀerent eﬃciencies 𝜂 on the minimum variance of the squeezed state.

of squeezing data, the squeezing parameter is rarely used by experimentalists. The most common parameter cited is (1 − VY 1) either on a linear or logarithmic scale, and this scales easily, as stated in Eq. (9.22). Note that a beamsplitter, or any other phase-independent loss, does not change the spectrum of the squeezing. Equation (9.22) holds equally well for all detection frequencies. Both the shape of the spectrum and the squeezing angle are not aﬀected by losses, and only the degree of squeezing is reduced. The squeezed light produced by some devices, for example, a below threshold OPO, can be minimum uncertainty state, a pure state with VY 1 VY 2 = 1. Does it remain a minimum uncertainty state after losses? The beamsplitter will mix in a fraction of the vacuum state, itself a minimum uncertainty state, and the combined output will be a mixed state, above the minimum uncertainty value. How much above? We can directly calculate VY 1out VY 2out = (𝜂VY 1in + (1 − 𝜂)) (𝜂VY 2in + (1 − 𝜂)) but do not obtain a simple answer. The result is easiest interpreted from a diagram. Figure 9.19 shows the values for the product of the variances as a function of 𝜂 for three degrees of squeezing: VY 1in = 0.5, 0.2, and 0.1. We see that the product rises quickly and for 𝜂 ≈ 0.5 reaches a maximum that depends on the initial degree of squeezing. For large attenuation all these states approach again a minimum uncertainty, the coherent state. If the light is too intense for the detector, we should not simply attenuate the light but reﬂect it oﬀ a cavity. If a cavity with smaller linewidth than the detection frequency is chosen, the cavity will fully reﬂect the noise sidebands whilst the carrier component is partially transmitted. In this way the optical power can be attenuated without losing the squeezing. An interesting extension would be to incorporate an active gain medium into the cavity [6]. This now becomes

329

9 Squeezed Light

3 VY1in = 0.1 VY1out * VY2out

330

2.5

2

VY1in = 0.2

1.5 VY1in = 0.5 1 0

0.2

0.4

0.6

0.8

1

Loss η

Figure 9.19 The uncertainty product for a squeezed state after loss. Three diﬀerent degrees of squeezing are shown.

a noiseless optical power ampliﬁer – it ampliﬁes the energy but leaves the noise sidebands outside the cavity linewidth unaﬀected. Alternatively we could detune the cavity and therefore rotate the squeezing ellipse, as discussed in the Section 9.4.4. Amplitude quadrature squeezing can be turned into phase quadrature squeezing and vice versa. We have all the components available to modify the squeezing parameters, but the only thing we cannot do is to increase the degree of squeezing.

9.4 Early Demonstrations of Squeezed Light 9.4.1

Four Wave Mixing

We are now discussing the pioneering squeezing experiments in detail. By the middle of 1984 several research groups were competing to measure the ﬁrst non-classical noise suppression – to detect the ﬁrst squeezed state. Each group favoured a diﬀerent non-linear process, and eventually all these attempts were successful. The ﬁrst group to report experimental observations of squeezing was led by R.E. Slusher at the AT&T Bell Laboratories in Murray Hill. Their experiment used 4WM in a Na atomic beam [15]. This is a resonant medium. The non-linearity was enhanced by an optical cavity. The ﬁrst proposal of squeezing with a 4WM mixing medium had been published as early as 1974 by H.P. Yuen and J.H. Shapiro [16]. The diﬃculty was to achieve suﬃcient 4WM gain whilst avoiding all other sources of noise. The analysis of squeezing by 4WM conﬁrmed that we require a non-degenerate situation [2]. That means the pump beam, which provided the energy for the process, and the signal and idler beams should have separate frequencies. This led to the classical interpretation of 4WM as correlating two sidebands, each detuned by ±Ω from the pump frequency, as discussed in Section 9.2.3

9.4 Early Demonstrations of Squeezed Light

One important noise source that could swamp any squeezing was spontaneous emission. Light from the pump beam, which was absorbed by the atoms, was re-emitted either at the same frequency or at the atomic resonance frequency. Such light produced ﬂuctuations in the photocurrent that raise the noise level above the QNL. It was necessary to detune the pump wave from the atomic resonance by many atomic linewidths. However, this reduced the non-linearity and a compromise between non-linearity and spontaneous emission noise had to be achieved. One technique to reduce the spontaneous emission, at a given detuning, was to reduce the Doppler broadening. Consequently most experiments were performed with atomic beams that are essentially Doppler-free as long as all optical beams are aligned at right angles to the propagation axis of the atoms. The atomic beams had to be optimized for high densities. In Slusher’s experiment an atomic absorption of exp(−0.5) on line centre was measured. The experiment consisted of a beam of Na atoms that was illuminated, pumped by an optical pump beam, and detuned by about 2 GHz away from the hyperﬁne components of the Na D1 transition at 590 nm. Figure 9.20 shows the experimental layout and Figure 9.21 the frequencies of all the optical beams in the experiment. A second beam, the probe, was aligned through the same atoms at

ϕLO

Dye laser

BS1 EO2 FB2

AO3,4

LO

AO1,2

vL+3vSC

vL+2vSC

PM1

BS2

SM2

DB

EO1 S1

SM1 Na

PM2

DA –

FB1

SA

Figure 9.20 Layout of the atom 4WM experiment. Source: After R.E. Slusher et al. [15].

vatom

vsignal

vprobe

vidler

vlock

Figure 9.21 The diﬀerent frequencies involved in the 4WM experiment. The cavity modes are shown as thin solid lines. They are shifted by the dispersion of the atoms.

331

332

9 Squeezed Light

a small angle to the pump beam. This probe beam was aligned into a heterodyne detection system together with a local oscillator beam that was split oﬀ from the pump beam. Two photodiodes detected the beams and generated a beat signal at the diﬀerence frequency between probe and pump, in this case at 595 MHz. In the absence of the atoms, and with the probe beam blocked, the noise level recorded was the QNL (1 Hoover). It can be interpreted as the noise due to the vacuum state entering the beamsplitter of the heterodyne detector. With the atoms present and the probe beam turned on, the 4WM medium generated a second sideband with exactly the opposite detuning to that of the probe. Now two beams were present and were detected together. In this way the probe beam was used to measure the classical 4WM gain and to align the detection system. The 4WM process can be interpreted as phase-sensitive ampliﬁcation: it increases amplitude modulation and reduces phase modulation sidebands. This gain correlates the two sidebands, as discussed in Section 9.2. Even with the probe beam switched oﬀ, this correlation exists. The ever-present quantum noise sidebands, which enter the medium on the axis that was deﬁned by the probe beam, will be correlated, resulting in a squeezed vacuum state. For a single transit of the Na beam, this eﬀect was far too small to be detectable. The 4WM process, and the resulting correlation, could be built up by a cavity that is tuned to both sidebands simultaneously. Each sideband was on resonance with a diﬀerent cavity mode. In this experiment they were not even adjacent modes, but they were separated by nine free spectral ranges (FSRs) (Figure 9.21). Again, the probe beam was used for the alignment of the cavity. A locking system kept the cavity on resonance. This used an optical beam, which probed the cavity at yet another optical frequency, avoiding the pump and the probe frequency. One practical problem was the fact that the atoms produced a strongly dispersive refractive index that shifts all cavity resonances. The cavity modes were asymmetric to the probe frequency, but by using acousto-optical modulators (AOMs), the locking beam could be shifted until both the probe and the newly generated beam were in resonance. The pump intensity was raised until it approached the level of the saturation intensity of the sodium vapour at a detuning of 2 GHz, equivalent to 400 atomic linewidths. and the pump beam was retro-reﬂected to provide an even higher pump intensity. Finally, the probe beam is blocked. The vacuum mode, which is collinear with the probe beam, experiences the correlation at the sidebands at +Ω and −Ω induced by the atoms inside the single-sided cavity. A squeezed beam is created that travels towards the detectors. The light is squeezed in a band of frequencies around the cavity resonance frequency. The strength and the quadrature of the squeezing are measured by scanning the phase of the local oscillator. This ﬁrst squeezing result is shown in Figure 9.22a. On the right-hand side, Figure 9.22b shows the result after continuous improvements 15 months after the ﬁrst observation [15]. The dotted line in Figure 9.22b gives the QNL, obtained with the cavity blocked. The measured noise increases 1.3 dB above and decreases −0.7 dB below the SQL. The electronic noise (ampliﬁer noise) is −10 dB below the SQL and contributes about 10% to the noise level. The detection frequency is 594.6 MHz, resolution bandwidth 300 kHz, and video bandwidth 100 Hz. The theoretical model predicts ±2 dB (solid curve) for an ideal measurement. The eﬃciency of the photodetectors is 𝜂det = 0.8, and the eﬃciency of the

9.4 Early Demonstrations of Squeezed Light

–58

Noise level (dBm)

–59

–60

–61

–62 0 (a)

(b)

π/2 ϕLO

π

3π/2

Figure 9.22 The ﬁrst squeezing results. (a) Photograph of the ﬁrst trace [15]. (b) Quantitative comparison between experimental results and theoretical predictions after months of further work. Source: After R.E. Slusher et al. [2].

interferometer is 𝜂vis = 0.81, resulting in a total eﬃciency of 0.65. Taking this and the electronic noise into account, the theoretically predicted noise is given as the dashed-dotted line in Figure 9.22b. It shows good agreement with the enhanced noise and moderate agreement with the squeezed noise quadrature, namely, −0.7 dB observed versus −1.05 dB predicted. One practical limitation is the mechanical stability of the interferometer required for the homodyne detection. Any mechanical vibration can wash out the noise suppression. This eﬀect, described as phase jitter, is cited as the reason for the disagreement with the prediction. It can explain this particular set of results. However, at higher intensities, the disagreement gets even larger, and other processes have to be considered. Non-linear coupling between the light used for the locking of the cavity to the laser frequency and the mode of the squeezed light is considered by the authors as a possible source of excess noise. This experiment clearly demonstrates the existence of a squeezed state of light, in this case a squeezed vacuum state. All predictions about the properties of the squeezed light and the detection systems were conﬁrmed qualitatively. The observed noise reduction is modest. Despite the predictions of squeezing of up to −10 dB, 4WM has never been really successful as a squeezing process. The technical diﬃculties with the atomic beam were frustrating, and this type of experiment has not been repeated by any other group. It remains a milestone in the early exploration of squeezed light. 9.4.2

Optical Parametric Processes

Some of the most successful results in the early period of squeezing came from the group led by H.J. Kimble at the University of Texas [17]. He used an

333

334

9 Squeezed Light

M2

M1 αp

A1 vp

Detunings

ζ

αd κp1 κd1 κp = Total loss at vp

Aout

Δp Δd

vd = 1/2 vp

κp2 κd2 κd = Total loss at vd

Figure 9.23 A parametric medium inside a cavity. The various input, output, and intra-cavity modes are labelled. This system has a threshold and can operate below threshold as a phase-sensitive ampliﬁer of the vacuum modes and above threshold as an oscillator. The subscript p refers to the fundamental frequency, and the subscript d refers to the subharmonic frequency.

optical parametric ampliﬁer (OPA) to generate squeezing. We have already seen in Chapter 6 that the OPA is ideally suited since it is phase sensitive and ampliﬁes one quadrature whilst it attenuates the other as shown in Figure 9.24. The trick to enhance the process is to build a cavity around the OPA as shown in Figure 9.23. The experiments with the non-linear OPO are carried out with a cavity containing a 𝜒 2 material, usually a crystal such as LiNbO3 . A single-mode laser illuminates the cavity. Following the derivations in Section 6.3.3, we know that a parametric oscillator below threshold will produce quite diﬀerent gain for the two quadratures, similar to 4WM, and therefore suppresses the noise selectively. Figure 9.24 symbolically shows how the quadratures are aﬀected and the conversion of a vacuum input state (circle) into a squeezed vacuum (ellipse). From Eq. (6.43) we can derive a noise spectrum and ﬁnd that the eﬀect is greatest at low frequencies (Ω ≪ cavity linewidth). For such an oscillator the crucial c , experimental parameter is given by the normalized pump amplitude d = 𝛼in ∕𝛼in c where 𝛼in is the input amplitude required to reach threshold. We wish to operate with d < 1 and can optimize the performance by approaching d → 1 by changing the input intensity. In terms of the cavity parameters, losses and non-linear gain, Figure 9.24 Symbolic representation of parametric gain in the phase diagram.

δX2

δX1

9.4 Early Demonstrations of Squeezed Light

1

S(Ω)

0.8 0.6 0.4 d = 0.5

0.2 0

4

0

2

2

4

Ω

Figure 9.25 The squeezing spectrum for an OPO below threshold for a ﬁxed value of d = 0.5. The modulation frequencies are scaled in units of the cavity bandwidth.

we ﬁnd d = 𝜒∕𝜅a , and consequently we can derive the variance from Eq. (6.35): V 2(Ω) = 1 −

4d (1 + d)2 + (2πΩ)2

(9.23)

which is identical to the squeezing spectrum, S(Ω) = V 2(Ω), since the noise suppression always occurs in the phase quadrature. An example is shown in Figure 9.25. At very low frequencies, Ω → 0, we get the optimum value V 2(Ω) =

(1 − d)2 (1 + d)2

(9.24)

and at the same time the noise enhancement in the orthogonal quadrature V 1(Ω) =

(1 + d)2 (1 − d)2

(9.25)

We see that the ﬂuctuations in the X2 quadrature tend to zero. On the other hand, the ﬂuctuations in the X1 quadrature diverge. The eﬀect for the two quadratures is shown in Figure 9.26. The system has a critical point at the 40

100

VY1

30 20 V (dB)

Gain

10 Amplitude

0

1 Phase 0.1 (a)

10

20 0

0.2

0.6

0.4 d2

0.8

VY2

10 1 (b)

0

0.2

0.6

0.4 d

0.8

1

2

Figure 9.26 Phase-dependent gain and resulting variances V1 and V2 for the two quadratures of a OPO below threshold as a function of the normalized input power d2 . This calculation is for very low detection frequencies less than the linewidth of the cavity Ω = 0 and for a ﬁnite quantum eﬃciency 𝜂 ≤ 1.

335

336

9 Squeezed Light

Ba2NaNb5O15 SHG

MgO:LiNbO3 0.53 μm OPO

1.06 μm Laser Nd:YAG

θlo +_

Figure 9.27 The schematic layout of the experiment by L.-A. Wu et al. [17] generating a squeezed vacuum state with a sub-threshold OPO.

threshold. Perfect squeezing is possible at threshold, at least in theory, but in practice the operation gets unstable and in most situations values no larger than d = 0.8 are chosen. Interestingly the noise suppression improves only slowly for intensities larger than half the threshold value (d2 = 0.5). The eﬃciency 𝜂 of the apparatus has to be taken into account, and the complete equation for the noise suppression is V 2(Ω) = 1 −

𝜂 4d (1 + d)2 + (2πΩ)2

(9.26)

The ﬁrst realization of this experiment was carried out by Wu et al. in 1986. The layout of their experiment is shown schematically in Figure 9.27. The light source is a continuous-wave (CW) Nd:YAG ring laser, in this case lamp pumped. The light is frequency doubled using an intra-cavity Ba2 NaNb5 O15 crystal. The laser cavity has two output beams, at 1064 and 532 nm, with orthogonal polarizations. The green light is sent to the parametric oscillator, which is a MgO-doped LiNbO3 crystal inside a linear cavity. When operated above threshold as a degenerate OPO, the cavity sends out a beam at exactly the laser frequency. This beam is recombined in a Mach–Zehnder interferometer with the infrared output from the laser cavity. This arrangement forms the homodyne detector for the output of the parametric oscillator. The laser provides the local oscillator. When operated below threshold, the output ceases, but the apparatus is still aligned. Now the OPO emits a squeezed vacuum state that is detected by the homodyne detector. All measurements were carried out at detection frequencies less than the cavity linewidth. The phase angle 𝜃 of the local oscillator is scanned. In this way the quadratures can be probed. The vacuum noise level is determined by blocking the beam from the OPO. The results are shown in Figure 9.28a. The minimum of the trace gives the optimum noise suppression. It was diﬃcult to stabilize the phase of the local oscillator; thus mainly scans of the projection angle 𝜃 were published by L.-A. Wu et al. [17]. We can investigate the properties of the OPA in diﬀerent ways. The noise suppression will improve when the threshold is approached. This has been tested and results are given in Figure 9.28a. The agreement with the theoretical predictions (Eq. (9.26)) is excellent. The best noise suppression observed is

9.4 Early Demonstrations of Squeezed Light

(ii) 2.0 (iv) V (θ) 1.0 (i) (iii) 0 (a)

θ0

θ0 + π

θ0

θ

θ0 + π

θ0 + 2π

θ0 + 3π

d2 = P2 /P0 0

0

0.5

1.0

S_(r) –0.5

–1.0

(b)

Figure 9.28 Experimental results. (a) Dependence of noise power on the local oscillator phase 𝜃. With the output of the OPO blocked, the vacuum ﬁeld entering the detector produces trace (i). With the OPO input present, trace (ii) exhibits phase-sensitive deviations both below and above the SQL. Trace (iii) is the ampliﬁer noise. (b) Noise suppression for diﬀerent normalized intensities d2 . Comparison of the inferred noise suppression V2(d2 ) = 1 + S− (r) after correction for the quantum eﬃciencies derived from measurements such as shown in (a) with optimized phase angle with those given by theory (solid curve). Detection frequency Ω is ﬁxed. Perfect squeezing corresponds to V2 = 0 and is indicated by the dashed line. Source: After L.-A. Wu et al. [17].

V1 = 0.33. Once the detector eﬃciencies and the losses are taken into account, the best inferred squeezing corresponds to a factor of 10, or −10 dB, for the light emitted by the cavity. Finally, this experiment is also a most impressive demonstration of the concept of a minimum uncertainty state. In Figure 9.28b both the minimum and the maximum variances for each input amplitude are plotted.

337

9 Squeezed Light

10.0

Squeezed states

Minimum uncertainty (1 + S+ )(1 + S_ ) = 1

(1 + S+ )

338

5.0

Vacuum state 1.0 0

0

0.5

1.0

1.5

(1 + S_ )

Figure 9.29 Variances V1 and V2 determined from measurements. The solid curve is the hyperbola V1 V2 = 1, which deﬁnes the class of minimum uncertainty states. Squeezed states are those states for which either V1 or V2 < 1 and that lie in the region bounded by the hyperbola and the dashed lines. Source: From L.-A. Wu et al. [17].

According to the theory, the product of the two variances should be unity, since the parametric ampliﬁcation preserves the minimum uncertainty area. The data points should lie on a hyperbola, and this is indeed the case. Note that this result can only to be expected for data describing directly the OPO, with perfect eﬃciency, since any loss would have turned these minimum uncertainty states into mixed states. The data for Figure 9.29 are inferred after removing the eﬀect of the detector loss. This was a landmark experiment. Some of the most striking features of squeezed light have been demonstrated in a single experiment: noise suppression, noise enhancement in the opposite quadrature, and preservation of the minimum uncertainty. And the light generated, a squeezed vacuum beam, is certainly not classical in any regard. The work of Liang Wu et al. is a remarkable achievement and remained unsurpassed for many years. The main advance has been in the reduction of losses inside the OPO. The limitations are now the eﬃciencies of the detector and technical ﬂuctuations in the phase of the local oscillator 𝜃 that wash out the very sharp minima. Work at Caltech, Konstanz, ANU, Shanxi University, Tokyo, Paris, and Hannover have established the OPO/OPA as the most reliable source for squeezed light.

9.4 Early Demonstrations of Squeezed Light

9.4.3

Second Harmonic Generation

The process of frequency doubling or SHG has been used to generate amplitude squeezed light. It is presently the most reliable and eﬃcient process to generate light with high intensity and good noise suppression. A qualitative description why SHG leads to amplitude squeezed light is obtained from the photon model. A coherent input beam can be thought of as a stream of photons with Poissonian distribution. The time separation between photons is random. This beam illuminates a non-linear crystal, which converts a fraction of the light into a new beam with twice the optical frequency. Two beams emerge, the remaining fundamental (f ) beam and an SH beam. The crystal converts pairs of photons from the incoming fundamental beam into single photons of the SH beam. If the probability for the conversion were constant, the statistics of the two emerging beams would be identical to that of the input beam. However, the conversion probability is proportional to 𝛼 2 𝜒 (2) , that is, the instantaneous intensity of the fundamental beam multiplied by the second-order non-linearity. This means the conversion is more likely when the pump intensity is high, when the photons in the pump are closely spaced. Closely spaced pairs of photons are most likely to be converted into a single SH photon. As a consequence the statistics of the SH photons is closely linked to the statistics of pairs in the fundamental beam. The SH photons come in a more regular stream than the pump photons. If the pump beam has a Poissonian photon statistics, the SH beam has a sub-Poissonian distribution. Similarly, the remaining fundamental light from which the pairs have been deleted has a more regular stream of photons; it also has a sub-Poissonian photon statistics. Similar arguments apply to non-linear absorption, which could also produce squeezing. This simple interpretation is suitable only for a single pass through the non-linear SHG material. In CW experiments the non-linearity is not large enough, and the crystal is placed inside a cavity to build up a mode with high intensity at the fundamental frequency. The ﬁrst successful observations were carried out by S.F. Pereira et al. [18] who detected a noise suppression of −0.6 dB in the fundamental beam. These early experiments were plagued by additional technical noise sources inside the build-up cavity. A breakthrough occurred when it became possible to polish and coat the non-linear crystal directly. In such a monolithic design, the internal losses can be kept to a minimum, the fundamental light is built up to increase the non-linearity, and the SH ﬁeld is reﬂected at one end to have all of the SH ﬁeld escape on one side. Care has to be taken that the fundamental and SH waves stay in phase, which is achieved by selecting the correct phase matching temperature. The coatings are designed such that phase shifts upon reﬂection preserve the optimum phase relation for a large number of round trips. Such a monolith was ﬁrst designed and operated by the group in München [4]. A. Sizmann et al. measured a noise suppression of −1.0 dB in the fundamental light in the presence of signiﬁcant electronic noise from which they inferred a noise suppression of at least −2.2 dB. This was one of the earliest applications of the balanced detector system (see Section 8.1). Unfortunately the system was not stable, but squeezing was observed only for milliseconds at a time. Systematic

339

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9 Squeezed Light

1064 nm

532 nm

High Refl 1064 High Refl 532

High Refl 1064 Low Refl 532

Figure 9.30 The monolithic design for second harmonic generation (SHG) using CW lasers. The fundamental ﬁeld (FF) is enhanced by the cavity, and the second harmonic ﬁeld is reﬂected at one end and forms one single output beam. Source: From R. Paschotta et al. [5].

PD1 + –

SA

BS PD2 532 nm Oven 110 °C Laser

F1

1064 nm

M1

1064 nm Lock-in 12 MHz

Figure 9.31 Typical layout of an SHG squeezing experiment. The monolithic resonator is pumped through an isolator (FI) by a single-mode ND:YAG laser at 1064 nm. The reﬂected second harmonic light is detected by a balanced detector, and the sum and diﬀerence currents from the two detectors (D1 and D2) are measured with a spectrum analyser (SA). The parameters used by R. Paschotta et al. are MgO:LiNbO3 , cavity length 7.5 mm, radius of curvature 10 mm, input reﬂectivities of RFH = 99.9%, RSH < 1%, and output reﬂectivities close to perfect for both ﬁelds. Source: After R. Paschotta et al. [5].

improvements were carried out by the group in Konstanz who achieved very reliable squeezing of considerable magnitude. They used a monolithic cavity (Figure 9.30). R. Paschotta et al. [5] measured −0.6 dB squeezing (−2.0 dB inferred) on the SH beam. This system showed great reliability (Figure 9.31). A quantitative analysis of the ﬂuctuations can be obtained from a description ̂ in at frequency of the cavity using quantum operators. The input mode A 𝜈L enters the cavity through a mirror with decay constant 𝜅f 1 . It builds up an intra-cavity mode â f . A mode â sh at frequency 2 𝜈L is generated via the second-order non-linearity of strength 𝜒 (2) and leaves the cavity via a mirror of decay constant 𝜅sh1 . The total decay of the internal fundamental mode is described by 𝜅f = (𝜅f 1 + 𝜅ﬂ ), which includes losses due to scattering, whilst that of the SH cavity is 𝜅sh = (𝜅sh1 + 𝜅shl ). A corresponding amount of vacuum noise is coupled in to each cavity. The quantum Langevin equations describing the

9.4 Early Demonstrations of Squeezed Light

–22

Noise power (dBm)

–24

Signal Shot noise

–26

–28

–30 0

10

20

30

Noise frequency (MHz)

Figure 9.32 Squeezing spectrum obtained by R. Paschotta et al. Source: From R. Paschotta et al. [5].

generation of the SH ﬁeld are given by √ dâ f √ ̂ﬂ ̂ in + 2𝜅ﬂ 𝜹A = −𝜅f â f − 2𝜒 (2) â †f â sh + 2𝜅f 1 A dt √ √ dâ sh ̂ sh + 2𝜅shl 𝜹A ̂ shl (9.27) = −𝜅sh â sh + 𝜒 (2) â 2f + 2𝜅sh1 𝜹A dt ̂ i are all vacuum ﬁelds. Equations like these were used by M. Collett where the 𝜹A and later R. Paschotta and co-workers [5, 19]. They concluded that the squeezing was best in the limit of low detection frequencies (Ω → 0) and that the variance of the SH beam could ideally reach the value of V 1SH = 0.11 or −9.5 dB. The squeezing extended up to frequencies of the order of the linewidth of the cavity. This model assumed that the laser was driven by a coherent state. However, in the experiment, the laser noise swamped the noise suppression at very low frequencies. The best noise suppression observed (Figure 9.32) was only V 1SH = 0.8, or −1.0 dB, at a frequency of 16 MHz. The full spectrum can be calculated using the formalism of linearization. It was shown that the spectrum of the SH light (assuming low scattering losses) is given by a noise transfer function [6]: V 1SH (Ω) = 1 −

8𝜅nl2 − 8𝜅nl 𝜅f (V 1las − 1) (3𝜅nl + 𝜅f )2 + (2πΩ)2

2

where 𝜅nl =

2𝜒 (2) |𝛼f |2 𝜅sh (9.28)

̂ in , the driving ﬁeld where 𝛼f = ⟨̂af ⟩ and V 1las is the amplitude noise spectrum of A of the laser at the fundamental. At low frequencies the large excess noise of the laser V 1las , derived in Chapter 6, dominates the measurement. An optimum detection frequency exists. The noise that masks the squeezing is not a technical imperfection, but it is intrinsic to the laser. It is possible to reduce the laser noise by placing a mode cleaner cavity between the laser and the squeezing cavity. This cavity will reﬂect the high

341

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9 Squeezed Light

Piezo A

PID

Mode cleaner (optional)

±

Lens

Mixer

Dichroic

Photodiode

Signal generator

Power monitor

λ 2

λ 4

Adder/subtracter

Monolithic frequency doubler

λ Faraday 2 rotator

Transmitted 1064 nm detector

Laser Slow control

Fast control

PBS

PBS PBS Reflected pump beam from doubler

28 MHz

45 MHz

BS

BS Reflected 1064 nm self-homodyne detector Highpass filter

Combiner

Laser controller

A

±

532 nm self-homodyne detector

±

PID

Figure 9.33 Schematic layout of the improved SHG experiment with mode cleaner. Source: After A.G. White et al. [20].

frequency noise for frequencies larger than its linewidth. The system can now be modelled by two sequential transfer functions, one for each cavity. This theoretical model was tested by A.G. White et al. [20]. The experimental layout shown in Figure 9.33 is similar to that used by R. Paschotta (see Figure 9.31). The light source is a diode laser-pumped monolithic Nd:YAG laser. The non-linear material is MgO-doped LiNbO3 . The crystal is phase-matched by tuning the temperature. The operating temperature is at about 110 ∘ C and has to be optimized to within ±2 mK. The SH light is separated by a dichroic mirror and detected by a balanced detector. The detectors receive optical powers of up to 15 mW each. The eﬃciency of the photodetectors (EG&G FND100) is low (𝜂det = 0.55). In a repeat of the experiment, a mode cleaner cavity, locked to the laser frequency, is placed between the laser and the crystal. The experimental noise spectra are shown in Figure 9.34 for the frequency range 2–34 MHz. The QNL was determined by measuring the diﬀerence of the currents from the two detectors. The electronic noise, typically −6 dB, has been subtracted. The experimental trace (a) shows the noise suppression for the monolithic cavity. The optimum of −0.6 dB is measured at 13 MHz. A theoretical prediction, trace (b), based on Eq. (9.28) and using measured values for the non-linearity, losses, intensities, and cavity reﬂectivities agrees very well. In a second experiment the mode cleaner cavity is inserted. As shown in trace (c), the high frequency noise from the laser (2–15 MHz) is substantially rejected. The best observed squeezing is increased to −1.6 dB at 10 MHz (3 dB inferred). Trace (d) shows the theoretical prediction obtained by including the transfer function for the mode cleaner cavity, which means replacing V 1las in Eq. (9.28) with the output variance from the cavity. The good agreement shows that all

10

8.36

7.5

6.03

5

3.81

(a)

1.78

2.5 (b) Quantum noise limit

0 (d)

–2.5

–1.45

(c)

(e)

0

Normalized noise power (dB) (measured)

Normalized noise power (dB) (inferred)

9.4 Early Demonstrations of Squeezed Light

–2.55

–5 2

7

12

17

22

27

32

Frequency, f (MHz)

Figure 9.34 Comparison between theory and experiment for the squeezing generated by second harmonic generation as shown in Figure 9.33. The linearized model describes the complete system in a modular approach through the use of transfer functions. (a, b) No cavity. (c, d) With mode cleaner cavity. (e) Model with no laser noise. Source: After A.G. White et al. [20].

eﬀects are included in the combined transfer function. How can these results be improved? Can we get closer to the limit of V 1SH = 1∕9? A systematic study has shown that the main obstacles are the limited ratio of non-linearity to absorption in available non-linear materials and competing non-linearities [21]. One appealing, and technically simple, alternative to the external cavity frequency doubler would be to incorporate the crystal into the laser cavity. Such internal, or active, systems show very high conversion eﬃciencies for SHG. Early theoretical studies predicted large squeezing in these systems, but a careful analysis by White et al. [20], including the intensity and dephasing noise of the laser, shows that in practice no squeezing can be expected. This prediction applies to a wide a range of lasers, and thus active doubling is not worth considering for experiments. 9.4.4

The Kerr Eﬀect

A third process for squeezing that had been proposed very early is the Kerr eﬀect. It was favoured by the group of M.D. Levenson and R.M. Shelby at IBM Almaden Research Laboratories. It led to some of the earliest published squeezing results and was later extended by H.A. Haus and K. Bergman at MIT to the pulsed regime where it achieved impressive −3.5 dB noise reduction. The responses of the Kerr medium was brieﬂy discussed in Section 9.1. Here we analyse it in detail. 9.4.4.1

The Response of the Kerr Medium

The classical description of light propagating through a non-linear Kerr medium has already been given in Eqs. (9.1) and (9.6). The ﬂuctuations of the quadrature

343

9 Squeezed Light

phase of the output ﬁeld can be described by quadrature operators, either in the time or as here the frequency domain: ̃ in ̃ out = X𝟏 X𝟏 2πn0 n2 L𝛼 2 (9.29) 𝜆 This output ﬁeld is detected using a homodyne detector by beating the output signal against a local oscillator. The detected quadrature is a linear superposition of 𝛿X1 and 𝛿X2 given by ̃ in + 2rKerr X𝟏 ̃ out = X𝟐 ̃ in X𝟐

with rKerr =

̃ ̃ cos(𝜃) + X𝟐 ̃ sin(𝜃) X(𝜃) = X𝟏

(9.30)

The variance is a measure of the noise in the quadrature and is calculated as shown in Section 4.1.1. It depends on the Kerr parameter rKerr and is normalized to the variance of a coherent state. Note that the amplitude remains unchanged and that for 𝜃 = 0, the amplitude quadrature variance is always equal to the QNL. This particular projection is independent of the non-linearity: 2 sin2 (𝜃) V (𝜃, rKerr ) = 1 + 2rKerr sin(2𝜃) + 4rKerr

(9.31)

where V (𝜃 = 0, rKerr ) = 1 is the QNL, and for V (𝜃, rKerr ) < 1 we have achieved squeezing. The angle 𝜃s for the best squeezing, the minimum of V (𝜃, rKerr ), can be found by evaluating d∕d𝜃(V (𝜃, rKerr )) = 0, which leads to ) ( 1 −1 (9.32) cot(2𝜃s ) = −rKerr or 𝜃s = arctan 2 rKerr In the limit of a small non-linear parameter (rKerr → 0), the best squeezing is observed for 𝜃s = −0.25π. For large squeezing parameters the angle approaches asymptotically the value −0.5π. Figure 9.35 shows an example of the normalized variance V (𝜃, rKerr ) for three values of the non-linearity, namely, rKerr = 0.25, 0.5, 0.75, and 1.0. It can be seen that the noise suppression is improving with larger non-linearity and that the angle of best squeezing is rotated. The noise suppression increases steadily with the non-linearity, and the optimum squeezing is given by √ 2 2 V (𝜃s ) = 1 − 2 rKerr 1 + rKerr + 2 rKerr (9.33) 4

1

3

rKerr = 0.5

2

rKerr = 0.25

rKerr = 0.75

0.8 V(θs)

V(θ)

344

1 0 (a)

0.6 0.4 0.2

0

π/4

π/2 θ (rad)

3π/4

0

π

(b)

0

0.5

1

1.5

2

rKerr

Figure 9.35 (a) V(𝜃) for three values of rKerr . (b) Dependence of optimum value V(𝜃) on rKerr .

9.4 Early Demonstrations of Squeezed Light

The calculation did not assume any speciﬁc detection frequency Ω. This is correct for a situation where the non-linearity does not depend on Ω. In practice, this applies only to the case of a travelling wave through a non-resonant Kerr medium. The two diﬀerent cases of (i) a broadband Kerr medium inside a cavity and (ii) of a resonant Kerr medium, such as atoms close to an absorption line, will require a description of n(𝛼 2 ) as a function of Ω and have to be evaluated for each detection frequency individually. It is useful to consider the Kerr eﬀect in view of quantum noise sidebands. At the core of the above calculation is the correlation between the quadratures stated in Eq. (9.29). This correlation enforces a link between the noise sidebands. Without such a correlation 𝛿EΩ and 𝛿E−Ω would be completely random. There would be independent contributions from 𝛿EΩ and 𝛿E−Ω to both 𝛿X1 and 𝛿X2. The correlation requires that for each component of 𝛿EΩ , there is also an additional component −2 rKerr 𝛿EΩ at the other sideband, and vice versa. The sidebands at the output are larger than at the input, as can be seen in Figure 9.4. In detection the correlation leads to a reduction of the noise in one quadrature. 9.4.4.2

Optimizing the Kerr Eﬀect

The Kerr eﬀect has been used in both CW and pulsed systems. It was actually one of the ﬁrst successful demonstrations of squeezing. The main practical problem is to achieve a large Kerr parameter rKerr , and this requires the largest possible values for the non-linear coeﬃcient 𝜒 (3) , interaction length L, and pump intensity I. There are four strategies for obtaining a large value for rKerr : (i) Optical ﬁbre. The coeﬃcient for the non-linearity is small, but the length of the medium and the intensity can be increased by using several hundred metres of single-mode ﬁbre. Values for rs in the range 0.1–0.5 have been obtained for the case of a CW squeezing and in the range 0.5–2.0 have been achieved using pulses (see Section 9.5). (ii) Non-resonant medium inside a cavity. The same material (quartz) as used in the optical ﬁbre can be used in a high ﬁnesse resonator. Whilst the physical length is short (typically 10 mm), the eﬀective length and the high intra-cavity intensity in a high ﬁnesse cavity could compensate this shortcoming. No results have yet been published. (iii) Resonant media. The non-linear coeﬃcient 𝜒 (3) can be enhanced several orders of magnitudes by operating close to an atomic resonance. Every absorption line shows a strong dispersive eﬀect, which means a strong linear refractive index. The spectral lineshape of the refractive index is aﬀected by the driving ﬁeld. The transition can be saturated and power broadened, which in turn provides a non-linear refractive index. In the experiments this eﬀect is again enhanced by a cavity around the medium. (iv) The use of cascaded 𝜒 (2) processes. It is possible to create a process that has the properties of the Kerr eﬀect by combining two 𝜒 (2) processes. It was shown [22] that the simultaneous occurrence of OPO and SHG is equivalent to one single process. Provided the system is degenerate, the excitation of the pump and down-converted mode will be coupled in a way that resembles a Kerr eﬀect for the pump mode. This was demonstrated in single-pass experiments [23] and for the CW case with a non-linear crystal in a cavity [24].

345

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9 Squeezed Light

These strategies suggest that large Kerr parameters should be achievable. However, when we analyse the experiments in detail, we will ﬁnd that the reality of squeezing experiments is not that simple. All four approaches have severe limitations since, in all experiments, there are extra sources of noise and competing non-linear eﬀects that will degrade the squeezing. In the actual experiments special tricks are used to suppress the extraneous noise, and a maximum of the non-linearity is traded oﬀ against a minimum of degradation due to competing eﬀects. 9.4.4.3

Fibre Kerr Squeezing

Fibre-optic quantum noise reduction was eventually developed to reliable and strong photon number noise reduction through three following generations of experiments [25]. After the pioneering experiments with CW laser light described here, the next generations used ultrashort pulses to achieve a larger non-linear phase shift with less Brillouin scattering noise in shorter ﬁbres. With short pulses the distortion of pulse shape and phase becomes an issue in coherent detection schemes. The elegant solution is to employ solitons that propagate free of chirp. For solitons, the local oscillator phase can easily be optimized for the entire pulse in the phase-sensitive detection of quadrature amplitude squeezing. No phase-matched local oscillator was needed in the two latest generations of experiments in ﬁbre-optic Kerr squeezing. Using the methods of spectral ﬁltering and asymmetric non-linear interferometers with picosecond or sub-picosecond solitons, directly detectable photon number squeezing was produced. These experiments are described in Section 9.5.4 on soliton squeezing. A comprehensive summary of ﬁbre-optic Kerr squeezing experiments is given in the tables of Ref. [25]. One of the pioneering experiments was carried out by M.D. Levenson, R.M. Shelby, and their collaborators at IBM in 1985 [13]. The idea was very straightforward: use a quartz ﬁbre with a Kerr coeﬃcient to turn a coherent state, generated by a krypton laser, into quadrature squeezed light. The experimental arrangement is shown in Figure 9.36. The detection system used the scheme of a detuned cavity that provides attenuation for the light without destroying the squeezing and provides a variable projection of the uncertainty area as a function of detuning. The details of this scheme are described in Section 9.3.3. Phase-dependent noise was readily observed. However, the initial squeezing results were rather disappointing. The apparatus appeared to be dominated by classical noise, frequently signiﬁcantly larger than the QNL. The reason is that optical ﬁbres show Brillouin scattering, which is driven by the thermal excitations of the solid-state material. This drives small ﬂuctuations in the refractive index. Inside the ﬁbre, which has a well-deﬁned refractive index guiding geometry, acoustic waves can be set up that have a complex spectrum of resonances. A consequence of these eﬀects is a broadband spectrum of phase noise, called guided acoustic wave Brillouin scattering (GAWBS) (Section 5.4.7). This adds to the noise in the squeezing quadrature and can mask all squeezing. The GAWBS noise can be partially suppressed by cooling the ﬁbre; thus the experiments by Levenson and co-workers

9.4 Early Demonstrations of Squeezed Light

DC

DVM

AC

100 Ω Laser

Spectrum analyser

D

Lens Faraday rotation isolation Phase modulator ωM

Amp

Cavity ωM

LHe Fibre

Figure 9.36 The schematic layout of the pioneering Kerr squeezing experiment in an optical ﬁbre at IBM. Source: After [26].

were performed at 4 K using liquid helium-cooled ﬁbres [26]. The GAWBS eﬀect also varies with the materials and the geometry used in the ﬁbre. Improved materials have more recently reduced the eﬀects of GAWBS [27]. It is possible to suppress many of these phase ﬂuctuations by cancelling the classical phase noise due to GAWBS in an interferometric arrangement. Here, the local oscillator and the squeezed beam both propagate through the ﬁbre in opposite directions, and both beams experience the same classical linear phase ﬂuctuations that then are cancelled in the detection scheme. Finally, there is stimulated Brillouin scattering in a ﬁbre. This is a non-linear phenomenon with a threshold above which a signiﬁcant fraction of light in the ﬁbre is backscattered. The transmitted light is extremely noisy, thereby limiting the intensity that can be used in a ﬁbre experiment. This eﬀect can be partially overcome using not a single-mode but a multimode input, where each mode is squeezed individually and has its own stimulated Brillouin threshold. The experiment has to be arranged such that all these squeezing features add up in phase. The results of the experiment are shown in Figure 9.37. The data clearly show squeezing at a squeezing angle of 28∘ . The best noise suppression observed is V 1(𝜃s ) = 0.85 (−0.7 dB), which corresponds to an inferred squeezing of V 1(𝜃s ) = 0.7 for the light emitted by the ﬁbre. What are the alternatives? For the non-resonant medium in a cavity, the main obstacle is the problem associated with achieving a suﬃciently high ﬁnesse and the eﬀects of losses in the cavity. The cavity will have to be locked actively to the laser frequency using the techniques described in Section 8.4. Squeezing can be achieved only at sidebands around each cavity resonance. That means the squeezing spectrum will show maxima at multiples of the FSR of the cavity. Any dispersion eﬀects have to be carefully avoided. Otherwise the FSR will change with the optical frequency, and the sidebands at plus and minus one FSR cannot be brought into resonance simultaneously.

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Figure 9.37 Early squeezing results from the ﬁbre Kerr squeezing experiment at IBM. Source: After [26].

2.5 Normalized noise level V

348

2.0

1.5

Standard quantum limit = 1.0 –100

–50

0

50

100

Phase shift θ (°)

How high a ﬁnesse is required? In the ﬁbre the interaction length is typically 100 m, the input intensity is 100 mW, and the waist size is 3 μm. In a resonator with the same input power, with a waist size of 10 μm and a material length of 10 mm, a ﬁnesse of about 100 000 would be required to achieve the same parameter rKerr for the interaction. For a single-port cavity, this requires a total loss of about 10−5 in optical power, including all absorption and scatter losses. Such low values require extremely pure and homogeneous materials that are not yet available. 9.4.4.4

Atomic Kerr Squeezing

The use of a resonant medium relies on the ability to tune the input light, which is to be squeezed, close to resonance with an absorption line of the medium. The non-linear refractive index is enhanced. However, for these small detunings, the spontaneous emission becomes an extraneous source of noise, as with 4WM (Section 9.4.1). The light absorbed and re-emitted by the atoms produces intensity ﬂuctuations that are detected alongside the ﬂuctuations due to the quantum noise of the input beam. This produces uncorrelated noise at the detection frequency and swamps the carefully arranged correlations between the sidebands that we want to observe. Therefore the squeezing experiment requires detuning larger than the atomic linewidth absorption. For thermal atoms this would mean the Doppler linewidth. To avoid this the experiments by A. Lambrecht et al.[28] were carried out in laser-cooled atoms. A calculation for the Kerr eﬀect generated by a sample of cold atoms inside an atomic trap predicts signiﬁcant noise suppression. In practice several complications arise: The high intensities inside the cavity lead to instabilities. The cavity with a non-linear medium displays bistability and irregular switching. In addition, real atoms have many states due to hyperﬁne and Zeeman splitting – the prediction of a simple two-level model can frequently not be achieved, and optical pumping has to be taken into account. After overcoming all these diﬃculties, good noise suppression of more than −1.8 dB below the QNL

9.5 Pulsed Squeezing

has been observed [28]. These are diﬃcult experiments, and despite an intensive optimization, such higher atomic densities in magneto-optical traps or even the proposal of using a Bose–Einstein condensate as the non-linear sample and much smaller detunings, the atoms are not a promising Kerr medium. The best results were probably achieved by the group led by P. Grangier and co-workers [29] who optimized the medium for the demonstration of the quantum non-demolition (QND) eﬀect, as described in Chapter 11. 9.4.4.5

Atomic Polarization Self-Rotation

A related mechanism for squeezing is the rotation of the polarization of the light induced by an atomic vapour that occurs at high intensities. Interaction of the near-resonant atoms with elliptically polarized light can cause the initially isotropic 𝜒 (3) non-linear medium to display circular birefringence: the two orthogonal circularly polarized components of the light will propagate at diﬀerent velocities. The result is a rotation of the linear polarization of a light beam. This in turn corresponds to a shear in phase space and the distortion of the initial symmetric Wigner function into an ellipse [30]. The eﬀect was demonstrated in rubidium and the experiment can be very simple: an atomic vapour cell, heated to achieve the correct atomic density and shielded from stray magnetic ﬁelds, is illuminated by carefully controlled polarized light. The output beam is separated with a polarizing beamsplitter into a squeezed vacuum beam and a local oscillator beam that are combined, after rotation of the polarization of the local oscillator, in a standard homodyne detector. Whilst a simple theory predicts squeezing of up to −6 to −8 dB [31], the experiment achieved so far a squeezing of −0.85 dB [30]. The limitations are largely due to self-focussing of the vapour, resonance ﬂuorescence, and phase mismatch. It remains to be seen if this tantalizingly simple experiment can achieve its full potential.

9.5 Pulsed Squeezing Some of the best squeezing results have been achieved using the very high intensities and non-linearities that can be achieved with optical pulses. 9.5.1

Quantum Noise of Optical Pulses

The concepts of quantum noise and squeezing can be extended to pulsed light sources. Consider a series of optical pulses, each pulse with a length tpulse repeated at time intervals trep ≫ tpulse , as shown in Figure 9.38a. The spectrum of the photocurrent will contain frequency peaks at the repetition frequency Ωrep and its higher harmonics. The width of each peak will depend on the length of the pulse train measured or the integration time of the detection. The distribution of power amongst, or the envelope of, the frequency peaks is set by the Fourier transform of the pulse shape of the individual pulses. This is schematically shown in Figure 9.38b. For the short pulses that can be generated with mode-locked lasers, (Ωrep typically 100 MHz, tpulse 0.2–100 ps), the envelope is very wide, but only

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9 Squeezed Light

tpulse

Intensity

trep

Time

(a) Intensity

350

(b)

Envelope due to pulse shape Quantum noise δΩ Ω rep

2 Ω rep

Frequency Ω

Figure 9.38 Optical pulses in the time domain (a) and the frequency domain (b). Quantum noise is present at frequencies between the fundamental components associated with the pulses.

a few of the fundamental peaks ﬁt within the detection bandwidth. In between these peaks the spectrum contains a quantum noise ﬂoor. This is equivalent to the quantum noise spectrum of a CW laser. Thus we can call this quasi-CW . One can interpret this noise as being the sum of many pairs of quantum noise sidebands, each beating with one of the fundamental peaks in the optical spectrum. Any part of the recorded spectrum at frequency Ω < Ωrep represents the Fourier component of the variation in the energy of many pulses. For example, the pulsed source could be intensity modulated at a frequency Ωmod . This would create spectral sidebands to all fundamental peaks of the optical spectrum, which means signals at nΩrep ± Ωmod . The quantum noise is the noise ﬂoor in such a sideband. In these experiments noise represents the lack of reproducibility of the pulses. An ideal classical light source would produce a sequence of perfectly reproducible pulses, and thus no noise outside the fundamental components would be detected. Coherent light cannot have exactly identical pulses, but the energy will vary from pulse to pulse. We are comparing the variation of one group of pulses with that of the next group – and observing the noise ﬂoor. This quantum noise is independent of the detection frequency, as long as it is less than the repetition frequency, and a QNL pulsed laser will have a ﬂat, white quantum noise spectrum between the fundamental peaks. The detection scheme for pulsed light is almost identical to that for CW laser light. The same photodiodes and SA are used. A single detector can reveal intensity ﬂuctuations. A balanced detector will be able to distinguish between classical and quantum intensity noise, and a balanced homodyne detector can detect noise of speciﬁc quadratures. However, three additional requirements have to be satisﬁed in the detection of pulses: (i) The detector should not respond to fundamental frequency components and their harmonics; otherwise saturation of the detector and the electronic ampliﬁers will occur even at very low optical powers. This can be achieved by using sophisticated ﬁlter networks with notch ﬁlters suppressing any

9.5 Pulsed Squeezing

response at Ωrep . Such detectors require extensive development. The only exception is the detection at very low frequencies, Ω ≪ Ωrep , where the detector sensitivity can simply be rolled oﬀ with low-pass ﬁlters. (ii) The pulses have to be carefully aligned in the time domain. For example, the pulses used as the signal and local oscillator beams in the homodyne detector have to arrive simultaneously, with a time diﬀerence of much less than the pulse length. This requires an accurate adjustment of all distances in the experiment. Any cavity would have to have round-trip times that are an integer of trep . (iii) The pulses contain very high intensities. Whilst this is an advantage in that the non-linear response is signiﬁcantly enhanced, it can introduce at the same time other non-linear processes, such as self-focussing, which limit the beam quality and thus the detectability of the light. The fundamental advantages of pulsed experiments are the high intensities and consequently the very large non-linear eﬀects that can be observed. The intensity during a pulse makes the build-up cavities, which are necessary in CW experiments, obsolete. Single-pass SHG or OPO operation is possible. Similarly, the Kerr eﬀect in single pass is suﬃciently strong to introduce a correlation between noise sidebands and therefore induce squeezing. However, there are some eﬀects that limit noise suppression. The squeezing occurs only during the pulse if we assume that the response of the material is instantaneous. When calculating the degree and the quadrature of the measurable squeezing, it is necessary to take the time development of the squeezing into account and to average over the temporal development of the noise properties and the squeezing during the duration of a pulse. Figure 9.39 shows an example where the degree of squeezing and the orientation vary during the pulse. The measurement averages the noise contributions, which are weighted by the intensity at the detector at any one time. Although the squeezing at the peak intensity gets the highest weighting, the observed total squeezing will be considerably less than predicted for the peak intensity alone. In addition the squeezing angle is again a weighted average, and squeezing in diﬀerent quadratures can partially cancel each other. Figure 9.39 The development of the squeezing during a light pulse, indicated by the change of the squeezing ellipse. Source: From H.A. Haus et al. [32]. Time

Squeezed axis

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9 Squeezed Light

Input 85/15 coupler

3 dB coupler

Squeezed vacuum

Local oscillator

Balanced detector –

Fibre (non-linear)

Output

Stabilization circuit (a)

15 10 Relative noise level (dBm)

352

5 Shot noise level

0 –5 –10 –15 –20 0

(b)

1

2

3

4

5

6

Pulse non-linear phase (rad)

Figure 9.40 The pulsed Kerr experiment. Source: After H.A. Haus et al. [32]. (a) The schematic layout. (b) Noise suppression results from the pulsed Kerr experiment.

9.5.2

Pulsed Squeezing Experiments with Kerr Media

A very successful series of squeezing experiments with optical pulses was carried out by the MIT group [32, 33] Figure 9.40b. Here the optical medium is a single-mode ﬁbre, and the non-linear process is the Kerr eﬀect. The light source is a mode-locked Nd:YLF laser generating 100 ps pulse with a repetition frequency of 100 MHz, Ωrep = 10 ns. The wavelength is 1.3 μm, which is the zero-dispersion wavelength of the ﬁbre and assures that the pulses maintain their pulse shape during the experiment. The experiment is shown schematically in Figure 9.40a. The experimental arrangement is equivalent to a non-linear Sagnac interferometer. The pulses from the laser enter the interferometer via a 3 dB ﬁbre coupler, equivalent to a 50/50 beamsplitter. There they are separated into two pulses of

9.5 Pulsed Squeezing

Time (s) Noise with swept mirror

Figure 9.41 The temporal trace of the noise from the pulsed experiment. Detection frequency is ﬁxed at Ω = 55 kHz. Source: After K. Bergman and H.A. Haus [33].

identical intensity, which move in opposite directions through the ﬁbre. After a full traverse, the pulses arrive simultaneously back at the coupler. They interfere and form a pulse that is moving back towards the laser. A pulse of squeezed vacuum light leaves through the other port of the interferometer. This pulse is detected with a homodyne detector. A fraction of the reﬂected light is used as a local oscillator, generated by the 85/15 coupler. The arrival time of the local oscillator has to be synchronized with the pulse of squeezed light. The phase of the local oscillator is set and stabilized with a piezo controller. The eﬀect is seen in diﬀerent experiments in two frequency bands: at low frequencies (80–100 kHz) and at high frequencies (≥ 100 MHz). The Sagnac interferometer is essential for such measurements. All ﬂuctuations induced by the ﬁbre, such as phase shifts and intensity variations due to mechanical vibrations, are experienced by both counter-propagating pulses. They are cancelled when the pulses interfere after one round trip and do not appear on the output. This also applies to phase noise induced by the ﬁbre, such as GAWBS. Equally, noise in the laser light frequency or intensity is compensated by the interferometer. In this particular example the suppression was so good that all noise with the exception of large relaxation oscillations were cancelled. What remains is the correlation between the quadratures, induced by the Kerr eﬀect. The squeezing in this case is so strong that they did not even need a SA, but it was directly detectable with an oscilloscope in the time traces of the laser intensity. Figure 9.41 shows alternating noise traces with the squeezing process turned on and oﬀ. The diﬀerence between these traces gives directly the amount of squeezing. A noise suppression of −3.5 dB was achieved at frequencies as low as 55 kHz. 9.5.3

Pulsed SHG and OPO Experiments

The high peak intensities of short laser pulses allow the realization of direct single-pass travelling-wave second harmonic generation (TWSHG) and the

353

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associated squeezing. Consider that in a single pass through a crystal of length , a beam of input power P(0) generates a beam of power PSHG () = ΓP(0). We call Γ the single-pass conversion eﬃciency. At the same time the beam will experience an eﬀective non-linear phase shift 𝜙eﬀ between the SH light generated at position z = and SH light generated at position z = 0 and propagating to z = . Best SHG is achieved with complete phase matching, or 𝜙eﬀ = 0. It was shown by the group of P. Kumar [22] that in this case the predicted noise suppression at low frequencies is S(Ω → 0) = 1 − Γ

(9.34)

This indicates the possibility of almost perfect squeezing provided that most of the light is converted into the SH. Even with a moderate amount of phase mismatch, the noise suppression should be very strong, and detailed calculations predict squeezing as large as 10 dB [22]. However, the corresponding experiments were so far not successful, but a noise suppression of only 0.3 dB has been observed [34]. This was achieved with a Na-doped potassium titanyl phosphate (KTP) crystal. An SHG eﬃciency of Γ = 0.15 was obtained at a peak intensity of about 250 MW/cm2 . The main technical challenge has been in isolating the squeezed from the fundamental light. Using a type II phase-matched crystal, it was possible to separate the SHG beam from the fundamental beam using polarization optics. But the achievable extinction ratio was too small. Excessive leakage of the input ﬁeld into the detector led to saturation of the detection system and limited the observed noise suppression. This type of squeezing requires further technical developments. Far more successful has been the generation of squeezed vacuum state using a degenerate optical parametric ampliﬁer (DOPA). The group of P. Kumar used a KTP crystal in single-pass operation. The ﬂuctuations are measured with a homodyne detector. Special attention was given to the matching of the local oscillator beam to the squeezed beam in both spectral and temporal domains. Noise suppression of 5.8 dB was observed [35], matching the best results from the corresponding CW cavity experiment. 9.5.4

Soliton Squeezing

One of the requirements for quadrature squeezing in the pulsed experiments is to maintain the temporal shape of the pulse throughout the medium. This is commonly achieved by working at the zero-dispersion wavelength of an optical ﬁbre. However, the material shows self-phase modulation that results in a chirp of the frequencies. Consequently a non-soliton pulse does not allow the local oscillator phase to be optimized for the pulse as a whole. This limited early pulsed ﬁbre squeezing experiments where Gaussian zero-dispersion pulses were used. A clever alternative was proposed by P. Drummond and his group [36] and in parallel by the group of H.A. Haus [37]. A special pulse form can be used to form temporal solitons that preserve their shape by a balance between dispersion and non-linear refractive eﬀects and propagate free of chirp. Solitons have almost particle-like properties, are robust against a variety of perturbations, and preserve their shape, energy, and momentum even after numerous interactions with

9.5 Pulsed Squeezing

each other [38]. The pulse shape is a hyperbolic secant. The pulse width tFWHM is dependent on the peak power Ppeak because shorter pulses experience stronger dispersive broadening and consequently require a higher peak power to balance dispersion. A fundamental soliton is obtained for a wide range of pulse energies and wide varieties of initial pulse shapes such as Gaussian or square pulses. After some initial self-reshaping, the exact balance of pulse width and peak power is eventually attained with a smooth soliton shape. 2 , It is convenient to introduce the pulse parameter N 2 = LD ∕LNL ∝ Ppeak tFWHM which indicates whether dispersive pulse broadening (N < 1) or non-linear pulse compression (N > 1) dominates the initial pulse evolution. The N parameter is the ratio of the length scale for dispersive pulse broadening, LD ∝ (tFWHM )2 , and the length per one radian non-linear phase shift in the pulse peak, LNL ∝ (Ppeak )−1 . These length scales are given by the dispersive and non-linear coeﬃcients of the ﬁbre. A fundamental soliton is also called a N = 1 soliton because it exactly balances dispersion with non-linear eﬀects. The quantum mechanical description, using a non-linear Schrödinger equation, was able to describe the soliton’s noise properties and unique quantum soliton eﬀects were found [37, 39]. The ﬁrst such eﬀect was amplitude quadrature squeezing, predicted by Carter et al. [40]. It was observed at IBM by M. Rosenbluh and R.M. Shelby [41], who used a colour centre laser to produce soliton pulses with a pulse length of 200 fs at 1500 nm propagating through a polarization-preserving single-mode ﬁbre. The Kerr eﬀect in the ﬁbre generated the squeezing. Noise suppression of −1.67 dB was measured with a homodyne detector. A second quantum eﬀect is the ‘entanglement’ of pairs of solitons. Their quadratures get linked, which can be exploited for QND experiments (Chapter 11) and quantum information experiments (Chapter 13). 9.5.5

Spectral Filtering

A third quantum eﬀect is soliton photon number squeezing by spectral ﬁltering, which was unexpectedly found by S.R. Friberg et al. [42] in 1996. The experimental method was fairly simple: a pulse launched into a ﬁbre was spectrally bandpass ﬁltered after emerging from the ﬁbre end and was then directly detected – an unusual idea since ﬁltering deliberately introduces losses to achieve squeezing. The eﬀect can be heuristically explained by considering the propagation of solitons in the ﬁbre. As they propagate, self-phase modulation and group velocity dispersion interact to cause a periodic broadening and shrinking of the soliton spectrum. For a ﬁxed length of ﬁbre the soliton energy, or photon number, determines the periodicity. When the soliton, after leaving the ﬁbre, is attenuated by a ﬁxed bandwidth spectral ﬁlter, the losses depend on the soliton’s pulse energy. The combination of propagation and ﬁltering results in a non-linear input–output relationship, i.e. non-linear absorption. At speciﬁc lengths, or pulse energies, the intensity noise is suppressed below the QNL – photon number squeezing is possible. These simple ideas were conﬁrmed by full quantum models [43]. The underlying squeezing mechanism can be best understood in terms of the multimode

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quantum structure of solitons. Spectrally resolved quantum noise measurements showed that anti-correlations in photon number ﬂuctuations emerge within the spectrum of the soliton as it propagates down the ﬁbre [44, 45]. If these anti-correlated spectral domains are selectively transmitted through a spectral ﬁlter and then detected simultaneously and coherently, their ﬂuctuations cancel each other in the observed photon ﬂux. This anti-correlation produces a photocurrent noise reduction far below shot noise. The pioneering experiment was carried out with 2.7 ps pulses with a repetition rate of 100 MHz centred at 1455 nm with a bandwidth of 1.25 nm. After propagating through 1.5 km of ﬁbre, the pulses emerge with a pulse width of 1.3 ps and a spectral bandwidth of 1.65 nm. They were ﬁltered with a variable bandpass ﬁlter to remove the high and low cut-oﬀ frequencies, resulting in the observation of −2.3 dB squeezing with N = 1.2 solitons [42]. The method was further investigated in Erlangen by A. Sizmann and G. Leuchs et al. with sub-picosecond solitons and a variety of ﬁlter functions, ﬁbre lengths, and pulse energies below and above the fundamental soliton energy [45]. When optimized in the experiment, up to −3.8(±0.2) dB of photocurrent noise reduction below shot noise was achieved in direct detection. If corrected for linear losses, a reduction in the photon number uncertainty by −6.4 dB can be inferred [45]. This technique is very elegant and has signiﬁcant practical advantages: It requires no coherent noise measurements and thus is insensitive to phase (GAWBS) noise. It is broadband and all that is required is a pulsed soliton source and suitable ﬁbre. By using an in-line ﬁlter, this device can be made exceedingly lossless and robust. 9.5.6

Non-linear Interferometers

The most promising and most successful method of photon number noise reduction with solitons so far are highly asymmetric non-linear interferometers (Figure 9.42a). The CW [46] and soliton-pulse analysis [47–49] show that the beam splitting ratio in the interferometer must be highly asymmetric in order to produce strong photon number noise reduction, in contrast to earlier experiments with symmetric non-linear ﬁbre interferometers that produced a squeezed vacuum. More than 10 dB of noise reduction was predicted. After the ﬁrst experimental demonstration in Erlangen by S. Schmitt et al. using 130 fs solitons, the interferometer parameters were optimized [48], and the observation of less than −5 dB in the photocurrent noise was reported (Figure 9.42c). This and parallel work reported by K. Bergman and co-workers [50] are the strongest noise reductions observed for solitons to date. When corrected for linear losses, the inferred photon number squeezing was −7.3 dB [51]. A simple model shows a steplike input–output energy transfer function. Around each minimum the transmission is non-linear, the output energy is stabilized at a ﬁxed value, and the photon number uncertainty is reduced below the shot noise limit. However, the squeezing mechanism can be better understood in terms of number–phase correlations induced by the Kerr non-linearity [52], similar to the CW case studied by M. Kitagawa and Y. Yamamoto [46]. A Kerr squeezed state from the high-power path of the interferometer is slightly rotated by the

9.5 Pulsed Squeezing

Fibre loop

X2

BS1 Laser (2) (1) BS2 + – Relative noise power (dB) Transmitted pulse energy (pJ)

(a)

120

(b)

α/(1–α) = 90/10 Lfibre = 6.40 m τFWHM = 126 fs fexp = 20 MHz

100 80 60 40 20 0 6 4 2 0 –2 –4

(c)

X1

Spectrum analysers

0

50

100

150

Total launched pulse energy (pJ)

Figure 9.42 Intensity squeezing of solitons with the non-linear interferometer. (a) Experimental arrangement. (b) The concept of projecting the squeezing ellipse after the Kerr eﬀect. (c) Experimental results with solitons of diﬀerent orders N. The top trace shows the transmission and the bottom trace the measured noise suppression. Source: From A. Sizmann and G. Leuchs [25].

addition of the weak ﬁeld from the low-power path of the interferometer, such that the amplitude quadrature of the total output ﬁeld is squeezed (Figure 9.42b). This squeezing mechanism implies that noise reduction grows with the non-linear phase shift in one arm of the non-linear interferometer [52]. This is in contrast to spectral ﬁltering where noise reduction originates from multimode photon number correlations, which does not grow with propagation distance. Today the non-linear loop interferometer is seriously considered in technical applications for stabilizing communication systems operating with Terabaud

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information rates. It is also employed in basic research as an eﬀective squeezing source for generating entangled beams in quantum information experiments (Chapter 13). To summarize, optical solitons in ﬁbres have provided the perspectives of building ﬁbre-integrated sources of noise reduction and given new insight into multimode quantum correlations with a terahertz bandwidth through spectrally resolved quantum measurements.

9.6 Amplitude Squeezed Light from Diode Lasers The simplest intensity squeezing experiment is to produce amplitude squeezed light directly with a laser of high eﬃciency that is driven by a sub-Poissonian current. The laser can be driven inside a high impedance circuit with a very quiet current (V idr ≪ 1). The conversion eﬃciency of the diode laser can be as high as 𝜂las = 0.5 − 0.75. An important distinction is the fact that a diode laser has an internal cavity to build up the radiation. The laser also involves both spontaneous and stimulated emission. Consequently, it has laser dynamics and spontaneous emission noise. The earliest models for diode lasers were developed by Y. Yamamoto and co-worker [53]. More recently linearized laser models have been developed [54]. It is possible to apply them to the diode laser system (see Chapter 6) and to calculate directly the transfer function between the ﬂuctuations of the pump source (idr ) and the light. As Eq. (6.17) shows, the noise of the laser intensity is completely dictated by the pump noise. For an electrically pumped laser, the eﬃciency of pumping is close to unity, and, in diode lasers, the output coupling is large. Thus the eﬃciency 𝜂las is basically given by 𝜅m ∕𝜅. The variances V 1 of the amplitude quadrature and VN of the intensity are equivalent, and it is common practice to use the Fano factor for this situation. With these assumptions, Eq. (6.17) turns into (1 − V 1las ) = 𝜂las (1 − V idr )

(9.35)

This is the same result as for any other passive device. The maximum noise reduction for a perfectly quiet drive current is V 1las = 1 − 𝜂las . Two further important assumptions have been made in this model, namely, that the laser emits all light into a single output mode and that all the current entering the laser is contributing to the lasing process. Bias currents have been neglected. In practice, however, both these assumptions are not easy to justify. An experiment by W.H. Richardson and Y. Yamamoto [55] showed a noise suppression of 6.5 dB. This is a most impressive result that was actually better than predicted by Eq. (9.35) and the overall eﬃciency of the laser used. No deﬁnite explanation has been given to date for this result. However, it is possible that the diode laser had signiﬁcant internal parallel currents that did not participate in the lasing process. Thus the actual laser eﬃciency 𝜂las might have been better than measured directly. Alternatively, some unknown process might have correlated the drive current with the light. Unfortunately, later experiments were not able to reproduce these early measurements, but they showed less noise reduction and tended to conﬁrm the prediction of Eq. (9.35). In particular Inoue observed noise reduction up to −4.5 dB in diode lasers cooled to 20 K [56].

9.6 Amplitude Squeezed Light from Diode Lasers

It has been observed that diode lasers will excite more than one laser mode. These might not contribute much to the intensity of the output beam, but they can inﬂuence the noise behaviour dramatically. A particularly systematic study was carried out by the group in France [57]. They showed that a free-running diode laser was emitting light into over 50 individual modes. The total intensity was found to be sub-Poissonian. However, by measuring only some of the modes, rejecting the others with a spectrograph, the noise increased drastically. The noise of the single central mode, which carried the vast majority of the power, was up to 40 dB above the Poissonian level. This suggests that the various modes have anti-correlated noise and the noise terms cancel each other when all modes are detected simultaneously. This behaviour can be predicted by a simple multimode laser model that includes the competition between the laser modes [57]. Only some of the modes will be operating well above threshold, whilst others will remain at threshold and only contribute ﬂuctuations. The competition incorporates a limitation of the total gain based on the available number of excited atoms. The division between the modes is not stochastic but balances the ﬂuctuations, similar to a conservation of the total number of available photons. The result is anti-correlation between the modes. More detailed theoretical models have been published to take mode competition properly into account [58], but further study is still required. It is possible to enforce single-mode operation, and simultaneous noise suppression, by either injection locking or optical feedback from an external cavity. When these techniques are optimized, the laser system will operate on a truly single-mode output and generate amplitude squeezed light. M.J. Freeman et al. [59] used injection locking, and H. Wang et al. [60] used external cavities to achieve single-mode operation. They achieved a noise suppression of 1.8 dB with lasers at room temperature. More recently, D.C. Kilper et al. observed a noise reduction by 4.5 dB in an injection-locked laser at 15 K. On the whole, these results ﬁt well to Eq. (9.35). An overview of all these results is given in Figure 9.46. However, it appears that not all diode lasers show reproducibly the same degree of noise suppression. The group led by Y. Yamamoto has succeeded for the ﬁrst time in designing and manufacturing diode lasers that operate in single mode due to their geometry and that reliably operate at −2.8 dB below the Poissonian limit [61]. In contrast, most commercial systems do not perform as well as earlier prototype versions of the same laser design. Large variations in the noise performance have been found even within one commercial series of lasers with otherwise identical speciﬁcations. The detailed reasons for this variation can be many. Finally, one experimental problem is that the maximum eﬃciency of the lasers occurs at the highest drive currents, well above threshold, where the output powers are large. For a measurement of the noise suppression, all the light has to be detected, unless complex structures are used to selectively attenuate the power but not the squeezed sidebands. Thus the beam cannot be attenuated or otherwise the noise suppression is lost. The limit of detectable power is set by the saturation properties of the detector; presently only a maximum of 50 mW can typically be detected with fast detectors capable of measuring ﬂuctuations at radio frequencies (RF) (≥ 10 MHz). This means that some of the best noise suppression cannot actually be detected.

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In summary, the expectation of simple and reliable diode lasers as low cost sources of non-classical light has not yet come to fruition.

9.7 Quantum State Tomography So far we have concentrated only on the measurements of the variances of quantum noise. It has been shown that this is fully suﬃcient for a description of squeezed states, or to be more general, any states that can be described by linearizable models. Figure 9.43 shows a very clear example of the noise suppression that was available already in 1996 after optimization of the experimental parameters. This type of scan provides a good knowledge of the properties of the quantum state. We saw in Section 4.5 that these states have 2D Gaussian Wigner functions. For a quadrature squeezed state, with unknown squeezing angle, we have to measure the variance V (𝜃) for a whole range of angles, and from these data the minimum value V (𝜃s ) is determined. We actually quote only this one number, and the entire squeezed state is described in this way. At least as long as it is a minimum uncertainty state, otherwise a second number, V (𝜃s + 90), is required to describe the state completely. On the other hand, techniques have been developed to measure the Wigner function. This is of fundamental interest since some quasi probability distributions (QPDs) exhibit unusual features and it would be nice to show diagrams of completely reconstructed QPDs. The key idea to these techniques is to measure higher-order moments of the noise statistics. Rather than measuring only the variance for a certain projection angle 𝜃, the full noise trace is recorded (see Figure 9.45). This was described by U. Leonhardt and H. Paul [63] and further elaborated by U. Leonhardt [64] and others. The idea was ﬁrst applied to pulsed –75 (a)

–80 Noise power (dBm)

360

–85

–90 (b) –95

–100 –0.5

0

0.5

1

1.5

2

ϕ (π rad)

Figure 9.43 Experimental data for CW squeezing using a monolithic OPO: the typical scan of the variances indicating squeezing and anti-squeezing. For comparison a theoretical curve with a noise suppression of −7.1 dB is shown. Source: After P.K. Lam et al. [62].

9.7 Quantum State Tomography

light. M.G. Raymer’s group at Oregon [65] recorded histograms of the energy of many pulses and in this way the full noise characteristics of the pulses. Many such histograms were recorded for diﬀerent angles 𝜃, and the underlying Wigner function can be reconstructed using tomographic techniques [65]. For CW experiments the process is similar. The full time history of the ﬂuctuations is recorded. This can be achieved by directly measuring the noise ﬂuctuations of the photocurrent and demodulating the signal with a ﬁxed single detection frequency. Alternatively an electronic SA is used, as shown in Figure 9.45a. By leaving the video bandwidth suﬃciently large, the full noise trace is recorded. For a squeezed state these traces vary dramatically with the projection angle, as shown Figure 9.45b. The many traces are then transformed into histograms of the ﬂuctuations for this speciﬁc projection angle and detection frequency Ω. The restriction to speciﬁc values of Ω is particularly important for those squeezing sources that have a distinct squeezing spectrum S(Ω). Here it is important to detect only those sidebands that are strongly squeezed. For each detection frequency a function W (Ω) can be reconstructed using the tomographic technique [63, 67]. Note that all the measured Wigner functions will be centred around zero and they show the ﬂuctuations that are independent of the mean value of the intensity. The quadrature of the squeezing is only determined as a rotation angle, which is measured either with a homodyne detector or in the case of a bright squeezed source with large intensity, in comparison with the phase of this beam. Apart from a rotation of the Wigner function, the phase and amplitude squeezed states look the same. In addition, the shape of the Wigner function is frequency dependent and can vary considerably with Ω. The same light source can simultaneously be squeezed at one frequency and noisy at another. The complete beam is described by a whole spectrum of Wigner functions (Figure 9.44).

X2 X1

Figure 9.44 The concept of quantum state tomography: complete noise traces are recorded for diﬀerent projection angles, and one such example is given here. These data can be recombined to reconstruct the Wigner function of the quantum state. Source: After U. Leonhardt [64].

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9 Squeezed Light

0.2 Local oscillator

Phase control θ Homodyne detector

Squeezed state generator Spectrum analyser

Noise amplitude (V)

Laser

i_(Ω,t,θ) Density matrix ρnm

0.1 0.0 –0.1 –0.2

Sampler, Reconstruction statistics algorithm P(xθ,θ) analyser

0

5

10

15

20

25

Time (ms)

(b)

(a)

1.0

0.4

0.8 0.6 0.4

p(n)

362

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0.2

0.0 0

2

4

6

8

10

0.0 0

(c)

(d)

2

4

6

8

10

12

Photon number n

Figure 9.45 Quantum tomography experiment. Part (a) shows the schematic layout. Results for a squeezed vacuum state are given: (b) the phase-dependent noise, (c) the squeezed two-dimensional Wigner function, and (d) the reconstructed photon probability distribution. Source: From S. Schiller et al. [66].

How does this compare with the pulsed situation that seems to provide one unique answer for the Wigner function? In this case the selection of the detection frequencies is made automatically; they are given by the Fourier components of the pulses, the pulse repetition frequency, and its higher harmonics. Only these speciﬁc detection frequencies contribute. Only one combined answer, a weighted average of several single frequency Wigner functions, was recorded in the measurements by M.G. Raymer’s group [65]. Using their OPO, the Konstanz group demonstrated the capability of tomography [66] very early on. They investigated a squeezed vacuum, measured the noise projections, and converted them into Wigner functions and photon distributions. One example is given in Figure 9.45. Their results, the complete squeezed Wigner function (c) and the photon statistics (d) with enhanced probabilities for even photon numbers, are a beautiful demonstration of quantum optics. This is now a routine diagnostic technique, both for the quality of the light that is generated and for the quality of the apparatus that is processing the squeezed light and using it for a variety of applications. The comprehensive review by A.I. Lvovsky and M.G. Raymer [68] provides many details of the experiments and the data analysis. It is interesting to note that we can also apply tomography usefully in situations where the light is not squeezed but at the QNL or weakly squeezed and contains

9.8 State of the Art of CW Squeezing

some well-known modulation. After testing the light at several locations in the apparatus, a comparison of the tomographic results will show quite clearly the performance and losses imposed by the apparatus. In this way the quality and capability of the apparatus can be tested and veriﬁed. This would, for example, mean that in the case of commercial production of components that should be able to handle squeezed light or entangled beams, tomography is a useful tool for quality testing and assurance of components and without the need to use non-classical light. As long as we stay within the limit of Gaussian states, tomography should in principle not reveal more than the measurement of the two orthogonal quadratures. However, can we use tomography to show even more exotic results? This depends not so much on the measurement techniques but on the types of states that we can manufacture, and this has not been easy. As pointed out in Section 4.5, we would require a system that cannot be linearized and ﬂuctuations that do not follow Gaussian statistics. One successful approach is to consider a state with a very small photon number, which is heralded by photon counting techniques. Alternatively photon number selective post-processing can be used to select the data for such a speciﬁc state of light imbedded in the optical beam. In these cases strongly non-classical Wigner functions have been recorded, as discussed in Chapter 13. An example is shown on the front cover of this book, where the non-Gaussian Wigner functions are made visible using a modiﬁed detection scheme, where the homodyne detection is conditional on a photon counting measurement. Alternatively we would require a CW system that is extremely non-linear, for example, close to an instability point, and where the coeﬃcients for the non-linearities (e.g. 𝜒 (2) or 𝜒 (3) ) change within the uncertainty limit of the state. These conditions are extremely diﬃcult to achieve since we do not have such non-linearities, and this remains a challenge for the future.

9.8 State of the Art of CW Squeezing Squeezing moved from the stage of a cute theoretical idea to the ﬁrst experimental demonstration in a remarkably short time. From 1980 to 1985 the details of possible squeezing were explored theoretically; then within one year several squeezing experiments were performed more or less simultaneously. Since the ﬁrst demonstration of squeezed states, we have seen a steady improvement in the size and the reliability of the noise suppression. During the ﬁrst few years, several unexpected new noise sources were discovered that hampered the progress. These had to be avoided, in most cases through improvements of the materials used. Examples are as follows: the new ﬁbres used by K. Bergman et al. had significantly less GAWBS noise than the earlier ﬁbres used by M.D. Levenson et al., the early noise problems encountered by H.J. Kimble et al. with external cavity SHG were avoided in the more recent monolithic devices, and the limitation in squeezing with the OPO was set by noise created inside the crystal, which can now be avoided by lower-loss materials. Very quiet monolithic diode-pumped Nd:YAG lasers have replaced the much noisier lamp-pumped lasers used in the earlier

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Squeezing history 3 0 Noise supression (dB)

364

–3

4WM/Kerr Pulsed SHG OPO/OPA Diode laser Twin beams TEM 01

–6 –9 –12 –15 1980 –18

1985

1990

1995

2000

2005

2010

2015

2020

Year

Figure 9.46 Best observed noise suppression in diﬀerent squeezing experiments in the time 1985–2016. The symbols describe diﬀerent processes: Boxes = CW vacuum squeezing using a sub-threshold OPO, diamonds = CW bright squeezed light (SHG, atoms, Kerr), triangles = diode laser noise suppression, crosses = pulsed squeezing experiments (OPO, SHG, solitons, etc.), and circles = twin-photon beam experiments.

experiments. And the detectors have been improved with a quantum eﬃciency of more than 0.95 now available for the visible (Si) as well as the near-infrared spectrum (InGaAs). All this technology had matured by about 2000. Some processes did not show suﬃcient non-linearities to produce signiﬁcant and reliable squeezing, including bistability [69] and atom–cavity interactions [70] and did not stand the test of time. New processes were found, such as the non-linear interferometer for pulsed squeezing. Now hundreds of diﬀerent experiments have been reported, too many to list separately. Instead, Figure 9.46 shows the historical development of the experiments. The observed noise suppression is plotted for several groups of experiments producing squeezed light. As can be seen, the OPO generating a squeezed vacuum state was the ﬁrst to show large squeezing and still remains the best process. At this point in time a suppression to −15 dB has reliably been achievable, albeit only by very few groups, since it requires an amazing attention to detail. If the long-term trend holds up, one could expect a suppression to −20 dB once all the components and materials are optimized. The CW experiments with atomic systems, both 4WM and strong cavity coupling, and with Kerr media were beautiful demonstrations of concepts and test of the theory. They were essential for our understanding but are less likely to play a role in applications. The singly resonant SHG was rapidly improving and is a simple and reliable source of bright amplitude squeezed light. Noise suppression down to −5 dB is a realistic expectation. These systems can be made very robust, small, and reliable, and with the use of periodically poled materials, we might see even simpler and better devices. For the pulsed experiments the results look equally impressive. Here the Kerr eﬀect, soliton squeezing, and the non-linear interferometer all have the

9.9 Squeezing of Multiple Modes

potential to achieve a noise suppression to −10 dB. The potential of new ﬁbres with engineered band-gap structures has not been explored. The only exception to this positive trend has been the diode lasers. Some outstanding results have been achieved very early on, but they could not be conﬁrmed. The best of the present results is supporting the direct theoretical prediction, and further understanding and reﬁnements of the manufacturing process could make −3 dB of noise suppression available in commercial low cost lasers. Squeezed vacuum states can be transformed into various types of squeezed states, the squeezing axis can be rotated, and coherent amplitude can be added through reﬂection oﬀ a cavity. We can control the squeezing independently of the coherent amplitude. In addition we can produce polarization squeezed light and spatially squeezed light. Fibre-based squeezing and diode lasers have the potential to be readily integrated into practical systems. In parallel the other main advance is reliability. In 1984 squeezing light was regarded as a ‘Hero’ experiment, which would normally work no earlier than 2 a.m. ‘Daylight squeezing’ during normal business hours became a norm by about 1996. Now we have systems squeezers that operate continuously for many days, ‘24/7’, for example, within a gravitational-wave observatory with both very high duty cycle and reliability. In addition we have single-pass squeezers as part of integrated optics that are directly built into custom-made devices. Technically this is all possible and commercialization is a clear option. The future use is limited only by our creativity for applications and the demand for such devices.

9.9 Squeezing of Multiple Modes 9.9.1

Twin-Photon Beams

Closely related to amplitude squeezed light is the generation of twin-photon beams. In this case two optical beams are generated by one above threshold OPO (Sections 9.4.2 and 6.3.3). The single input beam is converted, in a non-linear crystal inside a resonant cavity pumped above threshold, into two output beams that have identical statistical properties. In a simple picture one can imagine that each input photon generates two identical photons, one for each output beam, but unlike the parametric down-converter we have here not correlated pairs of photons but two single-mode beams with correlated photon ﬂux. Using type II materials, which means the two modes have orthogonal polarizations (Section 9.4.2), the beams can easily be separated and detected individually. The OPO in most cases is non-degenerate, which means the optical frequencies of the beams are diﬀerent but this has no consequences since they are not recombined optically. Theoretical calculations [71, 72] show that each beam individually has intensity ﬂuctuations that depend on the dynamics of the OPO, as discussed in Section 6.3.3. In particular, close to threshold, each beam has very large ﬂuctuations, V 1(Ω) ≫ 1, at four times the threshold the beams are at the QNL, and far above the threshold each beam should be squeezed, V 1(Ω) = 0.5. This has not yet been observed.

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9 Squeezed Light

However, if we detect the diﬀerence between photocurrents, these ﬂuctuations cancel because of their strong correlations, and we can observe a noise level well below the QNL. The QNL in this case corresponds to the noise measured on the diﬀerence between the intensities of ‘classical twin beams’, obtained by splitting any light beam on a 50∕50 beamsplitter. It is equal to the shot noise level for the combined photocurrent. This experiment was ﬁrst carried out with CW lasers by the Paris group, led by E. Giacobino and co-workers [73] who recorded a noise suppression of 27 % (V 1 = 0.73). Next came an experiment with mode-locked Q-switched lasers by O. Aytür and P. Kumar [74]. C.D. Nabors and R.M. Shelby [75] used a doubly resonant OPO to generate two beams of diﬀerent colours and reported a noise suppression of −3.1 dB. Systematic improvements of their apparatus by the group in Paris led to observations of noise suppression by −8.5 dB, after correction for the dark currents, which is a record that stood for a long time. It was beaten by the Taiyuan group [76] with a noise reduction of −9.2 dB and ﬁnally improved again by the Paris group to −9.5 dB [77]. Since these experiments involve only intensity measurements, and no local oscillators, they avoid the complications of mode matching and beam overlap, and the data are limited purely by the quantum eﬃciency of the detectors and the escape eﬃciency. Note also that twin-photon beams are obtained at the two outputs of a 50∕50 beamsplitter, which has at its two input ports a squeezed vacuum state and a local oscillator of appropriate phase. This is actually what is done in any homodyne measurement of squeezing. It is possible to generate one beam that has sub-Poissonian statistics by measuring one beam and using this information to control, via electro-optic feedforward, the intensity of the other. This was demonstrated by J. Mertz et al. [78] who achieved a noise suppression of −4 dB of one beam compared to the QNL of this individual beam. This new output beam can be directly used for all the applications of sub-Poissonian light, which forms a direct alternative to the quiet diode lasers mentioned previously. Twin-photon beams can also be directly used to beat the QNL in the spectroscopy of very weak absorption lines by −1.9 dB [79] and the measurement of very weak absorptions by −7.0 dB [76]. Finally, the Paris group showed that twin beams oﬀer the opportunity to introduce post-selection processes, commonly used in the photon counting regime, to the regime of continuous variables [77]. If the ﬂuctuations of the two photocurrents at a ﬁxed detection frequency Ωdet are recorded directly and stored, as it was described in the section on state tomography (Section 9.7), we can now produce a sequence of recordings, each containing short intervals of the ﬂuctuations. Thanks to the quantum correlations, we can use one beam to select intervals where V 1(Ωdet ) is within a narrow, well-deﬁned band and select from the other beam the corresponding intervals. The selected light describes a beam with unusual properties. For example, we could select information that describes a beam with strong sub-Poissonian distribution, which was demonstrated experimentally and produced an intensity noise reduction of −4.4 dB below the QNL. Alternatively the selection could be done with an electro-optical modulator. The resulting beam is now no longer continuous but intermittent, depending on the selection process that rejects the majority of the time intervals but has the advantage of representing a strongly non-classical quantum state.

9.9 Squeezing of Multiple Modes

9.9.2

Polarization Squeezing

The polarization of a beam of light is another property that is limited by quantum noise. In order to detect the polarization, we require a combination of two detectors. Each is producing quantum noise, and consequently one expects a QNL for the polarization. The polarization is well described by the Stokes parameters S0 , S1 , S2 , S3 (Section 3.5.2), which describe the light in terms of three degrees of freedom for linear and circular components and one normalization factor. The state of polarization can be described by a three-dimensional diagram, the 1 Poincaré sphere, deﬁned by the radius S = (S12 + S22 + S32 ) 2 . Each point on the sphere is one polarization, and the radius of the sphere is given by the intensity of the beam. An analysis of the quantum properties of polarization shows that they obey the cyclical commutation relation [Ŝ 1 , Ŝ 2 ] = 2iŜ 3 ,

[Ŝ 2 , Ŝ 3 ] = 2iŜ 1 ,

[Ŝ 3 , Ŝ 1 ] = 2iŜ 2

(9.36)

This means that the polarization has an uncertainty volume and the polarization is only deﬁned within a small sphere around each point on the Poincaré sphere, as shown in Figure 9.47A. The possibility of achieving polarization ﬂuctuations below the QNL was investigated extensively by the group around A.S. Chirkin in Moscow. In particular N.V. Korolkova and A.P. Alodjants investigated the possibility of using amplitude squeezed light to inﬂuence the polarization [81, 82]. It was shown that two squeezed beams can be combined on a beamsplitter and by selecting the correct polarization and phase between the two beams, the output beam will have one or more of the Stokes variables measurable below the QNL [83]. The uncertainty volume is compressed and can take the shape of a cigar, with the two Stokes parameters Ŝ 𝟏 Ŝ 𝟑 suppressed at the same time, or alternatively like a pancake, with suppression of only one parameter, Ŝ 𝟐 . The ﬁrst experimental demonstrations were made by the group at Bell labs [84] and the Aarhus group [85]. The ANU group, including R. Schnabel, W. Bowen, and P.K. Lam, achieved the goal to squeeze several Stokes parameters simultaneously and below the −3 dB limit. They generated two independent squeezed beams [80] with two seeded OPAs driven by one Nd:YAG laser and frequency doubler. The squeezed beams were combined on one beamsplitter. The apparatus is shown in Figure 9.48 as an example of the actual complexity of quantum optics experiments. The multitude of components, all individually aligned and optimized, is required to produce the pump, seed, and local oscillator beams, to produce the two squeezed beams, to combine them on a beamsplitter, and to detect the properties of the output state. The main challenges are to reduce the loss of all components, to optimize the mode match between all the beams, and to maintain the phase locking between all the cavities, such as lasers, mode cleaners, and OPOs. For this purpose this experiment contains 4 temperature control systems and 7 optical locking systems. By generating two bright squeezed beams with about 1 mW power and about −4 dB of squeezing each, they were able to achieve both the cigar and pancake states, with suppression of the Stokes parameters simultaneously by better than −3 dB at detection frequencies around 8 MHz. Figure 9.47B shows the experimental results in the form of reconstructed

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S3

(a)

Right-circularly (b) polarized

ˆ S 3

ˆ S 1

(A)

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S2

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45° polarized σS

(a) σ S3

σS

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Sˆ ⊥a

σS 3 σS

(e) σS 3

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(d) σ S3

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σS

2

3

σS

2

2

σS

σS

1

σS

σS 1

2

1

σS

σS 1

2

(B)

Figure 9.47 (A) Diagram of (a) the classical and (b) quantum Stokes vectors mapped on a Poincaré sphere. The ball at the end of the vector visualizes the quantum noise of the polarization states. (B) Measured quantum polarization noise at 8.5 MHz from diﬀerent combinations of input beams: (a) single coherent beam, (b) coherent beam and squeezed beam, (c) bright squeezed beams, (d) two phase squeezed beams, and (e) two amplitude squeezed beams. Source: After R. Schnabel et al. [80].

uncertainty areas on the Poincaré sphere. This type of light can now be used to couple the quantum properties of the light to the quantum properties of atoms, as discussed in Chapter 14. 9.9.3

Degenerate Multimode Squeezers

Following the lead of the polarization squeezer, it is possible to use the output of two squeezers within one experiment. We will see that, for example, teleportation requires two squeezed inputs, and for quantum gates and error correction, we might need up to eight squeezed inputs. This can be achieved by building several devices, each producing a separate beam, and these can be combined using a whole array of beamsplitters, resulting in several output beams that are detected individually. This technique has been developed into a ﬁne art by the group of A. Furusawa at Tokyo University, and many more details can be found in his book [86].

9.9 Squeezing of Multiple Modes

2011 ANU double squeezer, entangler, tweezer

1989 IBM Kerr squeezer QND

Figure 9.48 Squeezing experiments are complex. They combine low loss optical systems and nonlinear devices, high eﬃciency detectors and high quality opto-mechanical control. They are operated in well controlled environments, built on optical tables with minimum mechanical noise, excellent thermal and dust control. These photos shows two examples: on the right had side an experiment at IBM in 1989 for the demonstration of squeezing and QND in optical ﬁbres. On the left hand side an apparatus at ANU in 2011 with two OPO squeezers for entanglement, spatial mode squeezing and applications to tweezers with biological samples. The design principles remained, the complexity and performance of the devices had greatly improved. The future will see integrated opto-mechanical designs with low loss wave guides, custom manufactured materials and opto-mechanical systems providing reduced size, ruggedness and higher reliability. Source: After R.Schnabel et al. [80].

Alternatively, one could search for one device that can generate squeezing in several modes simultaneously. This is in principle possible and has been used for OPOs that operate simultaneously in both output directions and in this way produce two squeezed beams. This apparatus has considerably fewer components than the complex machine that has two separate squeezers, as shown in Figure 9.48. The goal of producing a device with many modes has for a long time been elusive. Early attempts of using degenerate OPOs in special cavity conﬁgurations to generate several spatial modes simultaneously were found to be too delicate. However, since 2010, there has been considerable progress, in particular with temporal modes, where more than 10 squeezed modes have been generated and detected by the Paris group in a femtosecond frequency comb [87] using one OPO and creating orthogonal modes where each is a diﬀerent combination of a large set of individual frequencies in the comb. This is now being applied to several ideas from quantum information, communication, and sensing. At the same time the team in Virginia led by O. Pﬁster has produced and veriﬁed 60 modes of carefully engineered combinations of frequencies within an OPO with two non-linear crystals [88]. Each combination represents a whole entity, or mode, which combines the simple quantum states at individual frequencies of the

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optical ﬁeld and their magnitudes and phases in such a way that the combined mode is part of an elaborate basis set. The apparatus is carefully designed and crafted to create and then let us test quantum correlations not only between two of these modes but a whole set of multimode correlations, following carefully set out rules for graph states. These states are the realization of complex rules designed to represent mathematical algorithms and the realization of quantum logic protocols.

9.10 Summary: Quantum Limits and Enhancement In summary, the tools for optical quantum engineering have now emerged. Squeezed light is now being used in a number of applications to improve the detection of light beyond the QNL associated with the photon statistics of the best classical light, with the details in Section 10.2. There are several other situations in which the apparatus is limited by the QNL, for example, attenuators and ampliﬁers. In all cases the QNL, which we experience with a coherent state of light produced by a perfect laser, represents the limit of the performance of the instrument. Fortunately there are ways in which these limits can be overcome through the use of quantum correlations of the optical ﬂuctuations created by non-linear optical systems. Table 9.1 provides an overview of these situations. Table 9.1 Other eﬀects that are related to squeezing. Limit of a coherent state

Quantum eﬀect

QNL

Squeezing

V1 = V2 = 1

Vsq < 1, Vantisq > 1

Attenuation

Squeezing

Vout = 𝜂Vin + (1 − 𝜂)

V1out = 𝜂V1in

Ampliﬁcation

Anti-squeezing

Vout = G Vin + 1

V2out = GV2in

Independent beams

Entangled beams

V1(A|B) V2(A|B) = 1

V1(A|B) V2(A|B) < 1

Measurement

Quantum non-demolition

Vout > Vin

V1out = V1in

Classical communication

Teleportation

V1out ≥ V1in + 2

V1in ≤ V1out < V1in + 2

V2out ≥ V2in + 2

V2in ≤ V2out < V2in + 2

Beam displacement

Quantum laser pointer

VΔx = Vdx 1

VΔx < Vdx 1

VΔ𝜃 = Vdx 2

VΔ𝜃 < Vdx 2

References

The ﬁrst column states the quantum limits as set by the coherent state in terms of the variances of the ﬂuctuations. One aim of quantum optics is to go beyond these limits. The second column of the table names the devices that use quantum correlations and states the limits that can be achieved. If only one squeezer is used, we see an improvement in one selected quadrature Vsq . By using two squeezed sources, we can arrange an improvement in both quadratures of the same mode, for example, in the case of entanglement between two beams and teleportation. The improvement applies to all aspects of the information that is encoded in these orthogonal quadratures. In the case of spatial measurements, the detection of a displacement or tilt in a transverse direction x is limited by the quantum noise in the quadratures of a separate orthogonal mode with quadratures Xdx 1 and Xdx 1. This mode contains all the information of the displacement and tilt, and squeezing this mode will yield improved measurements of this particular degree of freedom. The following chapter shows examples of the use of squeezed light for measurement applications. Presently the quality and reliability of the devices is being improved. After more creative engineering, the complexity and usefulness of the state generators will improve to a point where they are simply sources of a desired quantum correlation pattern. More and more complex tests are being devised and will be used for in-principle demonstrations and quality control. The results of today will appear to be just the beginning – and it is now time to consider real applications of muitlmode entanglement, based on squeezing of light, or an equivalent set of modes based on particles, on photons, or ions, or atoms, or other particle-like quantum eﬀects. We shall now progress to see in Chapter 10 the eﬀects such quantum correlations have on actual devices, real sensors, communication, and imaging and in Chapter 13 on quantum logic protocols and devices.

References 1 Levenson, M.D. and Kano, S.S. (1988). Introduction to Nonlinear Optics, 2e.

Academic Press. 2 Slusher, R.E., Yurke, B., Grangier, P. et al. (1987). Squeezed-light generation

by four-wave mixing near an atomic resonance. J. Opt. Soc. Am. B 4: 1453. 3 Pauli, W. (1980). General Principles of Quantum Mechanics. Springer-Verlag. 4 Sizmann, A., Horowicz, R.J., Wagner, E., and Leuchs, G. (1990). Observa-

5 6 7 8

tion of amplitude squeezing of the up-converted mode in second harmonic generation. Opt. Commun. 80: 138. Paschotta, R., Collett, M., Kürz, P. et al. (1994). Bright squeezed light from a singly resonant frequency doubler. Phys. Rev. Lett. 72: 3807. Ralph, T.C., Taubman, M.S., White, A.G. et al. (1995). Squeezed light from second-harmonic generation: experiment versus theory. Opt. Lett. 20: 1316. Tsuchida, H. (1995). Generation of amplitude-squeezed light at 431 nm from a singly resonant frequency doubler. Opt. Lett. 20: 2240. Caves, C.M. (1981). Quantum-mechanical noise in an interferometer. Phys. Rev. D 23: 1693.

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9 Xiao, M., Wu, L.-A., and Kimble, H.J. (1987). Precision measurement beyond

the shot-noise limit. Phys. Rev. Lett. 59: 278. 10 Yuen, H.P. (1976). Two photon coherent state of the radiation ﬁeld. Phys. Rev.

A 13: 2226. 11 Walls, D.F. (1983). Squeezed states of light. Nature 306: 141. 12 Shelby, R.M., Levenson, M.D., Walls, D.F. et al. (1986). Generation of

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26 27 28 29

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85 Soerernsen, J.L., Hald, J., and Polzik, E.S. (1998). Quantum noise of an atomic

spin polarization measurement. Phys. Rev. Lett. 80: 3487. 86 Furusawa, A. and vanLook, P. (2011). Quantum Teleportation and Entangle-

ment. Wiley-VCH. 87 Roslund, J., de Araujo, R.M., Jiang, S. et al. (2013). Wavelength-multiplexed

quantum networks with ultrafast frequency combs. Nat. Photonics. https://doi .org/10.1038/NPHOTON.2013.340. 88 Chen, M., Menicucci, N.C., and Pﬁster, O. (2014). Experimental realisation of multipartite entanglement of 60 modes of a quantum optical frequency comb. Phys. Rev. Lett. 112: 120505.

Further Reading Drummond, P. and Ficek, Z. (ed.) (2003). Quantum Squeezing. Springer. Furusawa, A. (2015). Quantum States of Light. Springer. ISBN: 978-4-431-55958-0. Schleich, W.P., Mayr, E., and Krähmer, D. (1997). Quantum Optics in Phase Space. Wiley-VCH. Walls, D.F. and Milburn, G. (1993 and 1997). Quantum Optics. Springer-Verlag.

377

10 Applications of Quantum Light Squeezed states and entangled pairs of photons allow us, in principle, to circumvent some of the quantum limits that apply to coherent states of light. Quantum measurements provide opportunities that classical systems do not have. This has practical applications and four areas stand out as the most likely applications: optical sensors beneﬁt from squeezed light in gaining sensitivity. Optical communication beneﬁts when information can be transmitted with non-classical light. Spatial eﬀects and imaging as well as timing are worth discussing as multimode examples for quantum enhanced measurements. Finally, interferometric detectors for gravitational waves (GWs) are the most impressive example for the use of squeezed light to date. For all these cases we will discuss the standard quantum limits (SQLs) that apply to coherent states and ways to overcome them. A diﬀerent class of applications based on quantum information concepts is discussed in Chapter 13.

10.1 Quantum Enhanced Sensors 10.1.1

Coherent Sensors and Sensitivity Scaling

An optical sensor is a device that monitors a property and translates a change of this property into an optical signal. The property under investigation could be a position, orientation, acceleration, temperature, pressure, concentration, chemical composition, light level, electric or magnetic ﬁeld, or even a GW, to name the most common. The sensor could be a local device, sometimes with very high microscopic spatial resolution, or it could be a remote sensing device allowing measurements of a property at a distance. Frequently, the sensor is used to produce the data for a 2D or 3D spatial or temporal map of the property. In all these cases we want to measure the property with the best possible sensitivity, low noise, and big dynamic range. We may wish to monitor the eﬀect continuously, and it is then normally an advantage to push the limit of sensitivity and signal-to-noise ratio (SNR). It is this goal where quantum optics makes a contribution (Figure 10.1). Technically speaking a coherent optical sensor can be regarded as a modulator (see Section 5.4.5), where the modulation of a beam of light is aﬀected by just one external parameter; let us call it ℘. The concept is that a change in ℘ leads to a A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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Parameter ρ (x,y,z,t) changing in time and space

Parameter ρ (x,y,z,t) changing in time and space

DET Input beam optical power A in ^2(t)

(a)

One- or two-photon excitation

Output A out ^2(t)

DET

(b)

Figure 10.1 Coherent optical sensor (a) and incoherent sensor based on ﬂuorescence (b).

change of the optical output power, or intensity for a ﬁxed beam size, which can then can be detected. Given an input mode â (t), we can write the output mode as √ √ â o (t) = 𝜖(℘(t))̂a(t) + 1 − 𝜖(℘(t))̂v(t) (10.1) where the eﬀective optical transmission, 𝜖, is a function of ℘ and v̂ (t) is the vacuum mode that inevitably couples for non-unit transmission. In a high sensitivity situation, we may typically want to estimate a small change in our parameter, 𝛿℘, away from a known value ℘o . Given that a small change in ℘ leads to a small change in the transmission, we can write ) ( √ K ′ 𝛿℘ â o (t) = â (t) 𝜖(℘o ) − √ 2 𝜖(℘o ) ( ) √ K ′ 𝛿℘ + v̂ (t) (10.2) 1 − 𝜖(℘o ) + √ 2 1 − 𝜖(℘o ) 𝜕𝜖 where K ′ = 𝜕℘ |℘o . We will discuss the ultimate limits of this type of sensor in Section 10.1.3. A related situation is where we want to use a sensor to continuously monitor small time-dependent changes 𝛿℘(t) of the parameter. The changes can be due to changes of ℘(x, y, z, t) in time at a ﬁxed location or due to scanning the sensor in space or both. The changes in the photocurrent at the output iout (t) = â †o (t)̂ao (t) are seen as a proxy for the changes in ℘(t). The design of the sensor can vary widely – we can have very direct links, such as the measurement of concentration of a compound by absorption or a displacement measurement in an interferometer. However, we also have some quite complex sensors, for example, for the measurement of magnetic ﬁelds, temperature, or pressure. We also assume that any change 𝛿℘(t) can occur with any time dependence, which means there could be harmonic signals with ﬁxed or variable frequencies, step functions, or bursts. Consequently we want to sample the data stream from the sensor at regular intervals, Δtsamp , fast enough to capture the dynamics of interest. On the other hand we assume that the underlying noise, from the input beam and from other sources such as the electronics, is of Gaussian nature.

10.1 Quantum Enhanced Sensors

As a concrete example let us take a simple case where our external parameter is a harmonic signal and there is a linear relationship between 𝜖 and the parameter, i.e. 𝜖 = K℘ = K(℘o + 𝛿℘o cos(2𝜋Ωsig t)), where K and 𝛿℘o ≪ ℘o are constants. ̃ = 𝛿℘o(𝛿(Ω + Ωsig ) + 𝛿(Ω − The Fourier transform of the ﬂuctuating term is 𝛿℘ ′ Ωsig )) and K = K. We may also assume that the ﬂuctuations and the signal are small compared to the average power such that the input mode can be writ̃ ̃ ̂ ten as â (t) = 𝛼 + 𝜹a(t) with |𝛼| ≫ 1 and hence A(Ω) = 𝛿(Ω)𝛼 + 𝜹A(Ω). Taking the Fourier transform of Eq. (10.2), substituting and neglecting small terms (see Section 5.4.5) gives √ √ ̃ o (Ω) = ℘o A(Ω) ̃ ̃ A + 1 − ℘o V(Ω) K 𝛼 𝛿℘ (10.3) + √ o (𝛿(Ω − Ωsig ) + 𝛿(Ω + Ωsig )) 4 ℘o where we have taken 𝛼 real. We see that the harmonically oscillating parameter maps onto frequency sidebands. The spectral power as measured via direct detection is proportional to the amplitude quadrature (see Section 4.5.1), so we ﬁnd ̃ o (Ω) + A ̃ †o (Ω)|2 ⟩ VI (Ω) = ⟨|A = 1 + ℘o (V 1A (Ω) − 1) K 2 𝛼 2 𝛿℘2o + (𝛿(Ω − Ωsig ) + 𝛿(Ω + Ωsig )) 4℘o

(10.4)

The signal to noise of, for example, the upper sideband is SNR =

K 2 𝛼 2 𝛿℘2o 4℘o (1 + ℘o (V 1A (Ωsig ) − 1))

(10.5)

If our input beam is in a coherent state, then V 1A (Ωsig ) = 1. However, we see that in this simple example amplitude squeezing, i.e. V 1A (Ωsig ) < 1, can improve the signal to noise of the sensor and hence improve the precision of our estimate of the parameter. Notice that in the coherent state case, the SNR scales with the input power Pin ∝ |𝛼|2 = n. This scaling is sometimes referred to as the quantum noise limit (QNL) for sensors. However, if we use squeezed light at the input, we ﬁnd that the quantum noise term is reduced, and, under ideal conditions, for the same parameter ℘, the signal now scales proportional to the optical power squared, 2 . This limit is known as the Heisenberg limit in the sensitivity. This can be Pin seen in the above example by setting ℘o ≈ 1 and assuming V 1A ≪ 1. We also require that the number of photons arising from the squeezing, ns ≈ 1∕(4V 1A ), is equal to the number of photons from the coherent amplitude, na = |𝛼|2 , such that na = ns = 1∕2 n ∝ 1∕2 Pin . Substituting these conditions into Eq. (10.6) gives 2

K 2 n 𝛿℘2o (10.6) 4 thus exhibiting the expected Heisenberg scaling. One test for the operation below the quantum limit is to measure the SNR as a function of the optical power Pin SNR ≈

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at the input, which would be proportional to Pin for the quantum noise lim2 for the quanited case and approach, but not exceed the proportionality to Pin tum enhanced case. The fact that this is the best possible scaling can be derived rigorously using the concept of information theory, Fisher information, and the Cramer–Rao Bound as will be discussed in Section 10.1.3. 10.1.2

Practical Examples of Sensors

In all of these cases, the sensor is measuring and tracking small changes in a property with high sensitivity. To reach its full potential, the sensor should be ﬁrst optimized to operate at the QNL, commonly referred to as the shot noise limit, and then the sensitivity can be improved by increasing the laser power. Once this is no longer possible, most likely through the onset of non-linear eﬀects, squeezing can provide an additional improvement. At the moment typically by a factor of 2 (or 3 dB) in the SNR. Potentially, this could be expanded to a factor 10 (or 10 dB) but at a tremendous cost in reﬁnement since even the smallest optical losses have to be avoided. Here the GW detectors stand out as the shining future example [1], since already all other improvements have been made and the scientiﬁc gain of even a factor of 2 improvement allows the detection in 3D from an eight times larger volume, which is very important for cosmology. The most notable examples of quantum enhanced sensors are described in more detail in the following. It has been shown that absorption spectra can be recorded with improved SNR [2]. Interferometers can be improved, as ﬁrst predicted by C.M. Caves et al. [3] and ﬁrst implemented for table top instruments [4, 5]. It is now a routine component of some of the large-scale GW detectors [6], as will be discussed in more detail in Section 10.3. Squeezed light improves the measurement of the polarization of a laser beam [7] and any eﬀect that introduces changes to the polarization, the macroscopic spin of a sample of atoms [8], and the displacement or position of a laser beam [9]. Squeezed light enhances the tracking of particles within living cells [10]. It can be used for the improvement of the uncertainty in timing pulses and frequency combs [11, 12]. Other technical applications will follow. Very early in the game, the concept was applied to the detection of absorption, see Figure 10.2(a,b). Amplitude squeezed light was used, as generated with quiet diode lasers (see page 213 or, from a second harmonic generator (SHG) experiment, Section 9.4.3. The experiments were identical to conventional absorption sensors or absorption spectroscopy with the only change being that the input beam is squeezed. The absorption has to be modulated at a frequency Ωem within the bandwidth of the squeezing spectrum. Care has to be taken that the overall eﬃciency of the apparatus and the detectors is very high; otherwise the degree of squeezing and the improvement in SNR rapidly disappears. One of the ﬁrst demonstrations was made by E.S. Polzik et al. [2], who demonstrated an improvement of 2 dB for the SNR for saturated absorption spectroscopy of a Cs vapour. They used an optical parametric oscillation (OPO) as a source for squeezed light, and the major challenge was to make this light source tunable. Another ﬁne demonstration was carried out by C.D. Nabors and R.M. Shelby [13] who showed that a modulation can be sensed with improved SNR using

10.1 Quantum Enhanced Sensors

SHG

Intensity squeezed laser

Absorber

Laser

OPO

Detector

Absorber

(a)

(b)

SHG

Laser PBS

Modulator

OPO Absorber

(c)

Intensity squeezed laser

Moving scattering particles

(d)

Figure 10.2 Diﬀerent schemes for using squeezed light for improved SNR in optical sensors. (a) Absorption using amplitude squeezed light. (b) Absorption using tunable OPO. Source: After E.S. Polzik et al. [2]. (c) Measuring DC absorption using twin photon beams. Source: After Hai et al. [14]. (d) Detecting scattered laser light for motion detection.

the bright squeezed light from an above threshold doubly resonant monolithic OPO. One of the diﬃculties is to modulate the absorption at a suﬃciently high frequency. The experiments by the group in Shanxi University [14] and in Paris and now many others use an elegant trick: twin photon beams generated by an above threshold OPO. One beam is used as a reference. The other is modulated at Ωmod and split into another pair of beams, probe 1 and probe 2, and only probe 1 is sent through the sample (see Figure 10.2c). By ﬁrst forming the sum of the two probe photocurrents and then the diﬀerence between the total probe and the reference, a current is generated that contains an AC signal at Ωmod , which is proportional to the strength of the absorption. In this way a slowly varying absorption can be observed with high sensitivity. A similar approach can be used to detect the intensity of scattered light (see Figure10.2d). It was shown that the light scattered by fast-moving smoke particles can be detected using an optical homodyne detectors and tracking the frequency sidebands. The Doppler shift provided the necessary frequency shift, and the spectrum was used to detect the density and the velocity distribution of the sample. This technique is known as Doppler velocimetry. Using amplitude squeezed light from a diode laser, Y.Q. Li et al. were able to demonstrate an increase in the SNR of 1 dB compared to an experiment with classical light [15]. More recently this was used, for example, for tracking particles inside biological samples with quantum enhancement (see page 418).

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10.1.3

Ultimate Sensing Limits

In Section 10.1.1 we found measurement precision limits from SNRs given a particular detection strategy, i.e. direct detection. Here we discuss a more general approach that ﬁnds the ultimate precision limit for any detection strategy. We also discuss recent suggestions for non-linear quantum sensors. As discussed earlier, in general, optical sensors use some interferometric arrangement as an actuator such that changes in a parameter of interest lead to a modulation of the intensity of a probe ﬁeld. Hence the beamsplitter interaction described by Eq. (10.2) can be considered a generic model of a quantum optical sensor, with diﬀerent types of sensors being characterized by diﬀerent functional dependences of 𝜖(℘) on the parameter of interest, ℘. Previously we used a simple signal-to-noise approach to analyse the precision of the parameter estimation. More generally the lower bound on the variance of an unbiased estimator, maximized over all possible measurements on the output probe, can be found by calculating the quantum Fisher information, H(𝜖). The variance of the estimator, V (℘), is bounded by the Cramer–Rao inequality [16], generalized to the quantum case [17]: V (℘) ≥ (

1 )2 𝜕𝜖 H(𝜖)N 𝜕℘

(10.7)

where N is the number of measurements taken. The Fisher information is the second moment of the classical logarithmic derivative of the likelihood function. The quantum Fisher information is further optimized to coincide with the best measurement strategy allowed by quantum mechanics. The quantum Fisher information can be evaluated via several diﬀerent methods. One often convenient ̂ 𝝈), ̂ between the two density operators 𝝆̂ method is via the quantum ﬁdelity, (𝝆, ̂ where and 𝝈, | [√√ √ ]||2 | = |Tr 𝝆̂ 𝝈̂ 𝝆̂ | | | | |

(10.8)

The quantum ﬁdelity is a well-known quantiﬁcation for the similarity of quantum states. For pure states, the ﬁdelity reduces to the overlap between states: = |⟨𝜓|𝜓 ′ ⟩|2 . The ﬁdelity can be used to calculate the quantum Fisher information via √ 8(1 − (𝝆̂ 𝜖 , 𝝆̂ 𝜖+𝛿𝜖 )) H(𝜖) = lim (10.9) 𝛿𝜖→0 𝛿𝜖 2 where 𝝆̂ 𝜖 is the output state of the probe after the interaction with the beamsplitter, whilst 𝝆̂ 𝜖+𝛿𝜖 is the output state after the interaction given a small perturbation, 𝛿𝜖, to 𝜖. For example, a coherent state probe remains a pure state after transmission through the beamsplitter, so the ﬁdelity is easily calculated as ( 2 2 ) √ √ |𝛼| 𝛿𝜖 ) (𝜌𝜖 , 𝜌𝜖+𝛿𝜖 ) = |⟨ 𝜖𝛼| 𝜖 + 𝛿𝜖𝛼⟩|2 = exp − (10.10) 8𝜖

10.1 Quantum Enhanced Sensors

Substituting Eq. (10.10) into Eq. (10.9) and hence into Eq. (10.7) gives the precision bound for a coherent probe: V (℘) ≥ (

𝜕𝜖 𝜕℘

1 )2 4|𝛼|2 N

(10.11)

Given that the variance of the estimator is inversely proportional to the SNR, we can note that Eq. (10.11) has the same 1∕n scaling as obtained from the signal-to-noise analysis. As we observed earlier a squeezed probe can do better, 2 leading ideally to a 1∕n scaling [18]. By using the general expression for ﬁdelity between Gaussian states (see Section 13.6.2), we can arrive at the same conclusion from the Fisher information analysis. Under more practical conditions, the addition of squeezing to a coherent probe can give a constant improvement factor as can also be shown from the Fisher information analysis. Determining the measurement basis that will saturate the quantum Cramer–Rao bound is more involved, but for Gaussian states it has been shown in general to be a displaced, squeezed number basis [19]. Recently it has been observed that if there is a strong non-linear coupling to 2 the probe, then variance scalings better than the 1∕n can be achieved [20], and an experimental example has been demonstrated [21]. For example, consider the following non-linear beamsplitter relationship: â o =

† † † † 1 1 (1 + ei𝜒∕2(̂a −̂v )(̂a−̂v) ) â + (1 − ei𝜒∕2(̂a −̂v )(̂a−̂v) ) v̂ 2 2

(10.12)

where, as in Eq. (10.1), â represents the probe mode and v̂ a vacuum mode. Here we wish to estimate the parameter 𝜒; however the transmission depends both on 𝜒 and the intensity of the probe in a non-linear way. A signal-to-noise analysis can be carried out by assuming that the ﬂuctuations and the signal are small compared with the average power such that the input mode can be written as ̂ with 𝛼 ≫ 1 and 𝜒 = 𝜒o + 𝛿𝜒 with 𝛿𝜒 ≪ 1 and linearizing the ﬂuctuâ = 𝛼 + 𝜹a ations. This leads to an approximate power spectrum for the ﬂuctuations of the output ﬁeld of 𝜒o 𝜒o 2 2 1 1 Vo = V𝜒 |𝛼|6 + Va |1 + ei 2 |𝛼| (1 + i𝜒o |𝛼|2 )|2 + |1 − ei 2 |𝛼| (1 + i𝜒o |𝛼|2 )|2 4 4 (10.13)

Now suppose we are looking for ﬂuctuations around the point 𝜒o = 0. Under these conditions, with a coherent probe, we ﬁnd SNR = V𝜒 |𝛼|6 ∕16, and so the 3 estimation precision scales as 1∕n , exceeding the Heisenberg scaling. These claims have generated some controversy. In particular it should be noted that: 1. The enhanced precision typically only applies locally, i.e. around the operating point (𝜒o = 0 in our example), and the range in which an enhancement is obtained may be too narrow to be useful [22]. 2. In order to have practical applications, an eﬃcient way to map some physical parameter of interest onto a non-linear parameter, such as 𝜒2 or 𝜒3 , is required. This is a technical challenge.

383

384

10 Applications of Quantum Light

Nevertheless, such non-linear devices are an interesting ﬁeld of future research in quantum optical sensors. 10.1.4

Adaptive Phase Estimation

Finally, we note that so far we have only discussed sensors that are measuring small variations starting with a known value. That means we do not need to estimate the full value of the parameter ℘. All we need to record in each time step is the change 𝛿℘(t). In contrast it is possible to consider sensors that estimate parameter values for which we have no previous knowledge, and thus we have to start in each time interval from the same point, e.g. ℘ = 0. For example, in an interferometric sensor we normally consider situations where we track the small changes in the phase 𝛿𝜙(t) of the light and compare it to the value of 𝜙(t − 𝜏) at the previous time step. In contrast one could start with no knowledge of 𝜙 in the range 0 to 2𝜋 in each time step. This task requires an adaptive approach and more resources and photons per time interval than a sensor, which constantly tracks small changes of the phase. A considerable amount of theory work has been done in this ﬁeld, in particular by H. Wiseman, G. Milburn, and H. Mabuchi and their respective teams [23, 24]. In more recent times experiments by a combined Tokyo and Canberra team, led by A. Furusawa and E. Huntington, have explored adaptive optical phase estimation using time-symmetric quantum smoothing [25] and combined with squeezed light [26], and they have veriﬁed the theoretical predictions. It is important to establish the correct quantum limit and minimum number of quantum resources, which are required to achieve the phase estimation. This team later studied the quantum-limited estimation of a mirror position. A detailed study was made to optimize which data should be used in post-processing. The authors concluded that using only data recorded before the measurement was not as eﬃcient as using the same number of data points recorded before and after the timing of the signal [27]. These studies are the beginning of more complex analysis and the design of devices that can be used in more sophisticated optical control situations at and below the QNL. This work has been successful with squeezed light and also with the corresponding experiments using single photons and the corresponding phase estimation algorithms. See, for example, the work at Griﬃths University by G.J. Pryde and co-workers [28], which summarizes the approaches to measurements and estimation and identiﬁes which are the most appropriate quantum states and how best to use them.

10.2 Optical Communication Optical communication has been one of the motivations for the development of quantum optics technology. In the last three decades, there has been a tremendous advance in optical communication technology: low loss ﬁbres, high quantum eﬃciency photodiodes, diode lasers, and laser ampliﬁers have all improved the system performance. The speed has increased relentlessly over the decades by a factor of 10 about every 4 years, as seen in Figure 10.3.

10.2 Optical Communication

1017 y ver 0e 1 × :

1016 1015 Capacity (bit/s)

1014

Capacity limit for current technology

nd Tre

1013 1012

s ear 4y

Space division multiplexing

WDM

1011 High spectral efficiency coding

1010 109 108 107 106 105 1980

EDFA Improved transmission fibres 1990

2000 Year

2010

Figure 10.3 Communication speed and multiplexing. Source: From Richardson et al. [29].

In optical systems the information is encoded on one or more channels or modes of light that are propagated via free space or more commonly via an optical ﬁbre and decoded at the receiver. The attraction of optical systems is the wide modulation bandwidth and the ability to multiplex the wavelengths, which means carrier frequencies, the polarization, and, very soon, spatial modes in the ﬁbre. The communication speed can be measured by bits per second (bits/s), and an overview of the communication speed can be seen in Figure 10.3. We have now systems approaching a speed of 1014 bit/s, which was hard to imagine only a few years ago, and this requires the simultaneous use of many advanced techniques. One trick is multiplexing, which means in our language that information is encoded in a series of orthogonal modes, with minimum crosstalk between the modes. Multiplexing technology is currently one of the major topics of international research and development. Combined with all the tricks used in coherent detection, using a known detection phase reference via a local oscillator, and homodyne detection, the information density and thus also communication bandwidth can be increased further. The technology of the fastest communication systems and quantum optics experiments is remarkably similar [29, 30], and we ﬁnd increasingly an overlap and a demand from the communication industry and engineers. It can be foreseen that we will soon reach the situation where the quantum properties of light become an important limit for the performance of optical communication systems. At that stage squeezed light might play a role in overcoming these limits and in enhancing the performance. In this chapter we want to estimate the size of such possible improvements. In principle, a communication system contains the components shown in Figure 10.4. Diﬀerent encoding and detection schemes are in use, with the simplest and most commonly used one direct incoherent modulation. In this case the transmitter is a diode laser, the

385

386

10 Applications of Quantum Light

Data in

Laser

Transmission line Demodulator

Fibre laser amplifier

Modulator

Data out

Coupler

Figure 10.4 Components of an optical communication system without coherent detection.

information is encoded by an amplitude modulator, and the receiver is a simple photodetector. More advanced systems use coherent schemes such as phase shift keying (PSK). This requires a phase modulator in the transmitter and a homodyne, or heterodyne, detector with a local oscillator beam that is phase-locked to the input mode. These systems can operate within a few decibels of the QNL. The question of how non-classical light can improve the performance of the system was the motivation for the early work by H. Takahashi [31] and H.P. Yuen and J.H. Shapiro [32]. The techniques for the encoding, propagating, and decoding of information on a quantum system were reviewed by Y. Yamamoto and H.A. Haus [33] and by C.M. Caves and P.D. Drummond [34]. Non-classical light can be used in several ways to improve components of the system: (i) the laser can produce sub-Poissonian or quadrature squeezed light and thus possibly allow a better SNR. This is particularly attractive and practical when diode lasers are used. (ii) The phase-insensitive laser ampliﬁer can be replaced by a phase-sensitive parametric ampliﬁer. (iii) The beamsplitter, or coupler, can be equipped with a squeezed vacuum input that reduces the noise introduced at this point or alternatively a quantum non-demolition (QND) device can be used [35]. In all cases the same detection system can be used. The overall improvement can be illustrated by considering the amount of information that can be encoded on a mode of light with given maximum optical power. For an incoherent system, where only intensity information is processed, this information content can be estimated by evaluating the number of states with X2

α

δX1

X1

X2

δX1

X1

Figure 10.5 Illustration of the number of distinguishable states for incoherent detection. Top, coherent states; below, squeezed states.

10.2 Optical Communication

fGW

Two rotating neutron stars or similar

Time

Figure 10.6 A typical source for gravitational waves and its eﬀect on space and time. The eﬀects observed by the end of 2017 include colliding black holes and spinning neutron stars.

diﬀerent intensities that can be distinguished. This is illustrated in Figure 10.5, where the sequence of distinguishable coherent states are shown, all with one phase = zero. Each diﬀerent state is represented by an uncertainty area, symmetric circles in the case of coherent states. An incoherent system recognizes diﬀerent intensities; the phase information is ignored. With sub-Poissonian light the uncertainty in the amplitude is smaller, and for given maximum power a larger number of states can be distinguished with the same SNR. Hence more information can be encoded Figure 10.6. A quantitative ﬁgure of merit for a communication system is the channel capacity, a concept that describes the maximum amount of information that can be transmitted based on statistical arguments. The Shannon capacity [36] of a communication channel with Gaussian noise of power (variance) N and Gaussian distributed signal power S operating at the bandwidth limit is ] [ S 1 (10.14) C = log2 1 + 2 N where C is in units of bits per symbol. Equation (10.14) can be used to calculate the channel capacities of quantum states with Gaussian probability distributions such as coherent states and squeezed states. Consider ﬁrst a signal composed of a Gaussian distribution of coherent state amplitudes all with the same quadrature angle. The signal power Vs is given by the variance of the distribution. The noise is given by the intrinsic quantum noise of the coherent states, Vn = 1. Because the quadrature angle of the signal is known, homodyne detection can in principle detect the signal without further penalty, and thus the measured signal to noise is S∕N = Vs ∕Vn = Vs . In general the average photon number per bandwidth per second of a light beam is given by 1 + 1 (V + V − ) − (10.15) 4 2 where V + (V − ) are the variances of the maximum (minimum) quadrature projections of the noise ellipse of the state. These projections are orthogonal quadratures, such as amplitude and phase, and obey the uncertainty principle V + V − ≥ 1. In the above example one quadrature is made up of signal plus n=

387

388

10 Applications of Quantum Light

quantum noise such that V + = Vs + 1, whilst the orthogonal quadrature is just quantum noise, so V − = 1. Hence n = 1∕4Vs , and so the channel capacity of a coherent state with single quadrature encoding and homodyne detection is [√ ] Cc = log2 1 + 4n (10.16) Establishing in an experiment that a particular optical mode has, this capacity would involve (i) measuring the quadrature amplitude variances of the beam, V + and V − , (ii) calibrating Alice’s signal variance, and (iii) measuring Bob’s signal to noise. If these measurement agreed with the theoretical conditions above, then Shannon’s theorem tells us that an encoding scheme exists that could realize the channel capacity of Eq. (10.16). An example of such an encoding is given in [37]. For photon numbers n > 2 improved channel capacity can be obtained by encoding symmetrically on both quadratures and detecting both quadratures simultaneously using heterodyne detection or dual homodyne detection. Because of the non-commutation of orthogonal quadratures, there is a penalty for their simultaneous detection that reduces the signal to noise of each quadrature to S∕N = 1∕2Vs . Also because there is signal on both quadratures, the average photon number of the beam is now n = 1∕2Vs . On the other hand the total channel capacity will now be the sum of the two independent channels carried by the two quadratures. Thus the channel capacity for a coherent state with dual quadrature encoding and heterodyne detection is [ ] [ ] S− 1 S+ 1 + log2 1 + Cch = log2 1 + 2 N 2 N (10.17) = log2 [1 + n] which exceeds that of the homodyne technique (Eq. (10.16)) for n > 2. The above channel capacities are the best achievable if we restrict ourselves to a semi-classical treatment of light. However the channel capacity of the homodyne technique can be improved by the use of non-classical squeezed light. With squeezed light the noise variance of the encoded quadrature can be reduced such that Vne < 1, whilst the noise of the unencoded quadrature is increased such that Vnu ≥ 1∕Vne . As a result the signal to noise is improved to S∕N = Vs ∕Vne , whilst the photon number is now given by Eq. (10.15) but with V + = Vs + Vne and V − = 1∕Vne where a pure (i.e. minimum uncertainty) squeezed state has been 2 assumed. Maximizing the signal to noise for ﬁxed n leads to S∕N = 4(n + n ) for a squeezed quadrature variance of Vne,opt = 1∕(1 + 2n). Hence the channel capacity for a squeezed beam with homodyne detection is ] [ Csh = log2 1 + 2n (10.18) which exceeds both coherent homodyne and heterodyne for all values of n. Although squeezing gives an improvement in channel capacity, this improvement is rapidly washed out by nonidealities in the system such as propagation loss and other ineﬃciencies. It still seems some way oﬀ before squeezing might be considered a viable option for increasing channel capacity. In principle, a ﬁnal improvement in channel capacity can be obtained by allowing non-Gaussian states. The absolute maximum channel capacity for a single mode is given by

10.3 Gravitational Wave Detection

the Holevo bound and can be realized by encoding in a maximum entropy ensemble of Fock states and using photon number detection. This ultimate channel capacity is CFock = (1 + n)log2 [(1 + n)] − nlog2 [n]

(10.19)

which is the maximal channel capacity at all values of n. Quantum optics can describe the ultimate limits of data communication, and we can ﬁnd ways to approach this limit. In 2018 such quantum limits ca

A Guide to Experiments in Quantum Optics Hans-A. Bachor and Timothy C. Ralph

Third Edition

Authors Hans-A. Bachor

The Australian Nat. University Research School of Physics Science Road Canberra ACT 0200 Australia

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

Dr. Timothy C. Ralph

University of Queensland Department of Physics St. Lucia QLD 4072 Australia

Library of Congress Card No.:

Cover

A catalogue record for this book is available from the British Library.

The artwork was prepared by B. Hage and is based on an experimental demonstration of photon number states published in: Chrzanowski H.M., Assad S.M., Bernu J., and Hage B. et al. (2013). Reconstruction of Photon Number Conditioned States Using Phase Randomized Homodyne Measurements, J. Phys. B At. Mol. Opt. Phys. 46: 104009.

applied for British Library Cataloguing-in-Publication Data

Bibliographic information published by the Deutsche Nationalbibliothek

The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliograﬁe; detailed bibliographic data are available on the Internet at . © 2019 Wiley-VCH Verlag GmbH & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microﬁlm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not speciﬁcally marked as such, are not to be considered unprotected by law. Print ISBN: 978-3-527-41193-1 ePDF ISBN: 978-3-527-68394-9 ePub ISBN: 978-3-527-68393-2 oBook ISBN: 978-3-527-69580-5

Adam-Design, Weinheim, Germany Typesetting SPi Global, Chennai, India Cover Design

Printing and Binding

Printed on acid-free paper 10 9 8 7 6 5 4 3 2 1

v

Contents Preface xv Acknowledgments xix 1 1.1 1.2 1.3 1.4 1.5

2

2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6 2.1.7 2.2 2.2.1 2.2.2 2.2.2.1 2.2.3 2.3 2.3.1 2.3.1.1 2.3.2 2.3.3 2.3.3.1 2.4 2.5

Introduction 1

Optics in Modern Life 1 The Origin and Progress of Quantum Optics 3 Motivation Through Simple and Direct Teaching Experiments 7 Consequences of Photon Correlations 12 How to Use This Guide 14 References 16 Classical Models of Light 19 Classical Waves 20 Mathematical Description of Waves 20 The Gaussian Beam 21 Quadrature Amplitudes 24 Field Energy, Intensity, and Power 25 A Classical Mode of Light 26 Light Carries Information 28 Modulations 30 Optical Modes and Degrees of Freedom 32 Lasers with Single and Multiple Modes 32 Polarization 33 Poincaré Sphere and Stokes Vectors 35 Multimode Systems 36 Statistical Properties of Classical Light 37 The Origin of Fluctuations 37 Gaussian Noise Approximation 38 Noise Spectra 39 Coherence 40 Correlation Functions 44 An Example: Light from a Chaotic Source as the Idealized Classical Case 46 Spatial Information and Imaging 50

vi

Contents

2.5.1 2.5.2 2.5.3 2.5.4 2.5.5 2.5.6 2.5.7 2.5.8 2.5.9 2.5.10 2.6

State-of-the-Art Imaging 50 Classical Imaging 52 Image Detection 55 Scanning 56 Quantifying Noise and Contrast 58 Coincidence Imaging 59 Imaging with Coherent Light 60 Image Reconstruction with Structured Illumination 60 Image Analysis and Modes 61 Detection Modes and Displacement 61 Summary 62 References 63 Further Reading 64

3

65 Detecting Light 65 The Concept of Photons 68 Light from a Thermal Source 70 Interference Experiments 73 Modelling Single-Photon Experiments 78 Polarization of a Single Photon 79 Some Mathematics 80 Polarization States 81 The Single-Photon Interferometer 83 Intensity Correlation, Bunching, and Anti-bunching 84 Observing Photons in Cavities 88 Summary 90 References 90 Further Reading 92

3.1 3.2 3.3 3.4 3.5 3.5.1 3.5.1.1 3.5.2 3.5.3 3.6 3.7 3.8

Photons: The Motivation to Go Beyond Classical Optics

4

Quantum Models of Light 93

4.1 4.1.1 4.1.2 4.1.3 4.1.4 4.2 4.2.1 4.2.2 4.2.3 4.3 4.3.1 4.3.2 4.3.3 4.3.4 4.3.4.1

Quantization of Light 93 Some General Comments on Quantum Mechanics 93 Quantization of Cavity Modes 94 Quantized Energy 95 The Creation and Annihilation Operators 97 Quantum States of Light 97 Number or Fock States 97 Coherent States 99 Mixed States 101 Quantum Optical Representations 102 Quadrature Amplitude Operators 102 Probability and Quasi-probability Distributions 104 Photon Number Distributions 108 Covariance Matrix 111 Summary of Diﬀerent Representations of Quantum States and Quantum Noise 112

Contents

4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.5 4.5.1 4.5.2 4.5.3 4.5.4 4.5.5 4.5.6 4.5.7 4.6 4.6.1 4.6.2 4.6.3 4.7 4.7.1 4.7.2 4.7.3 4.7.4

Propagation and Detection of Quantum Optical Fields 113 Quantum Optical Modes in Free Space 114 Propagation in Quantum Optics 115 Detection in Quantum Optics 117 An Example: The Beamsplitter 118 Quantum Transfer Functions 120 A Linearized Quantum Noise Description 121 An Example: The Propagating Coherent State 123 Real Laser Beams 123 The Transfer of Operators, Signals, and Noise 124 Sideband Modes as Quantum States 126 Another Example: A Coherent State Pulse Through a Frequency Filter 129 Transformation of the Covariance Matrix 130 Quantum Correlations 131 Photon Correlations 131 Quadrature Correlations 132 Two-Mode Covariance Matrix 133 Summary 134 The Photon Number Basis 134 Quadrature Representations 135 Quantum Operators 135 The Quantum Noise Limit 136 References 136 Further Reading 137

5

Basic Optical Components 139

5.1 5.1.1 5.1.1.1 5.1.2 5.1.3 5.1.4 5.1.5 5.1.5.1 5.1.6

Beamsplitters 140 Classical Description of a Beamsplitter 140 Polarization Properties of Beamsplitters 142 The Beamsplitter in the Quantum Operator Model 143 The Beamsplitter with Single Photons 144 The Beamsplitter and the Photon Statistics 146 The Beamsplitter with Coherent States 149 Transfer Function for a Beamsplitter 149 Comparison Between a Beamsplitter and a Classical Current Junction 151 The Beamsplitter as a Model of Loss 152 Interferometers 153 Classical Description of an Interferometer 154 Quantum Model of the Interferometer 155 The Single-Photon Interferometer 156 Transfer of Intensity Noise Through the Interferometer 156 Sensitivity Limit of an Interferometer 157 Eﬀect of Mode Mismatch on an Interferometer 160 Optical Cavities 162 Classical Description of a Linear Cavity 164

5.1.7 5.2 5.2.1 5.2.2 5.2.3 5.2.4 5.2.5 5.2.6 5.3 5.3.1

vii

viii

Contents

5.3.2 5.3.3 5.3.4 5.3.4.1 5.3.4.2 5.3.4.3 5.3.5 5.3.6 5.3.7 5.3.8 5.3.9 5.3.10 5.4 5.4.1 5.4.2 5.4.3 5.4.4 5.4.5 5.4.5.1 5.4.6 5.4.7 5.4.8

The Special Case of High Reﬂectivities 169 The Phase Response 170 Spatial Properties of Cavities 172 Mode Matching 172 Polarization 174 Tunable Mirrors 175 Equations of Motion for the Cavity Mode 175 The Quantum Equations of Motion for a Cavity 176 The Propagation of Fluctuations Through the Cavity 177 Single Photons Through a Cavity 180 Multimode Cavities 181 Engineering Beamsplitters, Interferometers, and Resonators 182 Other Optical Components 184 Lenses 184 Holograms and Metasurfaces 185 Crystals and Polarizers 187 Optical Fibres and Waveguides 188 Modulators 189 Phase and Amplitude Modulators 191 Spatial Light Modulators 193 Optical Noise Sources 195 Non-linear Processes 195 References 196

6

Lasers and Ampliﬁers 199

6.1 6.1.1 6.1.2 6.1.3 6.1.4 6.1.4.1 6.1.4.2 6.1.4.3 6.1.4.4 6.1.4.5 6.1.5 6.1.6 6.2 6.3 6.3.1 6.3.2 6.3.3 6.3.3.1 6.3.4 6.4 6.4.1 6.4.2

The Laser Concept 199 Technical Speciﬁcations of a Laser 201 Rate Equations 203 Quantum Model of a Laser 207 Examples of Lasers 209 Classes of Lasers 209 Dye Lasers and Argon Ion Lasers 209 The CW Nd:YAG Laser 210 Diode Lasers 213 Limits of the Single-Mode Approximation in Diode Lasers Laser Phase Noise 214 Pulsed Lasers 215 Ampliﬁcation of Optical Signals 215 Parametric Ampliﬁers and Oscillators 218 The Second-Order Non-linearity 219 Parametric Ampliﬁcation 220 Optical Parametric Oscillator 221 Noise Spectrum of the Parametric Oscillator 222 Pair Production 223 Measurement-Based Ampliﬁers 224 Deterministic Measurement-Based Ampliﬁers 225 Heralded Measurement-Based Ampliﬁers 228

213

Contents

6.5

Summary 230 References 231

7

Photon Generation and Detection 233

7.1 7.1.1 7.2 7.2.1 7.2.1.1 7.2.1.2 7.2.1.3 7.2.1.4 7.2.2 7.3 7.3.1 7.3.1.1 7.3.1.2 7.3.1.3 7.3.1.4 7.3.2 7.3.2.1 7.3.2.2 7.3.2.3 7.3.3 7.3.4 7.3.4.1 7.3.4.2 7.3.4.3 7.3.4.4 7.3.4.5 7.3.4.6 7.3.4.7 7.4

Photon Sources 236 Deterministic Photon Sources 239 Photon Detection 240 Detecting Individual Photons 240 Photochemical Detectors 241 Photoelectric Detectors 241 Photo-thermal Detectors 243 Multipixel and Imaging Devices 243 Recording Electrical Signals from Individual Photons 245 Generating, Detecting, and Analysing Photocurrents 247 Properties of Photocurrents 247 Beat Measurements 247 Intensity Noise and the Shot Noise Level 248 Quantum Eﬃciency 249 Photodetector Materials 250 Generating Photocurrents 251 Photodiodes and Detector Circuit 251 Ampliﬁers and Electronic Noise 252 Detector Saturation 254 Recording of Photocurrents 255 Spectral Analysis of Photocurrents 257 Digital Fourier Transform 257 Analogue Fourier Transform 258 From Optical Sidebands to the Current Spectrum 258 The Operation of an Electronic Spectrum Analyser 259 Detecting Signal and Noise Independently 260 The Decibel Scale 261 Adding Electronic AC Signals 262 Imaging with Photons 263 References 264 Further Reading 267

8

Quantum Noise: Basic Measurements and Techniques 269

8.1 8.1.1 8.1.1.1 8.1.2 8.1.3 8.1.4 8.1.4.1 8.1.5 8.1.5.1 8.2

Detection and Calibration of Quantum Noise 269 Direct Detection and Calibration 269 White Light Calibration 273 Balanced Detection 273 Detection of Intensity Modulation and SNR 275 Homodyne Detection 275 The Homodyne Detector for Classical Waves 275 Heterodyne Detection 279 Measuring Other Properties 280 Intensity Noise 281

ix

x

Contents

8.2.1 8.3 8.3.1 8.3.2 8.3.2.1 8.4 8.4.1 8.4.2 8.4.3 8.4.4 8.4.5 8.5

Laser Noise 281 The Intensity Noise Eater 282 Classical Intensity Control 282 Quantum Noise Control 285 Practical Consequences 289 Frequency Stabilization and Locking of Cavities 290 Pound–Drever–Hall Locking 292 Tilt Locking 293 The PID Controller 294 How to Mount a Mirror 295 The Extremes of Mirror Suspension: GW Detectors 296 Injection Locking 296 References 299

9

Squeezed Light 303

9.1 9.1.1 9.1.1.1 9.1.1.2 9.1.2 9.1.2.1 9.2 9.2.1 9.2.2 9.2.3 9.3 9.3.1 9.3.2 9.3.3 9.3.4 9.3.5 9.4 9.4.1 9.4.2 9.4.3 9.4.4 9.4.4.1 9.4.4.2 9.4.4.3 9.4.4.4 9.4.4.5 9.5 9.5.1 9.5.2 9.5.3 9.5.4 9.5.5

The Concept of Squeezing 303 Tools for Squeezing: Two Simple Examples 303 The Kerr Eﬀect 304 Four-Wave Mixing 307 Properties of Squeezed States 310 What Are the Uses of These Various Types of Squeezed Light? 312 Quantum Model of Squeezed States 314 The Formal Deﬁnition of a Squeezed State 314 The Generation of Squeezed States 317 Squeezing as Correlations Between Noise Sidebands 319 Detecting Squeezed Light 322 Detecting Amplitude Squeezed Light 322 Detecting Quadrature Squeezed Light 322 Using a Cavity to Measure Quadrature Squeezing 324 Summary of Diﬀerent Representations of Squeezed States 325 Propagation of Squeezed Light 325 Early Demonstrations of Squeezed Light 330 Four Wave Mixing 330 Optical Parametric Processes 333 Second Harmonic Generation 339 The Kerr Eﬀect 343 The Response of the Kerr Medium 343 Optimizing the Kerr Eﬀect 345 Fibre Kerr Squeezing 346 Atomic Kerr Squeezing 348 Atomic Polarization Self-Rotation 349 Pulsed Squeezing 349 Quantum Noise of Optical Pulses 349 Pulsed Squeezing Experiments with Kerr Media 352 Pulsed SHG and OPO Experiments 353 Soliton Squeezing 354 Spectral Filtering 355

Contents

9.5.6 9.6 9.7 9.8 9.9 9.9.1 9.9.2 9.9.3 9.10

Non-linear Interferometers 356 Amplitude Squeezed Light from Diode Lasers 358 Quantum State Tomography 360 State of the Art of CW Squeezing 363 Squeezing of Multiple Modes 365 Twin-Photon Beams 365 Polarization Squeezing 367 Degenerate Multimode Squeezers 368 Summary: Quantum Limits and Enhancement 370 References 371 Further Reading 376

10

Applications of Quantum Light 377

10.1 10.1.1 10.1.2 10.1.3 10.1.4 10.2 10.3 10.3.1 10.3.1.1 10.3.2 10.3.2.1 10.3.2.2 10.3.2.3 10.3.3 10.3.3.1 10.3.4 10.3.4.1 10.4 10.4.1 10.4.2 10.5 10.5.1 10.6

Quantum Enhanced Sensors 377 Coherent Sensors and Sensitivity Scaling 377 Practical Examples of Sensors 380 Ultimate Sensing Limits 382 Adaptive Phase Estimation 384 Optical Communication 384 Gravitational Wave Detection 389 The Origin and Properties of GW 389 Concept and Design of an Optical GW Detector 392 Quantum Properties of the Ideal Interferometer 393 Conﬁgurations of Interferometers 396 Recycling 397 Modulation Techniques 398 The Sensitivity of GW Observatories 400 Enhancement Below the SQL 402 Interferometry with Squeezed Light 405 Quantum Enhancement Beyond the SQL 410 Quantum Enhanced Imaging 411 Imaging with Photons on Demand 411 Quantum Enhanced Coincidence Imaging 412 Multimode Squeezing Enhancing Sensors 414 Spatial Multimode Squeezing 414 Summary and Outlook 419 References 419

11

QND 425

11.1 11.2 11.3 11.4 11.4.1

QND Measurements of Quadrature Amplitudes 425 Classiﬁcation of QND Measurements 427 Experimental Results 430 Single-Photon QND 432 Measurement-Based QND 434 References 437

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12

Fundamental Tests of Quantum Mechanics 441

12.1 12.2 12.3 12.3.1 12.3.2 12.3.2.1 12.3.2.2 12.3.3 12.3.3.1 12.3.3.2 12.4

Wave–Particle Duality 441 Indistinguishability 446 Non-locality 453 Einstein–Podolsky–Rosen Paradox 453 Characterization of Entangled Beams via Homodyne Detection 458 Logarithmic Negativity and Two-Mode Squeezing 459 Entanglement of Formation 460 Bell Inequalities 461 Long-Distance Bell Inequality Violations 466 Loophole-Free Bell Inequality Violations 466 Summary 468 References 468

13

Quantum Information 473

13.1 13.1.1 13.2 13.3 13.3.1 13.3.2 13.4 13.5 13.5.1 13.5.2 13.5.3 13.6 13.6.1 13.6.2 13.6.3 13.6.4 13.7 13.7.1 13.7.1.1 13.7.1.2 13.7.1.3 13.7.2 13.7.2.1 13.7.2.2 13.7.3 13.7.3.1 13.7.3.2 13.7.4 13.8

Photons as Qubits 473 Other Quantum Encodings 475 Post-selection and Coincidence Counting 475 True Single-Photon Sources 477 Heralded Single Photons 477 Single Photons on Demand 480 Characterizing Photonic Qubits 482 Quantum Key Distribution 484 QKD Using Single Photons 485 QKD Using Continuous Variables 489 No Cloning 492 Teleportation 492 Teleportation of Photon Qubits 493 Continuous Variable Teleportation 495 Entanglement Swapping 502 Entanglement Distillation 502 Quantum Computation 505 Dual-Rail Quantum Computing 506 Quantum Circuits with Linear Optics 507 Cluster States 511 Quantum Gates with Non-linear Optics 513 Single-Rail Quantum Computation 514 Quantum Random Walks 515 Boson Sampling 516 Continuous Variable Quantum Computation 518 Cat State Quantum Computing 519 Continuous Variable Cluster States 521 Large-Scale Quantum Computation 522 Summary 525 References 526 Further Reading 531

Contents

14

The Future: From Q-demonstrations to Q-technologies 533

14.1 14.2 14.3 14.3.1 14.3.2 14.3.3 14.3.4

Demonstrating Quantum Eﬀects 533 Matter Waves and Atoms 535 Q-Technology Based on Optics 537 Applications of Squeezed Light 537 Quantum Communication and Logic with Photons 539 Cavity QED 542 Extending to Other Wavelengths: Microwaves and Cryogenic Circuits 542 Quantum Optomechanics 542 Transfer of Quantum Information Between Diﬀerent Physical Systems 543 Transferring and Storing Quantum States 544 Outlook 544 References 545 Further Reading 547

14.3.5 14.3.6 14.3.7 14.4

Appendices 549

Appendix A: List of Quantum Operators, States, and Functions 549 Appendix B: Calculation of the Quantum Properties of a Feedback Loop 551 Appendix C: Detection of Signal and Noise with an ESA 552 Reference 554 Appendix D: An Example of Analogue Processing of Photocurrents 554 Appendix E: Symbols and Abbreviations 556 Index 559

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Preface The Idea Behind This Guide Some of the most interesting and sometimes puzzling phenomena in optics are those where the quantum mechanical nature of light is apparent. Recent years have seen a rapid expansion of quantum optics. Beautiful demonstrations and applications of the quantum nature of light are now common, and optics has been shown to be one of the key areas of physics to build quantum technologies based on the concepts behind quantum mechanics. The ﬁeld of photonics has already spawned major industries and changed the way we communicate, gather data, and handle information. Future applications of quantum information will enhance this trend and will introduce even more quantum aspects into our everyday activities. Why should we still write and read books at the age of search engines, blogs, and Wikipedia? This question did not arise when the ﬁrst edition of this book was published in 1998, which is a short step back in history but a long time ago in terms of the changes in information technology and the way we live, learn, and do research. You are likely to read this on a screen and can follow up links within seconds, tapping into an amazing wealth of information. This calls even more on this Guide to Experiment in Quantum Optics to explain the concepts, introduce reliable techniques, and help to select from the enormous amount of information that is available. We intend to guide you through the many experiments that have been published and to present and interpret them in one common style. This book provides a practical basis for exploring the ever-expanding technology in optics and electronics and provides a practical link to the rapidly expanding ﬁeld of quantum information, which is now being used in many diﬀerent platforms. Finally, this book is an attempt to communicate our personal insights and experiences in this ﬁeld of science. Science is driven by creative people, and some insight in the way researchers advance the ﬁeld should ﬁnd interest with you the reader. Many excellent textbooks are available on this topic. In the same way as this ﬁeld of research has been initiated by theoretical ideas, most of these books are written from a theoretical point of view. They provide fascinating, solid, and reliable descriptions of the ﬁeld. However, we found the literature frequently leaves out some of the important simpler pictures and intuitive interpretations of the experiments. In this guide we focus on both the ﬁrst and the most recent

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experiments. The guide explains the underlying physics in the most intuitive way we can ﬁnd. It addresses questions such as the following: What are the limitations of the equipment? What can be measured and what remains a goal for the future? The answers will prepare the readers to form their own mind map of quantum optics. It gives them the tools to make independent predictions of the outcome of their own experiments. For those who want to use quantum optics in practical ways, it tries to give an understanding of the design concepts of the machines that exploit the quantum nature of light. We assume that most of our readers will already have a fairly good understanding of optical phenomena and the way light propagates and how it interacts with components such as lenses, mirrors, detectors, etc. They can imagine the processes in action in an optical instrument. It is useful in the everyday work of a scientist to have practical pictures of what is going on inside the experiment or the machine. Such pictures have been shaped over many years of systematic study and through one’s own experimentation. Our guide provides such pictures, which can be ﬁlled in with many more details from theory and technology. In quantum optics three diﬀerent pictures, or interpretations, are used simultaneously: light as waves, light as photons, and light as the solution to operator equations. To these we add the picture of ﬂuctuations propagating through an optical system, a description using noise transfer functions. All of them are useful; all of them are correct within their limits. They are based on speciﬁc interpretations of the one formalism, and they are mathematically equivalent. In this guide all these four descriptions are introduced rigorously and are used to discuss a series of experiments. We start with simple demonstrations, gradually getting more complex and include most of the landmark experiments in single-photon and quantum noise detection, photon pair generation, squeezed states, quantum non-demolition, and entanglement measurements published to date. All four pictures – waves, photons, operators, and noise spectra – are used in parallel. In each case the most appropriate and intuitive interpretation is highlighted, and the limitations of all four interpretations are stated. In this way we hope to maintain and present a lot of the fascination of quantum optics that has motivated us and many of our colleagues, young and old, over so many years.

The Expansion of Quantum Optics and the Third Edition of This Guide The development of the ﬁeld has been rapid in recent years, and many new technical advances have been reported: vastly improved sources for photon pairs, squeezed and entangled states. We have better detectors and nonlinear media. With the ability to record and process much more information the era of analogue technology is gone. Post-processing is now the norm, quantum gates and quantum memories have been reported. We can generate and process multiple optical modes in frequency, space and time, demonstrate multimode entanglement and design and build reliable complex optical systems.

Preface

The ﬁeld of quantum information is growing rapidly, and new concepts for quantum communication, quantum logic, and quantum internet are emerging. Optics will play a major role in this. The ﬁeld has broadened to include both more fundamental concepts about the quantum nature of information and new ideas for the engineering of complex optical designs. Measurements using non-classical light are now well established and have become a routine operation in some select ﬁelds such as gravitational-wave detectors and sensors. Entanglement has grown from an abstract concept to a technical tool. At the same time commercial companies using quantum techniques for secure communication have already carved out a large market for themselves. The ideas in this guide, and the examples chosen, draw attention to both the techniques of single-photon detection and single-event logic and the techniques of continuous variable logic and the use of continuous coherent and squeezed beams. There are now far more practitioners of the former, and the latter has become a specialized niche, mainly used for metrology. However, both will play a role in the development of future applications and in the understanding the concepts of the quantum world. In addition, quantum optics based on pairs of photons has been combined with the technology of strong laser beams and continuous variables in a series of exciting hybrid experiments. This third edition focusses on the latest landmark experiments, and examples that are now outdated have been deleted. The coverage of quantum information concepts and protocols has been expanded to keep pace with the evolution of the ﬁeld. Reference is made to the state-of-the-art technologies, hinting at potential future developments. This guide follows the well-tested idea of providing independent building blocks in theory and experimental techniques that are used in later chapters to discuss complete experiments and applications. We found that most of them were still valid and thus remain unchanged. Optics is no longer dominant as the medium of choice for quantum technologies. Technical advances have allowed the same concepts now to be used in areas such as cryogenic electric circuits, superconducting systems, trapped ions and neutral atoms, Bose–Einstein condensates (BECs) and atom laser beams, and optomechanical systems, and other examples yet to be tested. This has posed a challenge: how to deﬁne the limits of this guide? To keep the book compact, we decided to focus entirely on optical systems and to point to links with other systems where required, but not to venture into the technical details of these ﬁelds. It is clear that future quantum technology will be using many complementary platforms. Optics will remain a major contributor and quite likely the most direct approach to the fascinating and sometimes counterintuitive eﬀects of the quantum world. Australia, 2018

Hans Bachor and Timothy Ralph

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Acknowledgments This book would not have been possible without the teaching we received from many of the creative and motivating pioneers of the ﬁeld of quantum optics. We have named them in their context throughout the book. HAB is particularly in debt to Marc Levenson, who challenged him ﬁrst with ideas about quantum noise. From him he learned many of the tricks of the trade and the art of checking one's sanity when the experiment produces results that seem to be impossible. TCR wishes particularly to thank Craig Savage for instilling the importance of doing theory that is relevant to experimentalists. Both of us owe a great deal to Dan Walls and Gerard Milburn who provided such beautiful vision of what might be possible. Over the years we learned from and received help for this book from a large international team of colleagues and students. Too many to name individually. All of you made important contributions to the our understanding and the way we communicate our knowledge and sight. We enjoy the feedback and are keen to discuss the mysteries and the past and future of quantum optics. We like to thank Aska Dolinska for the creative graphics work. Most importantly this Third Edition would not have been possible without the support of our loved ones. TCR wishes to thank his wife Barbara for her support and son Isaac for putting up with a distracted father. And HAB wishes to thank his wife Cornelia for the patience she showed with an absent-minded and many times frustrated writer. Hans Bachor Timothy Ralph 2018

1

1 Introduction 1.1 Optics in Modern Life The invention of the laser and the use of the special properties of laser light created one of the great technical revolutions of the twentieth century and has impact on many aspects of our modern lives. It allows us to manufacture the gadgets we love to use, including the computer with which this has been written. It allows us to communicate literally at the speed of light and with an information bandwidth that is unprecedented. Optics has been for a while the technology of choice for data storage, with DVD and Blu-ray still being used as a medium for data and 3D optical storage that might emerge in future. The modern alternatives, downloading and streaming, depend in many ways on photonics. In addition, there is a plethora of other applications of light, from laser welding in manufacturing to laser fusion for power generation and from laser-based microscopes to laser surgery. Imaging and display technologies are in many gadgets. The developments of optical technology have been relentless and will continue to be so for many more years. This has allowed us to access new levels of power, controllability, accuracy, complexity, and low cost that were deemed impossible only a few years ago. This is a convincing reason to study and teach quantum optics, as a scientist or engineer. Whilst all of the techniques in photonics use the quantum concept of stimulated emission and the laser [1], most of these applications still use quantum ideas in a quite simple and peripheral way. Most of the designs are based on conventional optics and can be understood on the basis of wave optics and traditional physical optics [2]. An interpretation of light as a classical wave is perfectly adequate for the understanding of eﬀects such as diﬀraction, interference, or image formation. Non-linear optics, such as frequency doubling or wave mixing, is well described by classical theory. Even most of the properties of coherent laser light can be understood and modelled in this way. There is a wealth or new ideas and applications emerging, exploiting the regimes of wave optics that were previously not accessible, with new theories and devices. This includes the many fascinating new devices made possible through new engineered materials, waveguide structures, photonic bandgap materials, linear and non-linear metamaterials. They all hold the promise of even higher performance, better speciﬁcations, and even new applications, such as cloaking or 3D storage, optical data processing at unheard A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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1 Introduction

speeds, or optical data processing. Conferences and research reviews all document that in 2018 a bright future is ahead in these areas of optics. However, which role will optics play in a future that will use quantum technologies? Firstly, the question of what ultimately limits the performance of optical devices lies outside the classical theory of light. We have to focus on a diﬀerent aspect of optics. We follow the ideas that come directly from the quantum nature of light, the full properties of photons, and the subtle eﬀects of quantum optics [3]. Initially, optics was recognized as a pretty ideal medium to study the ﬁner eﬀects of the quantum world. Photons in the visible spectrum were easily generated, detected with high eﬃciency, and not masked by thermal noise at room temperature, and any correlations could be recorded very well. Non-linear eﬀects allowed the generation of a variety of diﬀerent forms of light. This established optics as the medium of choice to explore the quantum world, and the early improvements in the technology in the 1980s led to great demonstrations, such as the Bell inequalities, signalling the beginning of experimental quantum optics. This exploration of the ‘weirdness’ of the quantum world is still today one of the frontiers in quantum optics and discussed in this guide. It was recognized that quantum eﬀects pose a limit to the sensitivity of many measurements, with the now successful detectors for gravitational waves the pre-eminent example. A closer analysis showed that there are ways of, at least partially, circumventing such limitations. Optical quantum metrology was born and is a major topic in this guide. It will continue to evolve – and we now see the ﬁrst routine applications of quantum-enhanced metrology, with few photons and with laser beams. The ideas are spreading and can be realized in other systems such as coherent groups of atoms or ions or in circuits. Quantum-enhanced metrology in its various forms is becoming more widespread, from the deﬁnition of the SI units to very practical situations. Optics technology and our knowledge of protocols and devices have steadily grown in time. Almost gone are the analogue devices of the 1990s, and we have direct access to digital data. For example, we can now detect, record, process, and store information in our experiments in much better ways, with A/D converters, multichannel analysers, memories of Terabit size, etc. This means that we can access much more information and process and analyse data more thoroughly. The next step might well include the use of artiﬁcial intelligence to optimize data analysis and also the performance of devices. The detection of quantum correlations are a case in hand: we can now directly explore the correlation properties, use diﬀerent approaches and measures to evaluate the quality of entanglement, and post-select data, or use heralded information, to select the best quantum data. This provides a path to optical quantum gates and more elaborate quantum logic. We can post-process the information and make it conditional on other measurements. This allows us to not only describe the state with tomography in great detail but also provide a path for looking inside the statistical properties of a beam of light. An example is the Wigner function shown on the front cover, which shows strong negative values that are contained in one beam from an entangled pair of beam – after

1.2 The Origin and Progress of Quantum Optics

conditional detection and letting information from one beam inﬂuence the reading from the other entangled beam [4]. Furthermore, so-called hybrid detection uses entangled beams, where the information from a single photon detector in one beam is used to post-select the most relevant data from stream of photon recorded from the other beam. Using this form of hybrid detection, it is possible to ﬁlter out states of light with speciﬁc quantum properties, for example, Schrödinger cat states [5]. They represent a state that is a quantum superposition of two orthogonal states and is one of the icons and hallmarks of quantum physics, as recognized in the 2012 Nobel Prize of Physics [6]. On the other hand, the last few years has seen an ever-increasing interest in the information that is contained within a beam of light, whether it be images, analogue pulses, or digital data encoded in the light. The emergence of the concept of quantum information has created a rising interest in the ability to communicate, store, and process more complex information than available in classical digital electronics, all by using beams of light or individual photons. This has created a great deal of interest, an expanding research ﬁeld, and a demand for practical quantum optics-based devices. Optics overlaps more and more with condensed matter and soft matter physics, materials science, and mechanical engineering. Ever-improving technology drives these ﬁelds, as well as now insights into quantum logic and quantum control. Optics is in many cases a precursor, sometimes a testing ground, and increasingly a component of such quantum devices.

1.2 The Origin and Progress of Quantum Optics Ever since the quantum interpretation of the black body radiation by M. Planck [7], the discovery of the photoelectric eﬀect by W. Hallwachs [8] and P. Lennard [9] and its interpretation by A. Einstein [10] has the idea of photons been used to describe the origin of light. For the description of the generation of light, a quantum model is essential. The emission of light by atoms, and equally the absorption of light, requires the assumption that light of a certain wavelength 𝜆, or frequency 𝜈, is made up of discrete units of energy each with the same energy hc∕𝜆 = h𝜈. This concept is closely linked to the quantum theory of the atoms themselves. Atoms are best described as quantum systems; their properties can be derived from the wave functions of the atoms. The energy eigenstates of atoms lead directly to the spectra of light which they emit or absorb. The quantum theory of atoms and their interaction with light has been developed extensively; it led to many practical applications [11] and has played crucial role in the formulation of quantum mechanics itself. Spectroscopy relies almost completely on the quantum nature of atoms. However, in this guide we will not be concerned with the quantum theory of atoms, but concentrate on the properties of the light, its propagation, and its applications in optical measurements. Historically, special experiments with extremely low intensities and those that are based on detecting individual photons raised new questions, for instance, the famous interference experiments by G.I. Taylor [12] where the energy ﬂux corresponds to the transit of individual photons. He repeated Th. Young’s double-slit

3

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1 Introduction

experiment [13] with typically less than one photon in the experiment at any one time. In this case, the classical explanation of interference based on electromagnetic waves and the quantum explanation based on the interference of probability amplitudes can be made to coincide by the simple expedient of making the probability of counting a photon proportional to the intensity of the classical ﬁeld. The experiment cannot distinguish between the two explanations. Similarly, the modern versions of these experiments, using the technologies of low noise current detection or, alternatively, photon counting, show the identity of the two interpretations. The search for uniquely quantum optical eﬀects continued through experiments concerned with intensity, rather than amplitude interferometry. The focus shifted to the measurement of ﬂuctuations and the statistical analysis of light. Correlations between the arrival of photons were considered. It started with the famous experiment by R. Hanbury-Brown and R. Twiss [14] who studied the correlation between the ﬂuctuations of the two photocurrents from two diﬀerent detectors illuminated by the same light source. They observed with a thermal light source an enhancement in the two-time intensity correlation function for short time delays. This was a consequence of the large intensity ﬂuctuations of the thermal light and was called photon bunching. This phenomenon can be adequately explained using a classical theory that includes a ﬂuctuating electromagnetic ﬁeld. Once laser light was available, it was found that a strong laser beam well above threshold shows no photon bunching; instead the light has a Poissonian counting statistics. One consequence is that laser beams cannot be perfectly quiet; they show noise in the intensity that is called shot noise, a name that reminds us of the arrival of many particles of light. This result can be derived from both classical and quantum models. Next, it was shown by R.J. Glauber [15] that additional, unique predictions could be made from his quantum formulation of optical coherence. One such prediction is photon anti-bunching, where the initial slope of the two-time correlation function is positive. This corresponds to greater than average separations between the arrival times of photons; the photon counting statistics may be sub-Poissonian, and the ﬂuctuations of the resulting photocurrent would be smaller than the shot noise. It was shown that a classical theory based on ﬂuctuating ﬁeld amplitudes would require negative probabilities in order to predict anti-bunching, and this is clearly not a classical concept but a key feature of a quantum model. It was not until 1976, when H.J. Carmichael and D.F. Walls [16] predicted that light generated by resonance ﬂuorescence from a two-level atom would exhibit anti-bunching, that a physically accessible system was identiﬁed that exhibits non-classical behaviour. This phenomenon was observed in experiments by H.J. Kimble et al. [17], where individual atoms were observed. This opened the era of quantum optics. It is now possible to track and directly display the evolution of the photon number inside a cavity, thanks to the innovation introduced by S. Haroche and coworkers. They have designed better and better superconducting cavities to store microwave photons and can carry out non-demolition experiments in a cavity with very long decay time. They can now directly observe the evolution of the photon number in an individual mode [18]. See

1.2 The Origin and Progress of Quantum Optics

for details Figure 3.7. This is a most beautiful demonstration of the theoretical concepts. The concept of modes will be a central feature in this guide. This concept has evolved in time; it now has several distinct meanings and is playing a pivotal role in quantum optics. For an experimentalist, a single mode can mean a perfect beam shape without any distortion. A single-mode laser means a device that is emitting a beam with exactly one frequency, in one polarization, and with a perfect beam shape. In the theory, a mode is the fundamental entity that can be quantized and be described by simple operators. We have a reliable and direct model for the behaviour of such a single mode, and up to recently most quantum optics were formulated in this single-mode model. At the same time, in optical technology, such as communication or imaging, the concept of a mode is associated with an individual channel of information. This could be a speciﬁc modulation frequency, as we are familiar with from radio transmission, or it could be a speciﬁc spatial image. In the world of optical sensing, we wish to detect and transmit changes of various parameters, such as lengths or position, magnetic or electric ﬁelds, pressure or temperature, to name a few. The evolution of these parameters are then encoded into the light and transmitted to the receiver. In this case, each parameter, and each degree of freedom for the change of the parameter, corresponds to one speciﬁc mode. It has now been recognized that a sensor can be optimized and will show the best possible performance, when the optical mode is fully matched to the information. This can be a simple mode, as mentioned in the case of the traditional single-mode laser, or it can actually be a much more complicated spatial and temporal distribution. The ﬁrst extension in quantum optics has been to multiple beams of light, each containing one mode each. We can have many separate beams, all originating from one complex optical apparatus. However, a single optical beam can also contain multiple modes, for example, spatial modes, temporal modes, or frequency modes. They correspond to separate degrees of freedom or separate information channels within the beam. Such a multimode beam might contain only a few, or the occasional, photon, or it might have the full ﬂux of a laser beam. In either case, the underlying physics is the same. The information transmitted is best described in terms of a set of independent orthogonal modes. Each can be quantized individually and has their own independent statistics. The signals, and the ﬂuctuations, transmitted by these modes do not interact with each other, and the full quantum description of these devices is straightforward using this concept. There has been great progress in the technology to generate individual photons in a controlled way. The technology has evolved to a stage where many of the initial thought experiments dealing with individual atoms can now be performed with real machines. In addition, a whole range of new types of single photon emitters has been developed and studied by many groups around the world. They include quantum dots that show the required quantum properties and are continuously improving in their robustness and reliability. They are now being used regularly in a variety of applications where small size matters. This includes applications for quantum communication with quantum dots placed in speciﬁc places or in the form of nanodiamonds imbedded as emitters in biological samples that respond to the local conditions in living cells.

5

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1 Introduction

Initially, these single photon eﬀects were a pure curiosity. They showed the diﬀerence between the quantum and the classical world and were used to test quantum mechanics. In particular, the critique by A. Einstein et al. and their EPR paradox [19] required further investigation. Through the work of J.S. Bell [20], it became possible to carry out quantitative tests. They are all based on the quantum concept of entanglement. In the simplest case, this means pairs of quantum systems that have a common origin and have linked properties, such that the detection of one allows the perfect prediction of the other. Such systems include not only individual photons or atoms, or coherent states of light, such as laser beams but also physical matter, such as was demonstrated with Bose–Einstein condensates. See Chapter 14. Starting with the experiments by A. Aspect et al. [21] with atomic sources, and later with the development of sources of twin photon pairs and experiments by A. Zeilinger [22] and many others, the quantum predictions have been extremely well tested. The concept of entanglement has now been demonstrated and tested with single photons and also with intense beams of light, over large distances of many kilometres and through optical ﬁbres. There has been a long series of experiments with the aim to demonstrate the validity and necessity of quantum physics beyond any doubt, closing step by step every loophole. It has even been possible to extend the concept of entanglement from two entities, such as two particles, to many particles, most prominently to many trapped ions. Such multipartite entanglement is at the core of plans to build quantum simulators and quantum logic devices. Similarly there has been an extension from two modes of light to several modes. This leads to the idea of multimode entanglement, which is a new topic of research, with both interesting physics and the potential for new applications. In summary, the progress in quantum optics, which initially was concerned with the quality of the quantum demonstrations, has recently led to the conquering of complexity and the study of the robustness, or decoherence, of such complex systems. The peculiarities of single photon quantum processes have led to many new technical applications, mainly in the area of communication. For example, it was found that information sent by single photons can be made secure against listening in; the unique quantum properties do not allow the copying of the information. Any eavesdropping will introduce noise and can therefore be detected. Quantum cryptography and quantum key distribution (QKD) [23] are now viable technologies, based on the inability to clone single photons or quantum modes. There is great anticipation of other future technological applications of photons. The superposition of diﬀerent states of photons can be used to send more complex information, allowing the transfer not only of classical bits of information, such as 0 and 1, but also of qubits based on the superposition of states 0 and 1. The coherent evolution of several such quantum systems could lead to the development of quantum logic connections, quantum gates, and eventually whole quantum computers that promise to solve certain classes of mathematical problems much more eﬃciently and rapidly [24]. Just as astounding is the concept of teleportation [25], the disembodied communication of the complete quantum information from one place to another. This has already been tested in experiments (see Chapter 13).

1.3 Motivation Through Simple and Direct Teaching Experiments

Optics is now a testing ground for future quantum information technologies and for data communication and processing. At the same time, the concepts and ideas of quantum optics are emerging in other related quantum technologies, for example, based on quantum circuits or mechanical systems, which are all quantized and show in principle the same physics. We can see the equivalence of the physics of photons, phonons, polaritons, and other particles that describe the excitation of quantum systems. Technology is moving very fast, and we can expect over the next decade to see machines that utilize any of these individual quantum systems or even combine several of them. The rules of quantum optics, as described in this guide, will become important in many diﬀerent other types of instruments. Optics could become one of the best ways to visualize the broader quantum concepts that apply to many other types of instruments.

1.3 Motivation Through Simple and Direct Teaching Experiments The best starting point for the demonstration of photons in current technology is to start with images that are generated by CCD (charge-coupled device) cameras. It is worthwhile to note that basically all information we have about light and optical eﬀects is based on either some form of photodetector or the human eye. The ﬁrst generates an electric signal, either pulses or a continuous photon current. Whilst the eye produces, in a clever and complex way, a signal in the neurones in the brain, even they can be interpreted as a form of electric signal. It is the characteristics of these signals and their statistics that provides access to the quantum properties of light. In the extreme case we would have a detector that is capable of detecting one individual photon at a time. This is now possible with special CCD cameras, where the sensitivity has been lifted to such a high level and thermal noise has been reduced so much that it hardly interferes with the observation. The number of dark counts, that is, clicks that are generated at random without photons present, can be neglected. In this way we can actually detect and record images at the single photon level as shown in Figure 1.1. The human eye is actually capable of seeing at this level, and many animals exceed our capability. At this level of illumination, the world looks very grainy. We can see individual events. The image is composed out of individual local detection events. An example, taken with a special CCD, is shown in Figure 1.1. At some level this can be regarded as trivial – obviously it will be like this. We have photons. On the other this image demonstrates several subtle points. We can see that the concept of intensity, bright and dark, is directly related to the concept of probability. Areas with a high local photon ﬂux, or high probability, correspond to high intensities, that is, a high amplitude of the electromagnetic ﬁeld. Conversely, dark areas of the image correspond to a low intensity. The central concept in quantum science of a probability amplitude become directly obvious. Two, or many, waves with diﬀerent amplitudes and relative phases can interfere, and this is a central concept of the classical wave picture of light. The example

7

1 Introduction

(a) 300 Number of photons

8

250 200 150 100 50 0 0

(b)

50

100

150

200

Pixel

Figure 1.1 Image of the fringes from a double-slit experiment recorded at the single photon level. (a) Distribution of individual photons. (b) Histogram of photons in each vertical line. The full movie can be seen at https://www.lcf.institutoptique.fr/Groupes-de-recherche/Gazquantiques/Membres/Permanents/Alain-Aspect. Source: After https://www.lcf.institutoptique .fr/Groupes-de-recherche/Gaz-quantiques/Membres/Permanents/Alain-Aspect.

shown in Figure 1.1 is a real interference fringe system, generated with coherent light and a double-slit apparatus. This is one of the simplest arrangements that shows the wave nature of light – and yet in this case it is detected by a number of individual photons. There is no contradiction here – the two explanations, waves and particles, are fully complementary. The shape of the fringes, the overall intensity function, is best described with the wave model. At the same time, the statistics of the light, the visibility that is observed, and thus the quality of the fringes can best be described with the photon model. This very simple example immediately shows the strengths and weaknesses of both models. It demonstrates how we want to use these two models for diﬀerent aspects of the experiment. We should choose the pictures in our mind that is most suitable to understand the speciﬁc aspect of the experiment we wish to investigate. The next level would be to quantify the statistics. What would be an appropriate statistical model for the ﬂuctuations, detected in one speciﬁc area of the image and averaged over many time intervals? Equivalently we could ask: what is the statistics of the number of photons detected in many separate areas of equal size, all receiving the same intensity? In the ﬁrst case one could imagine listening to the number of clicks being generated. It would sound like a random sequence of clicks, as we know them from a Geiger counter detecting 𝛼 or 𝛾 rays.

1.3 Motivation Through Simple and Direct Teaching Experiments

A proper analysis would reveal that we have a Poissonian distribution for the diﬀerent time intervals. The same result applies to the analysis of the number of photons in many image areas of equal size. The graininess we see in Figure 1.1 is a direct representation of the counting statistics, which can be derived from simple statistical arguments, assuming random processes. The randomness in the arrival times of the photons in a weak beam of light is so perfect that it is now used as the gold standard for randomness, exceeding in quality the widely used random number generators based on electronic processes or imbedded in computers. This has led to commercial activities where optically generated random numbers now can be accessed via the internet and the generators are for proﬁt. The demand is substantial – and already a small niche market for quantum random numbers has emerged. The 1960s saw a rapid development of new laser light sources and improvements in light detection techniques. This allowed the distinction between incoherent (thermal) and coherent (laser) light on the basis of photon statistics. The groups of F.T. Arecchi [26], L. Mandel, and R.E. Pike all demonstrated in their experiments that the photon counting statistic goes from super-Poissonian at the lasing threshold to Poissonian far above threshold. The corresponding theoretical work by R.J. Glauber [15] was based on the concept that both the atomic variables and the light are quantized and showed that light can be described by a coherent state, the closest quantum counterpart to a classical ﬁeld. The results are essentially equivalent to a quantum treatment of a classical oscillator. However, it is an important consequence of the quantum model that any measurement of the properties of this state, intensity, amplitude, or phase of the light, will be limited by these ﬂuctuations. This is the optical manifestation of Heisenberg’s uncertainty principle. Quantum noise can impose a limit to the performance of lasers, sensors, and communication systems, and near the quantum limit the performance can be quite diﬀerent. To illustrate this, consider the result of a very simple and practical experiment: use a laser, such as a laser pointer with a few milliwatt of power, and detect all of the light with a photodiode. The average number of detection events is now very large. Listening to the output, we can no longer distinguish between the individual clicks; we will hear a continuous noise. The level of noise will increase with the intensity, as the width of the Poissonian distribution increases with the average number of photons detected. This quantum noise is the direct experimental evidence of the quantum statistics of light. If we record the current from the photodetector on an oscilloscope, we would see a trace with random ﬂuctuations, above and below a mean level, representing the average intensity. The excursions have a Gaussian distribution around the mean value, which is fully consistent with the transition from a Poissonian to a Gaussian distribution. Relatively speaking this eﬀect gets smaller if we average the signal over more photons, that means using a brighter laser or measuring for a longer time. As we will see, this eﬀect cannot be avoided completely. Quantum noise is not an eﬀect that can be eliminated by good traditional engineering. Decades ago quantum noise was called shot noise and was regarded by many as a consequence of the photodetection process, as a randomness of the stream of electrons produced in the photodetector. This view prevailed for a long time,

9

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1 Introduction

Modulator A Laser B

X2

X1

Figure 1.2 Experiment to investigate the correlation between two parts of a laser beam, using a laser pointer, modulator, two detectors, and an x–y oscilloscope.

particularly in engineering textbooks. However, this view is misleading as we ﬁnd in the squeezing experiments described in this book. It cannot account for situations where non-linear optical processes modify the quantum noise, whilst the detector remains unchanged. In the squeezing experiments the quantum noise is changed optically, and consequently we have to assume it is a property of the light, not of the detector. Today it is straightforward to see these eﬀects of quantum noise with simple equipment, accessible to undergraduate students. The light source can be a laser pointer, carefully selected to operate in a single mode. The laser beam is split by a beamsplitter into two beams of equal power that are detected by two individual detectors, A and B, at diﬀerent locations. Each photocurrent is ampliﬁed, and they are both displayed via the x and y axes of one oscilloscope. We want to see both the average (DC) value of the photocurrent, which represents the time averaged intensity of the beams A and B, and the ﬂuctuations (AC) of the two photocurrents, as they represent any modulation and the ﬂuctuations, or noise, of the intensity of the two beams. This can easily be done by looking at the signals at a few megahertz, which is typically the bandwidth of the RF ampliﬁers used. This is shown in Figure 1.2. The laser can be modulated, through periodic variations of the laser current. Classical optics predicts that the intensity of both beams A and B will be modulated, and this will show up clearly as a line at 45∘ on the oscilloscope. That means the two photocurrents change synchronously in time, an increase in one is accompanied by an increase in the other etc. In practice some care is required to achieve a clear 45∘ line: there should be no delay in the signals from the detector to the oscilloscope and the size of the signals has been matched. This result shows a strong correlation between the two beams. A correlation of the photocurrents means a correlation of the ﬂux of photons. We can also look at the ﬂuctuations in more detail. If we look at beams A and B individually, we see ﬂuctuations of the current above and below the long-term average value. Closer inspection would show a Gaussian distribution of the excursions below and above, as we expect it for random noise in the beam. The magnitude of the ﬂuctuations will follow the gain curve of the RF ampliﬁers. With careful calibration of the frequency response of our equipment, we ﬁnd that the noise in the light is not frequency dependent; it is white noise. The magnitude of the ﬂuctuations is the same for a laser beam with or without a small modulation.

1.3 Motivation Through Simple and Direct Teaching Experiments

A

A

C

C

X2

X2

X1

X1

(a)

(b) A C

X2

X1

(c)

Figure 1.3 The noise of beams A and B displayed on an oscilloscope. (a) No beam, technical background noise. (b) The laser beam has no modulation, quantum noise. (c) The laser beam with modulation. The signal from the two detectors is displayed in the coordinates X1 and X2 , correlations appear in the direction C, and uncorrelated signal appears in both directions A and C.

The noise increases with intensity, and careful measurements show that the variance of the noise is proportional to the intensity. Is the noise from A correlated with the noise from B? The oscilloscope image immediately provides the answer. The noise from A and B results in a fuzzy blob on the screen. See Figure 1.3b. The ﬂuctuations in the two photocurrents are independent. The point on the screen of the oscilloscope traces out a random walk independently in both directions. This is in stark contrast to the modulation where the photocurrent ﬂuctuates synchronously, as shown in Figure 1.3c. The lack of correlation is immediately evident. This conﬁrms the special properties of quantum noise; it does not obey the classical rules. One simple explanation is that for small ﬂuctuations we would notice the eﬀect that one photon cannot be split into halves at the beamsplitter. It will appear in one or the other beam, with a random choice which beam it will join. We will see in Chapter 3 how this simple statement leads to an explanation of the properties of quantum noise in this situation. If we have no modulation, the blob is right in the middle of the screen. The blob is circular, given there is no additional noise in the system and the photon noise dominates over the electronic noise generated in the apparatus. For a laser beam with modulation, we will see a fuzzy area tilted at 45∘ , as shown in Figure 1.3c. This is a combination of the two eﬀects, a line

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1 Introduction

in direction C for the modulation and a fuzzy blob for the laser noise. Classical modulations and quantum noise have diﬀerent properties, even when they have similar magnitudes, and this simple experiment can show the diﬀerence.

1.4 Consequences of Photon Correlations It is worthwhile to explore the properties of quantum noise somewhat further. It was found that, in clear distinction to classical noise, no technical trick can eliminate quantum noise. This becomes evident from our experiment. A common conventional idea for noise suppression would be to make two copies of the signal, both containing the noise, and to subtract them for each other. We have all the necessary components in the apparatus used in Figure 1.2. We have made two copies of the beam. We can use one beam, say, A, for our experiment, record it with detector A, and then subtract the electronic ﬂuctuations generated by detector B. Based on our observations we now see that we can subtract the classical modulation, which is common to both beams. However, we cannot subtract the quantum noise, which is independent and uncorrelated. The subtraction works for any modulation, for harmonic modulation demonstrated here, and for random intensity modulations, such as those generated by a low-quality, noisy current source for the laser. That means we can distinguish between classical laser noise and quantum noise that is a consequence of the quantum properties of the photons. The result remains unchanged if the subtraction is replaced by a sum or if the two currents, or if the two currents are added with an arbitrary short time delay, such as introduced be an extra length of cable. The resulting noise is always the quadrature sum of the two-input signal. It is independent of the sign, or phase, of the summation. This is equivalent to the statement that the noise in the two photocurrents is not correlated. At this stage it is not easy to identify the point in the experiment where this uncorrelated noise is generated. One interpretation assumes that the noise in the photocurrents is generated in the photodetectors. A diﬀerent interpretation assumes that the beamsplitter is a random selector for photons and consequently the intensities of the two beams are random and thus uncorrelated. A distinction can only be made by further experiments with squeezed light, which are discussed later in this book. An alternative scheme for noise suppression is the use of feedback control, as shown in Figure 1.4. It achieves equivalent results to diﬀerence detection. The Modulator

Laser

Feedback control

Figure 1.4 The second attempt to eliminate laser noise: an improved apparatus using a feedback controller, or ‘noise eater’.

1.4 Consequences of Photon Correlations

intensity of the light can be controlled with a modulator, such as an acousto-optic modulator or an electro-optic modulator. Using a feedback ampliﬁer with appropriately chosen gain and phase lag, the intensity noise can be reduced, and all the technical noise can be eliminated [27]. It is possible to get very close to the quantum noise limit, but the quantum noise itself cannot be suppressed [28, 29]. This phenomenon can be understood by considering the properties of photons. As mentioned above, the quantum noise measured by the two detectors is not correlated; thus the feedback control, when operating only on quantum noise on one detector, will not be able to control the noise in the beam that reaches the other detector. This can be explained using a full quantum theory. The role of the photon generation process can be explored further. It was found that the noise of the light may be below the standard quantum limit if the pumping process exhibits sub-Poissonian statistics. This is particularly easy to achieve for LEDs, which are high eﬃciency light sources driven directly by electric currents, or with semiconductor lasers. The currents driving these devices are classical, at the level of the ﬂuctuations we are concerned with, and the ﬂuctuations can be controlled with ease to levels well below the shot noise level. For sources with high quantum eﬃciencies, the sub-Poissonian statistics of the drive current is transferred directly to the statistics of the light emitted. Such experiments were pioneered for the case of diode lasers by the group of Y. Yamamoto [30]. They showed that intensity ﬂuctuations can be suppressed in a high impedance semiconductor laser driven by a constant current. Similar work had earlier been carried out with LEDs by several groups. If we use this type of source in our experiment shown in Figure 1.3, the fuzzy blob would be smaller, in both directions, than the blob for a normal laser with the same output intensity. To explain the quantum noise and the correlations fully, we should not restrict ourselves to ﬂuctuations of the intensity of the light or the statistics of the photon arrival time. We will ﬁnd that there is noise both in the magnitude and phase of the electromagnetic wave. Consequently our quantum model requires a two-dimensional description of the light, with properties which we will call quadratures. Fluctuations, or noise, can be characterized by the variance in both the amplitude and phase quadrature. Almost a decade after the observation of photon anti-bunching in atomic ﬂuorescence, another quantum phenomenon of light was observed – the suppression, or squeezing, of quantum ﬂuctuations [31]. For a coherent state the uncertainties in the quadratures are equal and minimize the product in Heisenberg’s uncertainty principle. A consequence is that measurements of both the amplitude or the phase quadrature of the light show quantum noise. In a squeezed state, the ﬂuctuations in the quadratures are no longer identical. One quadrature may have reduced quantum ﬂuctuations at the expense of increased ﬂuctuations in the other quadrature. Such squeezed light could be used to beat the standard quantum limit. After the initial theoretical predictions, the race was on to ﬁnd such a process. A number of non-linear processes were tried simultaneously by several competing groups. The observation of a squeezed state of light was achieved in 1985 ﬁrst by the group of R.E. Slusher at Bell Labs in four-wave mixing in sodium atomic beam [32], soon followed by the group of M.D. Levenson and R. Shelby at IBM [33] the group of H.J. Kimble with an optical parametric oscillator [34]. In recent years a number of other non-linear processes have been used to

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1 Introduction

Non-linear medium Spectrum analyser

Laser

Squeezed light

Local oscillator + _ Homodyne detector

Figure 1.5 A typical squeezing experiment. The non-linear medium generates the squeezed light that is detected by a homodyne detection scheme.

demonstrate the quantum noise suppression based on squeezing [35]. A generic layout is shown in Figure 1.5. The experiments are now reliable, and practical applications are feasible. Quantum information is a key to many applications of quantum optics. Plans for using the complexity of quantum states to code information, to teleport it, to use it for secure communication and cryptography, and to store quantum information and possibly use it for complex logical processes and quantum computing have all been widely discussed and demonstrated to a greater or lesser degree. The concept of entanglement has emerged as one of the key qualities of quantum optics. As it will be shown in this guide, it is now possible to create entangled beams of light either from pairs of individual photons, which is now possible in technical applications and in teaching laboratories [37], or from the combination of two squeezed beams, and the demand and interest in non-classical states of light has sharply risen. We can see quantum optics playing a large role in future communication and computing technologies [23, 36]. For this reason, the guide covers both single photon and CW beam experiments parallel to each other. It provides a uniﬁed description and compares the achievements and tries to predict the future potential of these experiments.

1.5 How to Use This Guide This guide leads us through experiments in quantum optics, experiments that deal with light and demonstrate, or use, the quantum nature of light. It shows the practicalities and challenges of these experiments and gives an interpretation of their results. One of the current diﬃculties in understanding the ﬁeld of quantum optics is the diversity of the models used. On the one hand, the theory and most of the publications in quantum optics are based on a rigorous quantum model that is rather abstract. On the other hand, the teaching of physical optics and the experimental training in using devices such as modulators, detectors, and data acquisition systems are based on classical wave ideas. This training is extremely

1.5 How to Use This Guide

Experiment versus theory | a1>, | v > States

c1| a1> + c2| a2> + c3| v >

H Operators

(a)

|0> Operator and vacuum state

Complex state Prob. of det and correlation

Info SNR

(b)

Beams and modulation Experiment linear and non-linear intensity, phase components Signals and noise at

Loss

α2in, Φin, δX1(Ω), δX2(Ω), V1in (Ω), V2in (Ω) Quantum transfer function

Output beams

Detectors electronics

2

αout, Φout, V1out (Ω), V2out (Ω) Vv = 1 Vacuum beam

Coherent states variances at Ω

(c)

Figure 1.6 Comparison between an experiment (b) and the two theory descriptions for few photon states (a) and laser beams (c).

useful, but frequently does not include the quantum processes. Actually, the language used by these two approaches can be very diﬀerent, and it is not always obvious how to relate a result from a theoretical model to a technical device and vice versa. As an example compare the schematic representation of a squeezing experiment, shown in Figure 1.6, both in terms of the theoretical treatments for photons and laser beams and in an experimental description. The purpose of this guide is to bridge the gap between theory and experiment. This is done by describing the diﬀerent building blocks in separate chapters and combining them into complete experiments as described in recent literature. This guide starts with a classical model of light in Chapter 2. Experiments reveal that we require a concept of photons, Chapter 3, which is expanded into a quantum model of light in Chapter 4. The properties of optical components and devices are given in Chapter 5, followed by a detailed description of lasers and ampliﬁers in Chapter 6. Next is a detailed discussion of photodetection for single photons and beams in Chapter 7. On this basis we build with the discussion of complete experiments. The technical details required for reliable experimentation with quantum noise, including techniques such as cavity locking and feedback controller, are given in Chapter 8. The concept of squeezing is central to most attempts to improve optical devices beyond the quantum noise limit and is introduced and discussed in Chapter 9. It also describes the various squeezing experiments and their results are discussed; the diﬀerent interpretations are compared. Finally, the applications of squeezed light are described in Chapter 10 with a detailed description of the largest application,

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1 Introduction

the detection of gravitational waves. Similar to squeezing Chapter 11 discusses quantum non-demolition experiments. In Chapter 12 more and more complete experiments to test the fundamental concepts of quantum mechanics are discussed. Finally, the concepts of quantum information and the rapidly evolving state of art in using quantum optics, both single photons or CW beams, in quantum information devices are presented in Chapter 13. Quantum optics has helped to build the foundations of many other form of quantum technology, and the links to these many parallel lines of research is brieﬂy outlined in Chapter 14, which tries to look a little into the future. This guide can be used in diﬀerent ways. A reader who is primarily interested in learning about the ideas and concepts of quantum optics would best concentrate on Chapters 2–4, 9, 12, and 13 but may leave out many of the technical details. For these readers Chapter 5 would provide a useful exercise in applying the concepts introduced in Chapter 4. In contrast, a reader who wishes to ﬁnd out the limitations of optical engineering or wants to learn about the intricacies of experimentation would concentrate more on Chapters 2, 5, 6, and 8, and for an extension into experiments involving squeezed light, Chapters 9 and 10 can be added. A quick overview of the possibilities opened by quantum optics can be gained by reading Chapters 3, 4, 6, 9, 11–14. We hope that in this way our book provides a useful guide to the fascinating world of quantum optics.

References 1 Saleh, B.E.A. and Teich, M.C. (2007). Fundamentals of Photonics, 2e. Wiley. 2 Born, M. and Wolf, E. (1959). The Principles of Optics. Pergamon Press. 3 Haroche, S. and Raimond, J.-M. (2006). Exploring the Quantum. Oxford

University Press. 4 Hage, B., Janousek, J., Armstrong, S. et al. (2011). Demonstrating various

quantum eﬀects with two entangled laser beams. Eur. Phys. J. D 63: 457. 5 Ourjoumtsev, A., Jeong, H., Tualle-Brouri, R., and Grangier, P. (2007). Gener-

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ation of optical “Schrödinger cats” from photon number states. Nature 488: 784. https://doi.org/10.1038/nature06054. Haroche, S. (2012) Nobel Lecture 2012. http://www.nobelprize.org/ nobelprizes/physics/laureates/2012/haroche-lecture.html. Planck, M. (1900). Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum. Verh. Dtsch. Phys. Ges. 2: 237. Hallwachs, W. (1889). Ueber den Zusammenhang des Elektrizitätzverlustes durch Beleuchtung mit der Lichtabsorption. Ann. Phys. 273 (8): 666. https:// doi.org/10.1102/andp.18892730811. Lenard, P. (1902). Über die lichtelektrische Wirkung. Ann. Phys. 313 (5): 149. https://doi.org/10.1002/andp.19023130510. Einstein, A. (1905). Über einen die Erzeugung und Verwandlung des Lichtes betreﬀenden heuristischen Gesichtspunkt. Ann. Phys. 17: 132. Thorne, A., Litzen, U., and Johansson, S. (1999). Spectrophysics, Principles and Applications, 3e. Springer-Verlag.

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12 Taylor, G.I. (1909). Interference fringes with feeble light. Proc. Cambridge Phi-

los. Soc. 15: 114. 13 Young, Th. (1807). Course of Lectures on Natural Philosophy and Mechanical

Arts. London: Taylor and Walton. 14 Hanbury-Brown, R. and Twiss, R.Q. (1956). Correlation between photons in

two coherent beams of light. Nature 177: 27–29. 15 Glauber, R.J. (1963). The quantum theory of optical coherence. Phys. Rev.

Lett. 130: 2529. 16 Carmichael, H.J. and Walls, D.F. (1976). A quantum-mechanical master

equation treatment of the dynamical stark eﬀect. J. Phys. B 9: 1199. 17 Kimble, H.J., Dagenais, M., and Mandel, L. (1977). Photon antibunching in

resonance ﬂuorescence. Phys. Rev. Lett. 39: 691. 18 Guerlin, C., Bernu, J., Deleglise, S. et al. (2007). Progressive ﬁeld state collapse

and quantum non-demolition photon counting. Nature 448: 889. 19 Einstein, A., Podolsky, B., and Rosen, N. (1935). Can a quantum-mechanical

description of physical reality be considered complete?. Phys. Rev. A 47: 777. 20 Bell, J.S. (1964). On the Einstein-Podolsky-Rosen paradox. Physics 1: 195. 21 Aspect, A., Grangier, P., and Roger, G. (1982). Experimental realisation of

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Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: a new violation of Bell’s inequalities. Phys. Rev. Lett. 49: 91. Zeilinger, A. (2000). Quantum cryptography with entangled photons. Phys. Rev. Lett. 84: 4729. Kok, P. and Lovett, B.W. (2010). Introduction to Optical Quantum Information Processing. Cambridge University Press. Nielsen, M.A. and Chuang, I.L. (2000). Quantum Computation and Quantum Information. Cambridge University Press. Bennett, C.H., Brassard, G., Crepau, C. et al. (1993). Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70: 1895. Arecchi, F.T., Berne, A., and Sona, A. (1966). Measurement of the time evolution of the radiation ﬁeld by joint photocurrent distributions. Phys. Rev. Lett. 17: 260. Robertson, N.A., Hoggan, S., Mangan, J.B., and Hough, J. (1986). Intensity stabilisation of an Argon laser using an electro-optic modulator: performance and limitation. Appl. Phys. B 39: 149. Mertz, J. and Heidmann, A. (1993). Photon noise reduction by controlled deletion techniques. J. Opt. Soc. Am. B 10: 745. Taubman, M.S., Wiseman, H., McClelland, D.E., and Bachor, H.-A. (1995). Quantum eﬀects of intensity feedback. J. Opt. Soc. Am. B 12: 1792. Machida, S. and Yamamoto, Y. (1989). Observation of amplitude squeezing from semiconductor lasers by balanced direct detectors with a delay line. Opt. Lett. 14: 1045. Walls, D. (1983). Squeezed states of light. Nature 306: 141. Slusher, R.E., Hollberg, L.W., Yurke, B. et al. (1985). Observation of squeezed states generated by four-wave mixing in an optical cavity. Phys. Rev. Lett. 55: 2409.

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33 Shelby, R.M., Levenson, M.D., Walls, D.F. et al. (1986). Generation of

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squeezed states of light with a ﬁber-optics ring interferometer. Phys. Rev. A33: 4008. Wu, L.-A., Xiao, M., and Kimble, H.J. (1987). Squeezed states of light from an optical parametric oscillator. J. Opt. Soc. Am. B 4: 1465. (a) Kimble, H.J. and Walls, D.F. (ed.) (1987). Squeezed states of the electromagnetic ﬁeld. J. Opt. Soc. Am. B 4 (10 Special Issue); (b) Loudon, R. and Knight, P.L. (ed.) (1987). Squeezed light. J. Mod. Phys. Opt. 34 (Special Issue); (c) Yamamoto, Y. and Haus, H.A. (1986). Preparation, measurement and information capacity of optical quantum states. Rev. Mod. Phys. 58: 1001. Kok, P., Munro, W.J., Nemoto, K. et al. (2007). Linear optical quantum computing. Rev. Mod. Phys. 79: 135. Galvez, E.J. (2014). Resource Letter SPE-1: Single-photon experiments in the undergraduate laboratory. Am. J. Phys. 82 (11): 1019. https://doi.org/10.1119/1 .4872135.

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2 Classical Models of Light The development of optics was driven by the desire to understand and interpret the many optical phenomena that were observed. Over the centuries more and more reﬁned instruments were invented to make observations, revealing ever more fascinating details. All the optical phenomena investigated during the nineteenth century were observed either directly or with photographic plates. The spatial distribution of light was the main attraction. The experiments measured long-term averages of the intensity; no time-resolved information was available. The formation of images and the interference patterns were the main topics of interest. In this context the classical optical models, namely, geometrical optics, dating back to the Greeks and developed ever since, and wave optics, introduced through Huygens’ ideas and Fresnel’s mathematical models, were suﬃcient. Excellent technology emerged, largely in the form of imaging devices of any kind, from microscopes through photography to telescopes. In parallel spectroscopy emerged as a tool for atomic and molecular physics. A revolution came with the invention of the laser. This changed optics in several ways. It allowed the use of coherent light, bringing the possibilities of wave optics to their full potential, for example, through holography. Lasers allowed spectroscopic investigations with unprecedented resolution, testing the quantum mechanical description of the atom in even more detail. The fast light pulses generated by lasers allowed the study of the dynamics of chemical and biological processes. In addition, a range of unforeseen eﬀects emerged, for example, the high intensities resulted in non-linear optical eﬀects. One fundamental assumption of the wave model, namely, that the frequency of the light remained unchanged, was found to be seriously violated in this new regime. The sum, diﬀerence, and higher harmonic frequencies were generated. Finally, optics was found to be one of the best testing grounds of quantum mechanics. The combination of sensitive photoelectric detection with reliable, stable laser sources meant not only that interesting quantum mechanical states could be produced but also that quantum mechanical eﬀects had to be accounted for in technical applications. Light is now almost exclusively observed using photoelectric detectors, which can register individual events or produce photocurrents. A description of these eﬀects requires the ﬁeld of statistical optics. We need a description of the beam of light that emerges from a laser. A laser beam is a well-conﬁned, coherent electromagnetic ﬁeld oscillating at a A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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2 Classical Models of Light

frequency 𝜈 or angular frequency 𝜔 = 2π𝜈. This frequency is constant for all times and locations and can only be changed by the interaction of light with non-linear materials. Furthermore, the beam propagates and maintains its shape unperturbed by diﬀraction. The beam carries information in the form of modulation, spatial in the form of images or temporal in the form of signals or pulses. Transmitting information in various forms, such as intensity, phase, polarization, and spatial changes, is the biggest use of optics. In this chapter we develop classical models suited to describing light. In the following chapters we will see how these must be modiﬁed to account for quantum mechanical eﬀects.

2.1 Classical Waves 2.1.1

Mathematical Description of Waves

An electromagnetic wave in an isotropic insulating medium is described by the wave equation 1 𝜕2 E(r, t) = 0 c 𝜕t 2 with a solution for the electric ﬁeld vector E(r, t), given by ∇2 E(r, t) −

(2.1)

E(r, t) = E0 [𝛼(r, t) exp(i2π𝜈t) + 𝛼 ∗ (r, t) exp(−i2π𝜈t)] p(r, t)

(2.2)

The wave oscillates at a frequency 𝜈 in Hz. For now we consider only a single-frequency component, but in general the solution can be an arbitrary superposition of such components. The electric ﬁeld vector is perpendicular to the local propagation direction. The orientation of the electric ﬁeld on the plane tangential to the local wave front is known as the polarization and is described by the vector p(r, t). The dimensionless complex amplitude 𝛼(r, t) of the wave can be written as a magnitude 𝛼0 (r, t) and a phase 𝜙(r, t): 𝛼(r, t) = 𝛼0 (r, t) exp{i𝜙(r, t)}

(2.3)

The magnitude 𝛼0 (r, t) will change in time when the light is modulated or pulsed. It will also vary in space, describing the extent of the wave. The wave will typically be constant in space for dimensions of the order of several wavelengths. The direction and shape of the wave front are determined by the phase term 𝜙(r, t), and the spatial distribution of the 𝜙(r, t) describes the curvature of the waves. Simple examples of speciﬁc solutions are monochromatic plane waves and spherical waves. These represent the extreme cases in terms of angular and spatial conﬁnement. The monochromatic plane wave is the ideal case in which the complex amplitude magnitude 𝛼0 is a constant and the phase is given by 𝜙(r) = −k ⋅ r

(2.4)

The wave propagates in a direction and with a wavelength 𝜆 represented by the wave vector k, where k∕k is a unit vector that provides the direction of the local

2.1 Classical Waves

wave front and k = 2π∕𝜆. The wavelength is related to the frequency via 𝜆 = c∕n𝜈, where n is the refractive index of the medium in which the wave propagates and c is the speed of light in the vacuum. Thus we can write 𝛼(z) = 𝛼0 exp{−ikz}

(2.5)

where we have assumed that the wave propagates in the z direction. The wave has a constant amplitude in any plane perpendicular to this optical axis. Substituting Eq. (2.5) into Eq. (2.2) and assuming 𝛼0 as real, we obtain the familiar solution E(z, t) = E0 cos(𝜈t − kz) p(z, t)

(2.6)

Linear polarization means that p(z, t) has a ﬁxed direction perpendicular to k. Circular polarization means that in any one location p(z, t) the direction is rotating in time in the plane perpendicular to k. Equation (2.6) represents a wave that has no angular spread. In contrast the spherical wave originates from one singular point. All its properties are described by one parameter r, the distance to its point of origin. The amplitude magnitude reduces with 1∕r, and the phase is constant on spheres around the origin: 𝛼 (2.7) 𝛼(r, t) = 𝛼(r) = 0 exp{ikr} r 2.1.2

The Gaussian Beam

Both of these models are not physical, they are only crude approximations to a beam of light emitted by a real laser. Such a beam is conﬁned to a well-deﬁned region and should be as close as possible to the ideal of a spatially localized but non-divergent wave. It has to be immune to spreading by diﬀraction and therefore has to be a stable solution of the wave equation. An example of such solutions is the Gaussian modes. The simplest of these is circularly symmetric around the optical axis, thus depending only on the radial distance 𝜌 from the beam axis: 𝜌 2 = x2 + y 2 This special solution is the Gaussian beam or fundamental mode, with a complex amplitude distribution of ( ) i𝛼 −ik𝜌2 𝛼(r, t) = 0 exp (2.8) exp(ikz), q(z) = z + iz0 q(z) 2q(z) where z0 is a constant. The Gaussian beam is a good spatial representation of a laser beam. Laser light is generated inside a cavity, constructed of mirrors, by multiple interference of waves and simultaneous ampliﬁcation. These processes result in a well-deﬁned, stable ﬁeld distribution inside the cavity, the so-called cavity modes [1–3]. The freely propagating beam is the extension of this internal ﬁeld. It has the remarkable property that it does not change its shape during propagation. Both the internal and external ﬁelds are described by Eq. (2.8). By taking

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the square of the complex amplitude 𝛼(r, t), the intensity is found, resulting in a Gaussian transverse intensity distribution: ( ( ) ) W0 2 −2𝜌2 exp I(𝜌, z) = I0 W (z) W 2 (z) √ ( )2 z (2.9) and W (z) = W0 1 + z0 The properties of this Gaussian beam are the following: The shape of the intensity distribution remains unchanged by diﬀraction. The size of the beam is given by W (z). The beam has its narrowest width at z = 0, the so-called waist of the beam. It corresponds to the classical focus, but rather than pointlike, the Gaussian beam has a size of W0 . This value gives the radial distance where the intensity has dropped from I0 = I(0, 0) at the centre to I(W0 , 0) = I0 ∕e. At the waist the phase front is a plane, and 𝜙(𝜌) is a constant. Moving away from the waist, the beam radius W (z) expands, the intensity at the axis reduces, and, at the same time, the wave front curves. Along the optical axis the Gaussian beam scales with the parameter z0 , which is known as the Rayleigh length: z0 =

πW02

𝜆 It describes the point where the intensity on axis has dropped to half the value √ at the waist, I(0, z0 ) = 12 I0 , and the beam radius has expanded by a factor of 2. This can be regarded as the spatial limit to the focus region, and the value 2z0 is frequently described as the depth of focus or as the confocal parameter. Far away from the waist, for z ≫ z0 , the Gaussian beam has essentially the wave front of a spherical wave originating at the waist. Obviously, the Gaussian beam is still centred on the beam axis and has only a ﬁnite extent W (z). At these large distances, the size of the beam W (z) depends linearly on z; the beam has a constant divergence angle of Θ=

2𝜆 2πW0

(2.10)

In addition to the wave front curvature, a gradual phase shift occurs on the beam axis between a uniform plane wave with identical phase at z = 0 and the actual Gaussian beam. The phase diﬀerence increases to π∕4 at z = z0 and asymptotically approaches π∕2 (Table 2.1). Table 2.1 Properties of a Gaussian beam. z=0

z = z0

Beam waist

W = W0

√ W = 2W0

W (z) = z∕z0 W0

Intensity

I0

I0 ∕2

I0 (z0 ∕z)2

Wave front

Plane wave

Curved wave front

Spherical wave

Phase retardation

0

π∕4

π∕2

Location

z ≫ z0

2.1 Classical Waves

Figure 2.1 Gaussian beam propagation along the z axis.

I(y) I(x)

z W(z) W0 = W(z = 0) x x

As known from experiments with laser beams, the Gaussian beam is very robust in regard to propagation through optical components. Any component that is axially symmetric and the eﬀect of which can be described by a paraboloidal wave front will keep the Gaussian characteristic of the beam. This means the emerging beam will be Gaussian again, however, now with a new position z = z′ of the waist. All spherical lenses and spherical and parabolic mirrors transform a Gaussian beam into another Gaussian beam. Another important consequence of this transformation is the fact that Gaussian beams are self-reproducing under reﬂection with spherical mirrors (Figure 2.1). However, the Gaussian beam (Eq. (2.8)) is not the only stable solution of the paraxial wave equation. There are many other spatial solutions with non-Gaussian intensity distributions that can be generated by a laser and propagate without change in shape. One particular set of solutions is the Hermite–Gaussian beams. These beams have the same underlying phase, except for an excess term 𝜁l,m (z), which varies slowly with z. They have the same paraboloidal wave front curvature as the Gaussian beam, which matches the curvature of spherical optical components aligned on the axis. Their intensity distribution is ﬁxed and scales with W (z). The distribution is two-dimensional (2D), with symmetry in the x and y directions and usually not with a maximum on the beam axis. The intensity on the beam axis decreases with the same factor W (z) as the Gaussian beam along the z axis. The various solutions can be represented by a product of two Hermite–Gaussian functions, one in the x and the other in the y direction, which directly leads to the following notation: (√ ) ( ) (√ ) W0 2y 2x Gm Gl 𝛼l,m (x, y, z) = 𝛼0 Al,m W (z) W (z) W (z) )} { ( kr2 for l, m = 0, 1, 2, … − 𝜁l,m (z) × exp −i kz + 2R(z) (2.11) Here Al,m is a constant and Gl (u) and Gm (u) are standard mathematical functions, the Hermite polynomials of order l, m, respectively, which can be found in tables [2, 4]. In particular, G0 (u) = exp{−u2 ∕2}, and thus the solution 𝛼0,0 (x, y, z) is identical to the Gaussian beam described above. This set of solutions forms a complete basis for all functions with paraboloidal wave fronts. Any such beam can be described by a superposition of Hermite–Gaussian beams. All these components propagate together, maintaining the amplitude distribution. The only

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2 Classical Models of Light

00

10

20

30

01

11

21

31

02

12

22

33

Figure 2.2 Intensity distributions 𝛼l,m (x, y)2 of Hermite–Gaussian beams. These modes are labelled transverse excited mode TEMlm with the parameters l and m describing the horizontal and vertical order of the modes. Source: https://www.wikipedia.org/.Licensed Under CC BY SA 3.0.

component with circular symmetry is the Gaussian beam 𝛼0,0 (x, y, z). However, it is possible to describe other circularly symmetric intensity distributions. Consider, for example, the superposition of two independent beams with 𝛼1,0 (x, y, z) and 𝛼0,1 (x, y, z) and equal intensity; if these beams have random phase, their interference can be neglected and the sum of the intensities results in a donut-shaped circularly symmetric function, occasionally observed in lasers with high internal gain (Figure 2.2). 2.1.3

Quadrature Amplitudes

The description of a wave as a complex amplitude (Eq. (2.3)), with amplitude 𝛼0 (r, t) and phase distribution 𝜙(r, t), is useful for the description of interference and diﬀraction. As shown above, the phase distribution 𝜙(r, t) describes the shape of the wave front as well as the absolute phase, in respect to a reference wave. However, it is useful to separate the two to consider the absolute phase independently from the phase distribution or wave front curvature. An equivalent form of Eq. (2.2) can be given in terms of the quadrature amplitudes, X1 and X2, the amplitudes of the associated sine and cosine waves: E(r, t) = E0 [X1(r, t) cos(2π𝜈t) + X2(r, t) sin(2π𝜈t)] p(r, t)

(2.12)

The quadrature amplitudes are proportional to the real and imaginary parts of the complex amplitude: X1(r, t) = 𝛼(r, t) + 𝛼 ∗ (r, t) X2(r, t) = −i[𝛼(r, t) − 𝛼 ∗ (r, t)]

(2.13)

2.1 Classical Waves

Figure 2.3 Phasor diagram (a) for a constant single wave and (b) the representation of interference of several coherent waves.

X2

X2

E

α total

ϕ (a)

E

X1

(b)

X1

In this notation the absolute phase 𝜙0 of the wave is associated with the distribution of the amplitude between the quadratures X1 and X2: ( ) X2 (2.14) 𝜙0 = tan−1 X1 For X2 = 0 the wave is simply a cosine wave (Eq. (2.6)). We can regard this wave arbitrarily as the phase reference. A common description of classical waves whenever interference or diﬀraction is described is phasor diagram, a 2D diagram of the values X1 and X2 as shown in Figure 2.3a. In such a diagram, which represents the ﬁeld at one point in space and time, each wave corresponds to one particular vector of length 𝛼 from the origin to the point (X1, X2), and this vector could be measured as the complex amplitude averaged over a few optical cycles of length 1∕𝜈. The magnitude of the wave is given by the distance 𝛼 of the point from the origin, and the relative phase 𝜙0 is given by the angle with the X1 axis, since the absolute phase is arbitrary. It is useful to choose 𝛼 as real (𝜙 = 0), which means the vector is parallel to the X1 axis. If several waves are present at one point at the same time, they will interfere. Each individual wave is represented by one vector; the total ﬁeld is given by the sum of the vectors, as shown in Figure 2.3b. These types of diagrams are a useful visualization of the interference eﬀects. After the total amplitude 𝛼total has been determined, we can again select a phase for this total wave and choose 𝛼total as being real. All forms of interference can be treated this way and we obtain a value for the average, or DC, value of the optical amplitude. In addition, the wave can have ﬂuctuations or modulations in both amplitude and phase. Such variations correspond to ﬂuctuations in the quadrature values. We can describe the time-dependent wave with the complex amplitude 𝛼(t) = 𝛼 + 𝛿X1(t) + i 𝛿X2(t)

(2.15)

In the limit 𝛼 ≫ 𝛿X1(t), which means for waves that change or ﬂuctuate only very little around a ﬁxed value 𝛼, we see that 𝛿X1(t), 𝛿X2(t) describe the changes in the amplitude and phase, respectively. This is the regime that we use to describe beams of light detected by detectors that produce a photocurrent. We will call them the amplitude and phase quadratures. The direct link between variations in the quadrature values and those of the ﬁeld magnitude and phase is shown diagrammatically in Figure 2.4. 2.1.4

Field Energy, Intensity, and Power

The energy passing through a small surface area element dx, dy around the point x, y in a short time interval dt around the time t is obtained by integrating the

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Δϕ

X1 ΔX1

E

ΔE

ΔX2

t

X2

Figure 2.4 Fluctuations shown in phase space: the ﬁeld changes in time and moves from point to point in the phasor diagram. This corresponds to waves with diﬀerent amplitude and phase.

square of Eq. (2.2). Note that because the ﬁeld is propagating in the z direction, this is equivalent to the energy in the volume element dx, dy, c dt. We get (x, y, t) = E02 2 𝛼 ∗ 𝛼 dt dx dy

(2.16)

where we have assumed that dx dy dt is suﬃciently small that 𝛼 does not vary over the interval, but that dt ≫ 1∕Ω is satisﬁed and the time average will be over a large number of optical cycles. As a result only the phase-independent terms survive. The intensity of the beam is I(x, y) =

∫one second

E02 2 𝛼 ∗ 𝛼 dt

(2.17)

What is normally measured by a photodetector is the power P of the light, the total ﬂux of energy reaching the detector in a second, integrated over the area of the detector: P=

∫

I(x, y) dx dy

(2.18)

The energy in a small volume element (Eq. (2.16)) can also be expressed in terms of the quadratures as (x, y, t) = E02 (X12 + X22 ) dt dx dy where we have used Eq. (2.13). The parameter E02

(2.19) is proportional to the frequency

of the light, 𝜈, speciﬁcally

ℏ2π𝜈 (2.20) 2 where ℏ is Planck’s constant. It is the optical power P(t) in Eq. (2.18) that is measured in most experiments, and from this we infer the amplitude, the phase, and the ﬂuctuations of the beam. E02 =

2.1.5

A Classical Mode of Light

The idea of a phasor diagram can now be combined with the concept of spatial modes as described in Eq. (2.3). Let us concentrate on the simplest type of beam, a Gaussian beam. Equation (2.8) relates the relative amplitude and phase at any one point back to a point at z = 0 at the centre of the waist. Once the size W0 is given, the entire spatial distribution is well deﬁned. We can assign one overall optical

2.1 Classical Waves

P

W0

z ν, ϕ0

I0

Figure 2.5 Graphical representation of the parameters of a Gaussian mode of light.

power, or peak intensity I0 , or one overall phase to the entire spatial distribution. Thus the entire laser beam can be described by just six parameters. The spatial shape of the laser beam with its very narrow focus, see Figure 2.5 is the key for many applications of laser light. It allows us to deliver very high power to a very small volume, which is the key to most applications for cutting, ablating, or ionizing materials and many of the medical applications. The small size of the focus, limited by diﬀraction to a about half of a wavelength, allows high spatial resolution in microscopy or the coupling of light into a single-mode optical ﬁbre. Even more magically the high gradients in intensity in a laser focus can be used to exert forces onto particles that move inside the laser beam, and this can be used to trap particles, alas individual atoms or ions inside the focus, the physics behind optical tweezers invented by 2018 Nobel laureate A. Ashkin [5, 6]. This itself is the foundation of many applications in biology and also of many new quantum experiments with laser-cooled atoms, single atoms, and ions. Such a mode of light is strictly speaking constant in time, at least for short times. It corresponds to an idealized laser that has a perfectly stable, time-independent output, with an intensity distribution that is not changing at all in time. More realistically the laser output will not be perfectly constant, but there will be ﬂuctuations of the average intensity, the amplitude and phase, and all parameters listed in Table 2.2. Experiments conﬁrm that there are always ﬂuctuations (see Figure 1.2). They can be stabilized to some degree by technical means, but ultimately quantum mechanics limits how small the ﬂuctuations can be. This description of the light as a mode, which can be quantized, will be the key to an elegant quantum model. Table 2.2 The parameters of a mode of light. Parameters required to describe one mode of light

Intensity of the mode

I0

Frequency of optical ﬁeld

𝜈

Absolute phase of this mode

𝜙0

Direction of propagation

z

Location of the waist (z = 0)

r0

Waist diameter of the beam

W0

Direction of polarization

P

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2.1.6

Light Carries Information

Light is used to carry information. In many cases we think about images, where the information is encoded in the spatial domain, transverse to the propagation of light. In this case the magnitude and the phase of the light are a function of location r, the image is in the transverse x − −y plane, and the image can change along the z axis of propagation. Quite frequently this is constant in time – until the input image is changed. In many other situations we want to use light to measure some parameter or to communicate information from one place to another. In both cases we need a beam of light that can carry information, a simple constant beam of light that can be modulated in time. The physical characteristics of a freely propagating laser beam deﬁne such a mode of light. The shape of this mode is well determined. In all these cases the propagation of the light, be it in free space, through a complex apparatus, through an optical ﬁbre, etc., can be described by the mode of light, the simplest case being the Gaussian beam. We will select the appropriate degree of freedom that is being modulated, and it is useful to describe the entire beam, including all its time and frequency components, as one entity, as one mode. The beam can change in its appearance while it propagates through the apparatus – but it remains one entity. We will ﬁnd in Chapter 4 that this is the most convenient and practical way to quantize the light. However, the temporal dependence is most important, since it contains the information in the form of modulations. We describe this with a time-varying amplitude 𝛼(t) for this mode. The various aspects of the measurement and manipulation of laser beams occur at diﬀerent timescales and at their corresponding frequencies. Let us introduce the nomenclature for a set of natural and technical frequencies and timescales (Figure 2.6).

Δtusp

Δtsamp

Δtmeas

One sample Integrate interval for variance and average

Δtosc = 1/ Ωmod

Period of oscillation in periodic signal

Δtrec

T

Interval between recordings

Envelope s(t)

2π νopt

Ωsamp

Ωmod, Ωsens

Ωenv

High

Sampling of photon flux frequency

Modulator and signal carrier frequencies

Signal content frequency

CW

Long

Low

Figure 2.6 Deﬁnition of diﬀerent timescales and frequencies used in measuring and manipulating light.

2.1 Classical Waves

The natural starting point is the optical frequency 2π𝜈L of a laser beam. This is the highest frequency, typically 1015 Hz, and all the modulation frequencies Ωmod will be lower. We can sample the photon ﬂux at very high rates with the sampling frequency Ωsamp , which is between 2π𝜈L and Ωmod . The shortest technical timescale is Δtsamp = 1∕Ωsamp , which in many situations is achieved through the fast gating of either a photon counter or an analogue-to-digital converter. We require a number of samples to evaluate a variance for the ﬂuctuations or an average value for the photon ﬂux, which requires the time Δtmeas . Many such data points are required to impose a periodic signal at Ωmod or measure a periodic signal Ωsens from an external source. This requires a time interval Δtosc = 1∕Ωmod . In a sensing application we might want to take a whole series of readings, taken at time intervals Δtread , and we will have a full evaluation of the signal and noise for a considerable period of time T. In most cases we can assume that the performance of the instrument remains constant over a long time, that the noise is stationary and we can use samples of the noise to describe the performance, and that the laser parameters, such as optical power and optical frequency, are well stabilized. In this case we can record the envelope function from many readings and analyse the behaviour of a parameter we are sensing. We might ﬁnd periodic behaviour and typical envelope frequencies Ωenv . The absolute values for these timescales and frequencies can vary by several orders of magnitude and cannot be stated in general terms. The instruments can contain anything from femtosecond pulses to a constant laser. We can have short pulses of light where each pulse carries one piece of information. This will be particularly important when we think about individual photons in each pulse. Here the typical timescale Δtmeas for each measurement can be shorter than the time between two pulses, and with modern equipment this can be as short as a few picoseconds. In the other extreme we can have a continuous wave (CW) laser that is steady, and we measure for a time interval T and take many readings. The modulation and sensing frequencies can vary from single hertz for sensors such as environmental or astronomical detectors to gigahertz in communication systems. Still, there is a great deal of commonality that we will use in describing the properties of light, classical and quantum. Finally, there is the option of quasi-CW communication where we use trains of pulses with high repetition frequency, in particular to achieve high peak powers, and the measurement averages over a number of pulses. For the modulation and sensing, we can choose several degrees of freedom, such as phase, frequency, polarization, position and mode shape, etc., and in this way pack more information into one beam. Technically this is described as multiplexing. One example is the modulation of the polarization, which has 2 degrees of freedom, and we require two orthogonal polarization axes for one single beam. More degrees of freedom are possible, for example, additional optical frequencies or several additional co-propagating spatial modes. In addition we will encounter situations where we have several beams within one apparatus, for example, created by a series of beamsplitters. In all these cases we can deﬁne the available degrees of freedom, ﬁnd the corresponding orthogonal functions that describe

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these optical ﬁelds and their modulations, and describe this apparatus with a set of multiple modes. 2.1.7

Modulations

Information carried by a beam of light can be described in the form of modulation sidebands. Note that the equations used here are identical to those describing radio waves where amplitude modulation (AM) and frequency modulation (FM), or technically speaking phase modulation (PM), are the familiar, standard techniques. Many of the technical expressions, such as carrier for the fundamental frequency components and sidebands, are taken directly from this much older ﬁeld of technology. A beam of light that is amplitude modulated by a fraction M at one frequency Ωmod can be described by ( ) M 𝛼(t) = 𝛼0 1 − (1 − cos(2πΩmod t)) exp(i2π𝜈L t) 2) ( M exp(i2π𝜈L t) = 𝛼0 1 − 2 M + 𝛼0 [ exp(i2π(𝜈L + Ωmod )t) + exp(i2π(𝜈L − Ωmod )t) ] (2.21) 4 We see that the eﬀect of the modulation is to create new frequency components at 𝜈L + Ωmod and 𝜈L − Ωmod . These are known as the upper and lower sidebands, respectively, whilst the component remaining at frequency 𝜈L is known as the carrier (see Figure 2.7a). For completely modulated light, which means M = 1, α X2

νL − Ω2 νL − Ω1 νL + Ω1 (a)

X2

=

AM

νL + Ω2 ν

X2

Ω1

+

Ω2

FM

α

α X1

X1

(b)

δX2

δX1

X1 Ω1

Ω2 (c)

Ω

Figure 2.7 Three diﬀerent graphical representations of amplitude (AM) and frequency (FM) modulation at frequencies Ω1 and Ω2 , respectively. Part (a) shows the sidebands of the optical frequency at 𝜈L ± Ω1 and 𝜈L ± Ω2 , (b) shows the same modulations in the time domain and in a phasor diagram, and (c) shows the complete phasor diagram in frequency space.

2.1 Classical Waves

the sidebands have exactly 12 the amplitude of the fundamental component. The sidebands are perfectly correlated, and they have the same amplitude and phase, which means the two beat signals, generated by the sidebands at 𝜈L + Ωmod and 𝜈L − Ωmod beating with the carrier, are in phase. A second representation is in the time domain through oscillating vectors 𝛿𝛼 in the phasor diagram Figure 2.7b. The two vectors, 𝛿𝛼+Ω and 𝛿𝛼−Ω , rotate in opposite directions with angular frequency 2πΩ. A third representation is in frequency space in the phasor diagram shown in Figure 2.7c. Notice that for a given modulation depth M, the sideband amplitudes scale with the amplitude of the carrier, 𝛼0 . Thus even very small modulation depths, M, can result in signiﬁcant sideband amplitude if the carrier amplitude is large. The detected intensity is given by ( )2 M (2.22) I(t) = I0 |𝛼(t)|2 = I0 1 − (1 − cos(2πΩmod t)) 2 and oscillates at the frequency Ωmod around the long-term average I0 . The resulting photocurrent has two components, a long-term average iDC and a constant component at frequency Ωmod , labelled i(Ωmod ). A common (and useful) situation is when the modulation depth is very small, M ≪ 1. Then the resulting intensity modulation is linear in the modulation depth: I(t) ≈ I0 [1 − M(1 − cos(2πΩmod t)) ]

(2.23)

A light beam with modulated phase can be described in a similar way. The amplitude can be represented by 𝛼(t) = 𝛼0 exp[iM cos(2πΩmod t)] exp(i2π𝜈L t)

(2.24)

Clearly the intensity will not be modulated as |𝛼(t)|2 = |𝛼0 |2 . Expanding 𝛼t we obtain ) ( M2 2 𝛼(t) = 𝛼0 1 + iM cos(2πΩmod t) − cos (2πΩmod t) + · · · exp(i2π𝜈L t) 2 [( ) M2 = 𝛼0 1 − + · · · exp(i2π𝜈L t) 4 ) ( M + · · · (exp(i2π(𝜈L + Ωmod )t) + exp(i2π(𝜈L − Ωmod )t)) +i 2 ( 2 ) M − + · · · (exp(i2π(𝜈L + 2Ωmod )t) + exp(i2π(𝜈L − 2Ωmod )t)) 8 +· · · ]

(2.25)

Suppose we measure the quadrature amplitudes (X1 and X2) of this ﬁeld relative to a cosine reference wave (i.e. we take 𝛼 to be real). The X2 quadrature will have pairs of sidebands separated from the carrier by ±Ωmod , ±3Ωmod , … The X1 quadrature on the other hand will have sidebands separated from the carrier by ±2Ωmod , ±4Ωmod , … Again a simple, and useful, situation is for the modulation depth to be very small, M ≪ 1. The amplitude is approximately [ ] M 𝛼(t) ≈ 𝛼0 exp(i2π𝜈L t) + i [exp(i2π(𝜈L + Ωmod )t) + exp(i2π(𝜈L − Ωmod )t)] 2 (2.26)

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and the sidebands will only appear on measurements of the X2 quadrature. Exact expressions for larger M involve nth-order Bessel functions [7]. The phasor diagram corresponding to this double sideband approximation is shown in Figure 2.7b. The sidebands have the same magnitude. The relative phase of the amplitudes is such that the two resulting beat signals have the opposite sign and the beat signals cancel. Thus, as we have seen, phase modulation cannot be detected with a simple intensity detector. Another way of saying this is that the sidebands for phase modulation are perfectly anti-correlated. The treatment of frequency modulation proceeds in a similar way. Any information carried by the mode can be expressed as a combination of amplitude and phase modulation. Any modulation, in turn, can be expressed in terms of the corresponding sidebands.

2.2 Optical Modes and Degrees of Freedom 2.2.1

Lasers with Single and Multiple Modes

A single mode is only the simplest option for a laser beam output. The resonator inside the laser will have several resonance conditions. Optical waves with diﬀerent frequencies can be ampliﬁed, and also diﬀerent spatial amplitude and phase distributions can be in resonance. Each of these waves, with their speciﬁc set of parameters, can be considered as a diﬀerent mode, with distinctively diﬀerent values of the parameters listed in Table 2.2, and in the case of spatial modes diﬀerent amplitude and phase distributions. The details depend crucially on the geometry of the laser resonator [2] and the spectral response of the gain medium, and some of the details are summarized in Chapters 5 and 6. In many cases the laser might produce an output in several of these modes simultaneously – with all of these modes competing for gain. Examples of other laser modes are beams with the same shape but diﬀerent carrier frequencies, spaced in regular intervals and all linked by a common phase. This concept, known as a frequency comb, won the inventors a Nobel prize [8] and is the basis of many new applications in optical metrology of time signals and of new forms of spectroscopy and non-linear optics. Short laser pulses can be thought of as a frequency comb with a speciﬁc envelope function determining the number and strength of the individual components. These envelope functions are shaped to be the eﬀects inside the laser, or they can also be modiﬁed outside by instruments that have a frequency-dependent response, or dispersion, or with active means such as electro-optic pulse shapers. Frequently the aim of engineering is to produce a single-mode laser, which can be described by the simple set of parameters, with a waveform such as stated in Eq. (2.8). However, not every single-mode beam has to be this simple. A single-mode beam can be transformed, for example, with lenses or beamsplitters, or we could delete a part the beam, and during propagation diﬀraction would change the intensity distribution, possibly quite dramatically. The underlying idea of the mode remains the same in such a mode transformation, and the relevant question remains: can we ﬁnd one single mathematical description that contains all properties of the laser beam? Clearly, yes we can.

2.2 Optical Modes and Degrees of Freedom

This is correct for the spatially transformed single-mode beam. Another example is a carefully crafted pulsed laser system where techniques such as mode locking produce trains of pulses as short as femtoseconds, with correspondingly wide frequency combs, where all components of the comb are phase and amplitude linked. This entire ensemble of spectral components can be described as a single mode with one simple set of parameters. In this case the mode looks complex, in space or frequency, but we can still speak of one mode since all the information is still contained as a modulation of this speciﬁc entity. In each case we have to consider the number of degrees of freedom that are available, not the apparent complexity of the beam of light. 2.2.2

Polarization

Let us consider beams that require a description by two or modes. One clear example is the polarization of an otherwise single-mode beam. The polarization properties can be described by separately considering the amplitude of the ﬁeld oscillating in two transverse directions; let us call them horizontal H and vertical V . These are two orthogonal coordinates that could be in any physical directions. Mathematically we require two orthogonal basis vectors, H and V. A 2D complex vector 𝛼 p(r, t) with magnitudes |𝛼H |, |𝛼V | and phases 𝜙H , 𝜙V for the two directions provides the full description: 𝛼 p(r, t) = |𝛼H (r, t)| ei𝜙H H + |𝛼V (r, t)| ei𝜙V V = (𝛼H (r, t), 𝛼V (r, t))

(2.27)

The important point is that the properties, and also the information, in the two coordinates are completely independent. For example, a beam can have all its intensity in one or the other coordinate V and H, or it can have a shared amplitude in both. We would call a beam with 𝛼V = 0 horizontally polarized and with 𝛼H = 0 vertically polarized. A beam with 𝛼V = 𝛼H is diagonally polarized, D, while a beam with 𝛼V = −𝛼H is anti-diagonally polarized, A, orthogonal to D. We can make beams with the amplitude 𝛼V = i𝛼H , which we call right-hand circularly polarized (R), and with 𝛼V = −i𝛼H , which are left-hand circularly polarized (L). All of these descriptions, or projections on diﬀerent basis vectors, are equivalent, and we can transform the projection between any of them without gaining or losing information. We can choose the basis vectors that match best to the experiment, resulting in the simplest description. In all cases the projection is based on a 2D vector, and we have two orthogonal modes, both travelling within the same laser beam. Any transformation of the polarization in the apparatus, for example, through polarizers, wave plates as shown in Figure 2.8, through ﬁlters, glass plates, etc., can be described by 2D matrices Ξ, also known as Jones matrices, described in most textbooks about optics, such as [3]: 𝛼 p (r, t)out = Ξ 𝛼 p (r, t)in = Ξ1 Ξ2 Ξ3 … 𝛼 p (r, t)in

(2.28)

Here each component in the apparatus is described by a speciﬁc matrix Ξi , i = 1, 2, …, and light propagating through a sequence of such components can be described by multiplication of the individual matrices Ξ1 ,Ξ2 ,Ξ3 , … Examples for such Jones matrices include Ξprop for simple propagation, ΞH for a linear

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λ 2

V

PBS

R L

D

H

(a)

λ 4

A

(b)

PBS

(c)

PBS

Figure 2.8 Three examples of the polarization state of a single laser beam. The diﬀerent projections used are (a) horizontal H and vertical V, (b) diagonal D and anti-diagonal A, and (c) left-hand circular L and right-hand circular R polarization. In each case two detectors are used, each detecting one of the polarization components. The transformation from one basis to another is achieved by a combination of polarizing beamsplitter (PBS) and either quarter-wave or half-wave plate.

polarizer in direction H, ΞV for a linear polarizer in direction V , and Ξ𝜙 for a wave retarder by a phase angle 𝜙: ( ( ( ( ) ) ) ) 1 0 1 0 0 0 1 0 , ΞH = , ΞV = , Ξ𝜙 = Ξprop = 0 1 0 0 0 1 0 ei𝜙 (2.29) Note that the order of the matrices is important and has to be the same as the order of components in the experiment. Interchanging components, such as placing a polarizer in a diﬀerent section of the apparatus, can have dramatic eﬀects. A very important point is that the beam can be modulated diﬀerently in the coordinates H and V , and as long as we detect the light in this basis, the information streams that we receive in these coordinates are completely independent. The mathematical statement of orthogonality between H and V is fully equivalent to having no crosstalk between the coordinates. We can use this fact to optimize our experiment. For example, by modulating only 𝛼H and not 𝛼V and by using two detectors, set for the H and V polarization, respectively, we can ﬁnely tune the optics until we ﬁnd no signal coming from the V detector. We then change the modulation to V and should ﬁnd no signal coming from the H detector. We now have established two independent communication channels. The laser beam can carry twice as much information using polarization modulation than in amplitude modulation of an unpolarized beam. This shows the distinction of a two-mode beam from a single-mode beam: it has 2 degrees of freedom, it contains more information, and we can consider the question of correlation between the two modes. For example, if we take a beam with pure horizontal modulation, 𝛼V = 0. That means we only use one single mode and we have no correlation between the information in H and V. If we detect the same beam in the D and A coordinates rotated at 45∘ , we share the modulation in the two components and we have a strong correlation between 𝛼D and 𝛼A . However, there is no additional information, and the beam can still be described by a single mode. No information is added in the choice of a diﬀerent projection. In contrast, if we have independent modulations in both H and V, and that means no correlation between 𝛼H and 𝛼V , then we require a two-mode description. Detection in H and V would give us two independent channels and no

2.2 Optical Modes and Degrees of Freedom

crosstalk. In this case the detection in the basis D and A would create a correlation and would also mix up the information from the two channels. We still have a two-mode case, just but not an optimal description. 2.2.2.1

Poincaré Sphere and Stokes Vectors

It is useful to mention an alternative description of polarization. With the complex Jones vector, we have four components, and we can describe the state of the polarization by two real parameters: the magnitude ratio 𝛼H ∕𝛼V and the phase diﬀerence 𝜙H − 𝜙V . Alternatively, we might characterize the state of polarization by two angles, 𝜒 and 𝜓, which represent the orientation and ellipticity of the polarization ellipse, see Figure 2.9. We can now describe the polarization in the form of a three-dimensional (3D) diagram, the Poincaré sphere where each state is given by a point on the surface of the sphere of unit radius, with the spherical coordinates given by (r = 1, Θ = 90 − 2𝜒, Φ = 2𝜓). Each point on the sphere is a diﬀerent polarization state. For example, points on the equator, that is with 𝜒 = 0, represent linear polarization. The two points on the equator, u1 with 2𝜓 = 0, and u2 with 2𝜓 = 90 degree, represent linear polarization in the H and V axis. The north pole u3 with 2𝜒 = 90 and south pole with 2𝜒 = −90 degree represent left-handed and right-handed circular polarized light respectively. Other states on the sphere represent the various forms of elliptical polarization. The coordinates u1 , u2 , u3 of a point on the Poincaré sphere in Cartesian coordinates are also known as the ﬁrst three Stokes parameters (S1 , S2 , S3 ). The fourth parameter S0 is simply proportional to the intensity of the beam. For the full details, see Ref. [3]. This four-component vector, introduced by Stokes, is the original description of polarization. It contains the same information as the Jones vector with its two complex components. We ﬁnd that the latter is mathematically more elegant and ﬁts to the two-mode description, which is so powerful in describing quantum optics systems. However, the Poincaré sphere is a practical way to illustrate the eﬀects of polarizing components. It will appear explicitly when we discuss the eﬀect of polarization squeezing, where this description u3

y χ ψ x

1 1

2χ Polarization ellipse (a)

2ψ u1 (b)

u2

Poincaré sphere

Figure 2.9 Graphical representation of polarization (a) through the two angles, 𝜒 and 𝜓, which represent the orientation and ellipticity of the polarization ellipse and (b) the Poincaré sphere, in both spherical and Cartesian coordinates. Source: From B.E.A. Saleh and M.C. Teich 2007 [3].

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provides the most direct insight in the quantum noise. In addition it allows us to see the correspondence between polarization and other two-mode systems such as the spin of an atom or the time evolution of a two-level system. This is a beautiful generalized graphical representation that has its place in any description of atomic physics or non-linear optics. Finally, it is worthwhile to see the link between the two modes, which exist within the one laser beam, and the two separate beams, which are normally required for the simultaneous detection of the modes. A common example in polarization is the use of the polarizing beamsplitter, which sends all the light including modulation in the H polarization to one output beam and all the light and modulation in the orthogonal V polarization into a separate beam, as shown in left-hand side of Figure 2.8. In this case we would associate the beam travelling to one detector with one mode and the other beam with the other mode. Similarly we could have a beam with two diﬀerent carrier frequencies and then use a grating to separate the two modes into separate beams and modes.

2.2.3

Multimode Systems

Is it strictly necessary to physically separate the modes to detect them? Not necessarily. A counterexample would be a laser beam that has two separate spatial modes, like two orthogonal Hermite–Gaussian modes (see Figure 2.2), which are detected by a photodetector with many pixels. We can detect the two spatial modes independently by recording the photocurrents from all pixels of the detector and then doing some clever recombination of the currents to represent the individual modes. At no point are the modes spatially separated, but they propagate together. However, the mathematical description is the same as for a set of modes, and beams, travelling completely separate through space. This example shows the importance and need for a multimode description. And there are situations where the number of modes can be much larger than two. It is possible to create beams of light that have N independent modes, or information channels. For example, this could be beams with N diﬀerent carrier frequencies, with N diﬀerent spatial modes, or with N separate beams created by an apparatus with a complex components that are sensitive to polarization, frequency, spatial shape, etc. In all these situations we can create a basis of N orthogonal modes, which we can modulate and detect individually. For the case of spatial modes, two modes described by 𝛼i (r, t) and 𝛼j (r, t) are orthogonal if ∫

𝛼i (r, t)𝛼j∗ (r, t) dr = 0

(2.30)

We can have N independent information channels with no crosstalk between any of them, or we can create, through mode transformation, correlations between the modes. The analysis of a beam using a large set of orthogonal basis vectors allows us to determine the smallest possible number N of modes that is required for a speciﬁc situation: ( 𝛼1 (r, t), 𝛼2 (r, t), … , 𝛼(N−1) (r, t), 𝛼N (r, t) )

(2.31)

2.3 Statistical Properties of Classical Light

In practice this leads to the mathematical challenge of ﬁnding the lowest possible value of N or the lowest possible number of independent degrees of freedom for any given situation. There are many applications of this type of multimode optical physics. In optical communication it allows us to increase the information ﬂux that is communicated. The commonly used terminology is multiplexing. In optical sensing it allows us to isolate exactly the degrees of freedom of the laser beam we wish to detect and suppress other eﬀects on the beam we do not want to detect. In coherent image detection and analysis, we can deﬁne an object that we wish to detect as an optical mode and the changes to this object as orthogonal modes. That means each of the higher-order modes contains information about one speciﬁc degree of freedom, for example, a translation, rotation, or stretching of the object in the image. These changes can be tracked by detecting only the speciﬁc higher-order mode. One interesting example is the translation of a Gaussian beam in the x and y coordinates. The information about this change is stored in the mode that represents the derivative of TEM00 in these two directions, respectively, and corresponds directly to the modes TEM01 and TEM10 , which are shown in Figure 2.2. By recording the intensity changes in exactly these two modes, the position of the centre of the beam can be isolated and tracked with high precision [9]. Many other examples are being developed for precision sensing.

2.3 Statistical Properties of Classical Light 2.3.1

The Origin of Fluctuations

The classical model so far assumes that light consists of a continuous train of electromagnetic waves. This is unrealistic: atoms emitting light have a ﬁnite lifetime; they are emitting bursts of light. Since these lifetimes are generally short compared to the detection interval td , these bursts will not be detected individually. The result will be a well-deﬁned average intensity, which will ﬂuctuate on timescales > td . Similarly in most classical, or thermal, light sources, there will be a large number of atoms, each emitting light independently. In addition, the atoms will move, which leads to frequency shifts, and they can collide, which disrupts the emission process and leads to phase shifts. All these eﬀects have two major consequences: 1. Any light source has a spectral line shape. The frequency of the real light is not perfectly constant; it is changing rapidly in time. The description of any light requires a spectral distribution that is an extended spectrum, not a delta function. 2. Any light source will have some intensity noise. The intensity will not be constant. The amplitude of the light will change in time. There will be changes in the amplitude due to the ﬂuctuations of the number of emitting atoms but also due to the jumps, or discontinuities, in the phase of the light emitted by the individual atoms.

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The ﬁrst point leads to the study of spectral line shapes, an area of study that can reveal in great detail the condition of the atoms. The study of spectra is a well-established and well-documented area of research. However, in the context of this guide, we will not expand on this subject. The relevant results are cited, where required. For the analysis of our quantum optics experiments, we are not concerned as much with the spectral distribution of the light. We are more interested in the noise, and thus the statistical properties, of intensity, amplitude, and phase. Only the intensity ﬂuctuations can be directly measured in the experiments. However, with the homodyne detector, which will be explained in Section 8.1.4, we have a tool to investigate both the phase and amplitude of the light. The measurement of the statistical properties of light can be carried out in two alternative ways: either through a measurement of the ﬂuctuations themselves in the form of temporal noise traces or noise spectra or, alternatively, through the measurement of coherence or correlation functions. While requiring entirely diﬀerent equipment, the two concepts are related, and in this chapter predictions for both types of measurement will be derived. The emphasis here is on the results and the underlying assumptions and not on the details of the derivation. Many of the details can be found in the excellent monographs by Loudon [7], Louisell [10], and Gardiner [11]. In Chapter 8 it will be shown how these statistical properties can be measured and how they have been used to distinguish between classical and quantum properties of the light. 2.3.1.1

Gaussian Noise Approximation

In many of our discussions, we will be assuming that the ﬂuctuations have a Gaussian distribution over time. This assumption is certainly true for classical random systems that obey the central limit theorem. They are generally applicable for systems where the individual ﬂuctuations are small and the overall ﬁeld evolves smoothly in time. For example, the amplitude varies in small steps, and the phase evolves smoothly at the timescales at which we are observing. Thermal noise is a typical example. We will also ﬁnd that the quantum noise of laser beams does obey this limit. In these cases the variance, Eq. (2.34), is the only parameter of importance, and it gives a complete description of the ﬂuctuations. In many applications the Gaussian approximation is valid. However, there are situations in classical and quantum optics where non-Gaussian processes are important. One simple example would be a beam where the amplitude, or phase, is digitally modulated through an on–oﬀ switch or controlled by a function with several discrete values. Here the switching can be fast and abrupt. Or consider an apparatus that contains a component with a non-linear transmission function, such that the ﬂuctuations are clipped or limited at a certain level. In these cases we would see non-Gaussian ﬂuctuations. They require more complex mathematical descriptions, and the statistics would include higher-order moments, not just the ﬁrst-order variance. Non-Gaussian systems are not only interesting but also increasingly important in applications. While they are not common in metrology, we will ﬁnd that they play a major role in digital communication, in optical logic systems, and in digital data storage and data processing. We will have to consider the validity of the

2.3 Statistical Properties of Classical Light

Gaussian approximation when any of these applications are the motivation for our experiments in quantum optics. 2.3.2

Noise Spectra

A very direct way of measuring the ﬂuctuations, or the noise, of the light is to record the time history of the intensity I(t) and to evaluate the variance ΔI 2 (t) for a certain detection time. The variance of a classical value is deﬁned as ΔI 2 (t) = ⟨(I(t) − ⟨I⟩)2 ⟩ = ⟨I 2 ⟩ − ⟨I⟩2 = Var(I(t))

(2.32)

where ⟨I⟩ is the average intensity and the brackets ⟨· · ·⟩ denote averaging over a detection interval of length td . Similarly, we deﬁne the root mean square (RMS) value as √ √ RMS(I(t)) = Var(I(t)) = ΔI 2 (t) (2.33) As we have seen in Section 2.1.7, information can be carried on a light beam as small amplitude or phase modulations. The variance gives limited information about the level of noise at a particular modulation frequency and thus how successfully information can be carried. More useful is the spectral variance or noise spectrum, which gives the noise power as a function of frequency from the carrier. The spectral variance of a variable y is given by the Fourier transform of the autocorrelation function for that variable, ∞

V (Ω) =

1 ei2πΩ𝜏 ⟨y(t)y(t + 𝜏)⟩d𝜏 2π ∫−∞

(2.34)

and can be evaluated experimentally using a spectrum analyser (see Chapter 7). Most commonly we will be evaluating the spectral variances of the intensity, y = 𝛼 ∗ 𝛼, and the quadrature amplitudes, y = 𝛼 + 𝛼 ∗ and y = i(𝛼 − 𝛼 ∗ ). In many practical situations we can write our variable in the form of y(t) = y0 + 𝛿y(t)

(2.35)

where y0 is the average value of the variable and all the signals and noise are ﬂuctuations around the average, carried by the term 𝛿y(t), which is assumed to have zero mean (i.e. ⟨𝛿y(t)⟩ = 0). Under these conditions the spectrum is simply given by V (Ω) = ⟨|𝛿̃y(Ω)|2 ⟩

(2.36)

where the tilde indicates a Fourier transform. A simple example is illustrative at this point. Suppose the ﬂuctuations in our variable are made up of two parts: a deterministic signal, 𝛿ys = g cos(2πΩt), and randomly varying noise that we represent by the stochastic function 𝛿yn = f 𝜁 (t) (f and g are positive constants). The signal and noise are uncorrelated, so we can treat them separately. For the signal we use Eq. (2.36). The Fourier transform of a cosine is a pair of delta functions at ±Ω. Squaring and averaging over a small range of frequencies (say, 𝛿Ω) around ±Ω, we end up with two signals of size g 2 . For the noise it is easier to work from Eq. (2.34). We can take the

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log (ΔI2(Ω))

40

M(Ω2)

Signal SNR

Noise

Ω=0

Ω1

Ω2

Detection frequency, Ω

Figure 2.10 Example of a noise spectrum showing noise at many frequencies and two modulations at Ω1 and Ω2 . This plot has a logarithmic scale and the signal-to-noise ratio can be read oﬀ it directly if M(Ω) ≫ Var(I(Ω)).

autocorrelation of the noise to be ⟨𝜁 (t)𝜁 (t + 𝜏)⟩ = 𝛿(𝜏). That is, the noise is random on all timescales; there is no correlation between how it behaves from instant to instant. The Fourier transform of a delta function is a constant. Thus, after integration over the small bandwidth, the noise contributes 𝛿Ωf 2 to the spectrum at all frequencies. The signal-to-noise ratio (SNR) is deﬁned by SNR + 1 =

g2 𝛿Ωf 2

(2.37)

In Figure 2.10 an actual intensity spectrum with AM modulation is plotted. In this case the coeﬃcient g corresponds to the modulation M𝛼0 . The relative intensity noise RIN is the ratio of the variance to the average intensity. When stated as a function of the detection frequency Ω, the integration time td , or bandwidth, has to be taken into account. For a bandwidth of 1 Hz, it is given by RIN(Ω) = Var(I(Ω))∕⟨I⟩2

(2.38)

In classical theory it is in principle possible to make any noise arbitrarily small in the amplitude and phase such that the amount of information that can be carried by a beam of light is limited only by technical considerations. We shall ﬁnd that this is not true in the quantum theory of light. Strict bounds apply to the amount of information that can be encoded on light of a particular power. 2.3.3

Coherence

Light emitted by a pointlike source has properties closely resembling one single wave. The amplitudes at two closely spaced points are linked with each other, as part of the same wave. In contrast, the amplitudes at two very diﬀerent points or at two very diﬀerent times will not be identical. The question how far the detectors can be separated in space, or in time, before the amplitudes will diﬀer depends on an additional property of the light: its coherence. Any source of light has a coherence length and a coherence time. The amplitude at two points within the coherence length and coherence time is in phase and has the same ﬂuctuations. Interference can be observed. Light at two points separated by more than the

2.3 Statistical Properties of Classical Light

Figure 2.11 Stellar interferometer as proposed and developed by A. Michelson and F.G. Pease [12].

δθ

M3

M4 M2

M1 Lens

Detector

coherence length, or the coherence time, has independent amplitudes that will not interfere. One clear example of this concept is the technique of measuring the size of a star, which has a small angular size but is not a point source, using a stellar interferometer. The size of a distant star can usually not be resolved with a normal optical telescope. The starlight forms an image in the focal plane of the telescope, which is given by the diﬀraction of the light. The image depends on the geometrical shape and the size of the aperture of the telescope and has little to do with the actual shape and size of the object, the star. Diﬀraction limits the resolution. Only a bigger aperture of the telescope, larger than technically feasible, would increase the resolution. To overcome this limitation, A.H. Fizeau proposed the idea of a stellar interferometer, which was ﬁrst used by A.A. Michelson and F.G. Pease [12]. The concept of the interferometer is shown in Figure 2.11. In this apparatus the starlight is collected by two mirrors, M1 and M2 , which direct the light via the mirrors M3 and M4 to a telescope. There the two rays of light are combined in the focal plane. In addition we have optical ﬁlters in the instrument that select only a limited range of optical wavelengths. The mirrors M1 and M2 are separated by a distance d, and the apparatus is carefully aligned such that the distances from the focal plane to M1 and to M2 are identical. As a consequence we have interference of the two partial waves in the focal plane. An interference system will appear due to the fact that the mirrors M1 and M2 are not exactly at 45∘ to the axis of the telescope. This arrangement is equivalent to a Michelson interferometer with an optical wedge in one arm, and it measures the ﬁrst-order coherence of the light. For an extended source we get light from diﬀerent parts of the source reaching the telescope under slightly diﬀerent directions – let us say with a very small angular diﬀerence of 𝛿𝜃. Each wave will produce an interference system, but at a slightly diﬀerent location in the focal plane. This is due to the additional optical path Δl = d sin(𝛿𝜃) ≈ d 𝛿𝜃 inside the interferometer. We assume that the light from the diﬀerent parts of the star is independent. Consequently, it will interfere with light from the same part of the star, but not with light from other parts, and the intensity of the interference patterns from the diﬀerent directions is added. The interference pattern in the focal plane will be visible as long as the path

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diﬀerence is small compared to an optical wavelength. If the separation d between M1 and M2 is increased, the total interference pattern will eventually disappear, since the individual patterns are now shifted in respect to each other. An estimate for the separation at which fringes are no longer visible is d 𝛿𝜃 = 𝜆. In this situation the two individual patterns from the edges of the star are shifted by one fringe period, and all other patterns lie in between, washing out the total pattern. A more detailed analysis predicts the change in the visibility of the interference patterns as a function of the separation d. The interferometer measures the coherence of the light. In this case the coherence is limited due to the independence of the diﬀerent parts of the star, and the coherence length is the path diﬀerence for the light travelling from diﬀerent edges of the star to the Earth. The corresponding coherence time for the entire starlight is the diﬀerence in travel time. We ﬁnd that interferometry can be used to measure the coherence length and coherence time. Michelson and Pease built an instrument based on this idea and was able to measure star diameters down to about 0.02 arc seconds [12, 13]. There are improved versions of this instrument in operation, in particular an instrument called SUSI (Sydney University Stellar Interferometer), which have achieved resolutions as good as 0.004 arc seconds. However, there are technical limitations. It is very easy to lose the visibility of the fringes, already the wavelength is restricted to an interval of about 5 nm, and even then the path diﬀerence inside the interferometer has to be kept to less than 0.01 mm. A wider spectrum, containing more light, would require an even better tolerance. It is very diﬃcult to maintain such a small path diﬀerence for the entire time of the observation, while the mirrors track the star and the separation between the mirrors is changed by up to several metres. To avoid these complications, R. Hanbury Brown & R.Q. Twiss proposed an entirely diﬀerent scheme to measure coherence. Theirs is a very elegant approach that uses intensity ﬂuctuations rather than interference, an idea that had shortly before been used in radio astronomy. In this way they founded a new discipline based on the statistical properties of light. They set out to measure the correlation between the intensities measured at diﬀerent locations rather than the amplitude and therefore avoided all the complications associated with building an optical interferometer. In their experiment they replaced the mirrors M1 and M2 (Figure 2.11) with curved mirrors, which collected the light onto two diﬀerent photomultipliers, as shown in Figure 2.12 [14]. The photocurrents from these two detectors are ampliﬁed independently and then multiplied with each other, and the time average of the product is recorded. This signal reﬂects the correlation between the intensities. This apparatus does not require the mechanical stability of an interferometer. However, some questions arise: In which way is this experiment related to an interferometer? How can it provide similar information if we have disbanded the crucial phase measurement? To answer these questions, we remind ourselves that stellar interferometers measure the spatial coherence of light, transverse to the direction of observation. At any one point the phase of the light ﬂuctuates wildly, and so does the amplitude of light from a thermal source. Generally the amplitude and the phase are not directly linked, and both ﬂuctuate independently. However, for two points within

2.3 Statistical Properties of Classical Light

d

x Multiplier

Figure 2.12 Intensity interferometer proposed and developed by Hanbury Brown & Twiss. Source: Brown and Twiss 1956 [14]. Reproduced with the permission of Springer Nature.

a coherence length, the phase and the amplitude are linked to each other. The light will interfere and the intensity ﬂuctuations are correlated to each other. For two points separated by more than a coherence length, the interference will disappear, and so will the intensity correlation. Fortunately for Hanbury Brown and Twiss, and for us, it is a characteristic property of thermal light that the coherence length over which the phase is constant is approximately the same length in which the amplitude changes very little. Thus a measurement of the path length over which the intensity correlation disappears is in most cases identical to a measurement of the phase coherence length. Hanbury Brown and Twiss measured this distance and therefore were able to determine the size of the source. Already in 1956 they measured the diameter of the star Sirius. An improved apparatus was installed in Narrabri, Australia, with an extended baseline. The detectors are moved on a circular track with a diameter of 396 m [15], and the star diameters can be determined down to 0.0005 arc seconds. How does a simple multiplier reveal the correlation between the signal? We can describe the intensity at one detector as the sum of a constant average value, ⟨I⟩, and a zero mean ﬂuctuating term, 𝛿I1 (t) by I1 (t) = ⟨I⟩ + 𝛿I1 (t). Similarly the intensity at the other detector can be written as I2 (t) = ⟨I⟩ + 𝛿I2 (t). While both ⟨𝛿I1 (t)⟩ and ⟨𝛿I2 (t)⟩ are equal to zero, for correlated waves, we get ⟨𝛿I1 (t)𝛿I2 (t)⟩ ≠ 0. If the intensities are perfectly correlated, then 𝛿I1 (t) = 𝛿I2 (t) and the time average of the product will be ⟨I1 (t)I2 (t)⟩ = ⟨I 2 + 𝛿I1 (t)2 ⟩. If the separation of the sources exceeds the coherence length, then the intensities are not correlated and the product 𝛿I1 (t)𝛿I2 (t) randomly changes sign and the average ⟨𝛿I1 (t)𝛿I2 (t)⟩ vanishes. That means ⟨I1 (t)I2 (t)⟩ = ⟨I⟩2 , which is distinctly diﬀerent to the correlated case. A reduction in the correlation signal indicates that the coherence length has been reached. A study of the correlation signal for diﬀerent receiver separations allows a measurement of the second-order coherence of the light.

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2.3.3.1

Correlation Functions

After this introduction and motivation, we will now examine ﬁrst- and secondorder coherence, and the correlation functions that describe them, quantitatively. First-order coherence is important when two light beams are physically superimposed. As a generic example let us consider the case of two beams being combined on a 50 : 50 beamsplitter, e.g. a semi-transparent mirror, followed by direct detection of the intensity of one output beam. We allow a time delay, 𝜏, to be imposed on one of the beams before the beamsplitter. We can describe the input ﬁelds to the beamsplitter by their complex amplitudes 𝛼1 (t) and 𝛼2 (t + 𝜏). We assume the beams are in single identical spatial modes that are completely detected; thus we can ignore the spatial dependence of the amplitudes. For our purposes here we can assume that √ the action of the beamsplitter is simply to produce the output 𝛼out (t) = (1∕ 2)(𝛼1 (t) + 𝛼2 (t + 𝜏)), an equal linear superposition of the inputs. The detector produces a photocurrent proportional to the intensity of the output ﬁeld, which in turn is proportional to the absolute square of the amplitude, |𝛼out |2 . Our measurement result will then be proportional to |𝛼out |2 averaged over some measurement interval, say, td , such that 1 |𝛼out |2 dt td ∫ 1 = ⟨|𝛼1 (t)|2 + |𝛼2 (t + 𝜏)|2 + 𝛼1 (t)𝛼2∗ (t + 𝜏) + 𝛼1∗ (t)𝛼2 (t + 𝜏)⟩ 2 (2.39)

⟨|𝛼out |2 ⟩ =

where we have used ⟨· · ·⟩ to indicate the time average. Suppose that 𝛼1 and 𝛼2 both derive from the same continuous source oscillating at frequency 𝜈 such that 𝛼1 (t) = 𝛼2 (t) = 𝛼(t) exp(i2π𝜈t). Then ⟨|𝛼out |2 ⟩ = ⟨|𝛼|2 ⟩ + cos(2π𝜈𝜏 + 𝜙) ⟨𝛼 ∗ (t)𝛼(t + 𝜏)⟩

(2.40)

There is interference between the waves resulting in a modulation of the photocurrent as a function of the path diﬀerence. This basic eﬀect is the source of fringes in all ﬁrst-order interferometry including double-slit experiments. The temporal coherence of the light is quantiﬁed by the ﬁrst-order correlation function, g (1) (𝜏), deﬁned via g (1) (𝜏) =

⟨𝛼 ∗ (t) 𝛼(t + 𝜏)⟩ ⟨|𝛼|2 ⟩

(2.41)

To see that g (1) (𝜏) measures the coherence of the light, consider the depth of the modulation in the above example. This is given by the visibility , deﬁned by =

Imax − Imin Imax + Imin

(2.42)

Maximum modulation results in = 1, whilst no modulation results in = 0. For the output of Eq. (2.40), we ﬁnd |⟨𝛼 ∗ (t) 𝛼(t + 𝜏)⟩| ⟨|𝛼|2 ⟩ = |g (1) (𝜏)|

(𝜏) =

(2.43)

2.3 Statistical Properties of Classical Light

Thus, for this simple example, the ﬁrst-order correlation is equal to the visibility. If the source was a perfect sinusoid, then 𝛼 would be a constant in time and we would have g (1) (𝜏) = 1 regardless of the time delay. This corresponds to an inﬁnite coherence length. More realistically the source will have some bandwidth. Suppose that the source has a Gaussian spread of frequencies Δ𝜈 around 𝜈. This can be represented in Fourier (or frequency) space by 𝛼(Ω) ̃ = 𝛼c exp(−2π(𝜈 − Ω)2 ∕Δ𝜈 2 )

(2.44)

where 𝛼(Ω) ̃ is the Fourier transform of 𝛼(t) and 𝛼c is a constant. Taking the inverse transform gives Δ𝜈 𝛼(t) = 𝛼c √ exp(−πΔ𝜈 2 t 2 ) 2

(2.45)

for our amplitude in time space. Thus our ﬁrst-order correlation function becomes g (1) (𝜏) =

Δ𝜈 2 1 |𝛼c |2 exp(−πΔ𝜈 2 𝜏 2 ) exp(−πΔ𝜈 2 t 2 )dt td 2 ∫

1 Δ𝜈 2 |𝛼c |2 exp(−πΔ𝜈 2 t 2 )dt td 2 ∫ ) ( π (2.46) = exp − Δ𝜈 2 𝜏 2 2 a Gaussian function of 𝜏 with a width inversely proportional to the frequency spread. For a narrow frequency spread, Δ𝜈 small, we have a large coherence length, whilst for a broad spectrum, Δ𝜈 large, the coherence length becomes very short. It is this ﬁrst-order coherence length that is measured by the stellar interferometer. Now let us consider intensity correlation measurements. As a generic example we consider direct detection of the intensity of two beams that have not been optically mixed, but one of which has a variable time delay imposed on it before detection. The two photocurrents from the detectors are then sent into a correlator, which multiplies the photocurrents together. Suppose ﬁrstly that, as in our previous discussion, the two beams originate from the same source. The output of the correlator will then be proportional to the second-order correlation function, g (2) , which is deﬁned by g (2) (𝜏) =

⟨I(t)I(t + 𝜏)⟩ ⟨𝛼 ∗ (t)𝛼(t)𝛼 ∗ (t + 𝜏)𝛼(t + 𝜏)⟩ = ⟨I(t)⟩2 ⟨𝛼 ∗ (t)𝛼(t)⟩2

(2.47)

where the I are the detected intensities. From the symmetry of this deﬁnition follows the property g (2) (−𝜏) = g (2) (𝜏), and measurements need only to be made for positive values of 𝜏. Whilst the degree of ﬁrst-order coherence g (1) (𝜏) takes on values between 0 and 1 and is unity for 𝜏 = 0, the second-order correlation is not bounded above. Remembering that the variance of I is given by ΔI 2 = ⟨I 2 ⟩ − ⟨I⟩2 , we can write the second-order correlation at 𝜏 = 0 as g (2) (0) =

ΔI 2 +1 ⟨I⟩2

(2.48)

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Given that the variance obeys ΔI 2 ≥ 0, we ﬁnd that the value of the second-order coherence at zero time delay is g (2) (0) ≥ 1

(2.49)

and it is not possible to derive an upper limit. This proof is only valid for 𝜏 = 0 and cannot be applied to other delay times. The only restriction stems from the positive nature of the intensity, resulting in ∞ ≥ g (2) (𝜏) ≥ 0

for all

𝜏≠0

(2.50)

It is possible to relate the degree of second-order coherence at later times to the degree of coherence at zero delay time by using the Schwarz inequality in the form ⟨I(t1 )I(t1 + 𝜏)⟩2 ≤ ⟨I(t1 )2 ⟩⟨I(t1 + 𝜏)2 ⟩

(2.51)

For ergodic or stationary systems, two time averages depend only on the time diﬀerence, not the absolute time. For such systems we then have ⟨I(t1 )2 ⟩ = ⟨I(t1 + 𝜏)2 ⟩. Consequently we have ⟨I(t1 )I(t1 + 𝜏)⟩ ≤ ⟨I(t1 )2 ⟩ yielding g (2) (𝜏) ≤ g (2) (0)

(2.52)

For a perfectly stable wave (i.e. zero variance), the equal sign applies and g (2) (𝜏) = 1 for all delay times. Equation (2.52) means that the degree of coherence for any classical wave should always be less than g (2) (0). This is a result that has been contradicted for quantum states of light and has been used to demonstrate experimentally the quantum nature of light as shown in detail in Chapter 3.

2.4 An Example: Light from a Chaotic Source as the Idealized Classical Case In most practical cases classical light will be noisy. Only a perfect oscillator would emit an electromagnetic wave with perfectly constant amplitude and phase. There are some relatively simple models of realistic light sources. Chaotic light is the idealized approximation for the light generated by independent sources emitting resonance radiation. A practical example is a spectral lamp. In contrast, thermal light is an approximation of the light emitted by many interacting atoms that are thermally excited and together emit a broad, non-resonant spectrum of light. A practical example is a hot, glowing ﬁlament. Thermal light will be discussed in Chapter 3. The source of chaotic light is a large ensemble of atoms that are independent and thermally excited. They all emit light at the same resonance frequency. For simplicity, we assume they are suﬃciently slow so that the eﬀect of Doppler shifts can be neglected. The atoms will collide with each other; thus the waves of radiation emitted by each atom will be perturbed by collisions. The noise properties are best derived by analysing the total complex amplitude 𝛼(t) of the wave, which is the sum of the amplitudes emitted by the individual atoms. We can evaluate the coherence function of the total amplitude, and from there we can determine the

2.4 An Example: Light from a Chaotic Source as the Idealized Classical Case

intensity noise spectrum. For a large number M of atoms, the total wave amplitude is 𝛼(t) =

M ∑

𝛼j (t) = 𝛼0 exp(i2π𝜈0 t)

j=0

M ∑

(exp(i𝜙j )) = |𝛼(t)| exp(i𝜙(t))

(2.53)

j=0

where 𝜙j represents the phase of the individual contributions. For simplicity we assume that all atoms emit into the same polarization. For a large number of atoms, |𝛼(t)| and 𝜙(t) are eﬀectively randomly varying functions that describe the magnitude and the phase of the total amplitude at any time t. If the average time between collisions is 𝜏0 , one can derive [7] a probability (𝜏) that an atom is still in free ﬂight after a time 𝜏 since its last collision which is given by ) ( 𝜏 (2.54) (𝜏)d𝜏 = exp − 𝜏0 The time interval 𝜏0 is usually very short. If we assume that a single measurement takes a time larger than 𝜏0 , the coherence function can be evaluated as ⟨𝛼 ∗ (t)𝛼(t + 𝜏)⟩ = 𝛼02 exp(i2π𝜈0 𝜏) ⟨[exp(−i𝜙1 (t)) + · · · + exp(−i𝜙M (t))] [exp(i𝜙1 (t + 𝜏)) + · · · + exp(i𝜙M (t + 𝜏))]⟩ In multiplying out the large brackets, many cross terms appear, which correspond to waves emitted by diﬀerent atoms. Since the phase between the light from different atoms is random, they cancel and give no contribution to the total average. Thus ⟨𝛼 (t)𝛼(t + 𝜏)⟩ = |𝛼0 | exp(i2π𝜈0 𝜏) ∗

2

M ∑

exp[i(𝜙j (t + 𝜏) − 𝜙j (t))]

j=1

= M⟨𝛼j∗ (t)𝛼j (t + 𝜏)⟩ where 𝛼j (t) is the amplitude emitted by a single atom and thus the entire correlation is completely described by the single-atom correlation function. A single atom emits a steady train of waves until a random phase jump occurs due to a collision. If one neglects the small amount of light that is emitted during the collision (impact approximation), one gets the correlation terms ⟨𝛼j∗ (t)𝛼j (t + 𝜏)⟩ = |𝛼0 |2 exp(−i2π𝜈0 𝜏)P(𝜏) ( ) 𝜏 = |𝛼0 |2 exp −i2π𝜈0 𝜏 − 𝜏0 and the ﬁrst-order correlation function ( ) 𝜏 g (1) (𝜏) = exp −i2π𝜈0 𝜏 − 𝜏0

(2.55)

For all types of chaotic light, it is possible to derive the following relationship between the ﬁrst- and second-order correlation coeﬃcient [7]: g (2) (𝜏) = 1 + |g (1) (𝜏)|2

(2.56)

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2 Classical Models of Light

The exact form of both g (2) (𝜏) and g (1) (𝜏) depends on the type and the statistical distribution of collisions. For illustration a numerical example is given in Figure 2.13 showing the various properties and descriptions of a ﬂuctuating light beam. Here the chaotic light is simulated by calculating explicitly the emission from 20 independent sources, all with the same amplitude. The initial phase of these sources is selected at random. The initial conditions are given in Figure 2.13a: all twenty partial waves 𝛼i (t = 0) and the total amplitude 𝛼(t = 0) are shown. The phase evolution of the emitters is simulated by adding to each emitter a change in phase that can randomly vary in the range −0.13π to 0.13π for each time step. Thus each emitter has a random development of its phase, occasionally with a phase jump. For each time interval the total amplitude can be described as one point in a polar diagram, which leads to the distribution of points shown in Figure 2.13b. The vector describes a convoluted path; it moves from one point to another seemingly at random. The density of points can be interpreted as a probability of ﬁnding a certain amplitude. The points are scattered around the origin. While the amplitudes would average out, the most probable intensity is not zero. The full time development of the magnitude of the total amplitude |𝛼(t)|, the total phase 𝜙(t), and the scaled intensity I(t) = I0 𝛼 ∗ 𝛼 are all shown in Figure 2.13c–e. The magnitude |𝛼(t)| and the intensity I(t) show dramatic ﬂuctuations, occasionally dropping close to zero. The intensity can be described by the long-term average ⟨I⟩ and the RMS value, both included in Figure 2.13d. For this numerical example, the correlation functions can be calculated directly. Equations (2.41) and (2.47) have been evaluated, with the results shown in Figure 2.13f,g. The results are very close to the predictions for a chaotic source. Over the time interval shown, the ﬁrst-order correlation functions change smoothly from g (1) (0) = 1 down to 0.15. In the same time interval, g (2) (𝜏) declines from g (2) (0) = 1.8–1.0. There are some oscillations in g (2) (𝜏), which is typical for such a short statistical sample. The analysis of the longer time sample would approach the theoretically predicted smooth decline of the values of g (2) (𝜏). The link between g (2) and g (1) given in Eq. (2.56) is clearly reproduced in this example. In order to predict the intensity noise of the chaotic source of M atoms, we consider the instantaneous intensity I(t) and the averaged intensity ⟨I⟩. We ﬁnd 1 ⟨I⟩ = ⟨I(t)⟩ = E02 𝜖0 c|𝛼0 |2 2

⟨| M |2 ⟩ |∑ | 1 | exp(i𝜙 (t))| = E02 𝜖0 c|𝛼0 |2 M j | | 2 | j=1 | | |

(2.57)

since all the cross terms average out due to the randomness of the phase functions. The total average intensity is simply the sum of the average intensity emitted by a single atom. In order to evaluate the variance, we require a value for (

1 2 ⟨I(t) ⟩ = E 𝜖 c|𝛼 |2 2 0 0 0 2

)2

⟨| M |4 ⟩ |∑ | | exp(i𝜙j (t))| | | | j=1 | | |

(2.58)

2.4 An Example: Light from a Chaotic Source as the Idealized Classical Case

5 (a) Vector sum at one time (magnified)

(b) Time development of vector

10

0

0

10 5

5 10

0

5

10

0

10

(c) Magnitude 5

0 100

(d) Intensity

50 Average Int. 0 π

(e) Phase

0 −π 1

2

(g) g(2)(τ)

(f) g(1)(τ)

0.5

1

0 0

τmax

0

0

τmax

Figure 2.13 An example of the statistical properties of chaotic light. The light is simulated as the total ﬁeld emitted by 20 independent emitters, each with constant amplitude but randomly evolving phase. Part (a) shows the individual vectors at t = 0 and (b) the resulting total phase vector for various times. The time history is shown for the magnitude (c), the intensity (d), and the phase of the ﬁeld (e). This leads to the ﬁrst-order (f ) and second-order (g) correlation function.

49

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2 Classical Models of Light

The only terms that will contribute to this value are those where each factor is multiplied by its own complex conjugate. These terms result in ⟨( M )2 ( M )2 ⟩ )2 ( ∑ ∑ 1 ei𝜙j e−i𝜙j E2 𝜖 c|𝛼 |2 ⟨I(t)2 ⟩ = 2 0 0 0 j j ⟨M ⟩ M )2 ∑ ( ∑ 1 2 2 i(2𝜙j −2𝜙j ) i(𝜙j +𝜙k −𝜙j −𝜙k ) = e +2 e +··· E 𝜖 c|𝛼 | 2 0 0 0 j j≠k )2 ( 1 2 E0 𝜖0 c|𝛼0 |2 ⟨M + 2M(M − 1) + · · ·⟩ = 2 )2 ( 1 2 E0 𝜖0 c|𝛼0 |2 (M + 2M(M − 1)) = 2 where · · · represents terms that average to zero. With Eq. (2.57) we obtain ( )2 ) ( ⟨I⟩ 1 (2M2 − M) = ⟨I⟩2 2 − ⟨I(t)2 ⟩ = M M and using ⟨I(t)2 ⟩ = 2⟨I⟩2 for large M, the variance is ΔI(t)2 = (2⟨I⟩2 − ⟨I⟩2 ) = ⟨I⟩2

(2.59)

This is a simple and important result: for a chaotic source the variance of the time-resolved intensity is equal to the square of the average intensity. Calculations for more practical situations, which include Doppler broadening and realistic models of the line broadening, show basically the same result. The intensity noise spectrum will be conﬁned to the spectral linewidth. Outside the linewidth this classical description predicts an arbitrarily low intensity noise spectrum.

2.5 Spatial Information and Imaging 2.5.1

State-of-the-Art Imaging

Imaging is a widely used term. It transfers the spatial properties of an object, typically a 2D cross section, into an image that we either see directly or record in the form of paper or canvas and now more likely in an electronic format as a dataset that can be displayed or printed. Viewing something by eye or assisted with a lens, telescope, or microscope involves imaging, so is any form of recording with a camera. The quality and range of applications of imaging is improving and expanding all the time, with better spatial and temporal resolution, higher sensitivity, and the ability to transform a wider array of physical properties into images. Technology is improving at a rapid pace and with lower costs is making it much more widely accessible. A smartphone is an amazing imaging device for more and more purposes. We are surrounded by images. Figure 2.14 shows four examples from very diﬀerent situations. Figure 2.14a is a demonstration of the quantum nature of light using Young’s double-slit fringes recorded with increasing photon ﬂux. This is a teaching demonstration by T.L. Dimitrova and A. Weis for the classroom [16].

2.5 Spatial Information and Imaging

(a)

(b)

(d)

(c)

Figure 2.14 A variety of new imaging techniques: (a) teaching demonstration of Young’s double-slit experiment; (b) 3D image taken with short pulses, photon packets, and a depth-resolving fast detector; (c) scan of a brain tumour using PET-MRI, single-photon detection reconstruction, and false colour display; and (d) image of a live neurone in a rat brain created with scanning two-photon microscopy. Source: From [16–19].

Figure 2.14b shows a 3D reconstruction of an image where not only the position of the photons but also their arrival time is recorded using short light pulses, photon packets, and an optical time-of-ﬂight (TOF) technique. In this example from 2005, the depth resolution is less than a millimetre, and the resolution is 64 × 64 pixels [17]. From the data a 3D image is reconstructed, and this technology is constantly improving. Systematic progress is being made to add more pixels and enhance the speciﬁcations of the single-photon detector arrays, reaching 256 × 64 pixels by 2012 [20] and more since, by creating new applications such as TOF LIDAR imaging systems for autonomous devices. Figure 2.14c shows a ﬁctional slice through the brain of a person who has tumour. The image has been generated by multimodal PET-MRI [18]. This medical diagnostic technique uses time-resolved imaging of individual photons generated in scintillating crystals when hit by gamma rays, which are created in a nuclear decay. The image shows false colour coding, which scales with the local production rate of gamma rays and conversion into detectable photons. This signal is interpreted as an unusual activity in the brain. Finally, Figure 2.14d shows an in vivo scan of a live neurone in a rat brain. The neurone has been ﬁlled with a ﬂuorescent dye, and the dye is stimulated in a two-photon (2P) process with a focussed laser beam. The focal region is scanned in the three spatial dimensions,

51

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2 Classical Models of Light

many slices were taken, and a 3D image was reconstructed. This well-known confocal microscopy technique can be used to visualize the neurone, to modify the neurone with lasers beams [19], and in the future to record the electrical activity and the information ﬂow between neurones optically. It is clear from Figure 2.14 that the generation and detection of individual photons plays a major role in these state-of-the-art examples of imaging. Photons are a concept that we require to understand how the image is created, and in particular Figure 2.14a shows explicitly that the statistics of the photons inﬂuences the quality of the images. When the intensity, and thus the photon ﬂux, is low, we can clearly see the graininess of the images. The contrast improves with more light and more photons being detected. This is barely noticeable with our own eye in everyday situations but is rather pronounced when modern detectors with high sensitivity are used.

2.5.2

Classical Imaging

Imaging refers in general to the creation of a 2D intensity and colour distribution I(X, Y , 𝜆), a function that we see with our eyes or record for posterity and for data analysis. Most of these images involve colour, which means a dependence on the spectrum of wavelengths 𝜆. With our well-trained vision system, we can see images in three dimensions (3D) I(X, Y , Z, 𝜆), and most people can distinguish more than 100 000 hues of colour, which are mixtures of wavelengths. And we do this with a reasonable time resolution of a few tens of milliseconds, which can be described by a function with a total of 5 degrees of freedom, I(X, Y , Z, t, 𝜆). We all have the capacity to deal with such a complex situation with our vision system. And we all operate devices that record and process images, nowadays in digital and pixelated format, and that extend the capability of the human eye. Central to the description is an object distribution of complex optical amplitude 𝛼(x, y) = 𝛼0 (x, y) exp(i𝜙(x, t)) with 𝛼0 (x, y) the magnitude of the optical ﬁeld and 𝜙(x, y) the corresponding optical phase across the object plane. These are the important components if we deal with coherent imaging where the phase of the wave front is stable, propagating in well-deﬁned ways, and detectable. In practice this might require working with a very narrow bandwidth Δ𝜆 or a laser beam. Note that it is possible to build special imaging devices using interferometric tricks, which can make the phase distribution visible even for light with low coherence, for example, through phase contrast imaging and microscopy or white light interferometry. The object distribution is created by the illumination of the object plane from a light source, and various eﬀects create the light distribution, including straight transmission of the light, specular reﬂection or scattering of the object, ﬂuorescence of a concentration of dye, etc. We can treat each point in the object plane as a source of radiation, which follows through to the image system according to the rules of wave propagation. For incoherent illumination imaging we only require the intensity distribution Iobject (x, y). In the detection plane we normally form the intensity distribution Iimage (X, Y ) = |𝛼image (X, Y )|2 , the intensity distribution we are going to see. To have a reasonable intensity, we can integrate the intensity

2.5 Spatial Information and Imaging

over a small area in the object plane of the size 𝛿x times 𝛿y: 𝛿x,𝛿y

Iobject (x, y) =

∫−𝛿x,−𝛿y

|𝛼object (xi , yj )|2 dx, dy

(2.60)

Optics is generally concerned with the transfer of the light to see how the amplitude 𝛼object (x, y) in the object plane is transformed into the image intensity Iimage (X, Y ). For the propagation of this image through an optical system, we should keep track of both amplitudes and the optical phase terms. This will allow us to consider both coherent imaging, including speckle, holography, and spatial ﬁltering. It is common to describe the properties of an imaging system by the mathematical link between the object and the image, in particular the image of a single point, or pixel in the object plane, a delta function 𝛿(x0 , y0 ). The mathematical link between object and image is conventionally done in two steps – ﬁrst the scaling function s(x, y) of the image, which includes magniﬁcation, stretching, and distortion, and second a function that images each object point at location (x, y) in the object plane into a point at (X, Y ) in the image plane: Iimage (X, Y ) = Iobject [s(x), s(y)]

(2.61)

In the simplest case this could be a magniﬁcation by a factor M: (X − X0 ) = M(x − x0 ) and (Y − Y0 ) = M(y − y0 )

(2.62)

where (x0 , y0 ) and (X0 , Y0 ) are the origin of the coordinate systems in the two planes and are connected by the central ray, pointing in the direction of the propagation of the wave and perpendicular to the wave front. In many cases there will be distortions, which means that s(x, y) takes on a far more complex form. This can be simulated by ray tracing where we distribute the intensity at the point Iobject (x, y) into several components emitted in diﬀerent directions and each is traced through the imaging system, see Figure 2.15. We now have access to very powerful software routines that can do this for complex lens systems. In the ideal case the rays would meet again in a point. In reality they will form a distributed function, in the simplest form a symmetric Gaussian-like distribution. The image of a single point, or delta function, leads to a point spread function psf(X,Y ) for the amplitude or PSF(X,Y ) for the intensity distribution in the image plane around the point (X, Y ). In most cases we can assume that the point spread function is constant across the image plane and just one function PSF(0,0) (dX, dY ). The complete image can then be evaluated as a convolution of the object with one constant PSF: 𝛿X

Iimage (X, Y ) =

𝛿Y

∫−𝛿X ∫−𝛿Y

Iimage (X, Y )PSF(0,0) (dX, dY )dX, dY

(2.63)

This has the eﬀect of reduced spatial resolution. A great deal of engineering and ingenuity has gone into optimizing the PSF, and this is what we gain when we purchase a superior and expensive lens system, camera, or microscope. A very powerful method to describe the imaging process has been Fourier optics. It is very practical to describe the image quality and in particular the resolution of the image by analysing the object in terms of a series of orthogonal

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y Light source Image = detection plane

x

Image system Object amplitude

Object intensity

a (x, y) Φ (x, y)

I (x, y)

Coherent

Y Yk Xj

Incoherent

X

I (X,Y, Ω)

Wave propagation Ray approximation

Detector pixels digitization N (X j,Y k)

Point spread function Resolution

Figure 2.15 Schematic of classical imaging.

spatial harmonic functions. These are normally given in dimensionless units of oscillation per full frame of the image. The spatial frequencies describe the modulation of the amplitude across the image frame, starting with a constant average amplitude F0 and then contributions Fobj,k and Fobj,k at k = 1, 2, 3, … and l = 1, 2, 3, … oscillations per image frame in both directions. This reaches up to a maximum at the Nyquist limit of k = N∕2, l = M∕2 oscillations per image where N and M are the number of pixels considered. This series of harmonics, added up in both directions and all spatial frequencies, is equivalent description to the object amplitude: 𝛼object (x, y) =

N∕2 M∕2 ∑ ∑ k=0 l=0

( Fobj,l cos

2πk(x − x0 ) xmax

(

) Fobj,l cos

2πl(y − y0 ) ymax

)

(2.64) Alternatively, we can use a continuous Fourier transform. One of the reasons for the usefulness of this transformation in optics is that the Fourier analysis can be done directly by optical means. It is the distribution in far-ﬁeld diﬀraction plane and also in the plane of a perfect lens system. For collimated input beams in the focal plane of the lens, this Fourier description provides direct access to the understanding of the limits of resolution and the impact of diﬀraction on the imaging system. This concept, made popular by scientists such as Ernst Abbe in Germany and A. J. Fresnel in France, in the nineteenth century allowed the progress of modern lens design. They showed that the high spatial frequencies are essential to maintaining the resolution. An imaging system needs large numerical aperture in order to achieve high resolution. The notion of diﬀraction-limited resolution means that the actual PSF is very close to the

2.5 Spatial Information and Imaging

Figure 2.16 Phase distribution described by the low-order Zernike polynomials. 0

Z0

–1

1

Z1

–2

Z2

Z1

0

Z2

2

Z2

PSF given by the aperture size and geometry of the system. This theory was ﬁrst developed for incoherent light where the intensity spread function is considered. There is an extensive area of optics design that describes the propagation of the wave front with a diﬀerent set of functions, known as the Zernike polynomials. This is a diﬀerent set of orthogonal spatial functions that are used to describe the wave front when it propagates through the lens system. It is speciﬁcally chosen to describe the properties of the imaging process in terms of the most common distortions that can be observed. The ﬁrst six terms are shown in Figure 2.16. These polynomials are both a mathematical basis to describe the optics and a tool for the experienced expert to describe the quality and details of the imaging system, and one by one they describe tip in x, tilt in y, defocus, stigmatism, coma, etc. A quantitative description of the Zernike polynomials can be found in many optics textbooks, such as the classic text written by Born and Wolf. There is widespread literature about the use of these mode functions in lens design and vision correction. More recently these modes of light are playing an increasing role in the dynamic compensation for wave front distortions. With the use of fast spatial light modulators, wave front detection methods, and feedback algorithms, it is now possible to compensate dynamically for the ﬂuctuations that occur in the atmosphere above ground-based telescopes. The quality of the image and optical focussing can be improved using adaptive optics techniques. They can also be used to correct distortions that occur when imaging structures inside biological samples, particular for in vivo studies of brain tissue. These techniques have become much more accessible and powerful with the advent of fast multi-pixel spatial light modulators and more and more powerful control algorithms. 2.5.3

Image Detection

Finally, the intensity in the image plane is sampled by the active components of the eye or the electronic detector. In the case of the human eye, these are the receptors, cones, and rods, which are sampling the intensity and convert them into neurological signals to be processed by the optical nerve and the brain. In technical applications we now have multi-pixel detectors, commonly now with thousands of pixels in both directions, N > 1000, M > 1000. In each pixel the amount of light is integrated across a pixel and also in time due to the ﬁnite response time of the detector, and this is converted into a photocurrent, ideally proportional to the intensity or averaged over time and stored as an electric

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charge. The abundance of photodiodes, with fast response time, charge-coupled devices (CCD), or complementary metal–oxide–semiconductor (CMOS) detectors now allows optimization for every imaging task. More than 10 megapixels and less than 10 ms exposure time are now easily achieved. For practical reasons there is a strong demand for image compression techniques; otherwise the information content could not be easily handled, in particular for video recordings. There are several algorithms, and those most widely used, JPEG for still images and MPEG for video images, are based on a projection on a set of 2D basis functions. In the case of JPEG, this is a set of wavelet functions that are related but not identical to the harmonic functions in Eq.(2.64). Normally, only the magnitude of these contributions is stored. The resolution of the stored image depends on the order to which this projection is extended. In most cases a trade-oﬀ is made between the resolution limits due to data storage capacity and the properties of the image system. The technical capacities are increasing, and with this the quality is improving, leading to the developments of new standards such as ultra high deﬁnition (UHD) and 4K. There is now a convergence of the standards that deﬁne detection and displays, resulting in very realistic images. 2.5.4

Scanning

An alternative imaging scheme is to scan a small well-deﬁned spot, for example, to project an image or to record absorption or ﬂuorescence as shown in Figure 2.17. Many diﬀerent projection displays have been developed over the last decade, initially incoherent, using various forms of modulators, lamps, or more recently LEDs. The data from the recorded image are used to control the spatial intensity modulators. As an alternative, coherent laser beam can be used, which are scanned either with mechanical mirror scanning systems, such as galvanometers or spinning prisms as used in some laser printers, or alternatively with spatial phase modulators and optical arrangements using concepts similar to holography. These types of laser projection imaging systems are now becoming widespread in commercial cinema displays with the UHD standard or even higher pixel count. An equivalent imaging technique is laser scanning for imaging in LIDAR, the optical equivalent of RADAR that uses the detection of the reﬂected light synchronized to the scanning to recreate an image. Here the scanning is over a wide angular section, and the detector might contain just one pixel, a bulk detector, or alternatively a chip with many pixels. Short pulses, fast scanning speed, and excellent synchronization are important in order to measure distances with a resolution of centimetres or even millimetres. Suﬃcient contrast is achieved by developing detectors with a high dynamic range. LIDAR used to be a scientiﬁc speciality for atmospheric research or speciﬁc defence applications. However, since 2015, the need for vision systems in autonomous vehicles has created a rapid development and expansion of this imaging technique. At the microscopic scale the equivalent scanning techniques are used in scanning beam microscopy and confocal microscopy, where the ability to focus a laser down to the size of about one wavelength with a high-quality high numerical aperture lens system is used to produce high-resolution images. The basic idea is

2.5 Spatial Information and Imaging

Laser Scanning unit

Laser beam focussed at the mask

Beamsplitter

Time recording, synchronized with scanner

Object plane, with mask

I1 ( t)

or fluorescent dye

Bulk detector with one element

Figure 2.17 Schematic of a scanning imaging system. In this example a mask is illuminated with a scanned focussed laser beam, and all the intensity is recorded with a large single-element bulk detector. Alternatively a structure ﬁlled with a ﬂuorescent dye could be placed in the object plane, and all the light emitted at the focus is recorded by this detector. A time series is recorded and stored as a stack of slices from which a 2D or 3D image can be reconstructed.

that the sample under investigation contains a ﬂuorescent dye that is excited by the laser beam. One dye molecule requires a single photon with matching energy for excitation and emits one other photon, with lower energy, into an arbitrary direction. This ﬂuorescent light is detected with a large numerical aperture lens, in order to increase both sensitivity and resolution. The spatial resolution is given by the size of the laser beam, typically a Gaussian beam with a focus inside the sample. However, the ﬂuorescence can come from many parts of the beam, and this technique works well with a thin sample. In order to produce more localized images, 2P excitation is used. Here the probability of excitation is proportional to the square of the beam intensity, or local photon ﬂux, and thus the ﬂuorescence is strongly concentrated in the location of the laser beam focus. In this way spatial resolution close to the diﬀraction limit can be achieved in the x and y direction, with typically 10 times lesser resolution in the z direction. One elegant technique to scan the beam is holographic projection with a spatial phase modulator that allows fast scanning in all three dimensions. After recording and storing multiple x, y slices, each with a speciﬁc value of z, it is possible to reconstruct a complete 3D representation of the ﬂuorescence in the sample, which can be stored and displayed from any point of view. This technique has widespread use in biology and in particular neuroscience, as shown in ﬁgure 2.14d where one individual neurone is being imaged in a living

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2 Classical Models of Light

sample of brain tissue. This is very powerful and in future can be extended to functional imaging of the activities inside an organ or a cell. For example, with the help of a dye or other substances, which are sensitive to the local electric ﬁelds, we can not only display the spatial structure of the neurone but also its electrical activity. The action potential and information ﬂow can be imaged. Considerable progress is being made in developing more and more advanced image scanning techniques. 2.5.5

Quantifying Noise and Contrast

Spatial and temporal resolution is one speciﬁcation of image quality. How can we describe the quality of the object and image distributions in terms of clarity, noise, and contrast? One universal approach is to analyse the SNR. Let us consider this ﬁrst for a scanned image, where the entire information is contained in a long time series. The total time signal can be described, after Fourier analysis, by a set of signal components at modulation frequencies Ωj with amplitudes 𝛼j for each component. In order to describe the quality, we also need to consider the random ﬂuctuations or noise in the time series. This could be due to statistical eﬀects in the light intensity or due to noise generated in the detection device. To make the eﬀect of noise visible, we compare recordings of the image, which includes both signal and noise, [S + N], and of the noise alone. Assuming that the noise is Gaussian in its statistical properties, we can quantify the noise by evaluating the variance VN for a long time interval. We might ﬁnd noise that is stronger at certain frequencies Ωj , or we might ﬁnd white noise with the same strength at all frequencies. Then we can apply the same process to the signal and determine [S + N](Ωj ). For a strong high contrast image, each simple harmonic signal component at Ωj contributes a signal that is proportional to the square of the modulation amplitude. For such a large modulation amplitude 𝛼j (Ωj ), we can approximate the SNR as SNR(Ωj ) ≈ Var([S + N](Ωj ))∕Var(N(Ωj )) ∝ |𝛼j (Ωj )|2

(2.65)

This is used as a description of the quality of the time series, and once we place the scans together in an image, it can also describe the quality of the 2D images. For a very simple example, such as the fringes shown in Figure 1.1a, which have a cosine intensity distribution, we can obviously use this process, now replacing the time with the spatial coordinate x and the time frequency Ω with a spatial frequency 𝜈. The SNR(𝜈) for such a single harmonic spatial function describes the quality, or contrast, of this particular image of the modulation. Let us expand this to all other spatial frequencies and consider the SNR separately for each spatial Fourier component 𝜈j resulting in a spectrum SNR(𝜈, 𝜇), where 𝜈 and 𝜇 describe the spatial frequencies along the x and y axes. We can describe the quality of the object function by such a 2D contrast function and is linked directly to the contrast or visibility of the object. The formal process includes an extension of Eq. (2.65) into two dimensions. The equivalent of white noise required is now independent of the position anywhere in the image plane. Towards high spatial frequencies 𝜈, 𝜇, the signal will get smaller, and the contrast will drop oﬀ due to diﬀraction eﬀects. Setting a practical minimum value

2.5 Spatial Information and Imaging

for the SNR(𝜈, 𝜇), we can deﬁne the spatial resolution of the object function. In practice this is tested with a variety of object functions, for example, in the various forms of test charts, which are widely used in photography or optical testing. They include features of smaller and smaller size, corresponding to a series of spatial frequencies. An important speciﬁcation in the design, manufacture, and testing of imaging systems is the transmission of the SNR as a function of spatial frequencies. Low-quality optics lowers the contrast and the resolution. 2.5.6

Coincidence Imaging

There is also an alternative process of imaging where two beams of light correlated in both intensity and direction of propagation are used and the image is formed through coincidence detection. Each beam is directed onto separate detectors, in this case by the use of a beamsplitter and mirror (see Figure 2.18). One detector (2) has spatial resolution. The other is a bulk detector with only one spatial element and records the total intensity of the beam (1). The object to be imaged is inserted into the beam (1). Is it possible to obtain spatial resolution despite the use of a bulk detector and the imaging system being on the wrong beam? By recording only the correlated component of the photocurrents between the two photodetectors, a clear image can be created. By recording the coincidence counts for all outputs, simultaneously or in a scanning sequence, the information from the bulk detector (1) will provide the required information of how much light has been absorbed in any speciﬁc location of the mask. This might appear to be a complex arrangement. However, it has the major technical advantages that it might not be possible to build the imaging system Mirror

Imaging system Detector 2 with spatial resolution Y Yk

Beamsplitter

ed lat re m and r Co bea ns s tio itie ec ns dir inte

Xj

X

I2 (X,Y, Λ1)

Object plane, with mask

I1 ( Λ2)

Detector 1 with one element

Correlation processing

Figure 2.18 Schematic of coincidence imaging using pairs of optical rays with correlated directions, one single-element bulk detector, and one detector with spatial resolution.

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around the mask, for example, because it is inside a small container, in a vacuum system, or an environment that does not allow direct optical access. This type of imaging has been given the popular name of ghost imaging, presumably because there is nothing tangible in front of the conventional imaging system. It has been shown that this system works with coherent light such as laser beams, as well as with light that has low coherence and with light that has thermal ﬂuctuations. The real strength comes once we manage to have beams with diﬀerent wavelengths fully correlated. For example, beam 1 could be at a wavelength, such as infrared, where no detector with spatial resolution is available, while beam 2 can be conveniently imaged. In all these situations we can make use of the rules of classical optics to synthesize an image from the coincidence counts. First, demonstrations have been reported [21], and the goal is to create images of otherwise invisible objects, for example, clouds of gas such as methane or carbon dioxide that have absorption lines only in the infrared where it is diﬃcult to build tunable light sources. 2.5.7

Imaging with Coherent Light

Note that in the case of coherent imaging, for example, with a monochromatic laser beam, the propagation is slightly more complex due to the diﬀraction processes than the predictions from ray optics. This would require full analytic solutions and can also be summarized in the form of propagating closed forms of laser modes, such as the TEM modes shown in Figure 2.2. One frequent situation is the focussing of a laser beam into a second waist at locations that do not simply follow the normal lens equations used in incoherent illumination. A second area where coherence will show dramatic eﬀects is the illumination and imaging of irregular objects, such as rough surfaces or liquids with scatter centres, which will create a very complex diﬀraction pattern known as speckle. This pattern is predictable but also responds to the slightest changes in alignment and thus appears unsteady due to the inﬂuence of mechanical vibrations. Thus speckle is normally seen as an unwanted side eﬀect of coherent illumination. 2.5.8

Image Reconstruction with Structured Illumination

Images can be formed using a single-pixel photodetector by illuminating the object with a range of speciﬁcally chosen patterns and recording the reﬂection from each of these patterns. The general principle is that the total reﬂection will be diﬀerent for each pattern and that with a clever algorithm it should be possible to reconstruct the spatial details of the object. Clearly great care has to be taken in selecting the set of patterns, which should be orthogonal in the sense that the information created by each individual pattern is as independent from the others as possible. Various sets of patterns have been tested over time; starting in the 1970s [22], the algorithms have been optimized, and it is now possible to create useful images. The system can be portable and fast. An impressive demonstration comes from the team led by Miles Padgett at Glasgow of the real-time detection of methane gas leaks using a single-pixel camera [23]. In this case 256 diﬀerent masks are used, and imaging with an equivalent of

2.5 Spatial Information and Imaging

16 × 16 pixels at 25 frames per seconds is demonstrated, all with a single-pixel detector. With improved computing power, many more such applications of image reconstruction will emerge [24]. 2.5.9

Image Analysis and Modes

In many situations the imaging process is not completed when the intensity distribution, or photon ﬂux, has been recorded. The image is analysed for its content. For our own vision we have a very sophisticated processor for this in our optical nerve and the brain. This can analyse images with remarkable speed and accuracy. In technical terms we can analyse one image through its characteristic features, for example, we can enhance or suppress the contrast or certain colours or make edges more visible. This can be done by processing the photocurrents from the various pixels of the detector. It is also possible to ﬁlter out certain shapes in the image, which we could achieve by adding the photocurrents from a group of pixels. The analogue equivalent is that we consider a certain mode of light, which correspond to intensity distribution and wave front shape. We can expand the actual image at the detector plane over a whole set of such modes. A mode can describe a speciﬁc feature we are looking for. It can be a local feature, for example, a face or a feature in a medical image, which can be represented by a speciﬁc weighted sum of pixels. Or it could be a recurrent feature, such as regular set of windows in the image of a high-rise building or a speciﬁc set of tracks in the image of an electronic circuit. The detection mode is a very versatile concept. Once deﬁned we can track it from image to image. Face recognition, ﬁngerprint recognition, and similar biometric schemes are one form of image analysis. It can be implemented directly by combining the outputs of many pixels in a clever way. 2.5.10

Detection Modes and Displacement

This approach allows us to detect spatial changes within the detection plane. We can determine the movement of the mode, how far and how fast. We can measure how it rotates, expands, or shrinks. That means we can detect changes in specifically designed degrees of freedom by comparing images or analysing a whole string of images recorded like a video stream. In this way spatial measurement with beams of light can be seen as a special form of imaging by deﬁning a speciﬁc mode u(X, Y ) and tracking its changes. One of the simplest examples is to use a round beam of light. This is our mode that we place on a detector with two pixels, or two groups of pixels, and we measure the diﬀerence of the photocurrents. The mode that comes from the object is round. The detection mode is a plane cut in half. If the beam moves, we will get a strong signal for the displacement as the symmetry between the top and bottom half is broken. This simple device can measure the displacement with remarkable accuracy and high intensities; this can be much less than the diﬀraction limit, only limited by the SNR. In the language of modes, we can deﬁne a mode that represents the displacement of the beam, with the new mode for displacement in X deﬁned as uΔX (X, Y ) = 𝜕u(X, Y )∕𝜕X.

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The device should now have a detection mode udet , which is matched well to uΔX . The split detector mentioned above is a good but not perfect approximation. Let us use the example of a coherent laser beam to derive the optimum detection scheme. The beam produced by a perfect laser can be described by the mode that is shaped inside the laser by the resonator where a set of mirrors build up a powerful internal standing wave selectively ampliﬁed by the gain of the pumped laser medium. The resulting output beam is the TEM0,0 mode (see Figure 2.5), the lowest order of the set of Hermite–Gaussian modes described in Eq. (2.11) and shown in Figure 2.5. A beam perfectly aligned on the centre of the split detector beam is described by 𝛼0,0 (x, y, z), with the beautiful property that it is transformed into another TEM0,0 mode when imaged with a lens, in the limit of the paraxial approximation. We can easily propagate a laser beam through an imaging system and will have a TEM0,0 mode in the detection plane. When we now consider a shift of the beam, using Eq. (2.11), we ﬁnd that the ﬁrst derivative 𝜕𝛼0,0 ∕𝜕x = 𝛼1,0 . This means that all the information of the movement of the beam is contained in a TEM1,0 mode. Measuring this mode is the best and complete way of measuring the shift of the beam. The optimum device is a detector that has the detection mode TEM1,0 . Similarly we can detect shifts in the y direction. We can expand this idea further and ﬁnd that each of the TEMl,m modes, shown in Figure 2.2, represents one independent degree of freedom, such as tilt and expansion. Generalizing this concept further, we can see that imaging can be interpreted as a transformation from one optical mode into another. We start with an illumination mode that enters the object plane and creates the mode uobj that contains the amplitude and phase distribution. This is a new mode, which is projected through the imaging system to the detector. There we have a detector mode udet , which is deﬁned by the shape of the detector and its pixels and the wiring and combination of pixels that is used. The detection process is now a projection, or overlap, of the image of uobj onto udet . With a properly chosen detection mode, we can measure the change of the spatial properties of uobj in one well-speciﬁed degree of freedom. It is quite possible to do this for several degrees of freedom at the same time, by monitoring several detection modes, ideally all deﬁned orthogonal to each other. A straightforward example is displacement and expansion in two orthogonal directions, or rotation in opposite directions [9].

2.6 Summary We have now a set of tools to describe the properties of a beam of light: it is a mode with a set of DC values and we can describe the power that can be measured both in space and in time. In addition, we can communicate information with the beam through modulations using the AC sidebands. Communication of data is now the most widespread use of light; it has led to a whole industry of photonics. Noise limits the quality of communication. We can describe the ﬂuctuations of the light as a spectrum or alternatively through correlation functions. This noise is intrinsic to the way the light is generated, and we distinguish between incoherent light, such as sunlight, coherent light from a laser, and all cases in between.

References

Correlation measurements have opened new ways in determining the properties of the light, with many applications ranging from astronomy to microscopy. Finally, we have so many applications of such a diverse range of techniques in imaging. Initially in the domain of human sight and photography, this is expanding rapidly with the ever-improving technology for detection, image processing, and reconstruction. Optical engineering has found many alternative approaches to image the very large and the very small and many ways to show us the dynamics of objects and eﬀects and processes that were invisible before [25]. The ﬁeld of classical optics is wide, versatile, and beautiful. However, all of this relies on photons and their special properties.

References 1 Kogelnik, H. and Li, T. (1966). Laser beams and resonators. Appl. Opt. 5:

1550. 2 Siegman, A.E. (1986). Lasers. University Science Books. 3 Saleh, B.E.A. and Teich, M.C. (2007). Fundamentals of Photonics, 2e. Wiley. 4 Abramovitz, M. and Stegun, I. (1972). Handbook of Mathematical Functions.

New York: Dover. 5 Askin, A. (1970). Acceleration and trapping of particles by radiation pressure.

Phys. Rev. Lett. 24: 156. 6 Askin, A. (1978). Traping of atoms by resonance radiation pressure. Phys. Rev. 7 8 9

10 11 12 13 14 15 16 17

Lett. 40: 729. Loudon, R. (1973). Quantum Theory of Light. Oxford: Oxford University Press. Hänsch, T. (2005) Nobel Prize Lecture 2005. http://www.nobelprize.org/nobel prizes/physics/laureates/2005/hansch-lecture.html. Delaubert, V., Treps, N., Harb, C.C. et al. (2006). Quantum measurements of spatial conjugate variables: displacement and tilt of a Gaussian beam. Opt. Lett. 35: 1537. Louisell, W.H. (1973). Quantum Statistical Properties of Radiation. New York: Wiley. Gardiner, C.W. (1991). Quantum Noise. Berlin: Springer-Verlag. Michelson, A.A. and Pease, F.G. (1921). Measurement of the diameter of 𝛼 orionis with the the interferometer. Astrophys. J. 53: 249. Pease, F.G. (1931). Interferometer methods in astronomy. Ergeb. Exakten Naturwiss. 10: 84. Brown, R.H. and Twiss, R.Q. (1956). Correlation between photons in two coherent beams of light. Nature 177: 27. Brown, R.H. (1964). The stellar interferometer at Narrabri observatory. Sky Telescope 28: 64. Dimitrova, T.L. and Weis, A. (2008). The wave particle duality of light: a demonstration experiment. Am. J. Phys. 76: 137. Niclass, C. and Charbon, E. (2005). A single photon detector array with 64 × 64 resolution and millimetric time depth accuracy for 3D imaging.

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18 19 20

21

22 23 24 25

In: ISSCC. 2005 IEEE International Digest of Technical Papers. Solid-State Circuit Conference, 364. https://doi.org/10.1109/ISSCC.2005.1494020. Charbon, E. (2014). Single photon imaging in complementary metal oxide semiconductor processes. Trans. R. Soc. A 372: 20130100. Go, M.A., Choy, J.M.C., Colibaba, A.S. et al. (2016). Targeted pruning of neuron’s dendritic tree via femtosecond laser dentrotomy. Sci. Rep. 6: 19078. Niclass, C., Ito, K., Soga, M. et al. (2012). Design and characterisation of a 256×64-pixel sinle-photon-imager in CMOS for a MEMS-based laser scanning time-of-ﬂight sensor. Opt. Express 20: 11863. Kim, C.C. and Kanner, G. (2010). Infrared two-color ghost imaging using entangled beams. Proceedings Volume 7815, Quantum Communications and Quantum Imaging VIII; 781503. Pratt, W.K., Kane, J., and Andrews, H.C. (1969). Hadamard transfer image coding. Proceedings of IEEE57, 58. https://doi.org/10.1109/PROC.1969.6869. Gibson, G.M., Sun, B., Edgar, M.P. et al. (2017). Real-time imaging of methane gas leaks using a single-pixel camera. Opt. Express 25: 002998. Altmann, Y. et al. (2018). Quantum-inspired computational imaging. Science 361: 2298. SPIE International Year of Light (2015). Celebrating Light.

Further Reading In this area is numerous. Highly comprehensive reference books are: Born, M., and Wolf, E. (1959). The Principles of Optics. London: Pergamon Press. Mandel, L., and Wolf, E. (1995). Optical Coherence and Quantum Optics. Cambridge: Cambridge University Press.

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3 Photons: The Motivation to Go Beyond Classical Optics Although classical optics is very successful in describing many aspects of light, it is not all inclusive. Many fascinating eﬀects are due to the quantum nature of light; they are the foundation of quantum optics. This chapter introduces the concept of the photon as the smallest detectable quantity of light and develops some of the basics required for modelling quantum optics experiments.

3.1 Detecting Light So far we have considered light to be just one form of electromagnetic waves, much like radio waves. The detection of radio waves and of light is fundamentally diﬀerent. Radio waves are detected with an antenna, which is a macroscopic structure made out of many atoms. The waves induce an electric ﬁeld in the antenna, a movement of many electrons. This leads to an electric signal that can be ampliﬁed and processed. The signal is proportional to the magnitude of the electric ﬁeld. In contrast, the detection of light relies on the interaction with individual atoms. The microscopic, or atomic, properties of the material of the detector determine the detection process. A photodetector does not measure the ﬁeld; it cannot follow the frequency 𝜈 of the oscillations, typically of the order of 1015 Hz. It can, however, determine a transfer of energy from the radiation ﬁeld to the atoms in the detector. And from this process we can conclude properties of the light intensity. What are the experimental details of the process of photodetection? One fundamental fact is the resonance-like property of the interaction. Atoms interact only with radiation that matches their eigenfrequencies. The absorption and emission of light results in very speciﬁc spectra. For free atoms, as those in a gas, these spectra have a resonant structure; they consist of several narrow, well-deﬁned lines. Thus, free atoms are not suitable for a general-purpose detector. Consequently, we use a material with many interacting atoms with a broad absorption spectrum. But the fact remains – individual atoms absorb the radiation that matches their individual eigenfrequencies. One well-established detector is the photographic plate. The atoms in the photosensitive emulsion can absorb light. One individual absorption process can lead to the chemical change of an entire grain in the emulsion and therefore A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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create one dark dot on the plate. This technique has one major technical problem: long exposure times. Not every photon incident leads to the change of a grain. It requires a lot of intensity to expose a plate. It is also far more convenient to deal with electric signals. Thus we use the photoelectric eﬀect. Very early experiments with this eﬀect showed that radiation interacts with such a detector by the absorption of multiples of the energy h𝜈 [1] (Figure 3.1). The probability det per unit time of detecting one individual photoelectron averaged over the detection time interval tmeas scales with the integral of the intensity det =

D h𝜈tmeas ∫0

tmeas

I(t ′ )dt ′

(3.1)

This equation applies to any form of the photoelectric eﬀect. The response time is very similar. The conversion eﬃciency D depends on the frequency of the detected light and on the material used. Each individual detection process results in one individual electron. This is not suﬃcient to produce a signal that can be processed. Several techniques have been invented to turn one electron into a measurable pulse of many electrons. This has led to the development of a range of modern photodetectors, such as photomultipliers, photodiodes, and charge-coupled devices (CCDs) and complementary metal-oxide semiconductor (CMOS) sensors [2]. For technical detail see Chapter 7. Whilst diﬀerent in their technology, they all produce a photocurrent that is proportional to the light intensity, and the detector circuitry averages over a time tmeas , which is much longer than a cycle of the oscillation. The link between the photocurrent i(t) and intensity I(t) is given by i(t) =

D′

tmeas

tmeas ∫0

I(t ′ )dt ′

(3.2)

where D′ describes the sensitivity and the gain of the detector. Detection times as short as 10−11 seconds have been achieved. Such a detector can only record changes of the intensity that occur on timescales longer than tmeas . The electric signal will contain a direct current (DC) term, proportional to the long-term average intensity, and alternating current (AC) components with frequencies up to 1∕tmeas . Please note that in our description so far, there has been no need for the quantization of the radiation. We need a quantized detection process, but Eq. (3.1) could also be used for a classical electromagnetic ﬁeld. And indeed, at high intensities, such a classical interpretation is valid. However, at low intensities, we ﬁnd experimental evidence that the light ﬁeld has to be quantized and that we need the Photo-current i(t)

Beam of light

t Sequence of photons

Figure 3.1 Photoelectric detection.

3.1 Detecting Light

concept of particles of light, of photons. For example, consider a situation where a pulse of light contains energy less than h𝜈. In this case the classical equation (3.1) would predict a probability that is larger than zero, whilst the experimental evidence shows that there is no detection at all. This can be taken into account by choosing a quantized theory of the detection process [3] that replaces Eq. (3.1) with t

meas D′ ′ ̃ † A⟩dt ̃ ⟨A (3.3) h𝜈 tmeas ∫0 ̃ and A ̃ † are operators that describe the complex amplitude of the electric where A ﬁeld and its conjugate. We will ﬁnd that there are some peculiar features of these operators compared with the classical functions. For example, the sequence of the operators in the integral is important. The sequence in Eq. (3.3) corresponds to the normal ordering that has been chosen such that absorption of photons is the detection process and a diﬀerent ordering would lead to completely unrealistic eﬀects. This quantum model is described in Chapter 4. One motivation for introducing quantized radiation stems from the question: how quickly, or with what response time, can low intensity light pulses be detected? In a classical theory there would have to be a delay in order to accumulate suﬃcient energy in one atom to release an electron. The response time tmin for light with a photon ﬂux density of N photons/m2 s and for an atom with cross section A m2 can be estimated as tmin = 1∕(NA). This is the time required for the energy equivalent to one photon to be transmitted through one atom; it would be a minimum for the response time. Is this realistic? Let us consider sunlight as an example. Green light, with a wavelength of 500 nm, is selected with an optical ﬁlter that transmits about 1% of the entire spectrum. This green light has a photon ﬂux density of about 1019 photons/m2 s as can be calculated from Planck’s radiation law using the well-known data for the sun, with a surface temperature of 6000 K and diameter of 32 arc minutes. Assuming an atomic cross section of 10−18 m2 , the detection time is about 0.1 second. For less intense beams the time would increase to fantastic values, whilst we know that our eye can be fairly fast even at intensities much lower than full sunlight. Actually, we can almost detect individual photons with our eyes. The latest studies have shown that we can sense the presence of individual photons and that the actual sensitivity depends on the actual conditions, including what has been seen immediately beforehand [4]. In order to see detectable features, the sensitivity of the human eye for green light can be estimated as about 5 × 10−18 W, which corresponds to a photon ﬂux of about 12 photons/s into the eye or a photon ﬂux density of about 109 photons/m2 s . This low level is suﬃcient to release photoelectrons, and the detection process certainly does not take longer than one second. If the simple classical model was correct, we could not see low intensity images, such as the impressive display of star patterns in a dark night. Furthermore, the vision process would be entirely diﬀerent to our experience. The main diﬀerence between bright and dim objects would be the time it takes to see them; the detected intensity would always appear to be similar. Thus we conclude that our personal experience of vision requires the concept of photons.

det =

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3 Photons: The Motivation to Go Beyond Classical Optics

It was pointed out that the simpliﬁed model of the atom interaction overestimates the response time. A faster response can be estimated by assuming that atoms can draw energy from a wider area. This would be the case when the atoms are separated or when only some of the densely packed atoms contribute to the detection process. Estimates for such a situation have been made for classical oscillators by Lorentz [5] and can be combined with the assumptions of modern atomic theories [6, 7]. These calculations show the ability of atoms to draw energy from a wide area. One physical interpretation of this eﬀect is that the incoming ﬁeld induces a dipole moment in the atom, which in turn emits a wave that destructively interferes with the incoming wave. In this way the energy is concentrated in the atom. For the case of sunlight, a response time of about 10−7 seconds, and for the sensitivity limit of the eye, a response of 10−1 seconds can be calculated. Whilst these values are short enough to be comparable with the response time of the human eye, they are still orders of magnitudes too long compared with experiments. Fast response times were measured even before the advent of modern short pulse laser technology. In 1927 Lawrence and Beams [8] used an electric spark to generate a light pulse, which was detected with a photocathode that could be gated using an additional internal grid. The timing of the grid could be delayed by propagating the electric gating pulses along various lengths of cable, and it was observed that the photoelectrons from the detector appeared simultaneously with the light pulse, within an accuracy of 10−9 seconds. Forrester et al. [9] used the detection of beat signals between two monochromatic thermal sources and concluded that the detectors can follow the electromagnetic ﬁeld with accumulation times less than 10−10 seconds. Nowadays even shorter pulses, femtoseconds, and higher modulation frequencies are routinely used in optical instruments. All these observations show the need for a concept of light as a quantized system – the need for photons.

3.2 The Concept of Photons We really need more than one concept of photons. So far we have taken the view of an experimentalist who would talk about detecting photons from a propagating beam of light. Photons have properties quite diﬀerent from those of classical particles. They have energy h𝜈, zero rest mass, momentum ℏk, and spin ℏ, and they are non-interacting particles following the Bose statistics. A laser beam contains a large number of these particles (see Table 3.1), and the measurement of the beam intensity corresponds to a measurement of the ﬂux of photons. The photons that were created by the emission process, that is, the de-excitation of the atoms via stimulated or spontaneous emission, are destroyed in the detection process. This in turn creates an electron–hole pair in the detector material. For the experimentalist photons are particles that can be localized, for example, at the detector. The experimentalist takes the consequence of the detection, the electron created or the grain in a photographic plate that is darkened, as the evidence of the photon. Figure 3.2 shows two examples of images that are typical for the detection of individual photons in one laser beam at very low light intensity. In this case the

3.2 The Concept of Photons

Figure 3.2 Two images from multi-pixel sensors, such as a CCD or CMOS camera sensor, with low intensity illumination. Each image has an array of individual detectors, or pixels, where the individual photons trigger electrical events that are recorded and displayed. The number of photons recorded in each pixel is colour-coded, from no photon as black to several photons as red or organ yellow to even higher numbers as green. These images show graininess or randomness, with no speciﬁc pattern. The images would be similar for thermal and laser light. The information can be used for imaging or for the measurement of photon statistics or could be added up as a total photon ﬂux. Table 3.1 Examples of the photon ﬂux and the photon density of typical optical ﬁelds.

Type of light

Intensity I (W/m2 )

White light (T = 6000 K)

103 4

Spectral lamp

10

5

Electric ﬁeld E (V/m)

Photon density (m−3 )

Photons/mode

103

1013

10−4

3

14

10−2

15

3 × 10

4

10

CW laser

10

10

10

1010

Pulsed laser

1013

108

1023

1018

For simplicity it is assumed that all ﬁelds have the same wavelength of 𝜆 = 500 nm, and thus all photons have the same energy h𝜈 = 2.510−19 J. In the case of the pulsed laser, the photon density inside the pulse has been evaluated. The table shows that we can consider a mode to contain a large number of photons only for laser beams.

detector array has pixels that are smaller than the size of the laser beam, which is in the form a Gaussian single-mode beam. A small faction of the beam illuminates the array of pixels shown here, each creating a number of detection events during the measurement interval chosen. The two images could be representative of two samples in time at the same location with the same average intensity or alternatively of two spatial samples in the one laser beam recorded at the same time. In either case the selection of events is random, and all the photons belong to the same mode. There is one probability distribution for detection for the entire mode. It is tempting from this point of view to envisage a beam of light as a stream of photons travelling at the speed of light. But this is incorrect and, as we shall see in the following sections, cannot successfully explain many experimental results. A diﬀerent concept for photons, perhaps more familiar to a theoretician, is needed to describe the propagation of light. Here we would talk about the photon number of a mode, where ‘mode’ refers to the radiation ﬁeld modes described in Section 2.1. Each mode has energy states like a harmonic oscillator, and the

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state of excitation of a mode is said to be the number of photons in the mode. In this concept photons are distributed, they do not exist at any speciﬁc location, and they are a property of the entire mode. The only possible processes are the annihilation and creation of a photon in which the entire mode is aﬀected, and to describe this we need the formal rules of quantum mechanics. The wonder of quantum mechanics is that the apparently contradictory points of view – localized, particle-like emission and detection but delocalized ﬁeld-like propagation, often referred to as wave–particle duality – can coexist. A laser beam can be described by a single mode propagating from the laser, with physical properties much like the Gaussian mode in Eq. (2.8). The photon number of the laser mode is linked to the ﬂux of photons in the external laser beam, which can be detected. Laser modes typically contain a large number of photons, and the ﬂux is high. For thermal light, on the other hand, the number of possible modes can exceed the photon number, and on average there will be less than 1 photon in any mode (see Table 3.1). A more detailed review of the concept of photons can be found in books by H. Paul [7] and R. Loudon [10].

3.3 Light from a Thermal Source We will now look at the example of light emitted by a broadband thermal source. We will assume that the emission properties of the atoms are quantized, as ﬁrst proposed by M. Planck [11], who tried to explain new observations of the black-body radiation or Schwarzkörperstrahlung, which had been made at the newly founded Physikalische-Technische Reichsanstalt (PTR) in Berlin, precursor to the modern PTB (Physikalische Technische Bundesanstalt), by O. Lummer and E. Pringsheim in 1900. These beautiful technical and theoretical achievements deserve description in some details. See also [12]. In 1885 W. Wien and O. Lummer followed an idea by G. Kirchhoﬀ and proposed to build an isothermal resonator that should produce inside the cavity the so-called black-body radiation. The light can be observed through a small hole in the cavity and was expected to match the predictions of the new Wien’s law [13]. In 1900 the team of O. Lummer, F. Kurlbaum, and H. Rubens constructed at the PTR a unique electrically heated resonator that could reach a temperature of 1600 ∘ C, and they recorded the radiation spectrum from 1 to 6 μm, as shown in Figure 3.3), using a new bolometer design. With even more technical developments of the radiation detectors, they were able to make remarkable improvements and extended the range of measurements to 18 μm with suﬃcient precision. The data showed a deviation from Wien’s radiation law of up to 50% [14]. The experimentalist reported the result personally to Max Planck, who was Kirchoﬀ ’s successor at the Universität in Berlin. On the very same day, Planck discovered an empirical formulation of the radiation law. He reported his results a few days later at a meeting of the DPG (Deutsche Physikalische Gesellschaft) following lecture of the experimentalists. Together, they found a close ﬁt to the observed radiation data. Two months later, in December 1900, Planck presented a full theoretical deduction of his equation using an atomistic–probabilistic description of entropy introduced by L. Boltzmann,

3.3 Light from a Thermal Source

E 140 130 120 110 1646 K (1653 K)

100 90 80 70 60

1460 K

50 40 30

1259 K

(b)

20 1095 K

10 (a) 0

998 K 904 K 723 K

1

2

3

4

5

6

λ

Figure 3.3 (a) Absolute measurements of the spectral energy distribution carried out in 1900 by O. Lummer and E. Pringsheim on the electrically heated black body over a spectral range of 1–6 μm. The broken line was calculated with Wien’s radiation law. The continuous line represents the measurements that increasingly deviate from Wien’s law with higher temperature and increasing wavelength, but were in good agreement with Planck’s radiation law. (b) The apparatus is also shown. Source: From PTB historical records and summary by J. Hollandt [12].

which he had rejected until then. In an ‘act of desperation’, Planck only allowed certain discrete states of energy. Using the experimental data he was able to determine the free variables in his equation. Whilst now regarded as the hour of birth of quantum mechanics, neither Planck himself nor his audience were aware of the widespread consequences of this discovery of Planck’s constant in the new century ahead. The term photon was not used at the time. We will use it to describe the fact that the light is emitted and absorbed in ‘lumps’ or ‘quanta’ by atoms, molecules, and solid materials. Using a semi-classical approach, light emitted by a thermal source can be analysed by considering a ﬁnite volume of space with zero ﬁeld boundary conditions. The region will support a discrete number of standing waves of the electromagnetic ﬁeld. We ﬁrst derive the number and the spectral density of these modes, and from there we determine the distribution of photons in the light emitted by the source. If the region has the physical dimension L, it can support a variety of modes with the wave vectors kx = 𝜋𝜈x∕L, ky = 𝜋𝜈y∕L, kz = 𝜋𝜈z∕L, resulting in a mode density: 𝜌𝜈 d𝜈 =

8𝜋𝜈 2 d𝜈 c3

(3.4)

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These modes are thermally excited according to a Boltzmann distribution: exp(−En ∕kB T) (En ) = ∑∞ n′ =0 exp(−En′ ∕kB T)

(3.5)

Assuming the condition En = (n + 12 )h𝜈 for the energy of a mode, we can derive a probability distribution for n photons in one mode: n (𝜈, T) =

1 − exp(−h𝜈∕kB T) exp(n h𝜈∕kB T)

(3.6)

We can now evaluate the average number n(𝜈, T) of photons in a mode: n(𝜈, T) =

∞ ∑ n=0

nn (𝜈, T) =

1 exp(h𝜈∕kB T) − 1

(3.7)

which is the well-known black-body formula. It explains the existence of the black-body spectrum of a solid at temperature T. The ﬂuctuations in a single mode of this radiation can be rather large. For normal thermal conditions and for radiation in the visible or near infrared, the average excitation is very low. For example, at T = 300 K and for 𝜈 = 1015 Hz, the average excitation is about 7 × 10−4 photons in each mode. The probability of ﬁnding a ﬁxed number of photons, for example, n = 0, 1, 2, …, in one mode follows a Bose–Einstein distribution that actually has a maximum at n = 0. However, this calculation is only valid if we count photons in a single spatio-temporal mode. More typically, for the detection of light from a thermal source with measurable power, we have to consider many modes at diﬀerent frequencies. They all contribute to the measured intensity. We have to evaluate the probability m that m photons are emitted in the detection interval td . The interval is chosen so large that m ≫ 1. Further we assume that the average intensity of the light is not time dependent. The light source sends out a stream of photons, and at all times we measure the same average intensity and the same average number of photons. The result from classical statistics for many independent sources is t ⟨I⟩ ⟨m⟩m exp(−⟨m⟩) with ⟨m⟩ = d (3.8) m! h𝜈 which is known as the Poissonian distribution. One important property of this distribution is Δm2 = ⟨m⟩, which states that the width of the distribution, measured by the variance, is equal to the mean value of the distribution. The probability distribution in Eq. (3.8) describes the ﬂuctuations of the intensity emitted by a thermal source. The nature of these ﬂuctuations is such that they are independent of the time intervals from which the photon number data is collected. These time intervals can be close and regularly spaced or widely separated and intermittent; the results will be the same, provided a suﬃciently large sample is taken. Noise that behaves in this way is known as stationary noise. As a consequence g (1) (𝜏) and g (2) (𝜏) are independent of 𝜏. In particular, the noise spectrum I(Ω) (averaged over a small frequency range 𝛿Ω about Ω) is independent of the detection frequency Ω, i.e. all frequency components carry the same intensity (I(Ω) is ﬂat). This type of noise m (td ) =

3.4 Interference Experiments

is referred to as white noise. The value of the variance is ﬁxed for a given average intensity and depends only on the length of the detection interval: Var(I(Ω)) =

⟨I⟩h𝜈 td

(3.9)

This statistical result can be applied to any other situation where events generated by independent processes are counted and the average number of such events is not dependent on time. Examples are the measurement of particles from radioactive decay or the number of raindrops, particles of rain, in a steady downpour. The spectrum of the noise for all these cases is equivalent to the sound of a heavy rain shower on a metal roof. It was this analogy, namely, the dropping of lead shot on a metal surface, that was the origin for the name of this statistical phenomenon: shot noise. The classical description provides the tools for describing the statistical properties of light. Chapter 4 gives a complete quantum mechanical description. The classical description can serve as a reference, and the interesting question will be: how do the classical predictions compare with the more complete results from quantum theory? We will ﬁnd that the predictions for thermal light, that is, light emitted by many independent atoms, are identical in both the classical and the quantum models. The lesson here is that it is sometimes suﬃcient to quantize the emitting (and absorbing) atoms whilst still retaining a classical description of the light (this is often referred to as a semi-classical description). In contrast, the results from a quantum mechanical description of the laser will be quite diﬀerent from the results for a classical oscillator. A detailed comparison between the two models will show up several features that are uniquely due to the quantum nature of the light.

3.4 Interference Experiments The measurement of interference, made accessible through the use of instruments such as a Michelson interferometer, clearly demonstrates the wave nature and the coherence properties of light. However, the same type of interference experiments carried out at very low intensities also documents some of the quantum properties. This was done in very early experiments by Taylor [15]. These experiments used photographic plates and extremely long exposure times – up to three months. The experiments showed that the visibility of the fringes remained constant, at a given total exposure, independent of how long it took and thus how many photons could have been in the instrument at any one time. It should be noted that all these interference experiments are most impressive achievements. The interferometers are very sensitive devices, and nothing would have been easier than wiping out the interference fringes through a small perturbation of the apparatus. More recent experiments [16, 17] using photoelectric detection conﬁrmed these results. In addition, some of these experiments used interferometers with total optical paths far exceeding the coherence length of the light source. For example, in the experiment by Janossy, each arm of the Michelson interferometer

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3 Photons: The Motivation to Go Beyond Classical Optics

was 14.5 m long, requiring a very quiet and temperature-stable (ΔT < 0.001 K) environment, which was achieved in an underground tunnel. Again, the visibility remained unchanged – independent of the actual intensity. Complementary results were obtained with the synchrotron radiation emitted by a storage ring. Here the number of electrons generating the radiation can be determined separately from the intensity of the radiation – and again the visibility was shown to be independent of the photon number. All these experiments support the statement by P. Dirac: The new theory, which connects the wave function with probabilities for one photon, gets over the diﬃculty by making each photon go partly into each of the two components. Each photon then interferes only with itself. Interference between two diﬀerent photons never occurs. [18] The beamsplitter in the interferometer does not divide the light into smaller parcels – any detection of the light shows quanta of the same amount of energy. This result cannot be derived if we think of photons as small local parcels of energy – miniature billiard balls – with the properties of classical particles. It reveals one of the consequences of quantum mechanics, namely, that a conclusion about the speciﬁc outcome of an experiment cannot be drawn before the experiment is carried out. This consequence of quantum mechanics becomes very clear by considering the various possible outcomes of an interference experiment where detectors and counters are used in an attempt to measure the presence of the photon in the one or the other of the interferometer arms. Any such attempt would fail, as it has been so elegantly expressed by R. Feynman et al. in his lecture notes [19]. The duality of the wave–particle model is a clear conclusion from these experiments: light is neither exclusively a wave nor exclusively a stream of particles. The type and condition of the experiment determines which one of these models will dominate and which one will give the simpler interpretation. All the time light is something more inclusive, more complicated than either of these models would suggest. The statement that ‘a photon will only interfere with itself’ should not be taken too literally in the sense that it could be tested through the observation of individual events. In the case of the photographic recording, each detection event produces one clear dark dot on the photographic plate. This dot could be part of an interference pattern or of a random distribution of dots. It is the distribution of many dots – a frequent repeat of the same measurement – that reveals the result of the experiment. Theoretically, one single dot measured exactly at the location on the plate where there should be absolutely no intensity according to the calculation would have an enormous signiﬁcance. Such an event would prove the calculation wrong (Figure 3.4). But we have to note that in reality there will always be some other imperfection, such as an imbalance of the beamsplitter, ﬁnite coherence length, or background noise, which could have resulted in this particular single event. It is the distribution of the dots, or of the counts in the electronic instrument, that is crucial for the test of a model. As is common

3.4 Interference Experiments

4 1

3

2

Figure 3.4 Historical example: interferometer at the turn of the century. Source: From the notes of A.A. Michelson and E.W. Morley [20]).

for all tests of quantum mechanics, a large number of individual experiments are required to allow a valid conclusion. The nature of the light seems to depend on the type of experiment: a single beamsplitter can often be well described by the photon picture, and a complete interferometer requires the wave picture. An interesting extension of the demonstration described above are the so-called delayed choice experiments. Here we are trying to trick the light into the wrong behaviour by delaying the decision of which type of experiment is being performed until it has already passed the ﬁrst beamsplitter. Will the photon be able to predict the type of experiment that is lying ahead and react in the appropriate way? Whilst this view is certainly naive, it poses a valid question: does it matter whether we have a delay built into the interferometer? One example for such an experiment [21], shown in Figure 3.5, uses pulses of light of short duration (150 ps). The intensity was chosen suﬃciently low that on average only every ﬁfth pulse contained one photon. Pulse 150 ps

n = 0.2 photons

Electro-optic shutter

Polarizing beamsplitter

5m Fibre

Det 1

5m

Det 2

Figure 3.5 Schematics of a delayed choice experiment. Source: After Hellmuth et al. [21].

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3 Photons: The Motivation to Go Beyond Classical Optics

Both arms of the interferometer contain 5 m of single-mode ﬁbre. One arm contained an electro-optical switch that allowed this arm to be blocked and the interference to be turned oﬀ through an electronic signal. In order to put the photons in the above-mentioned dilemma, the interferometer was operated such that the switch was operated after the pulse had passed through the ﬁrst beamsplitter. However, the delay had no eﬀect on the visibility of the interference pattern. The outcome of the experiment depended only on the conﬁguration of the experiment that was present at the time of detection, independent of the earlier state of the apparatus. A more thorough discussion of these types of experiments is given in Chapter 12. This raises the question as to whether photons from two completely independent sources can interfere with each other. It is well known that light from diﬀerent sources, or from diﬀerent parts of the same source, does not interfere when observed for a long time. This is a consequence of the fact that the phase of the electromagnetic wave emitted by the two sources has two independent phase terms. In a normal interferometer, which has wave front splitting mirrors, this does not matter as long as the path length diﬀerence is less than the coherence length. Similarly, for independent sources, the fringes should also appear if the measurement takes less time than the coherence time tcoh . During such a short time interval, the phase will not vary enough to reduce the visibility. This has been shown in an experiment with two independent lasers [22], in this early experiment two ruby lasers. Spatial interference patterns from single pulses of the lasers were recorded, clearly showing the interference between photons from diﬀerent sources. An alternative is to measure the beat between two lasers; this was ﬁrst demonstrated for two independent He–Ne lasers [23]. The experiment operated continuously, and the narrow linewidths of the gas lasers made the observations straightforward. The major problem was the drifts in the laser frequency due to mechanical and temperature variations. Once this was overcome, this experiment demonstrated clearly that interference exists between beams from independent sources, provided that the detection time t1 is less than the coherence time tcoh . With modern technology, such as tunable diode lasers and radio wave spectrum analysers, this type of experiment is almost trivial. The eﬀect is common knowledge for a laser physicist. However, the interesting question remains: should this interference eﬀect between independent lasers persist in the limit of very small intensities? The ﬁrst answer might be No, since we cannot be certain of the relative phase during the long measurement time required for the weak beams. However, if we select only data measured at times when the relative phase is within a set small range, one could expect an interference pattern to become visible. Exactly this experiment was carried out by Radloﬀ et al. [24]. Two He–Ne lasers were used that had been built in one mechanical structure in order to minimize the technical noise and the phase ﬂuctuations due to vibrations. The beat frequency between the beams was continuously monitored. A very small fraction of the light was selected and formed a spatial interference pattern. This was gated by an electro-optic shutter and only opened when the frequency was within a preset range. The photon ﬂux was as low as 105 photons/s with a typical shutter time of 10−5 seconds.

3.4 Interference Experiments

After exposures of about 30 minutes, clearly recognizable interference patterns became visible. These patterns were present even at arbitrarily low intensities. The particle picture is not at all suitable for this situation. It is extremely unlikely that one photon from each laser will pass together through the shutter. The wave model on the other hand provides a very simple explanation. The only problem arises when we want to understand the quantization of the energy together with interference. These results for the interference of photons from independent sources are in some contradiction to the statement by P. Dirac, to repeat: ‘Each photon interferes with itself. Interference between two diﬀerent photons does not occur’. Dirac made this statement whilst considering only interference experiments known at his time. It should not be applied to the experiments involving independent sources. Nevertheless, attempts have been made to rescue Dirac’s statement by evoking additional processes that guarantee that photons are emitted simultaneously by both lasers. But these attempts are really not all that helpful in understanding the quantum nature of light. In summary, we can say that the only conditions under which interference can be observed are those where it is impossible to detect which path the photon has taken. Any attempt to measure the path, or even to reconstruct the path after the event, would wipe out the interference pattern. A discussion of these types of experiments has led to the postulate of Heisenberg’s uncertainty principle. For this particular case the principle states that ΔxΔp ≥ h∕4𝜋. It enforces that we can measure either the interference pattern, by determining the position x of the photon, or alternatively the path of the photon, as measured by its momentum p. But we cannot measure both simultaneously [7]. Discussion about these results has played a prominent role in the development of quantum mechanics as a whole. A related idea was behind the design by J.D. Franson, who designed an interferometer that can be used to distinguish between single- and two-photon interference [25]. His design is shown in Figure 3.6. In this apparatus let us consider two coincident photons, emitted by a source, which individually travel in opposite directions towards two identical interferometers. Each interferometer contains θ2

θ1

γ2

D2

Source

D′2

γ1

D1

D′1

Figure 3.6 The Franson interferometer. Source: Franson 1991 [25]. Reproduced with permission of APS.

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3 Photons: The Motivation to Go Beyond Classical Optics

a shorter and a longer path, and the diﬀerence Δttrans in transit times over the two paths is taken to be much larger than the coherence time of the photons. In addition, it has individual phase settings Θ1 and Θ2 . That means each interferometer by itself would not produce fringes individually and the visibility of a fringe due to a change of the phase Θ1 and Θ2 would be extremely low for such a large path length imbalance. However, interference between the quantum mechanical amplitudes for the photons to have both travelled the shorter paths and the longer paths produces a modulation in the coincidence counting rate R of photons from the two sets of detectors, given by cos2 [(Θ1 + Θ2 + 2𝜋𝜈 Δttrans )∕2]. This type of interferometer would only show a modulation due to the quantum mechanical wave function of the photons, which is sensitive to indistinguishability of the paths. That means an interference can occur for the quantum mechanical amplitudes and thus for the counting rates. This would be a clear distinction between two types of interference. Moreover, if we illuminate this interferometer with an entangled pair of photons, there is a link between the probability amplitudes on both sides of this interferometer, and we can measure the visibility of the fringes due to the settings Θ1 and Θ2 . This should allow us to learn about the quantum properties of the photon pairs and will be discussed in Chapter 13.

3.5 Modelling Single-Photon Experiments We now turn to a more idealized description of single-photon experiments in order to see how they may be modelled. We will introduce a number of basic optical elements as ‘black boxes’, describing and contrasting their operation for classical light waves and single photons. Descriptions of how these optical elements can be constructed will be found in later chapters. Similarly, the simple quantum mechanical models introduced here will also later be expanded. The average number of photons per second in a beam of light is proportional to its intensity. However, the quantization of the ﬁeld introduces a degree of freedom not present in the classical theory, the photon statistics. Suppose we had a ‘photon gun’, that is, a light source that emits just one photon when we pull the trigger. Such a photon gun can be produced through pulsed excitation of a single molecule, though presently not conveniently. Such a source would share some characteristics with the highly attenuated sources described in Section 3.4. However, whilst an attenuated laser may produce one photon on average in some time interval, there will be ﬂuctuations: sometimes there will be two or more photons, and sometimes there will be none. On the other hand, we will always detect just one photon in the measurement interval coming from our photon gun. There is no way to describe this diﬀerence in the photon statistics with a classical description of light. We must introduce a new concept, that of the quantum state of the light. The quantum state describes all that is worth knowing about a light beam. The photon gun produces light in a single-photon number state, which we will write as |1⟩. Studying the behaviour of such a light state in simple situations illustrates some of the basics of the quantum description of light.

3.5 Modelling Single-Photon Experiments

3.5.1

Polarization of a Single Photon

Let us ﬁrst consider how a single polarized photon behaves. We will describe this behaviour using single-photon polarization states. We will see how it is possible to build a consistent description of light phenomena in spite of wave–particle duality. To do so, however, we will need to introduce some quite diﬀerent concepts to those found in classical theory. Recall that in our classical description the polarization of the ﬁeld was described by the vector p(t), which gave the direction of the electric ﬁeld tangential to the wave front. For linearly polarized light p will be ﬁxed. For circularly polarized light p will rotate at the optical frequency. It is possible to design optics that will decompose a light ﬁeld into two orthogonal modes. The amplitudes of the two modes will be equal to the projection of p onto the chosen axes. For example, a horizontal/vertical polarizing beamsplitter will project the ﬁeld onto a horizontally polarized mode and a vertically polarized mode. These two modes will exit through diﬀerent ports of the beamsplitter. Similarly, one can use a diagonal/anti-diagonal polarizing beamsplitter to decompose the ﬁeld into linearly polarized ﬁelds at 45∘ and −45∘ away from horizontal. With slightly more eﬀort it is also possible to create a left/right circular polarizing beamsplitter that decomposes the ﬁeld into clockwise and anticlockwise rotating circular polarizations. Suppose we ﬁre a string of single photons at a horizontal/vertical polarizing beamsplitter. As we have noted, classically such a device will direct horizontally polarized light in one direction and vertically polarized light in the other. Thus we will deﬁne photons that exit through the horizontal port of this beamsplitter as being either in the horizontal state or the vertical state as |H⟩

or |V ⟩

(3.10)

By ‘exit through the horizontal port’, we mean that a photon detector placed at the output of the horizontal port will detect a photon, or equally a photon detector placed at the vertical port does not detect a photon. These situations are illustrated in Figure 2.8. The symbol for the state, |−⟩, is referred to as a ket. These deﬁnitions are sensible because a photon that exits through the horizontal port of the beamsplitter and is passed through another polarizing beamsplitter will certainly exit through the horizontal port. A similar deﬁnition applies to vertical photons. The diagonal single-photon state, |D⟩, and the anti-diagonal state, |A⟩, can be deﬁned in a similar way by analysing the beams with a diagonal/anti-diagonal polarizing beamsplitter. So far this is straightforward. The physics of classical polarized beams and single photons looks the same. Things become more interesting when we start to mix up the polarizations. If we send a diagonally polarized classical beam of light into a horizontal/vertical polarizing beamsplitter, then half the beam will exit through the horizontal port and half will exit through the vertical port. What will happen if we send single photons in the state |D⟩ through this beamsplitter? As we have discussed in Section 3.4, the photons cannot be divided, in the sense that a

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detector placed at one of the output ports will either detect a whole photon or no photon. Instead they must go one way or the other. To be consistent with the classical result for many photons, it must be that they go one way 50% of the time and the other way the other 50% of the time. But can we tell in which direction an individual photon will go? To answer this ﬁrst remember how |D⟩ photons behave at a diagonal/anti-diagonal polarizing beamsplitter: they all exit through the diagonal port. That is, they all behave perfectly predictably and identically. Yet when these identically behaving photons are sent into a horizontal/vertical beamsplitter, we have argued some must take one path and others take the other. Thus the path an individual photon in the state |D⟩ takes through the horizontal/vertical beamsplitter is not speciﬁed. All that is speciﬁed is that on average half the photons will go one way and half will go the other. This example illustrates the fact that in quantum mechanics it is probabilities of outcomes, not particular outcomes, that are predicted. Even though we can say with certainty that a photon in the state |D⟩ will emerge from the diagonal port of a diagonal/anti-diagonal beamsplitter, its reaction to a horizontal/vertical beamsplitter is as random as a ﬂip of a coin. It is tempting to think that maybe there are other variables (perhaps hidden to us) that do determine in a precise way the polarization behaviour of individual photons under all conditions. However such a possibility can be ruled out experimentally (see Chapter 12). We are forced to accept the intrinsic indeterminacy of the quantum world. Continuing our discussion of polarization states, we can also introduce the right circular single-photon polarization state, |R⟩, and the left circular state, |L⟩, in an analogous way to the other states. A photon in state |R⟩ will randomly take one port or the other when sent into either a horizontal/vertical or a diagonal/anti-diagonal beamsplitter. A photon in state |L⟩ will behave in the same way. Similarly a photon in state |H⟩ will give a random result for both diagonal/anti-diagonal and right/left circular polarizing beamsplitters and so on for the other states. We will now outline the mathematical formalism for treating these situations. 3.5.1.1

Some Mathematics

The state kets we have introduced are vectors. They live in a vector space called Hilbert space. Hilbert space is a complex vector space: when we add vectors, we are allowed to do so using complex coeﬃcients. The dimensions of the vector space are equal to the number of degrees of freedom of the property being described. In the polarization example our vector space is discrete and has dimension 2. That is, in all the experiments we have described, there are only two possible outcomes: the photon comes out from one port or the other. It can only take two discrete values. This is in stark contrast to the classical case in which the amount of the ﬁeld that comes out of one or other port can vary continuously between all and nothing. In general, discrete systems can have any number of dimensions, and continuous systems can also be described. A continuous variable can give any outcome in some range, and their vector space will have inﬁnite dimensions. We will look at such examples in optics in Chapter 4. As for other vector spaces, we can reach any vector in our space through linear superpositions of n orthogonal basis kets, where n is the dimension of the system.

3.5 Modelling Single-Photon Experiments

To deﬁne orthogonality we introduce bra vectors, which are written ⟨x|. Bras and kets are related by the Hermitian conjugate, an operation similar to complex conjugation and indicated by the ‘dagger’ symbol, †. Thus we can write z∗ ⟨x| = (z|x⟩)† .

(3.11)

where z is just a complex number. Vectors |x⟩ and |y⟩ are orthogonal if ⟨x| × |y⟩ ≡ ⟨x|y⟩ = 0

(3.12)

In general the object ⟨x|y⟩ is called a bracket and is similar to the dot product in real vector spaces. A ket is said to be normalized if ⟨x|x⟩ = 1. We will generally assume this property. Simple physical questions can be asked in the following way. If we prepare our system in the state |𝜓⟩ and then use an analyser to ask the question ‘Is my system in the state |𝜎⟩?’, the probability, P, that we get the answer ‘Yes’ is given by P = |⟨𝜓|𝜎⟩|2

(3.13)

The bracket ⟨𝜓|𝜎⟩ is referred to as the probability amplitude and will in general be a complex number. The orthogonality condition then says that if we prepare our system in |𝜎⟩, our analyser will never ﬁnd it in |𝜓⟩ if these two states are orthogonal. Normalization reﬂects the fact that if we prepare our state in |𝜓⟩, we will certainly (P = 1) ﬁnd it there with our analyser. We will try in this book to minimize the mathematics by introducing concepts on a need-to-use basis. However, a deep understanding of the workings of quantum optics requires a good understanding of the mathematical structure of quantum mechanics. Readers unfamiliar with the basics of quantum mechanics are encouraged to study other texts such as Refs. [26] and [27].

3.5.2

Polarization States

The concepts introduced above allow us to identify our polarization states |H⟩ and |V ⟩ as orthogonal states. That is, ⟨H|V ⟩ = ⟨V |H⟩ = 0, which means that a photon in state |H⟩ never comes out the vertical port and vice versa. We will assume also that our states are normalized such that ⟨H|H⟩ = ⟨V |V ⟩ = 1. This means that we should be able to express all our polarization states as linear superpositions of the horizontal and vertical states. Consider ﬁrst the diagonal states. From our discussion in Section 3.5.1, we know that they must have the properties ⟨D|A⟩ = ⟨A|D⟩ = 0 |⟨D|H⟩|2 = |⟨D|V ⟩|2 = 0.5 |⟨A|H⟩|2 = |⟨A|V ⟩|2 = 0.5

(3.14)

or the diagonal states are orthogonal and a diagonal state placed through a horizontal/vertical polarizing beamsplitter has a 50% probability of exiting through

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3 Photons: The Motivation to Go Beyond Classical Optics

either port. It is straightforward to check that 1 |D⟩ = √ (|H⟩ + |V ⟩) 2 1 |A⟩ = √ (|H⟩ − |V ⟩) 2

(3.15)

satisfy the conditions of Eq. (3.14). Now consider the circularly polarized states. We need to satisfy ⟨R|L⟩ = ⟨L|R⟩ = 0 along with |⟨R|H⟩|2 = |⟨R|V ⟩|2 = 0.5 |⟨L|H⟩|2 = |⟨L|V ⟩|2 = 0.5 and |⟨R|D⟩| = |⟨R|A⟩|2 = 0.5 2

|⟨L|D⟩|2 = |⟨L|A⟩|2 = 0.5

(3.16)

This looks impossible until we remember that Hilbert space is a complex vector space. Thus we are allowed to form the complex coeﬃcient superpositions 1 |R⟩ = √ (|H⟩ + i|V ⟩) 2 1 |L⟩ = √ (|H⟩ − i|V ⟩) 2

(3.17)

which satisfy the above requirements. It follows that an arbitrary polarization state can be represented by |𝜔⟩ = x|H⟩ + ei𝜙 y|V ⟩

(3.18)

where x2 + y2 = 1 and 𝜙, x, and y are real numbers (absolute phase factors have no physical signiﬁcance). Notice that such a representation is not unique. We could equally well write all our states as superpositions of diagonal and anti-diagonal states or circular states. Any two orthogonal states can span the Hilbert space. We have seen that in order to describe the behaviour of a quantized light ﬁeld, we have had to introduce two new concepts: (i) the state of the light and (ii) that predictions are made of probabilities of events, not actual events. The fact that these concepts lead us to represent, for example, a diagonal photon as an equal superposition of horizontal and vertical photon states may seem relatively benign at this point. That such a superposition has no analogue for classical particles will be demonstrated more graphically in Section 3.5.3. In Chapters 12 and 13, we will discuss in detail photon experiments that use polarization as their degree of freedom. In particular, we will investigate the unique communications and information processing abilities these quantum states can exhibit.

3.5 Modelling Single-Photon Experiments

3.5.3

The Single-Photon Interferometer

We will now describe a simple single-photon interference experiment using a Mach–Zehnder interferometer, similar in concept to the experiments described in Section 3.4. A more general description of this device can be found in Section 5.2. Here we will illustrate the basic physics of the arrangement in the extreme situation of only a single photon present in the interferometer at a time and how the mathematics introduced in the previous sections can describe it. A Mach–Zehnder interferometer consists of a 50/50 beamsplitter, which divides the input light into two equal parts, followed by a series of mirrors, which direct the two beams back together again on a second 50/50 beamsplitter that recombines them. The path length diﬀerence between the two arms of the interferometer can be adjusted. This then corresponds to the situation discussed in Section 5.2, and thus classically we expect the amount of light exiting one port to change as a function of the path length diﬀerence. In particular, if our path length diﬀerence remains well within the coherence length, we will be able to pick a path length such that all the light exits from a single port. This set-up is depicted schematically in Figure 3.7. Let us now describe the behaviour of this interferometer when single photons pass through it. We will assume that the path length diﬀerence is held in the situation described above such that for classical beams all the light enters and exits through a single port. We will also assume that our beamsplitters and mirrors are polarization independent so that the polarization of our single photons is irrelevant, and we can simply label a single photon in a beam by the number state ket |1⟩. There are however two ways in which the photon can enter the interferometer: via the beam path labelled a in Figure 3.7 or by the beam path labelled b. Suppose we in fact send our photon in by beam path a. We can describe this initial situation by the state ket |𝜎⟩ = |1⟩a |0⟩b

(3.19)

meaning there is one photon in beam a and no photon in beam b. There is an even probability of the photon exiting from either port of the ﬁrst beamsplitter. The output state from the beamsplitter can thus be written as 1 |𝜎 ′ ⟩ = √ (|1⟩a |0⟩b + |0⟩a |1⟩b ) 2 Figure 3.7 Schematic of single-photon interferometer.

50/50

(3.20)

Pathlength adjustment

a

b 50/50

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3 Photons: The Motivation to Go Beyond Classical Optics

The path length condition means that, apart from an overall phase, this state remains the same as it propagates through the interferometer. The time symmetric nature of light propagation through a beamsplitter means that when the beams reach the second beamsplitter, they must return to their original state of a photon in a and no photon in b (see Eq. (3.19)). Thus we just get the photon equivalent of the classical result – the photons always exit through a single port. Now let us consider the physical signiﬁcance of our description. In particular, what does Eq. (3.20) mean? It says that the photon is in a superposition of being in each path. It is tempting to think that this just means the photon is in one or the other path with a 50% probability. Certainly the probability of ﬁnding the photon in path a Pa = |⟨1|𝜎 ′ ⟩a |2 = 0.5 is 50%. But is the interference eﬀect, i.e. the fact that the photons always exit from the a port (see Figure 3.7), consistent with the photons taking one path or the other? Well, the second beamsplitter is identical to the ﬁrst one so our ﬁrst puzzle is: why should the photon only exit through one port? We might imagine that it picks up some kind of ‘label’ at the ﬁrst beamsplitter that tells it what to do at the second. So suppose we ﬁnd our photon in beam a in the middle of the interferometer, but then we ‘release’ it and let it ﬁnish its journey to the second beamsplitter. Experimentally we ﬁnd that now there is no interference and the photon is just as likely to come out from either of the output ports. Quantum mechanically, because we now know that the photon is in beam a, the state inside the interferometer becomes |𝜎 ′ ⟩m = |1⟩a |0⟩b , and so the photon just behaves as it did at the ﬁrst beamsplitter and has a 50/50 chance of emerging from either port. We might argue that when we ‘caught’ the photon, it forgot its label and so just behaved randomly at the second beamsplitter. So consider another possibility. Suppose we look for the photon in beam b but we do not ﬁnd it. Now quantum mechanically the situation is the same as above; we know that the photon must be in beam a so the state again becomes |𝜎 ′ ⟩m = |1⟩a |0⟩b and the interference should disappear. But any common- sense description would say this cannot be. We have not interacted with the photon, so any label it has must be unaﬀected. Thus we would expect the interference to still appear. Experiments, however, conﬁrm the quantum mechanical result (see Chapter 12). We are forced to conclude that the superposition state of Eq. (3.20) represents in some objective sense the presence of the photon in both paths of the interferometer. Any experiment that causes the photon to manifest itself in one or the other path, whether an actual detection occurs or not, destroys this superposition and in turn wipes out the interference. This is wave–particle duality and we will encounter it in many guises throughout this book.

3.6 Intensity Correlation, Bunching, and Anti-bunching The quantum properties of light can be investigated by using the techniques of photon counting. We will ﬁnd that the statistics of photons are considerably different from the statistics of independent particles. Again, we have to combine

3.6 Intensity Correlation, Bunching, and Anti-bunching

the particle picture with the wave picture. For these statistical experiments it is necessary to measure quantitatively the correlation between diﬀerent beams of light. This approach has already been discussed in Section 2.3.3. In particular the discussion of the Hanbury Brown and Twiss experiment for thermal light, in their case starlight, showed a way for determining the correlation between ﬂuctuating currents ia (t) and ib (t) by recording and analysing the product ia (t)ib (t) of these currents. This technique can be used for any type of light, including laser light, and it also applies to any intensity of light. At very low intensities the photodetector becomes a photon counter. For a photon counter the probability of a count is proportional to the intensity of the light. Now the number n(t, tdec ) of counts recorded at time t within a time interval tdec represents the intensity of the light. The detection time interval tdec should be shorter than the coherence time tcoh of the light source; otherwise the ﬂuctuations are simply averaged out. As before we determine the intensity correlation by using detectors Deta and Detb , measuring the photon counts na (t, tdec ) and nb (t, tdec ) independently, and then forming the product na (t, tdec )nb (t, tdec ). Actually, such medium intensities, ranging from a few to several tens of counts per detection interval, are very challenging. No particularly satisfactory technology for this range has yet emerged, though progress is steadily being made (see Chapter 12). On the other hand, at very low intensities, where the probability of more than a single count per detection interval can be neglected, good and reliable technology exists (see Chapter 7). A moment’s thought shows that the probability of coincidences, which means events where both detectors register a photon within a short time interval, is proportional to the value of na (t, tdec )nb (t, tdec ). It is technically very attractive to build a circuit that only responds to such coincidences. This technique shows an increase of the coincidence rate under the same conditions where big correlations are measured. Going back to the Hanbury Brown and Twiss experiment, the coincidence rate is high if the separation between the detectors is suﬃciently low. This is an excess of coincidences – compared to the numbers one would expect from completely random counting events. In other words, whenever a photon has reached detector Deta , the probability for another photon to be at Detb is enhanced. This could not be understood by regarding photons naively as independent particles, thereby ignoring the wave properties of the light. The photons emitted from the diﬀerent part of the source would not know of each other. Their arrival time would not be correlated. In contrast, the wave picture predicts that the instantaneous intensity on the detector is inﬂuenced by all atoms in the source. Each part of the source is making a small contribution. Thus both detectors see related contributions where light is originated from one atom in the source. Again, this means we cannot consider individual photons, each with one well-deﬁned birthplace. Atoms cannot be localized in that way. These observations are concerned with spatial eﬀects. Intensity information can also be gained from temporal eﬀects within a single beam. To measure the time correlation of a beam of light, we can use a trick that was used by Hanbury Brown and Twiss as a precursor to the stellar observations. We can use a beamsplitter to generate two light beams with each one being detected individually, as

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3 Photons: The Motivation to Go Beyond Classical Optics

na (t, tdec)

Deta

nb (t, tdec)

Detb

X

Figure 3.8 Schematic layout of an optical intensity correlator.

Correlator (product)

shown in Figure 3.8. The coincidence rate can now be directly measured using detectors that are gated. Only events that happen within the gate time at both detectors are recorded. By moving one of the detectors either further away or closer to the beamsplitter, we introduce a delay time 𝜏 and can determine the coincidence rate as a function of the delay time. Alternatively, we can leave the detectors stationary and use electronic delays. For thermal light the coincidence rate for 𝜏 < tcoh should be higher than for 𝜏 > tcoh . This is a direct analogy to the properties of the second-order correlation coeﬃcient. It corresponds to g (2) (𝜏 < tcoh ) > g (2) (𝜏 > tcoh )

(3.21)

as shown in Eq. (2.47). Since the coincidence rate for 𝜏 ≫ tcoh can only be those events that are completely random, we simply have to look for excess coincidences. The long delay time measurements serve as a normalization. One practical diﬃculty is that the detection time tmeas has to be less than the coherence time if we are to be within the single-mode approximation. For normal thermal light the linewidth is very large, the coherence time is extremely short, and the response of the detectors is too slow. A direct measurement of the coincidences for thermal light is not possible. In contrast, laser light has a small linewidth and long coherence time, and the experiment is feasible. One clever experimental trick is to simulate thermal light through scattering of laser light on rapidly and randomly moving objects. This was ﬁrst done by Arrecchi et al. [28]. The experiments used light from a gas laser with long coherence time, which was scattered by polystyrene particles of various sizes suspended in water and detected by two detectors. The coincidence rate, or second-order correlation coeﬃcient, was measured. For comparison the laser light itself was investigated, and the results show a clear excess of coincidences. The shape of g (2) (𝜏) ﬁts properly to the prediction of a thermal light source with a Gaussian linewidth, and the diﬀerence between thermal light and coherent laser light is clearly demonstrated. An alternative interpretation of these results is to say that the photons from the thermal light arrive in bunches and the probability for photons to arrive together, in coincidence, is large for short delay time intervals. This phenomenon is known as photon bunching. It is typical for thermal light that has naturally large ﬂuctuations. In contrast, for coherent light there is no preferred time interval between photons. Coherent light is not bunched. The coincidences show random Poissonian statistics on all timescales. Can there be anti-bunched light, where the photons keep a distance, in time, to each other? This would be light that clearly violates the classical conditions. The second-order correlation function would be less than 1, which is not possible for classical statistics, as

3.6 Intensity Correlation, Bunching, and Anti-bunching

Microscope objective

g2 (τ) 3 Optical excitation 2

Timer Start

Phototube Stop

1 Computer

25 (a)

(b)

50 Delay τ (ns)

75

100

Figure 3.9 The experiment by Kimble et al. [29]. (a) Schematic outline. (b) Measured values for g(2) (𝜏) derived from the data. The broken curve shows the theoretically expected form for a single atom, arbitrarily normalized to the same peak.

already shown in Section 2.3. A measurement of anti-bunched light would be a clear demonstration of quantum behaviour, of non-classical light. Well in fact the single-photon quantum state we discussed in the last section would be just such a state. Recall from Eq. (3.20) that such a beam will send a photon in one or other direction at a beamsplitter. Thus, at zero time delay, it is impossible to record a coincidence from such a state in the set-up of Figure 3.8. Hence we have g (2) (0) = 0. A pioneering experiment by Kimble et al. succeeded in 1977 in demonstrating the generation of non-classical light [29]. They used the ﬂuorescence of single atoms to generate light with the statistical properties of anti-bunching. The experiment was carried out with Na atoms, which were ﬁrst prepared into a two-level system and then excited by two resonant continuous-wave (CW) laser beams, as shown in the schematic layout in Figure 3.9a. The ﬂuorescence light from a very small volume, containing on average not more than a single atom, was imaged through a 50/50 beamsplitter on to two photodetectors, D1 and D2 . Upon arrival of a photon, the detectors generated pulses that were processed, via counting logic, and produced the probability function (t, t + 𝜏) for the arrival of photons at diﬀerent detectors with a time delay 𝜏. This information was converted into the second-order correlation function g (2) (0) (see Section 2.3). The excitation of individual atoms is determined by the quantum mechanical evolution of the atomic wave function. After the single atom has been excited by a laser beam, it will emit one photon, let us say at time t = 0. The atom cannot immediately emit another photon; it requires time for re-excitation. Consequently the probability for the generation of pairs of photons with small time separation is very small. The ﬂuorescence light has a value of g (2) (0) close to zero, as shown in Figure 3.9b, which is not possible for classical light. This light clearly shows quantum properties – it is anti-bunched. Since this ﬁrst demonstration, several experiments, for example, reported by J. Cresser et al. [30], J.G. Walker and E. Jakeman [31], and P. Grangier et al. [32], have been able to reveal the properties of similar quantum systems in more and more detail.

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More recently, the observation of anti-bunching has become the method of choice for identifying single quantum emitters. For example, ﬂuorescent light emitted by nitrogen vacancy (NV) centres in diamond can be analysed, and when strong anti-bunching is observed, one can be sure that the observed ﬂuorescence comes from a single NV centre in diamond [33]. See Chapter 13 for details.

3.7 Observing Photons in Cavities Many researchers pointed out that the most direct evidence of the quantization of the radiation ﬁeld, namely, the discrete nature of the photocurrent, could also be explained by a classical description of the radiation ﬁeld, provided that the detector is a linear quantum system. Do we have better evidence of the quantum properties of the radiation ﬁeld? Researchers set out to ﬁnd a more direct proof for the discreteness of the ﬁeld and analysed the light interacting with atoms contained inside a cavity. The atom can be approximated as a system with just two energy levels, a two-level system. The coupling of this system with a single mode of the radiation ﬁeld leads to a dynamical evolution of the atomic excitation in the form of nutations at the Rabi frequency. This simple model for the interaction is referred to as the Jaynes–Cummings model [34], well described in the literature [35], and extensively used in coherent optics with light or microwaves (nuclear magnetic resonance, NMR). In a strong coherent ﬁeld, which means a beam with many photons, the coupling leads to a nutation at a frequency associated with the mean photon number n𝛼 , which is a continuously variable number. However, for a weak ﬁeld (n𝛼 ≈ 1), the Rabi frequencies should be discrete and linked directly to the photon numbers in the ﬁeld. By placing the atoms inside a resonator, interesting properties of the coupled atom–light system can be studied. This idea led to a series of beautiful experiments with the micromaser built at Garching in the group around H. Walther. A similar technology of cavity atom coupling has been perfected by a team in Paris around S. Haroche and coworkers [36]. In these systems so-called Rydberg atoms, that is, atoms in excited states with very high quantum numbers (n > 40) and that have very simple, long-lived states, are placed inside a microwave cavity of extremely high quality. The cavity encloses the atoms almost completely, and it can be cooled to exclude thermal noise, and the coupling between the internal ﬁeld, the photons, and the probe in the form of the atoms can be made to dominate over all other eﬀects. The atoms couple to the microwave ﬁeld, and the coupled systems evolve in an elegant way determined by quantum mechanics. By observing the atoms that leave from the resonator, conclusions can be drawn about the state of the light–atom ﬁeld inside. These systems can show features such as the collapse and revival of the wave function of the atom, as reviewed, for example, early by P. Meystre and M. Sargent III [26] and more recently by S. Haroche and coworkers [36]. We will come back to these experiments when we discuss diﬀerent ways of generating strong non-classical states of light in Chapter 12 (Figure 3.7). Early experiments with Rydberg atoms by G. Rempe et al. [38] in a micromaser built in Garching have revealed an oscillation of the atomic population in a

0.5

76 Ph 5 4 oto nn 3 2 um be 1 0 r, n

0

Probability

7 6 5 4 3 2 1 0

5.0 〈n〉

1.0

Photon number, 〈n〉

3.7 Observing Photons in Cavities

4.5 4.0 0.06 0.07 0.08 Time (s)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 100 Time (s) 80 60 r, N e 40 mb Direct observation of 20 nu m dynamics of photons in a cavity o t A

Figure 3.10 Direct observation of photons in a cavity via QND detection by S. Haroche team. The left side shows the progressive collapse of the ﬁeld into a photon number state in the cavity by observing more and more atoms. The photon number probabilities are plotted versus the photon number n and the atom number N. The histograms evolve, as N increases from 0 to 110 atoms, from a ﬂat distribution into n = 5. The right-hand side shows the dynamics. The mean photon number ⟨n⟩ is continuously monitored via QND over 0.7 s. The inset zooms into the n = 5 to 4 jump, showing that it is detected with a time resolution of about 0.01 s. Source: Haroche and coworkers 2007 [37]. Reproduced with permission of Springer Nature.

thermal ﬁeld with (1.5 < n𝛼 < 3.8) and in the micromaser ﬁeld. Similarly, the series of experiments in Paris led by S. Haroche and coworkers has achieved extreme conditions [36]. Here the Rydberg atoms traverse a cavity where both atom–cavity interaction time and cavity damping time are very long. This requires a cryogenic cavity (T = 0.8 K) of very high quality (Q = 7 × 107 ) and a very stable source of slow atoms. The average delay between successive atoms was set to 2.5 ms, much longer than the cavity decay time. In the cavity the radiation ﬁeld and the atoms are coupled together, resulting in a time evolution that is dominated by oscillations at the Rabi frequency, and this determines the state of the atom at the end of its travel through the ﬁeld in the resonator. After leaving the cavity the atoms were detected by state-selective ﬁeld ionization. From these data, accumulated for many atoms, the dynamics of the atom–cavity evolution was reconstructed, as shown in Fig. 3.10. This showed a mixture of oscillations in time. Taking the Fourier transform of the traces at low average photon numbers, the experiments clearly showed discrete frequency components that correspond to the discrete Rabi oscillations associated with the photon numbers n = 1, 2, 3, 4, …. The discreteness of the radiation ﬁeld is directly evident. The technology has been systematically improved. More recent experiments by the Paris group have reﬁned this approach to the point where it is now possible to probe the ﬁeld inside the cavity repeatedly – in this way ﬁrst preparing and then analysing discrete photon number states in the cavity [37, 39]. It is now possible to carry out continuous quantum non-demolition (QND) measurements by using the coupling of the light ﬁeld to the excitation of atoms in the cavity, and we can

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3 Photons: The Motivation to Go Beyond Classical Optics

directly observe the establishment of the photon number. More details of this approach are given in Chapter 11. The major improvement over the earlier experiments is an increase of about 2 orders of magnitude in the quality of the cavity, pushing the damping time up to t = 0.13 s. This means that an average photon will travel over 10 000 km, back and forth within the cavity, before escaping. Given that successive atoms can now be sent across the cavity and measured every 0.5 ms, the trapped cavity ﬁeld can now be probed with thousands of individual atoms before it decays. Quantum optics predicts that if a near stationary cavity ﬁeld is measured to have a particular number of photons, then successive measurements will ﬁnd the same number of photons in the ﬁeld, as described in Chapter 4. This is exactly what is observed, with hundreds of successive atomic measurements returning the same discrete photon number. These experiments unambiguously show that the discreteness of optical energy measurements is a property of the ﬁeld, not of the atoms in the measurement device. S. Haroche shared the Nobel Prize for physics in 2012 for this work, and his lecture provides a fascinating overview of the search for the evidence of photons.

3.8 Summary We have compelling experimental evidence that photons exist and that the classical wave picture has to be extended. There is no doubt that we require quantum description of light for many current experiments and even technologies. Some of these observations are linked directly to the detection process, whilst others show in detail the quantum nature of light. This is introduced formally in Chapter 4. For many years optical experiments in the visible spectrum have had the strong advantage that they combine high eﬃciency light generation and detection, can be operated at room temperature without much impact from thermal noise, and, importantly, are directly convincing to the observer. Optical experiments continue to play an important role of testing and verifying the quantum theory and its many predictions. At the same time optical techniques have been a driver of many, many technologies such as communication, sensing, data storage, and even entertainment. We are now busy in creating new applications based on quantum ideas.

References 1 Lenard, P. (1902). Über die lichtelektrische Wirkung. Ann. Phys. 8: 149. 2 Rieke, G. (2003). Detection of Light: From the Ultraviolet to the Submillimeter,

2e. Cambridge: Cambridge University Press. 3 Glauber, R.J. (1962). Coherent and incoherent states of the radiation ﬁeld.

Phys. Rev. 131: 2766. 4 Tinsley, J.N., Molodtsov, M.I., Prevedel, R. et al. (2016). Direct detection of a

single photon by humans. Nat. Commun. 7: 12172. 5 Lorentz, H.A. (1910). Alte und neue Fragen der Physik. Phys. Z. 11: 1234.

References

6 Paul, H. and Fischer, R. (1983). Light absorption by a dipole. Usp. Fiz. Nauk

141: 375. 7 Paul, H. (1995). Photonen, eine Einführung in die Quantenoptik. Stuttgart:

Teubner. 8 Lawrence, E.O. and Beams, J.W. (1927). The instantaneity of the photoelectric

eﬀect. Phys. Rev. 29: 903. 9 Forrester, A.T., Gudmundsen, R.A., and Johnson, P.O. (1955). Photoelectric

mixing of incoherent light. Phys. Rev. 99: 1691. 10 Loudon, R. (2000). The Quantum Theory of Light, 3e. Oxford: Oxford Univer-

sity Press. 11 Planck, M. (1900). Zur Theorie des Gesetzes der Energieverteilung im Nor-

malspektrum. Verh. Dtsch. Phys. Ges. 2: 237. 12 Hollandt, J. (2012). 125 Years of Research On and with the Black Body,

Metrology Throughout the Ages, vol. 3, no.3, 87. PTB Mitteilungen. 13 Wien, W. (1896). Über die Energievertheilung im Emissionsspectrum eines

schwarzen Körpers. Ann. Phys. 294: 662. 14 Lummer, O. and Pringsheim, E. (1900). Über die Strahlung des schwarzen

Körpers fur lange Wellen. Verh. Dtsch. Phys. Ges. 2: 163–180. 15 Taylor, G.I. (1909). Interference fringes with feeble light. Proc. Cambridge Phi-

los. Soc. 15: 114. 16 Janossy, L. and Naray, Zs. (1958). Investigation into interference phenomena

17 18 19 20 21 22 23 24 25 26 27 28

at extremely low light intensities by means of a large Michelson interferometer. Il Nuovo Cimento 9: 588. Hariharan, P., Brown, N., and Sanders, B.C. (1993). Interference of independent laser beams at the single-photon level. J. Mod. Opt. 40: 113. Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4e. Oxford University Press. Feynman, R., Leighton, R.B., and Sands, M. (1963). The Feynman Lectures on Physics, vol. 3. Addison and Wesley. Michelson, A.A. and Morley E.W. (1927). On the relative motion of the earth and the luminiferous ether. Am. J. Sci. 1887 XXXIV (203). Hellmuth, T., Walther, H., Zajonc, A., and Schleich, W. (1987). Delayed-choice experiments in quantum interference. Phys. Rev. A 35: 2532. Magyar, G. and Mandel, L. (1963). Interference fringes produced by the superposition of two independent maser light beams. Nature 198: 255. Javan, A., Ballik, E.A., and Bond, W.L. (1962). Frequency characteristics of a continuous wave He-Ne optical maser. J. Opt. Soc. Am. 52: 96. Radloﬀ, W. (1971). Zur Interferenz unabhängiger Lichtstrahlen geringer Intensit á́t. Ann. Phys. 26: 178. Franson, J.D. (1991). Violation of a simple inequality for classical ﬁelds. Phys. Rev. Lett. 67: 290. Meystre, P. and Sargent, M. III (1989). Elements of Quantum Optics, 448. Springer-Verlag. Sakurai, J.J. (1985). Modern Quantum Mechanics. Addison-Wesley. Arrecchi, F.T., Berne, A., and Sona, A. (1966). Measurement of time evolution of the radiation ﬁeld by joint photo-current distributions. Phys. Rev. Lett. 17: 260.

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29 Kimble, H.J., Dagenais, M., and Mandel, L. (1977). Photon antibunching in

resonance ﬂuorescence. Phys. Rev. Lett. 39: 691. 30 Cresser, J.D., Haeger, J., Leuchs, G. et al. (1982). In: Dissipative Systems in

Quantum Optics (ed. R. Bonifacio), 21. Springer-Verlag. 31 Walker, J.G. and Jakeman, E. (1985). Photon-antibunching by use of a photo-

electron event triggered optical shutter. Opt. Acta 32: 1303. 32 Grangier, P., Roger, G., Aspect, A. et al. (1986). Observation of photon anti-

bunching in phase-matched multiatom resonance. Phys. Rev. Lett. 57: 687. 33 Scgirhagle, R., Chang, K., Loretz, M., and Degen, C.L. (2014).

34

35 36 37 38 39

Nitrogen-vacancy centers in diamond: nanoscale sensors for physics and biology. Annu. Rev. Phys. Chem. 65: 83. Jaynes, E.T. and Cummings, F.W. (1963). Comparison of quantum and semiclassical radiation theories with application to the beam maser. Proc. IEEE 51: 89. Cohen-Tannoudj, C., Diu, B., and Laloë, F. (1980). Quantum Mechanics. Wiley. Brune, M., Schmidt-Kaler, F., Maali, A. et al. (1996). Quantum Rabi oscillation: a direct test of ﬁeld quantization in a cavity. Phys. Rev. Lett. 76: 1800. Guerlin, C., Bernu, J., Deleglise, S. et al. (2007). Progressive ﬁeld state collapse and quantum non-demolition photon counting. Nature 448: 889. Rempe, G., Walther, H., and Klein, N. (1987). Observation of quantum collapse and revival in a one-atom maser. Phys. Rev. Lett. 58: 353. Deleglise, S., Dotsenko, I., Sayrin, C. et al. (2008). Reconstruction of non-classical cavity ﬁeld states with snapshots of their decoherence. Nature 455: 510.

Further Reading Haroche, S. (2012). Nobel Lecture 2012. www.nobelprize.org/nobel prizes/physics/laureates/2012/haroche-lecture.html. Haroche, S. and Raimond, J.-M. (2006). Exploring the Quantum. Oxford University Press. Kingston, R.H. (1995). Optical Sources, Detectors and Systems. Academic Press. Kwiat, P.G., Mattle, K., Weinfurter, H. et al. (1995). New high-intensity source of polarization entangled photon pairs. Phys. Rev. Lett. 75: 4337. Mandel, L. and Wolf, E. (1995). Optical Coherence and Quantum Optics. Cambridge: Cambridge University Press. Zeilinger, A., Weihs, G., Jennewein, T., and Aspelmeyer, M. (2005). Happy centenary, photon. Nature 433: 230.

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4 Quantum Models of Light 4.1 Quantization of Light 4.1.1

Some General Comments on Quantum Mechanics

In this chapter we will develop quantum mechanical models for light that will help us understand the various experimental results and techniques we will encounter in this book. Let us begin by reviewing the types of predictions one can make about experimental outcomes of quantum mechanical systems and how these are obtained from the mathematics. In classical mechanics, an observable quantity is represented by a variable such as x, the position of a particle. This is just a number that may vary as a function of time and/or other parameters. The outcome of an experiment that measures position will just be the value of x. In general, for quantum mechanical systems, the result of an experiment will not be unique. Instead, a range of possible outcomes will occur with some probabilities. Consequently, a single number is insuﬃcient to describe all the possible outcomes of a particular experiment. Thus, in quantum mechanics, variables are ̂ the position operator, describing the type of meareplaced by operators such as x, surement to be made, and states, represented by kets, |𝜎⟩. As we saw in the last chapter, these contain the information about the possible outcomes. Each operator representing an observable quantity will have a set of eigenstates associated with it whose eigenvalues represent the spectrum of possible results that a measurement of this observable can produce. Typically a system will be in a state that is a superposition of various eigenstates of some observable. A special case is when the system is actually in an eigenstate of the observable, in which case a deﬁnite result is predicted. Most generally the system will be in a statistical mixture of various quantum states. Quantum theory makes predictions about the results of large ensembles of measurements. The average value of an observable is given by the expectation value of the observable’s operator: ̂ = ⟨𝜎|x|𝜎⟩ ̂ ⟨x⟩

(4.1)

Equation (4.1) predicts the average value of measurements performed on a large ensemble of systems all prepared in the same state, |𝜎⟩, of the observable ̂ We can also consider expectation values of higher represented by the operator x. A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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4 Quantum Models of Light

moments such as ⟨x̂ 2 ⟩, which naturally leads us to deﬁne the variance of an observable as ̂ 2 ⟨Δx̂ 2 ⟩ = ⟨x̂ 2 ⟩ − ⟨x⟩

(4.2)

and similarly for higher-order properties. In classical mechanics a non-zero variance, or noise, is associated with imperfect preparation of the system with the desired property. In quantum mechanics even ideally prepared systems can exhibit non-zero variance due to the probabilistic nature of the theory. Such non-zero ﬂuctuations are thus often referred to as quantum noise. The characteristics of a system with respect to a particular observable can be completely characterized by the expectation values of all the moments of the observables’ operator. However, in many cases of interest in this book, the systems will exhibit Gaussian statistics. In such cases, the system is completely characterized by just the ﬁrst- and second-order moments. A powerful link between classical variables and quantum operators exists in the procedure of canonical quantization, ﬁrst pioneered by Dirac. If a system is described classically by the canonically conjugate variables x and p (in the sense of Hamilton’s equations [1]), then the correct quantum mechanical description can be obtained by simply replacing x and p with the operators x̂ and p̂ in the Hamiltonian and replacing the classical Poisson bracket with the commutation ̂ p] ̂ = x̂ p̂ − p̂ x̂ = iℏ. In the following section we will use this procerelation [x, dure to convert the classical description of light introduced in Chapter 2 into a quantum mechanical description. 4.1.2

Quantization of Cavity Modes

The classical description of light in Section 2.1 assumed as a model for a laser beam a single frequency wave. Although an idealization, for classical light it is not an unreasonable one as it would neglect modulations and ﬂuctuations of the light. Anticipating this we start our discussion by considering the discrete optical modes in a ring cavity. A suﬃciently high ﬁnesse cavity will only support ﬁelds with wavelengths 𝜆 according to 𝜆 = L∕n,

n = 0, 1, 2, 3, …

(4.3)

where L is the cavity round-trip length. This discrete frequency approximation is valid in both the classical and quantum domains. Recall ﬁrst from Section 2.1.4 that the energy in a classical light ﬁeld is proportional to the sum of the squares of the quadrature amplitudes X1 and X2. Speciﬁcally for polarized modes in a high ﬁnesse cavity, the energy is given classically by ℏ𝜔k (4.4) (X12k + X22k ) 4 where k labels the diﬀerent frequency modes and 𝜔k = 2π𝜈k is the angular frequency. Moreover we can identify suitably scaled versions of the quadratures H = Σk

4.1 Quantization of Light

as the canonical variables of the classical light wave. Speciﬁcally qk and pk are canonical variables where √ ℏ𝜔k qk = X1k 2 √ ℏ pk = − X2 (4.5) 2𝜔k k We can now apply canonical quantization by introducing the operators q̂ k and p̂ k with [q̂ k , p̂ l ] = iℏ𝛿kl . This immediately implies that the operator equivalents of ̂ k and X𝟐 ̂ k obey the commutator the quadrature amplitudes X𝟏 ̂ l ] = 2i𝛿kl ̂ k , X𝟐 [X𝟏 The energy operator or Hamiltonian for the system is just ∑ ℏ𝜔k ̂ 2) ̂ 2 + X𝟐 ̂ = (X𝟏 H k k 4 k 4.1.3

(4.6)

(4.7)

Quantized Energy

The Hamiltonian is composed of a sum of components with a whole range of frequencies. These diﬀerent waves are independent in the sense that diﬀerent frequency components commute with each other (see Eq. (4.6)) – much like we are used to considering classical waves of diﬀerent frequencies to be independent. Like in a classical cavity the frequency is a discrete variable, and only frequencies 𝜔 associated with the spatial modes are possible. In fact the Hamiltonian is mathematically equivalent to a sum of uncoupled harmonic oscillators, one for each frequency 𝜔. Each term describes one mode. For each mode we can solve the operator equation. Before carrying out the calculations in detail, let us consider the overall properties of the quantized harmonic oscillator. One important consequence of the quantization of the oscillator is the discreteness of the possible energy states of this system. The system has a series of possible energies that are equally spaced as illustrated in Figure 4.1: ) ( 1 ℏ𝜔 (4.8) n = n + 2 Similarly, the quantized ﬁeld has discrete energies, n . The energy of the ﬁeld can only be increased by multiples of the energy quantum of ℏ𝜔. This quantization suggests that the ﬁeld is made up of particles – photons. However, one has to be careful to use only those properties of particles that can be directly derived from the quantized ﬁeld equations. Care has to be taken with this picture to avoid an over-interpretation and to regard photons too much as particles behaving like small classical particles. It is worthwhile to contemplate one peculiar case, the lowest possible energy 0 = 12 ℏ𝜔, known as the zero point energy. The value of 0 is not zero. There is remaining energy. However, this does not upset our measurements since it is not

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4 Quantum Models of Light

Figure 4.1 Energy levels of a harmonic oscillator.

Energy

96

(n + 1/2) ħω

possible to extract this energy from the system. It is the lowest energy that can be achieved with this particular mode of light. All measurements of the energy of the mode refer back to it. The fact that we can only measure changes in the energy, not the absolute energy, means that the ground state is not directly accessible. Further discussions on this may be found in E.A. Power [2]. It is noteworthy that the value of 0 is unique to the optical frequency. A ﬁeld that consists of many modes has a total zero point energy of 12 ℏ𝜔 per mode. Since in theory there is no upper bound on the frequencies, the energy of the ground state is inﬁnite – this creates a conceptual problem. Mathematically this can be overcome by renormalization [3], and in practice any real experiment will have a ﬁnite detection bandwidth and the inﬁnity is avoided. It is possible to measure the eﬀect of the zero point energy directly. One can create an experimental situation where the number of allowed modes in a multimode system can be changed, and as a consequence the total zero point energy changes. This increase in energy results in a force that tries to change the system towards the lowest total zero point energy. Such eﬀects were investigated by H.B.G. Casimir and D. Polder [4]. Consider a cavity with length d. The cavity supports modes for all wavelengths 𝜆 = 2d∕n = L∕n. The longest supported wavelength is L. The size of the cavity determines the energy of the allowed low frequency modes and thus the total zero point energy. That means that for a smaller cavity, shorter L, the total zero point energy is smaller than for a larger cavity. The change in total zero point energy is proportional to the change in number of modes. H.B.G. Casimir and D. Polder [4] worked this out for a system of two conducting plates. In this case the total zero point energy varies with L−3 , and the resulting force Fcas is given by Aπhc [N] (4.9) 4.810−20 d4 where A is the area of the plates and d = L∕2 is the separation. This force does not depend on the electron charge, and it does not require any electric ﬁeld between the plates. It is a quantum mechanical eﬀect that applies to two plates in vacuum. The forces can be signiﬁcant for very small plate separations. As an example values of A = 10−4 m2 and d = 0.01 mm result in Fcas = 1.3 × 10−11 N. This pure quantum force was demonstrated in careful measurements where all other forces were excluded [5]. Without elaborating, it is worth noting that similar forces are part of the explanation of the well-known van der Waals forces [2]. These forces are essential for the explanation of a range of intermolecular eﬀects and include Fcas =

4.2 Quantum States of Light

such practical eﬀects as the stability of colloids in paint suspensions. The zero point energy can play an important role. 4.1.4

The Creation and Annihilation Operators

Just as, given certain assumptions about the spatial modes, we could describe the essential characteristics of a classical light ﬁeld via its complex amplitude, 𝛼 (see Chapter 2), so its quantized equivalent, â , plays a similar role in a quantum description of light. The operator â and its adjoint, â † , raise and lower the excitation of a speciﬁc mode by one increment, respectively. It follows from Eq. (4.6) ̂ = i(̂a† − â ) that they obey the Boson ̂ = â + â † and X𝟐 and the relationships X𝟏 commutation rules [̂a𝜔 , â †𝜔′ ] = (̂a𝜔 â †𝜔′ − â †𝜔 â 𝜔′ ) = 𝛿𝜔𝜔′

(4.10)

[̂a𝜔 , â 𝜔′ ] = [̂a†𝜔 , â †𝜔′ ] = 0

(4.11)

and We can rewrite the Hamiltonian operator of the radiation ﬁeld of a single mode in terms of these operators as ̂ = 1 ℏ𝜔(̂a† â + â â † ) H 2 ( ) ( ) 1 1 = ℏ𝜔 n̂ + (4.12) = ℏ𝜔 â † â + 2 2 where n̂ = â † â is the photon number operator. The Hamiltonian above represents the number of photons in one mode multiplied by the energy of one photon in that mode. For a multimode ﬁeld the total Hamiltonian would be the sum of several terms such as given in Eq. (4.12), one for each frequency component. Note the speciﬁc ordering of the operators â † and â . The ordering is important since the operators do not commute. ̂ p̂ and the creation Finally, the link between the conjugate variable operators q, and annihilation operators â † , â is given by √ ℏ𝜔k q̂ = (̂a + â † ) 2 √ ℏ p̂ = −i (̂a† − â ) (4.13) 2𝜔k It is again worth noting that the operator algebra of the quantized electromagnetic ﬁeld is identical to that of the quantized simple harmonic oscillator.

4.2 Quantum States of Light 4.2.1

Number or Fock States

The Hamiltonian for a single mode of the radiation ﬁeld (Eq. (4.12)) has the eigenvalues ℏ𝜔(n + 12 ). The corresponding eigenstates can be represented by the

97

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4 Quantum Models of Light

notation |n⟩. These are known as the number or Fock states and are eigenstates of the number operator ̂ = n|n⟩ â † â |n⟩ = n|n⟩

(4.14)

The ground state, also known as the vacuum state, is deﬁned by â |0⟩ = 0

(4.15)

It cannot be lowered any further. The energy of the ground state has already been discussed in Section 4.1.3 and is given by 1∑ ̂ ⟨0|H|0⟩ = ℏ𝜔k (4.16) 2 k where we sum over all frequencies. The operators â and â † are raising and lowering operators on a series of equally spaced eigenstates: √ √ (4.17) â |n⟩ = n|n − 1⟩ and â † |n⟩ = (n + 1)|n + 1⟩ They can be used to generate all possible number states starting with the ground state |0⟩ by a successive application of the creation operator (̂a† )n |n⟩ = √ |0⟩ n!

(4.18)

They are orthogonal and complete, i.e. ∑ ⟨n|n′ ⟩ = 𝛿nn′ and |n⟩⟨n| = Î

(4.19)

n

where Î is the identity operator. The number of photons n is exactly known for a number state; hence the variance of the photon number is zero for a number state: ̂ ̂ 2=0 Vn = ⟨Δn2 ⟩ = ⟨n|n̂ n|n⟩ − ⟨n|n|n⟩

(4.20)

We encountered the single photon number state in Section 3.5 where we found that a consideration of their properties in an interferometer led us to a radically diﬀerent view of light from the classical one. We can illustrate this further by contrasting the average value of the quadrature amplitudes for classical light and light in a number state. Recall that in a classical description of light, an arbitrary quadrature amplitude, X 𝜃 , can be described as a function of the phase angle, 𝜃, according to X 𝜃 = cos[𝜃]X1 + sin[𝜃]X2

(4.21)

We can convert this equation to the correct quantum mechanical form by simply replacing the quadrature variables by quadrature operators. The average value of the quadrature amplitude for light in a particular quantum state is then found by taking the expectation value over the state of interest. However what we ﬁnd for a number state is that ̂ 𝜃 ⟩n = ⟨n|X ̂ 𝜃 |n⟩ = 0 ⟨X

(4.22)

4.2 Quantum States of Light

where the orthogonality of the number states ensures that all terms like ⟨n|̂a|n⟩ and ⟨n|̂a† |n⟩ are zero (see Eq. (4.17)). The results in Eqs. (4.21) and (4.22) could hardly be more diﬀerent. In the classical case the quadrature amplitude oscillates as a function of the quadrature angle, 𝜃, whilst for the number state the average value is always zero, independent of the quadrature angle. It is clear that number states represent quite exotic states of light, quite dissimilar to what is produced directly by a laser. We will return to consider them again later but for now we look for light states with more familiar behaviour, which, in turn, turn out to be those most naturally produced. For now we use the number states as a basis set and ﬁnd more realistic superpositions. However, one particular number state plays a major role in all quantum optical experiments: the vacuum state |0⟩. 4.2.2

Coherent States

The number states do not represent an optical state with a well-deﬁned phase. To describe laser light we require a state that has a more precisely deﬁned phase. This is the coherent state |𝛼⟩, which is the closest quantum approximation to the ﬁeld generated by a laser [6]. Consider, in an idealized way, how a light wave is produced in the classical picture. Initially the energy in the electric and magnetic ﬁelds is zero. Then, by some means, the electric ﬁeld is displaced to some non-zero value, say, |𝛼|. This in turn displaces the magnetic ﬁeld and so on, producing a propagating light ﬁeld. Suppose we follow the same idealized recipe in quantum mechanics. This suggests that we should displace the quantum mechanical vacuum state to some non-zero value. We might hope that this will produce an oscillating ﬁeld. However, because the quantum mechanical vacuum has non-zero energy, we also expect uniquely quantum features. ̂ acting on the vacuum ket has the eﬀect of displacThe displacement operator D ing it by an amount 𝛼, where 𝛼 is a complex number that can be directly associated with the complex amplitude in classical optics (see Chapter 2). We deﬁne ̂ |𝛼⟩ = D(𝛼)|0⟩ The displacement operator is given by ̂ D(𝛼) = exp(𝛼 â † − 𝛼 ∗ â )

(4.23)

It is a unitary operator, i.e. ̂ −1 (𝛼) = D(−𝛼) ̂ ̂ † (𝛼) = D D and has the properties ̂ ̂ † (𝛼)̂aD(𝛼) = â + 𝛼 D † † ̂ ̂ D (𝛼)̂a D(𝛼) = â † + 𝛼 ∗

(4.24)

The family of states |𝛼⟩ is called coherent states, and, as we will now show, they behave in many ways like classical laser light. Due to the unitarity of the displacement operator, they are normalized to unity: ⟨𝛼|𝛼⟩ = 1

(4.25)

99

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4 Quantum Models of Light

Also the coherent states are eigenstates of the annihilation operator â , as can be veriﬁed from ̂ † (𝛼)̂aD(𝛼)|0⟩ ̂ ̂ † (𝛼)̂a|𝛼⟩ = D = (̂a + 𝛼)|0⟩ = 𝛼|0⟩ D ̂ Multiplying both sides with D(a) results in the eigenvalue condition â |𝛼⟩ = 𝛼|𝛼⟩

(4.26)

Using Eq. (4.26) we now see that the expectation values for the quadrature amplî 𝜃 ⟩𝛼 , are identical to the classical ̂ 𝜃 |𝛼⟩ = ⟨X tude operator in a coherent state, ⟨𝛼|X variables, provided we identify the displacement 𝛼 with the classical complex amplitude. That is ̂ 𝜃 ⟩𝛼 = cos[𝜃]X1 + sin[𝜃]X2 ⟨X

(4.27)

Hence we see that the average value of the ﬁeld oscillates as a function of quadrature angle. Similarly the expectation value for the photon number in a coherent state is ⟨̂a† â ⟩𝛼 = 𝛼 ∗ 𝛼

(4.28)

which again is the same as the classical value. Thus we see that the average values of the observables for a coherent state are identical to those for a classical ﬁeld. However, unlike a number state or a classical ﬁeld that contains a deﬁnite amount of energy, a coherent state does not contain a deﬁnite number of photons. This follows from the fact that a coherent state |𝛼⟩ is made up of a superposition of number states: ∞ ∑ |n⟩⟨n|𝛼⟩ (4.29) |𝛼⟩ = n=0

Using Eq. (4.18) and the completeness of the number states, Eq. (4.19), one obtains ) ( 1 𝛼n (4.30) ⟨n|𝛼⟩ = √ exp − |𝛼|2 2 n! which is the contribution of the various number states. Now consider the relationship between two diﬀerent coherent states. Using Eqs. (4.29) and (4.30), we ﬁnd that their scalar product is ) 2 ( 1 | | |⟨𝛼1 |𝛼2 ⟩|2 = | exp − (|𝛼1 |2 + |𝛼2 |2 ) + 𝛼1∗ 𝛼2 | | | 2 2 = exp −|𝛼1 − 𝛼2 | (4.31) which is not zero. That means two diﬀerent coherent states are not orthogonal to each other. However, for two states with signiﬁcantly diﬀerent photon numbers, that is, |𝛼1 − 𝛼2 | > 1, the product is very small. Such states are approximately orthogonal. One consequence is that the coherent states form a basis that is over-complete. But despite this, the coherent states can be used as a basis on which any state |Ψ⟩ can be expanded [7]. In summary, the coherent states have properties similar to those of classical coherent light and in fact represent its closest quantum mechanical

4.2 Quantum States of Light

approximation. Just as under certain conditions laser light can be approximated as idealized classical coherent light, so in situations where a quantum mechanical description is needed, laser light can often be well approximated by a coherent state. We will ﬁnd in Section 5.1 that coherent states are stable in regard to dissipation, which means one coherent state is transformed into another coherent state when attenuated. Consequently, the coherence properties of the light remain similar after attenuation. Coherent states will be used extensively in our modelling of laser experiments. Being a complex number, the range of possible values of 𝛼, and thus the range of possible coherent states, requires a two-dimensional representation. This is only natural since we can distinguish diﬀerent states of light experimentally by two parameters, intensity and phase. But we note that only one of them, the intensity, has a simple quantum mechanical equivalent (i.e. the number operator). A similarly simple quantum mechanical operator for the phase is not available. Phase operators can only be approximated in a rather complex way through a limit of a series of operators [7–9], and their deﬁnition has stimulated a lively debate amongst theoreticians. However, the real and imaginary parts of the annî and X𝟐, ̂ are both hilation operator â , i.e. the quadrature amplitude operators X𝟏 well-behaved operators. This leads to a pair of Hermitian operators that describe the ﬁeld. 4.2.3

Mixed States

So far we have only considered pure ﬁeld states. More generally we can have ﬁelds that are in an incoherent mixture of diﬀerent pure states. This is referred to as ̂ a mixed state. A mixed state is normally represented by a density operator, 𝝆, such that 𝝆̂ = 𝑤1 |𝜎1 ⟩⟨𝜎1 | + 𝑤2 |𝜎2 ⟩⟨𝜎2 | + · · · represents a mixture of the states |𝜎1 ⟩, |𝜎2 ⟩, … with fractional weightings 𝑤1 , 𝑤2 , …, respectively. It is important not to confuse mixtures of states and superpositions√ of states; they are very diﬀerent. Consider the equal superposition state |𝜙⟩ = 1∕ 2(|a⟩ + |b⟩). Written as a density operator, this state is 𝝆̂ p = |𝜙⟩⟨𝜙| 1 1 1 1 = |a⟩⟨a| + |a⟩⟨b| + |b⟩⟨a| + |b⟩⟨b| 2 2 2 2 In contrast an equal mixture of the states a and b is just

(4.32)

1 1 |a⟩⟨a| + |b⟩⟨b| (4.33) 2 2 The coherences between the states that are present for the superposition are absent in the case of the mixture. Expectation values for mixed states are given by 𝝆̂ m =

̂ = Tr{x̂ 𝝆} ̂ ⟨x⟩ where the trace, Tr, is taken over a complete set of eigenstates. An important example of a mixed state in optics is the thermal state, which can be represented

101

102

4 Quantum Models of Light

as a mixture of number states by ) ( ) ( 1 G−1 G−1 2 𝝆̂ th = |2⟩⟨2| + · · · |1⟩⟨1| + |0⟩⟨0| + G G G

(4.34)

where G is a parameter that ranges from one for a zero temperature thermal ﬁeld (the vacuum) up to a high temperature thermal ﬁeld for G → ∞.

4.3 Quantum Optical Representations 4.3.1

Quadrature Amplitude Operators

Let us now look in some detail at the quadrature amplitude operators deﬁned by ̂ = â + â † X𝟏

̂ = i(̂a† − â ) and X𝟐

(4.35)

We will sometimes refer to them as the amplitude quadrature and the phase quadrature, respectively, for reasons that will become apparent when we discuss bright ﬁelds (Section 4.5). A variety of nomenclature for the quadrature ̂ X𝟐} ̂ ≡ amplitude operators exist in the literature – some common ones are {X𝟏, + ̂− ̂ ̂ ̂ ̂ p}. ̂ {X , X }; {X, P}; {x, ̂ We can deﬁne eigenstates of these operators via X𝟏|x1⟩ = x1|x1⟩ and similarly ̂ X𝟐|x2⟩ = x2|x2⟩. The eigenvalues xi form an unbounded continuous set. Because ̂ and X𝟐, ̂ it is not possible to of the non-zero commutation relation between X𝟏 ̂ ̂ be in a simultaneous eigenstate of X𝟏 and X𝟐. The resulting uncertainty principle constrains the variances of the operators in any state to ̂ 2⟩ ≥ 1 ̂ 2 ⟩⟨ΔX𝟐 ⟨ΔX𝟏

(4.36)

where the quadrature ﬂuctuation operators are deﬁned: ̂ = X𝟏 ̂ − ⟨ΔX𝟏⟩ ̂ ΔX𝟏

̂ = X𝟐 ̂ − ⟨ΔX𝟐⟩ ̂ and ΔX𝟐

(4.37)

̂ or X𝟐 ̂ are unphysical as inﬁnite uncertainty It follows that exact eigenstates of X𝟏 in an unbounded spectrum implies inﬁnite energy. This is analogous to the situation for position and momentum of a free particle. We saw in the previous section that the expectation values of the quadrature operators in a coherent state were equal to their classical values. It is straightforward to show that the uncertainty product for the quadratures in a coherent state is given by ̂ 2 ⟩𝛼 = 1 ̂ 2 ⟩𝛼 ⟨ΔX𝟐 ⟨ΔX𝟏

(4.38)

which means a coherent state is a minimum uncertainty state. Equally important is the fact that the uncertainty of the coherent state is split equally between the two quadratures. The uncertainty relationship is symmetric: ̂ 2⟩ = 1 ̂ 2 ⟩ = ⟨ΔX𝟐 ⟨ΔX𝟏

(4.39)

Hence the coherent state can be thought of as the best allowed compromise for a simultaneous eigenstate of the two quadratures. The quadrature operators allow us to represent a beam of light graphically. These so-called phasor diagrams are

4.3 Quantum Optical Representations

very popular in quantum optics – ever since their introduction by C. Caves [10]. Any state of light can be represented on a phasor diagram of the operators, that ̂ versus ⟨X𝟏⟩. ̂ is, a plot of ⟨X𝟐⟩ In such a diagram a particular coherent state |𝛼⟩ is ̂ ̂ equivalent to an area centred around the point given by the value (⟨X𝟏⟩, ⟨X𝟐⟩) for 2 ̂ ̂ this state. The size of the area is given by the variances ⟨ΔX𝟏 ⟩ and ⟨ΔX𝟐2 ⟩ and ̂ 2 ⟩, and the area of the is usually represented just by a circle with diameter ⟨ΔX𝟏 circle describes the extent of the uncertainty distribution. This phasor diagram is a qualitative description that shows several important results: 1. For a coherent state the uncertainty area is symmetric. Any measurement of the ﬂuctuations will yield the same result, independent of the projection angle 𝜃 chosen. 2. The uncertainty area is of constant size, independent of the intensity |𝛼|2 of the state. ̂ X𝟐 ̂ 3. Two states are only distinguishable if their average coordinates in the X𝟏, ̂ 2 ⟩. plane are separated by more than ⟨ΔX𝟏 The last result is signiﬁcant. It gives us a practical rule when two states are measurably diﬀerent. The uncertainty area sets a minimum change in amplitude or phase that we have to make before we can distinguish two states. In a classical system such an uncertainty region would be associated with noise, due to technical imperfections. However, here the uncertainty area is a consequence of the quantum rules – not technical imperfections. Consequently, the uncertainty area represents the eﬀect of quantum noise on a measurement, whether it is amplitude, phase, or any other property of the light. Experimental measurements cannot be more precise than the quantum noise. The uncertainty area is two-dimensional. One can imagine the entire parameter space being covered, or tiled, with circular uncertainty areas of uniform size (see Figure 4.2). The number of tiles, which are not all overlapping, approximately determines the number of states that can be distinguished. As a practical example, consider the light in an optical communication system. The information is transmitted by changing either the intensity or phase of the light at one modulation frequency. We assume the intensity is limited. How many distinguishable Figure 4.2 Uncertainty areas can cover the entire parameter space like tiles.

ˆ 2 ΔX2

X2

ˆ 2 ΔX1

X1

103

104

4 Quantum Models of Light

states of light, that is, diﬀerent pieces of information, could we send? The light is represented by a two-dimensional plane with a circle that indicates the largest possible intensity Imax . Inside this circle we can place our tiles. The number of tiles estimates the upper limit to the number of states we can use, that is, the number of diﬀerent pieces of information that a beam of light modulated at one frequency can contain. As described in Section 10.2, in 2018 many more systems are approaching this quantum resolution, and ultimately technology will reach this limit in most cases. 4.3.2

Probability and Quasi-probability Distributions

The uncertainty area is a simple, useful tool that can, under certain circumstances, be used to completely characterize the state. To understand under what conditions this is possible, we need to look at the probability distributions ̂ and X𝟐. ̂ Such measurements can be carried out associated with measuring X𝟏 using the technique of balanced homodyne detection, which will be described in detail in Chapter 8. The probability of obtaining a result in a small interval, 𝛿x, ̂ performed on a system around a particular value x1 from a measurement of X𝟏 in the state |Ψ⟩ is given by PΨ (x1 ) = |⟨x1 |Ψ⟩|2 𝛿x

(4.40)

̂ the probability of obtaining results in a and similarly for a measurement of X𝟐 small interval around x2 is PΨ (x2 ) = |⟨x2 |Ψ⟩|2 𝛿x

(4.41)

In other words |⟨xi |Ψ⟩| is the probability density at the point xi . These distributions can be evaluated for various states using the relationships 2

d ̂ ⟨x1 |X𝟐|Ψ⟩ = −i ⟨x |Ψ⟩, dx1 1

d ̂ ⟨x2 |X𝟏|Ψ⟩ =i ⟨x |Ψ⟩ dx2 2

(4.42)

For example, for a coherent state, we can show ⟨x1 |̂a|𝛼⟩ = 𝛼⟨x1 |𝛼⟩ 1 ̂ + iX𝟐|𝛼⟩ ̂ = ⟨x1 |X𝟏 2( ) d 1 ⟨x1 |𝛼⟩ x1 + = 2 dx1

(4.43)

Solving the relevant diﬀerential equation then gives 2 1 |⟨x1 |𝛼⟩|2 = √ e−(x1 −x1 ) π

(4.44)

where x1 = (𝛼 + 𝛼 ∗ )∕2 is the real part of 𝛼. We see that our x1 results form a Gaussian distribution. The width or variance of the distribution is 1 as expected. Similarly we ﬁnd 2 1 |⟨x2 |𝛼⟩|2 = √ e−(x2 −x2 ) π

(4.45)

4.3 Quantum Optical Representations

for the probability distribution of x2 results. Here x2 = i(𝛼 − 𝛼 ∗ )∕2 is the imaginary part of 𝛼. Again, we ﬁnd a Gaussian distribution of width 1. Because the coherent states are Gaussian, they can be completely characterized by the ﬁrst̂ and X𝟐 ̂ and thus the uncertainty area of the preand second-order moments of X𝟏 vious section. This property leads to an elegant approach based on the covariance matrix, which will be discussed in Section 4.3.4. It turns out that many experimentally accessible optical quantum states have Gaussian quadrature distributions and therefore are quite easily characterized. This is not in general true however. For example, the quadrature distributions for a single photon number state are given by |⟨x1 |1⟩|2 =

2 1 √ 2x2 e−x1 1 π

(4.46)

and similarly 2 1 |⟨x2 |1⟩|2 = √ 2x22 e−x2 π

(4.47)

These distributions are clearly non-Gaussian. A two-dimensional probability distribution for a state |Ψ⟩ is given by the Q-function deﬁned as |⟨𝛼|Ψ⟩|2 π which is positive and bounded by QΨ (𝛼) =

(4.48)

1 π Physically the Q-function represents the probability density of obtaining the ̂ and result 𝛼 = (x1 + ix2 )∕2 from an optimal simultaneous measurement of X𝟏 ̂ ̂ ̂ X𝟐. Because X𝟏 and X𝟐 do not commute, such a measurement is necessarily nonideal. By optimal we mean that the resolution of the measurement is as good as the uncertainty principle allows. Such an optimal measurement can be achieved using the techniques of heterodyne or dual homodyne detection, which will be discussed in Chapter 8. For a coherent state |𝛽⟩, the Q-function is simply QΨ ≤

exp(−|𝛼 − 𝛽|2 ) |⟨𝛼|𝛽⟩|2 = (4.49) π π It is a Gaussian function centred around the point (e(𝛽), m(𝛽)). The width of the Gaussian is two, that is, twice that of what we found when the quadratures were measured individually. The additional noise is a result of the nonideal measurement in which one extra unit of quantum noise is added. Note that this function has a width that is independent of the average photon number of the state. A strong coherent state has the same-sized Q-function as a weak ﬁeld – just displaced further from the origin. The Q-function for a number state is quite diﬀerent to that of a coherent state and is given by Q𝛽 =

Qn =

|𝛼|2n exp(−|𝛼|2 ) |⟨𝛼|n⟩|2 = π πn!

(4.50)

105

106

4 Quantum Models of Light

Figure 4.3 Uncertainty area of a number state.

X2

X1

This distribution has the shape of a ring (see Figure 4.3). The maximum probability is a circle around the origin, and the distribution in the radial direction is Gaussian centred at the average photon number with a width that decreases very rapidly, with n! That means for large photon numbers the radial distribution is approaching a delta function. A number state contains no phase information at all. The ring shape does not favour any particular phase. A two-dimensional distribution more closely related to our homodyne distributions is the Wigner function, which can be written as WΨ (𝛼) =

̂ D(−𝛼)|Ψ⟩ ̂ ̂ 2⟨Ψ|D(𝛼) 𝚷 π

(4.51)

̂ is the parity operator. The parity operator measures the symmetry of where 𝚷 the probability amplitude distribution of a state. The number states are parity eigenstates with eigenvalues +1 for n even and −1 for n odd. So the action of the parity operator on an arbitrary superposition of number states is ̂ 0 |0⟩ + c1 |1⟩ + c2 |2⟩ + · · ·) = (c0 |0⟩ − c1 |1⟩ + c2 |2⟩ − · · · ) 𝚷(c This is eﬀectively a π rotation on the phasor diagram. Thus the Wigner function is the overlap between the state, displaced by −𝛼, and itself, also displaced by −𝛼 but then also rotated by 180 degrees. For a coherent state |𝛽⟩, we will obtain 2⟨𝛽 − 𝛼|𝛼 − 𝛽⟩ 2e−2|𝛼−𝛽|2 = (4.52) π π This is a symmetric Gaussian with width 1, centred at the point (e(𝛽), m(𝛽)). Its projections along the X1 and X2 directions correspond with the homodyne probability distributions of Eqs. (4.44) and (4.45). In this way the noise introduced by the quantum ﬂuctuations can be interpreted as producing ‘classical’ noise distributions. However, the Wigner function is a quasi-probability function. Although for Gaussian states it is always positive and generally well behaved, it can become negative for non-Gaussian states, such as the number state, and thus cannot be interpreted as a true probability distribution. For such states the quantum ﬂuctuations do not have a classical’ noise interpretation. Producing and analysing exotic states with negative Wigner functions is technically challenging but is now W𝛽 (𝛼) =

4.3 Quantum Optical Representations

Figure 4.4 The randomness of the quadrature values represented by a weighted random distribution of dots. Each dot represents one particular instantaneous value. The histogram of all ̂ and X𝟐 ̂ dots both in the X𝟏 directions corresponds to the projection of the Wigner function.

ˆ X2

ˆ X1

achieved routinely by several groups. We will look at some examples of negative Wigner functions in Chapter 12. The uncertainty areas that are shown in the phasor diagrams are simpliﬁed representations of the Wigner function. They are contours drawn at the half maximum value of the full two-dimensional Wigner function (see Figures 4.4 and 4.5). This line is the edge of the uncertainty area. It is described by only two values, the size of the minimum and the maximum axes. From an experimental point of view, we use the measured variance, for a certain angle 𝜃, as a value for the uncertainty of this particular quadrature. That means we select one 𝜃, measure the variance of this quadrature, and obtain two points. We can repeat this for many angles, and in a real experiment we would actually do this to check the theory and reduce the experimental errors. All these points together describe the edge of the uncertainty area. Whilst the Wigner function of a coherent state is circularly symmetric, this is not the only option. The width of the distribution in orthogonal directions could be diﬀerent, and it is possible for one of the variances to fall below 1. The only requirement is that the product of the two widths, that is, the product of the variances, satisﬁes the uncertainty relationship. We shall see in Chapter 9 that such states can be generated in optical experiments, which are known as squeezed states. It follows from the special properties of Gaussian functions that for such Figure 4.5 A two-dimensional Gaussian distribution and its contour. This is the origin of the circular uncertainty area.

X2

X1

107

108

4 Quantum Models of Light

a state the contour at the half maximum value is always an ellipse. The diameter along the minor and major axes corresponds to the minimum and maximum quantum noise. The various distributions we have looked at can be generalized to mixed states such that PΨ (xi ) = ⟨xi |𝝆̂ 𝚿 |xi ⟩ QΨ (𝛼) =

⟨𝛼|𝝆̂ 𝚿 |𝛼⟩ π

and 2 ̂ ̂ ̂ Tr{D(−𝛼) 𝝆̂ 𝚿 D(𝛼) 𝚷} (4.53) π where in each case 𝝆̂ 𝚿 is the density operator of the state to be characterized. For example, the Wigner function for a thermal state (see Eq. (4.34)) is given by WΨ (𝛼) =

2

2e−2|𝛼| ∕𝜎 (4.54) π which is again a symmetric Gaussian with width 𝜎 = 2G − 1. Finally, let us make a comment on other possible representations. Another obvious representation would generate a delta function for any particular coherent state. A representation with this property is the P-representation. A plot of the P-function would immediately show which coherent state is present. However, there are a number of mathematical problems that arise with the P-function once, for example, squeezed states are considered. Since for a squeezed state the variance of one of the quadratures is smaller than for a coherent state, and since a delta function cannot be made any narrower, the P-function will get negative values and even singularities. However, regularized versions of the P- function have been developed, which avoid some of these mathematical problems, and these can form a useful way of theoretically analysing quantum optical systems. It can actually be shown that all three representations (Q, P, and W ) are only limiting cases of a whole continuum of possible representations. Wth (𝛼) =

4.3.3

Photon Number Distributions

In the previous section we looked at results of measurements of the quadrature amplitudes. We now consider photon number measurements of a single optical mode. These are modelled as measurements of the observable n̂ = â † â . In general the probability distribution, 𝜌 (n), of a state described by the density operator 𝝆̂ is given by its diagonal elements in the photon number basis, i.e. ̂ 𝜌 (n) = ⟨n|𝝆|n⟩

(4.55)

For a photon number state, there is a unique photon number; it will be exactly the same in every measurement, and hence Eq. (4.55) has only one non-zero element. However, for a coherent state, one expects to ﬁnd a spread in the results of many measurements of the photon number. We can use the expansion in Eq. (4.29) to determine the probability 𝛼 (n) of ﬁnding n photons in the coherent state |𝛼⟩: 𝛼 (n) = |⟨n|𝛼⟩|2 =

|𝛼|2n exp(−|𝛼|2 ) n!

(4.56)

4.3 Quantum Optical Representations

0.2 nα = 4

0.1

nα = 1000

nα = 36

0.05

2Δn = 64

P(n)

P(n)

0.15

0.035 0.03 0.025 0.02 0.015 0.01 0.005

0 0

10

20

30

40

50

60 n

950

1000

1050

1100

Figure 4.6 Poissonian distribution of photons for three coherent states: (i) n𝛼 = 4, (ii) n𝛼 = 36, and (iii) n𝛼 = 1000.

This is a Poissonian distribution with a mean value of |𝛼|2 , which represents the average number n𝛼 of photons in this mode. The intensity ﬂuctuates according to this distribution. The noise is not a result of technical imperfections of the experiment. This is a quantum property of the light. Mathematically speaking it is a direct consequence of the quantum mechanical commutation rules. One important property of any Poissonian distribution is that the variance of the distribution is equal to the√mean value. Consequently, the width of the distribution Δn is equal to Δn = n𝛼 . Examples of three such distributions for n𝛼 = 4, n𝛼 = 36, and n𝛼 = 1000 are shown in Figure 4.6. Evaluating the relative uncertainty of√the mean photon number of a coherent state results in Δn∕n = √ n𝛼 ∕n𝛼 = 1∕ n𝛼 , which decreases with n𝛼 . A state with a high photon number is better deﬁned than that of a state with a small number of photons. We can interpret this result in the photon picture: a photodetector will not see a regular stream of photons, but it will see a somewhat randomized stream such that the measurement of the absorption events – for a perfect detector equal to the number of photons incident – has a distribution given by 𝛼 (n). Each event creates a ﬁxed number of electrons, and the stream of electrons leaving the detector forms the photocurrent. The distribution 𝛼 (n) is directly represented by the variation of the photocurrent. The intensity noise properties of the light could be expressed in terms of the distribution function 𝛼 (n). Thinking in the time domain, this means that the photons arrive in unequal time intervals as shown in Figure 4.7d. For large intensities 𝛼 (n) tends to a normal Gaussian function. In contrast, for low intensities, the Poissonian distribution diﬀers markedly from a Gaussian function. A convenient way to classify the photon number distribution, particularly for bright ﬁelds, is via its width relative to Poissonian. In many practical situations light has more ﬂuctuations than a coherent state, and its photon distribution is wider than a Poissonian function. In this case we would call it super-Poissonian. On the other hand, we will ﬁnd that non-linear processes can be used to make the distribution narrower. This we will call sub-Poissonian light. In order to quantify the noise one can deﬁne the Fano factor, which is given by the ratio of the actual width of the distribution to the width of a Poissonian distribution with the same mean photon number. For some arbitrary state 𝜌𝛽 with

109

110

4 Quantum Models of Light

δX2

δX2

δX2 α2 δX1

α1

(a)

X

δX1

(d) I(t)

P(n)

I2

[I2]

[I1]

Vnα1

n nα2

nα1

(b)

t (e) I(Ω)

Counts

α2 1

α1 t

(c)

0

Ω (f)

Im(α(Ω)) Re(α(Ω))

(g)

Ω

Figure 4.7 Diﬀerent representations of a coherent state of light, shown for states |𝛼1 ⟩, |𝛼2 ⟩ with diﬀerent intensities. (a) Phase space diagram, (b) photon number distribution, (c) arrival times of photons, (d) Wigner function, (e) time evolution of the intensities, (f ) variance of the intensities, and (g) the noise at diﬀerent frequencies.

4.3 Quantum Optical Representations

mean photon number n𝛽 and distribution V n𝛽 , we deﬁne the Fano factor as f𝛽 =

V n𝛽 n𝛽

(4.57)

which classiﬁes the states in the following way: f f f f

=0 1

number state sub-Poissonian state coherent state super-Poissonian state

Note that the Fano factor is determined only by the variance. In fact the Fano factor is identical to the normalized variance of the photon ﬂux, which, for many practical situations, is equivalent to the variance V 1 of the quadrature amplitude, as we shall see in Section 4.5. In much of this guide, we will use V 1 interchangeably with the Fano factor. Another important way to classify photon number distributions, particularly for low intensity ﬁelds, is the second-order coherence function, g (2) , previously introduced for classical ﬁelds, which we will discuss in detail in Section 4.6.1. For zero time delay, we have the following relationship between the second-order coherence function and the Fano factor: 1 g (2) (0) = f + 1 − (4.58) n 4.3.4

Covariance Matrix

The second-order quadrature moments can be arranged into a 2 × 2 matrix V, referred to as the covariance matrix [11]: ( ) 1 ̂ X𝟐 ̂ + ΔX𝟐Δ ̂ X𝟏⟩ ̂ ̂ 2⟩ ⟨Δ X𝟏Δ ⟨ΔX𝟏 2 V= 1 ̂ (4.59) ̂ + ΔX𝟐Δ ̂ X𝟏⟩ ̂ ̂ 2⟩ ⟨ΔX𝟏ΔX𝟐 ⟨ΔX𝟐 2 where the quadrature ﬂuctuation operators are deﬁned in Eq. (4.37). Similarly, the displacement vector, d, can be deﬁned via the ﬁrst-order moments: ( ) ̂ ⟨X𝟏⟩ d= (4.60) ̂ ⟨X𝟐⟩ For Gaussian states the covariance matrix plus the displacement vector completely characterizes the state. For simplicity, the covariance matrix is often presented in its standard form: ( ) ̂ ′2 ⟩ 0 ⟨ΔX𝟏 ′ V = (4.61) ̂ ′2 ⟩ 0 ⟨ΔX𝟐 ̂ + sin 𝜃 X𝟐 ̂ and X𝟐 ̂ ′ = cos 𝜃 X𝟐 ̂ − sin 𝜃 X𝟏 ̂ are rotated ver̂ ′ = cos 𝜃 X𝟏 where X𝟏 sions of the quadratures with 𝜃 speciﬁcally chosen to make the oﬀ-diagonal

111

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4 Quantum Models of Light

terms vanish. Such a 𝜃 can always be found. Physically this corresponds to rotating the quadratures until they line up with the major and minor axes of the uncertainty ellipse (see Section 4.3.2). In other words, in the standard form, ̂ ′2 ⟩ is minimized, or vice versa. ̂ ′2 ⟩ is maximized and V 2 = ⟨ΔX𝟐 V 1 = ⟨ΔX𝟏 Many useful properties about Gaussian states can easily be extracted from the covariance matrix. This will be particularly important when we consider multimode states and their correlations. A simple single-mode example is that the uncertainty principle can be expressed as DetV ≥ 1. Here DetM is the determinant of the matrix M, reducing to the familiar Heisenberg form where the covariance matrix is in its standard form (Eq. (4.61)). 4.3.4.1 Summary of Diﬀerent Representations of Quantum States and Quantum Noise

We have examined a number of representations of quantum optical states, several based on quadrature amplitudes, plus the photon number distribution. All representations are complete, so the choice of which to use for a particular problem may depend on mathematical tractability, the physics one wishes to highlight, or simply taste. Being quasi-probability distributions, these representations often suggest interpretations in terms of noise. Quantum noise is the most easily observable consequence of the quantization of light. It is a direct result of the use of operators for describing the amplitude of the electromagnetic ﬁeld and of the commutation rules applied to these operators. In other words, the minimum uncertainty principle applied to light leads to the quantum noise that aﬀects many optical measurements. For Gaussian states a noise interpretation is always possible for the quadrature amplitudes; however in general such an interpretation is at odds with photon number measurements. For non-Gaussian states this fails even for quadrature measurements as negative probabilities’ can result. A useful quantum state, sometimes seen as lying on the boundary between classical and quantum states, is the coherent state. It is an idealized state of light representing the output of a perfect laser source. In mathematical terms it is the eigenstate of the annihilation operator. A physical consequence of this is that attenuation changes one coherent state into another coherent state, just with lower energy. The characteristic properties of the coherent state are preserved. In many situations the coherent state can be interpreted as a classical light ﬁeld with the addition of stochastic noise. The coherent states can be used as a basis set, and all types of light can be described as a superposition of these basic states. This expansion can be represented by a probability distribution for measuring particular 𝛼, the Q-function, which includes the intrinsic measurement penalty. The characteristic properties of a coherent state of light can be summarized in Table 4.1. The corresponding graphical representations of quantum noise are shown in Figure 4.7.

4.4 Propagation and Detection of Quantum Optical Fields

Table 4.1 Properties of coherent states. Properties of coherent states

Average complex amplitude

𝛼 = |𝛼|ei𝜙

Average photon number

n𝛼 = |𝛼|2

Quadrature amplitude operators

̂ X𝟐 ̂ corresponding to a complex Operators X𝟏, amplitude 𝛼

Wigner function

Quasi-probability distribution function is given by a two-dimensional Gaussian function of unity width. Contour of the Wigner function is a circle

Fluctuations of the quad. amplitude

Variance of the operator is given by V𝛼 (𝜃) = V 1𝛼 = 1, independent of quadrature and detection frequency. Standard quantum limit (SQL)

Intensity noise

V n𝛼 = n𝛼 , which is equivalent to the shot noise formula. The intensity noise spectrum is ﬂat, independent of detection frequency

Photon number distribution

Poissonian: 𝛼 (n) = |⟨n|𝛼⟩|2 =

Photon statistics

Time sequence of photon arrival is irregular as shown in Figure 4.7c. The statistics of the number of photons arriving in a time interval times follows a Poissonian distribution

|𝛼|2n n!

exp(−|𝛼|2 )

Fano factor

f𝛼 = 1 for all detection frequencies

Classical ﬁeld

Carrier component has an amplitude |𝛼|, with optical phase arg(𝛼). It has completely uncorrelated noise sidebands

4.4 Propagation and Detection of Quantum Optical Fields Up till now we have been implicitly considering a stationary single frequency mode in a cavity and the behaviour of observables of that mode without specifying how they would be measured. Now we wish to address the experimentally relevant situation of propagating ﬁelds and photodetection. We begin with a quite general discussion of the problem whilst still limiting our discussion to that of single-mode ﬁelds. In free space there is an inﬁnite continuum of frequency modes that must all be quantized. The approach described in this section can be a satisfactory approximation in situations where, over the detection bandwidth, the frequency spectrum of the source and the frequency response of the optical elements are both ﬂat. That is, although there are many modes present, they are all doing the same thing’. A single mode can also be constructed from a superposition of frequency modes as we shall describe in the ﬁrst part of this section. The approach of this

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section is also appropriate if the optical elements act coherently on a single mode of any kind to map it to another single mode. In the following section we will discuss a more speciﬁc approach in which the multimode eﬀects of non-uniform spectral responses are included. 4.4.1

Quantum Optical Modes in Free Space

A single propagating mode is deﬁned quantum mechanically in an analogous way to the classical approach (see Section 2.2): ̃k dk e−i𝜔t F(k, x)A (4.62) ∫ ̃ k are continuum annihilation operators. They obey the commuThe operators A tation relation ̃ † ] = 𝛿(k′ − k) ̃ k′ , A (4.63) [A â (x, t) =

k

and the mode function is normalized such that ∫

dk|F(k, x)|2 = 1

(4.64)

As a result it is easy to show that [̂a(x, t), â † (x, t)] = 1

(4.65)

as required for Boson creation and annihilation operators. Two such modes, say, ̂ are said to be independent or orthogonal if they commute, i.e. â and b, [̂a(x, t), b̂ † (x, t)] = 0

(4.66)

Notice that if the mode function for the mode b̂ is G(k, x), then this implies ∫

dkF(k, x)G(k, x)∗ = 0

(4.67)

which is identical to the deﬁnition of independent classical modes in Section 2.2.3. Indeed all the properties of classical modes discussed there transfer across to the quantum realm – a feature exploited in many of the experiments we will discuss later in the book. An idealized quantum optical mode that is nonetheless a good approximation in many situations is the plane wave Gaussian pulse: â (z, t) = where

√ ̃𝜔 = A

∫

̃𝜔 d𝜔e−i(𝜔t−kz) F(𝜔)A

−2zo k⃗ 2 ko ⃗ ko A ̃k dke √ 4zo π ∫

(4.68)

(4.69)

and k⃗ = (kx , ky ), k ≡ kz is the direction of propagation and 𝜔 = 2π𝜈 = |k|∕c. Also, ko is the average wave number of the beam. The parameter zo is deﬁned in Section 2.1.5. Note that ̃ † ′ ] = 𝛿(𝜔 − 𝜔′ ) ̃ 𝜔, A (4.70) [A 𝜔

4.4 Propagation and Detection of Quantum Optical Fields

Physically, Eq. (4.68) describes a pulse of light in a transverse Gaussian mode (see Sections 2.1.5 and 2.2.3) on axis and close to its waist. The pulse shape as a function of frequency is given by F(𝜔). The relatively simple form of Eq. (4.68) comes from the fact that the wave fronts of a Gaussian mode are approximately plane waves close to their waist such that the transverse and longitudinal degrees of freedom decouple. It may seem very restrictive to have the requirement that we are close to the waist of the beam; however, lenses can be used to focus a Gaussian beam down to its waist at any spatial location. Typically in experiments it is arranged that optical beams interact with optical elements at their waists. A simple case of a pulse shape that is easy to work with is if the pulse shape is also Gaussian: √ 1 −(𝜔−𝜔o )2 (4.71) F(𝜔) = √ e 2𝑣 𝑣π where 𝜔o = ko ∕c is the average frequency of the beam and 𝑣 is the variance around this mean. With this pulse shape, the following conditions on the parameters for Eq. (4.68) to be a good approximation close to the waist: √ are required √ z, 1∕𝑣 ≪ zo and 𝑣 ≪ 𝜔o . Now consider the quantum mode ̂ t) = b(z, where

√ B̃ 𝜔 =

d𝜔e−i(𝜔t−kz) F(𝜔)B̃ 𝜔

(4.72)

−2zo k⃗ 2 ko ⃗ ko eixo kx A ̃k dke √ 4zo π ∫

(4.73)

∫

Equations (4.68) and (5.47) describe two pulses with identical spectral characteristics co-propagating in parallel Gaussian modes separated by a distance xo . √ ̃ 𝜔 , B̃ † ′ ] = 0 and the two modes will be indepenProvided xo ≫ ko ∕zo , then [A 𝜔 dent. Often when we talk about independent modes, we will be thinking of this type of situation with spatially separated optical beams; however, as discussed in Section 2.2, there are many other degrees of freedom that can be used to create independent modes. In the rest of this section, we will work at the level of the mode operators without being speciﬁc about the type of modes involved. 4.4.2

Propagation in Quantum Optics

The propagation of quantum optical modes through optical elements is modelled ̂ † . Two distinct pictures ̂ where U ̂ −1 = U via unitary transformation operators, U, can be employed: (i) the Schrödinger picture, in which the evolution of the state of the mode |Ψ⟩ → |Ψ′ ⟩ is calculated via ̂ |Ψ′ ⟩ = U|Ψ⟩ and the system observables remain stationary, or (ii) the Heisenberg picture, in which the evolution of a mode observable â → â ′ is calculated via ̂ ̂ † â U (4.74) â ′ = U

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and the state of the system is stationary. The two pictures are equivalent in the sense that all expectation values are equal in the two pictures because ̂ ̂ † â U|Ψ⟩ = ⟨Ψ|̂a′ |Ψ⟩ ⟨Ψ′ |̂a|Ψ′ ⟩ = ⟨Ψ|U

(4.75)

However, depending on the particular situation, the two pictures oﬀer diﬀerent levels of computational complexity and diﬀerent physical insights. For propagation through stationary elements (that do not change with time), we can write ̂ = ei Ĥ 𝜏∕ℏ U

(4.76)

̂ is the interaction Hamiltonian characterizing the particular optical elewhere H ment and 𝜏 characterizes the strength of the interaction or the time for which it acts. For the latter case it is easy to show that the equation of motion for an arbitrary Heisenberg operator is then 1 dâ ̂ = [̂a, H] (4.77) dt iℏ The Heisenberg picture most resembles the classical treatment thanks to canonical quantization that requires that the equations of motion of the annihilation operator â are identical to those of the classical complex amplitude 𝛼. Thus in many situations it is straightforward to write down the propagation transformations in the Heisenberg picture, and one is able to retain a considerable amount of classical intuition in the quantum treatment. This is particularly true for Gaussian states as will be discussed in the next section. However, this is not always the case, and in certain situations the higher-order moments of the annihilation and creation operators may have equations of motion quite dissimilar from their classical counterparts. The Schrödinger picture on the other hand gives a uniquely quantum mechanical view of the transformations and as such tends to be most useful when quantum eﬀects dominate, such as in photon counting experiments. In general it is not trivial to convert from a potentially straightforward analysis of an optical circuit in the Heisenberg picture to the corresponding Schrödinger state transformation. An exception is propagation through linear optical elements, that is, passive elements that do not change the photon number in the modes. Clearly the action on the vacuum mode of unitary operators representing such elements will be the identity, that is, ̂ lin |0⟩ = |0⟩ U

(4.78)

Now suppose we write out the state we wish to transform in the number basis: |𝜓⟩ = c0 |0⟩ + c1 |1⟩ + c2 |2⟩ + · · · ) ( 1 †2 † = c0 + c1 â + c2 √ (̂a ) + · · · |0⟩ 2

(4.79)

4.4 Propagation and Detection of Quantum Optical Fields

where we have used Eq. (4.18). Now we ﬁnd ( ) 1 † † 2 ̂ lin c0 + c1 â + c2 √ (̂a ) + · · · U ̂ |0⟩ ̂ lin |𝜓⟩ = U ̂† U U lin lin 2 ( ) 1 ′′† ′′† 2 = c0 + c1 â + c2 √ (̂a ) + · · · |0⟩ 2

(4.80)

̂ aU ̂ † is the inverse Heisenberg picture annihilation operator, where â ′′ = Û obtained by inverting the standard Heisenberg equations of motion. This technique is easily generalized to optical elements that mix more than one mode and, where applicable, is a convenient tool for moving between the two pictures. 4.4.3

Detection in Quantum Optics

The basic detection tool in quantum optics is photodetection. One way or another almost all measurements of optical observables rely on photodetection. For bright ﬁelds photodetection measures the intensity of the ﬁeld; for weak ﬁelds it measures the photon number of the ﬁeld. Although quite distinct technologies must be employed in the bright and weak ﬁeld limits, the process can be modelled in both cases as the measurement of the photon number observable â † â . More sophisticated detection models will be discussed in later chapters. As for propagation, detection in quantum optics is modelled diﬀerently in the Heisenberg and Schrödinger pictures. In the Heisenberg picture photodetection can be viewed as the transformation of the quantum mode â ′ into the classî a represents the photocurrent or, more gen̂ a = â ′† â ′ . The mode N cal mode N erally, the information generated by the detector and is classical in the sense that ̂ †a ] = 0. Nevertheless N ̂ a is an operator and must be so to correctly predict ̂ a, N [N ̂ a can be investigated by calculating expectation all behaviour. The statistics of N values of its various moments: ̂ a ⟩ = ⟨Ψ|N ̂ a |Ψ⟩ ⟨N j

j

(4.81)

where |Ψ⟩ is the initial state of the mode â . Using the technique of homodyne detection (see Section 8.1.4), photodetection can also be used to measure the quadrature observable â ′ ei𝜃 + â ′† e−i𝜃 . This then can be modelled as a transfor̂ 𝜃a = â ′ ei𝜃 + â ′† e−i𝜃 . mation of the quantum mode â ′ into the classical mode X 𝜃 ̂ Once again, although Xa represents a classical mode, it must be modelled as an ̂ a the statistics of the quadrature amplitude can be calculated operator. As for N by taking expectation values over the initial state of mode â . The operators representing the photocurrents can be manipulated as they might be in an experiment. For example, correlation measurements can be modelled by adding, subtracting, or multiplying photocurrent operators representing the measurement of diﬀerent modes.

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In the Schrödinger picture photodetection can be viewed as projection onto the eigenstates of â † â , i.e. the number states |n⟩. Thus we can predict the probability that a particular result n will occur via Pn = |⟨n|Ψ′ ⟩|2

(4.82)

Average values, variances, etc. can be calculated from the probability distribution thus generated. Similarly if the experiment is conﬁgured such that a quadrature observable is measured, then the projection is onto the eigenstates of â ei𝜃 + â † e−i𝜃 , |x𝜃 ⟩, and the probabilities are given, as was used in Section 4.3.2, by PΨ (x𝜃 ) = |⟨x𝜃 |Ψ⟩|2 𝛿x

(4.83)

It is important to note that the predictions about the photocurrent generated in the Schrödinger picture are just numbers, unlike the operators generated in the Heisenberg picture. A major conceptual diﬀerence arises between the two approaches when multimode systems are considered. Suppose a system initially describable by a pure state in a single mode interacts with a number of other modes, also initially in pure states, during propagation. In the Heisenberg picture there will now be a number of output modes that are all a function of the input modes. Now suppose we have lost’ all but one of these modes, â ′ . Nevertheless we proceed as before by ̂ a = â ′† â ′ , just as we would with classical modes. This analysing the properties of N scenario looks quite diﬀerent in the Schrödinger picture however. After propagation we are left with a state that describes the global properties of all the modes. If we have lost the other modes, we must describe that explicitly by averaging over all the possible detection results the other modes may have given. This is achieved by tracing over all the lost modes such that 𝝆̂ a = Trc≠a {𝝆̂ ′ }

(4.84)

where 𝜌 is the global state of the system, whilst 𝝆̂ a is the reduced density operator that describes the state of only mode â . The subscript c ≠ a indicates that the trace is over all modes except â . Under certain conditions the trace will not change the state of mode â . The state is then said to be separable as it indicates that the portion describing mode â can be factored out. More generally though the state will be inseparable or entangled, meaning no factorization is possible. Then inevitably 𝝆̂ a will be a mixed state, and ′

Pn = ⟨n|𝝆̂ a |n⟩

(4.85)

will give the probability of a particular detection result n. If the results of measurements on the other modes are known, then we must insert them explicitly so |Ψa ⟩ = {⟨nb |⟨nc |⟨nd | …} |Ψ′ ⟩

(4.86)

where now |Ψa ⟩ is the conditional state of the system given that the results nb , nc , nd , … were obtained for the other modes. 4.4.4

An Example: The Beamsplitter

To illustrate some of the ideas introduced in this section, we consider a simple example: a single photon number state passing through a beamsplitter. A more

4.4 Propagation and Detection of Quantum Optical Fields

general discussion of beamsplitters will be given in Chapter 5. The beamsplitter is assumed to have transmittance 𝜂. The other mode entering the beamsplitter is unoccupied. We label the mode containing the single photon state â and the unoĉ Thus the initial state can be written as |1⟩a |0⟩b . Let us consider the cupied mode b. Heisenberg approach ﬁrst. As described in Section 5.1, the classical equations for mixing two modes with amplitudes 𝛼 and 𝛽 on a beamsplitter can be written as √ √ 𝛼out = 𝜂 𝛼 + 1 − 𝜂 𝛽 √ √ 𝛽out = 1 − 𝜂 𝛼 − 𝜂 𝛽 (4.87) Canonical quantization then allows us to immediately write √ √ â ′ = 𝜂 â + 1 − 𝜂 b̂ √ √ b̂ ′ = 1 − 𝜂 â − 𝜂 b̂

(4.88)

Hence, if we photodetect â , we obtain ′

̂ a = â ′† â ′ N = 𝜂 â † â + (1 − 𝜂)b̂ † b̂ +

√ 𝜂(1 − 𝜂)(̂a† b̂ + b̂ † â )

(4.89)

The average photon number detected is then ̂ a ⟩ = ⟨1|a ⟨0|b â ′† â ′ |0⟩b |1⟩a n = ⟨N ̂ b+ = 𝜂⟨1|a â † â |1⟩a + (1 − 𝜂)⟨0|b b̂ † b|0⟩ √ ̂ b + ⟨0|b b̂ † |0⟩b ⟨1|a â |1⟩a ) 𝜂(1 − 𝜂)(⟨1|a â † |1⟩a ⟨0|b b|0⟩ =𝜂

(4.90)

where we have used the relationships of Eq. (4.17). In a similar way the variance is ̂ 2a ⟩ − ⟨N ̂ a ⟩2 Vn = ⟨N

= 𝜂(1 − 𝜂)⟨1|a â † â |1⟩a ⟨0|b b̂ b̂ † |0⟩b = 𝜂(1 − 𝜂)

(4.91)

The result for the average photon number can be understood from a classical ﬁeld point of view: the intensity is reduced according to the transmission of the beamsplitter; the non-zero variance after some loss at the beamsplitter reﬂects the particle aspect of light. More on this is said in Chapter 5. Now let us consider the same problem in the Schrödinger picture. Using the technique of Eq. (4.80), we ﬁnd the state evolution through the beamsplitter is given by |Ψ′ ⟩ = â ′′ |0⟩a′ |0⟩b′ √ √ = ( 𝜂 â ′ + 1 − 𝜂 b̂ ′ )|0⟩a′ |0⟩b′ √ √ = 𝜂|1⟩a′ |0⟩b′ + 1 − 𝜂|0⟩a′ |1⟩b′

(4.92)

In order to consider only detection of the â mode, we need to trace over the b̂ mode as per Eq. (4.84). Writing 𝝆̂ ′ = |Ψ′ ⟩⟨Ψ′ |, we ﬁnd 𝝆̂ a′ = ⟨0|b′ 𝝆̂ ′ |0⟩b′ + ⟨1|b′ 𝝆̂ ′ |1⟩b′ = 𝜂|1⟩a′ ⟨1|a′ + (1 − 𝜂)|0⟩a′ ⟨0|a′

(4.93)

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4 Quantum Models of Light

Using Eq. (4.85), we ﬁnd that detection of this state gives the probabilities P1 = ⟨1|𝝆̂ a′ |1⟩ = 𝜂 P0 = ⟨0|𝝆̂ a′ |0⟩ = 1 − 𝜂

(4.94)

which leads to the same average value and variance as predicted from the Heisenberg result. Now, instead of losing’ the b̂ mode, let us suppose that we have photodetected it. If the photodetection of the b̂ mode gave the result 0’, then the conditional state of the â mode will be √ (4.95) |Ψ⟩a′ = {⟨0|b′ }|Ψ′ ⟩ = 𝜂|1⟩ √ indicating that a photon will be found in mode â with certainty. The factor of 𝜂 indicates that the probability of this state being prepared √ (i.e. the probability of the measurement of the b̂ mode giving the 0’ result) is ( 𝜂)2 = 𝜂. Similarly, if the measurement of the b̂ mode gave the result 1’, then the conditional state of the â mode would be √ |Ψ⟩a′ = {⟨1|b′ }|Ψ′ ⟩ = 1 − 𝜂|0⟩ indicating that it is certain that no photon will be found in the â mode. How could we predict this latter result in the Heisenberg picture? Well, we found that ̂ one was not found in â and vice versa. whenever a photon appeared in mode b, Another way of saying this is that the two modes are anti-correlated. To test for anti-correlation in an experiment, we might add the photocurrents obtained from the photodetection of the two modes and look for reduced noise (variance). Adding the photocurrents, we would obtain (using Eq. (4.88)) ̂ b = â ′† â ′ + b̂ ′† b̂ ′ ̂a+N N = â † â + b̂ † b̂

(4.96)

which has zero variance indicating perfect anti-correlation as expected. Although the results match in the two pictures, as they must, the calculational techniques and the physical pictures suggested are quite diﬀerent.

4.5 Quantum Transfer Functions As we saw in Section 4.3, in general a complete description of a state of light, including its quantum properties, can be carried out through various probability and quasi-probability distributions [7]. For the description inside an optical cavity, a single function will suﬃce. In contrast, for a freely propagating beam, the situation is more involved [12]. We can expand our beam in a basis of single frequency modes. We will then ﬁnd that we need a whole continuum of distribution functions to describe these modes. These functions can be derived in the Schrödinger picture as the solutions of complex stochastic diﬀerential equations, in many cases master equations [7, 13, 14]. Whilst complete, this approach is often more general than required in practice. Here we present an alternative route based on the Heisenberg picture in which we obtain operator equations that describe how the individual frequency modes transform – the

4.5 Quantum Transfer Functions

quantum transfer functions. We restrict our attention to systems that are linear or can be linearized. Linearization can be done if the modulations and the ﬂuctuations we are concerned with are small compared with the amplitude of the laser beam. This is usually correct for situations with optical powers larger than microwatts. A continuous-wave (CW) laser beam is composed of many independent single frequency modes and so is inherently a multimode situation. In contrast a single pulse from a pulsed laser may be in a single mode described by a coherent superposition of many frequency modes. In either case a complete description of the quantum evolution of the beam is available via the quantum transfer functions. Working in the Heisenberg picture means the operator equations are very similar to the classical equations encountered in Section 2.3.2, and similar interpretations can be made. This approach is not directly applicable to experiments that resolve single photons using photon counting technology. However, in Chapter 12, we will adapt this approach to treat such experiments. 4.5.1

A Linearized Quantum Noise Description

Consider a propagating CW laser beam in a single spatial mode. We will represent ̂ where we use capitals to remind us that the beam by the continuum operator A, this is a continuum operator with units s−1∕2 . We can break this operator up into two contributions: (i) the mean value of the amplitude, given by one complex number 𝛼, and (ii) the ﬂuctuations and modulations, described by the operator ̂ 𝜹A(t), representing small ﬂuctuations around the mean. We thus write ̂ = 𝛼 + 𝜹A(t) ̂ A

(4.97)

Note the similarity here with the classical treatment of small ﬂuctuations, Eq. (2.15), the diﬀerence being that now the ﬂuctuating term is represented as an operator. Note also that Eq. (4.97) is really just a special case of a ﬂuctuation ̂ = 𝛼. We use small 𝛿 here to emphasize that operator (see Section 4.3) where ⟨A⟩ these ﬂuctuation operators are assumed small compared to their expectation ̂ is values. If the beam is detected by a photodetector, then the photocurrent N given by ̂ ̂ † (t)A(t) ̂ =A N ̂ ̂ † (t))(𝛼 + 𝜹A(t)) = (𝛼 + 𝜹A ̂ † (t) + 𝛼 𝜹A(t) ̂ + 𝜹A ̂ † (t) 𝜹A(t) ̂ = 𝛼 2 + 𝛼 𝜹A ̂ = 𝛼 2 + 𝛼 𝜹X𝟏(t)

(4.98)

where we have taken the arbitrary phase of 𝛼 real and linearized by assuming the ﬂuctuations are small and hence neglecting higher than ﬁrst-order terms in the ﬂuctuations. By small’, we mean that ultimately the contribution to the calculated expectation values due to these terms will be negligible. We see that from the quantum point of view, we are eﬀectively measuring the in-phase ̂ is often referred ̂ = 𝜹A ̂ ̂ † (t) + 𝜹A(t). For this reason 𝜹X𝟏 quadrature operator 𝜹X𝟏 to as the amplitude quadrature operator. By introducing another bright beam as a reference, it is also possible to measure the ﬂuctuations in the phase of the

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beam (see Chapter 8). Given the linear approximation, these turn out to be ̂ = i(𝜹A ̂ † (t) − 𝜹A(t)), ̂ proportional to the out-of-phase quadrature, 𝜹X𝟐 which is thus often referred to as the phase quadrature. This link is illustrated in ̂ and 𝜹X𝟐 ̂ have continuous eigenvalue spectra. This Figure 4.7a. Recall that 𝜹X𝟏 implies that under such measurement conditions, discrete single-photon events cannot be observed, regardless of the resolution of the detectors. The discussion is more naturally continued in frequency space. The Fourier transform of the operator in Eq. (4.97) is ̃ ̃ A(Ω) = 𝛼 𝛿(Ω) + 𝜹A(Ω)

(4.99)

where 𝛿(Ω) is the Dirac delta function. Recall that 𝛼 is deﬁned in a frame rotating at the mean optical frequency 𝜈o (see Section 2.1). Thus Ω = 2π(𝜈o − 𝜈)

(4.100)

is the frequency diﬀerence from the mean optical frequency. The mean coherent amplitude of the ﬁeld appears as a delta function spike, the carrier, 𝛼 𝛿(Ω), which is only non-zero for Ω = 0, i.e. at the mean optical frequency. We can consider the Fourier transformed ﬂuctuation operator, in some small frequency interval around radio frequency Ω, as approximately describing a single quantum mode at this frequency. Such modes obey the commutation relations ̃ ̃ † (Ω′ )] = 𝛿(Ω − Ω′ ), [A(Ω), A

̃ ̃ ′ )] = 0 [A(Ω), A(Ω

(4.101)

This is a useful and practical picture because experimentally we can perform Fourier analysis on the photocurrent produced by photodetection using a spectrum analyser, discussed in Chapter 8. We can also place information on the laser beams in the form of modulation at various diﬀerent frequencies Ωmod , which is measured in Hz, and each frequency can be thought of as an independent information channel. From Eq. (4.98), we ﬁnd that the operator describing the photocurrent in Fourier space is ̃ out (Ω) ̃ out = 𝛼 2 𝛿(Ω) + 𝛼 𝜹X𝟏 N

(4.102)

Now the ﬁrst term describes the average power, concentrated in a DC spike, and the second term describes the measurement of the amplitude quadrature at frequency Ω. The variance of the photocurrent at frequency Ω is given by 2 ̃ out (Ω)|2 |𝛼⟩ = 𝛼 2 ⟨𝛼||𝜹X𝟏 V Nout (Ω) = 𝛼out out

V 1out (Ω)

(4.103)

We have a direct link between the measured values V Nout (Ω) and the property V 1out (Ω) of the state. In particular we see that ultimately the intensity noise of a bright laser is determined by the quantum states of its frequency sidebands. If the laser mode is subjected to amplitude or phase modulations (either deterministic signals or stochastic noise), these can be incorporated into the description as c-number variables. Thus we can modify Eq. (4.99) to read ̃ ̃ ̃ A(Ω) = 𝛼𝛿(Ω) + 𝜹A(Ω) + 𝛿 S(Ω)

(4.104)

where the ̃ indicates the Fourier transform of a classical variable. If the power ̃ spectra of 𝛿 S(Ω) are much larger than those associated with the quantum part

4.5 Quantum Transfer Functions

̃ (𝜹A(Ω)), then the beam will behave classically. However, once again, if all classical noise is suppressed, we will still be left with ﬂuctuations due to the quantum noise. If we are trying to measure or send information via small modulations of the beam, then the signal to noise associated with this measurement or communication will likewise be determined by the quantum noise. 4.5.2

An Example: The Propagating Coherent State

Let us illustrate the preceding discussion with an idealized and then more realistic example. In Section 4.2.2 we described a coherent state as the displacement of a single cavity mode initially in a vacuum state. We may think of a propagating coherent state in a similar way as a beam with all its power at a single frequency. Notice that this implies a CW laser, as a single frequency corresponds ̃ to a constant output in time. This also means that the quantum state of the 𝛿 A(Ω) sideband modes is the vacuum. From Eq. (4.103) and using Eqs. (4.15) and (4.101), we immediately ﬁnd 2 ̃ out |2 |0⟩ ⟨0||𝜹X𝟏 V Nout (Ω) = 𝛼out ̃ 2 + (𝜹A ̃ † )2 + 𝜹 A ̃ † 𝜹A ̃ + 𝜹A𝜹 ̃ A ̃ † |0⟩ = 𝛼 2 ⟨0|(𝜹A out

2 = 𝛼out

(4.105)

where the absolute square is taken because we are in Fourier space. The laser intensity exhibits Poissonian statistics at all frequencies. In a similar way it follows that the quadrature variances are at the quantum noise limit (QNL) for all angles and all frequencies. These results are as might be expected for a travelling ﬁeld generalization of a coherent state. Of course this is an idealization. Real lasers have a ﬁnite linewidth and may have structure in their ﬂuctuation spectra (see Chapter 6). Nevertheless, for lasers oscillating on single longitudinal and spatial cavity modes, so-called single-mode lasers, at suﬃciently high frequencies, the power in the sidebands will become negligible. At or above such frequencies, the laser will behave like a coherent state, and it is in this sense that lasers are often referred to as approximately producing a coherent state. A typical solid-state laser will be coherent’ at frequencies above about 10 MHz. Another way of saying this is that if we sample the ﬁeld on short timescales, the laser can be thought of as producing a coherent state. On the other hand, as we go to low sideband frequencies (long timescales), eventually the power in the sidebands will become an appreciable fraction of the central peak power and our linearization assumption will break down. 4.5.3

Real Laser Beams

We will ﬁnd in Chapter 6 that a real laser beam contains much more noise than a coherent state. Figure 4.8 shows an example of a typical laser that has an intensity noise spectrum V 1(Ω) well above the QNL V 1(Ω) = 1 of the coherent ̃ state. This ﬁgure shows the link between the values for the quadratures 𝜹X𝟏, ̃ 𝜹X𝟐 and the actual measured intensity noise. In addition such a laser beam will contain modulation signals at certain frequencies. We need a technique to derive

123

124

4 Quantum Models of Light

ˆ (Ω) δX1

ˆ (Ω) δX2

V1(Ω)

its 1

r un

e oov

H

0 Ω

Ω

Figure 4.8 The Fourier components of the noise spectrum for a typical real laser. On the right the ﬂuctuations for six diﬀerent detection frequencies Ω are represented by their uncertainty areas. On the left is the corresponding continuous intensity noise spectrum V1(Ω).

such spectra and to describe how the noise and the signals will change whilst the light propagates through the apparatus. 4.5.4

The Transfer of Operators, Signals, and Noise

We are now in a position to describe a complete optical system and to evaluate the propagation of the noise and the signals through the system. Each optical component of the system can be thought of as having several inputs, described ̃ in1 , by their coherent amplitudes 𝛼in1 , 𝛼in2 , … etc., their ﬂuctuation operators 𝜹A ̃ in2 , … , etc., and their states |Φin1 ⟩, |Φin2 ⟩, … , etc. The corresponding spectral 𝜹A variances for each input are ̃ inj (Ω)|2 |Φinj ⟩ V 1inj (Ω) = ⟨Φinj ||X𝟏 ̃ inj (Ω)|2 |Φinj ⟩ V 2inj (Ω) = ⟨Φinj ||X𝟐

(4.106)

where j = 1, 2, … Inside the optical components the modes are combined. The algebraic equations for the coherent amplitudes are solved to give the output coherent amplitude. We usually choose the arbitrary overall phase of the input ﬁelds such that the output coherent amplitude is real. After linearization the equations of motion for the ﬂuctuation operators can be solved in frequency space. The ﬂuctuation operators in the frequency domain for one output mode

4.5 Quantum Transfer Functions

can then be described by equations of the form ̃ in1 (Ω) + c1c (Ω) 𝜹A ̃ † (Ω) ̃ out (Ω) = c1a (Ω) 𝜹A 𝜹A in1 ̃ ̃ † (Ω) + · · · + c2a (Ω) 𝜹Ain2 (Ω) + c2c (Ω) 𝜹A in2

̃ in1 (Ω) + c∗ (−Ω) 𝜹A ̃ † (Ω) ̃ † (Ω) = c∗ (−Ω) 𝜹A 𝜹A 1c 1a out in1 ̃ in2 (Ω) + c∗ (−Ω) 𝜹A ̃ † (Ω) + · · · + c∗ (−Ω) 𝜹A 2c

2a

in2

(4.107)

and similarly for all other output modes. The coeﬃcients ckl (Ω) are complex numbers that describe the frequency response of the component. Note that the response is not necessarily symmetric about the carrier. The cka and ckc are linked via commutation requirements. For linear systems the ckl do not depend on the beam intensities and all ckc = 0. We can also treat the case of propagation through non-linear media, such as 𝜒 (2) and 𝜒 (3) media. This includes non-linear processes such as frequency doubling, parametric ampliﬁcation, the Kerr eﬀect, or non-linear absorption. Equation (4.123) remains linear, but in these cases the coeﬃcients c1a , c1c , c2a , c2c , … are dependent on the average optical power of the beams. In various examples throughout this book, we will show how the coeﬃcients cij can be derived from the physical properties of the component, ̃ out (Ω) for such as reﬂectivities, conversion eﬃciencies, etc. Once the operator 𝜹A the output mode in question has been found, the variances can be determined using Eq. (4.106). Let us consider some general cases: (1) The simplest case is where all the inputs are independent: ̃ inl (Ω)⟩ = 𝛿kl . This will be true if all the input beams represent ̃ ink (Ω)𝜹X𝟏 ⟨𝜹X𝟏 beams from independent sources or diﬀerent vacuum inputs. Secondly, we require that the coeﬃcients obey c∗ij (−Ω) = cij (Ω), i.e. magnitude and phase of the frequency response are symmetric. This will be true if there are no detunings or strong dispersion in the optical system. In this case we obtain the simplest type of transfer function: V 1out (Ω) = |c1a (Ω) + c1c (Ω)|2 V 1in1 (Ω) + |c2a (Ω) + c2c (Ω)|2 V 1in2 (Ω) + · · · V 2out (Ω) = |c1a (Ω) − c1c (Ω)|2 V 2in1 (Ω) + |c2a (Ω) − c2c (Ω)|2 V 2in2 (Ω) + · · ·

(4.108)

Notice that for a linear system (c1c = c2c = · · · = 0), the transfer functions for both quadratures are identical. (2) The coeﬃcients still obey c∗ij (−Ω) = cij (Ω), but we allow for inputs that are not independent. Now there can be correlations between the quadrature variances of diﬀerent inputs. However, we can always regain the form of Eq. (4.108) by breaking down each of the inputs into a function of its own inputs until we end up with all the inputs being independent. (3) The coeﬃcients obey c∗ij (−Ω) = c∗ij (Ω), i.e. the magnitude of the frequency response is symmetric but not the phase. This will occur if the ﬂuctuations

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4 Quantum Models of Light

suﬀer a diﬀerent phase rotation to that of the coherent amplitude. In this case we get cross-coupling between the transfer functions of V 1out and V 2out such that now V 1out (Ω) = |c1a (Ω) + c1c (Ω)|2 cos2 (𝜃)V 1in1 (Ω) + |c1a (Ω) − c1c (Ω)|2 sin2 (𝜃)V 2in1 (Ω) + · · · V 2out (Ω) = |c1a (Ω) − c1c (Ω)|2 sin2 (𝜃)V 2in1 (Ω) + |c1a (Ω) + c1c (Ω)|2 cos2 (𝜃)V 1in1 (Ω) + · · ·

(4.109)

where 𝜃 is deﬁned by cij (Ω) = |cij (Ω)|e . We have assumed that V 1 and V 2 are the quadratures of minimum and maximum ﬂuctuations. (4) The coeﬃcients obey c∗ij (−Ω) ≠ c∗ij (Ω). This will occur when the frequency response of the optical components is asymmetric around the carrier frequency, for example, due to detunings. It is not possible to represent the output as a transfer function of quadrature variances in this case. i𝜃

The simplest example of a transfer function describes the inﬂuence of loss terms only and has the form V 1out (Ω) = |c1a |2 V 1in1 (Ω) + |c2a |2 + |c3a |2 + · · ·, where all the inputs apart from V 1in1 (Ω) are vacuum inputs that have the numerical value of 1. Such a transfer function has identical eﬀects on both quadratures. An example of case (2) would be where either a modulation is common or two or more of the input modes have correlated noise. This is the case when the beams originate from one source, such as two beams formed by a beamsplitter, or where two beams are generated by an above threshold optical parametric oscillation (OPO). Examples of case (3) can be as benign as mixing beams on a beamsplitter where the coherent amplitudes of the inputs have diﬀerent phases, or components with strong dispersion where the refractive index changes within a frequency interval of Ωdet , or as complex as single-ended cavities with optical detuning. The simplest example of case (4) is a double-ended optical cavity with detuning. An appropriate name for Eq. (4.108) is quantum noise transfer function. This equation is very similar to the classical transfer functions used by communication engineers. The classical function can be derived using a description of the various inputs as classical waves with modulation at Ω. Noise behaves exactly in the same way as the modulations. Quantum noise is not included. The propagation and interference of these waves are determined, and the result is the size of the modulation at the output. In contrast, Eq. (4.108) includes all the quantum eﬀects of the light, in particular the eﬀects of quantum noise. We can directly see how the signals vary in comparison to the quantum noise and can evaluate the coeﬃcient for the transfer of the signal-to-noise ratio from input to output. This approach has been developed since the 1980s by a number of research groups, very prominently by the groups in France who pioneered squeezing [12]. 4.5.5

Sideband Modes as Quantum States

We will now outline how we can relate the quantum transfer functions to the detailed quantum state of the ﬁeld at diﬀerent frequencies. In particular we will

4.5 Quantum Transfer Functions

consider the example of modulation of a quantum noise limited beam and see how this can be viewed as producing a coherent state at the modulation frequency. In Section 2.1.7 we saw how small modulations of a classical light beam resulted in coherent power being injected symmetrically into upper and lower sidebands around the carrier. In Section 4.5.1 we described classical modulations in the Heisenberg picture as evolution of the quantum ﬂuctuation operator ̃ ̃ ̃ 𝜹A(Ω) → 𝜹A(Ω) + 𝛿 S(Ω)

(4.110)

This Heisenberg evolution is equivalent to the Schrödinger evolution of displacement (see Section 4.2.2), i.e. ̃ ̃ A(Ω))|𝜓⟩ |𝜓⟩Ω → D(𝜹 Ω

(4.111)

Now consider the amplitude quadrature operator in Fourier space (Figure 4.9). It ̃ † (Ω), is is easy to show that in general the Fourier transform of a conjugate, e.g. A † ̃ , via a change in sign related to the conjugate of the Fourier transform, e.g. A(Ω) † ̃ ̃ † (Ω) = A(−Ω) . Using this property of the Fourier transform, we can of Ω, e.g. A observe that ̃ † (Ω) ̃ ̃ 𝜹X𝟏(Ω) = 𝜹A(Ω) + 𝜹A † ̃ ̃ = 𝜹A(Ω) + 𝜹A(−Ω)

(4.112)

Thus we see that a quadrature measurement actually probes both the upper and lower sidebands. Suppose the initial state of the sidebands is a vacuum. Using Eq. (4.111) we ﬁnd that modulation produces the coherent sideband state ̃ ̃ |𝜓 ′ ⟩ = |𝛿 S(Ω)⟩ Ω |𝛿 S(−Ω)⟩−Ω

(4.113)

where, in response to Eq. (4.112), we have written the state explicitly in terms of an upper and lower sideband part. We are ignoring the low frequency contributions from the carrier. For an arbitrary mix of ideal amplitude and phase modulation, we have ̃ 𝛿 S(Ω) = 𝛽(𝛿(Ω + Ωmod ) + 𝛿(Ω − Ωmod ))

(4.114)

The complex number 𝛽 = Ma ∕4 + iMp ∕4, where Ma and Mp are the amplitude and phase modulation depths, respectively, and Ωmod is the modulation X2

E

vL–Ω

vL+Ω

X1 Random distribution in time (a)

vL

(b)

Random components at all frequencies

v

Figure 4.9 Comparison between (a) the phase space diagram and (b) the sideband picture. The quantum properties are encapsulated in noise sidebands at all modulation frequencies above and below the laser frequency 𝜈L .

127

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4 Quantum Models of Light

frequency (see Section 2.1.7). Averaging over a small range of frequencies around Ωmod , we can write our state as |𝜓 ′ ⟩ = |𝛽⟩Ωmod |𝛽⟩−Ωmod |0⟩Ω≠±Ωmod

(4.115)

A pair of identical coherent states has been created at the upper and lower sideband frequencies. As quadrature measurements probe the sidebands symmetrically (see Eq. (4.112)), the results of quadrature measurements are identical to those for a single frequency coherent state. For example, the expectation value of X1 in the Schrödinger picture is given by † ̃ ̃ ̃ + ⟨𝛽|𝜹A(−Ω ⟨𝜹X𝟏(Ω mod )⟩ = ⟨𝛽|𝜹A(Ωmod )|𝛽⟩Ω mod ) |𝛽⟩−Ω mod

mod

= 𝛽 + 𝛽∗

(4.116)

This is of course the same as the result that would be obtained in the Heisenberg picture. This result generalizes to other states provided the transfer functions remain symmetric about the carrier. Most interesting are states for which the upper and lower sideband modes become entangled as this has no classical analogue. Squeezed states, which are described in Chapter 9, are an example of such states. Using a homodyne detector it is possible to measure the variances VΘ (Ω) at all frequencies Ω. The properties of the light in a small interval around each frequency can then be described by one Wigner function W (Ω). The interval is assumed small enough such that the frequency response is ﬂat. In this way the entire beam can be described as a series of quantum states, one for each small frequency interval. This is a Schrödinger picture view of the travelling ﬁeld, which complements very well the actual technology used in optical communication. Here each of these frequency intervals is used as a separate communication channel. The variance VΘ (Ω) can be determined mathematically by ﬁrst projecting all parts of the Wigner function onto one axis at angle Θ. This produces a series of one-dimensional distribution functions DF(Θ) that describe in the form of a histogram the time-varying photocurrent in a small frequency range. In a second step the variances of these functions are evaluated, which provides information about the width of DF(Θ). In general we would need to evaluate the moments of DF(Θ) to all orders. However, as we have seen, for the restricted class of Gaussian functions, a direct link exists between the Wigner functions and the variances, and a complete description is given by Vmin (Ω), Vmax (Ω), and Θmin (Ω), as shown in Section 4.3.2. It is also possible to obtain the average photon number in the sidebands from the quadrature phase measurements. Consider the sum of the amplitude and phase quadrature variances written in terms of annihilation and creation operators. We have † ̃ † ̃ ̃ ̃ 𝜹A(Ω) + 𝜹A(−Ω)𝜹 A(−Ω) ⟩ V 1(Ω) + V 2(Ω) = 2⟨𝜹A(Ω) † † ̃ ̃ ̃ ̃ 𝜹A(Ω) + 𝜹A(−Ω) 𝜹A(−Ω)⟩ +2 = 2⟨𝜹A(Ω)

(4.117)

Hence we can write 1 † ̃ ̃ 𝛿N(|Ω|) = ⟨𝜹A(|Ω|) 𝜹A(|Ω|)⟩ = (V 1(Ω) + V 2(Ω) − 2) 4

(4.118)

4.5 Quantum Transfer Functions

For a coherent state (of the entire beam), we expect V 1 = V 2 = 1 at all frequencies, and so from Eq. (4.118) there are no photons in the sidebands as discussed in Section 4.5.2. This discussion requires a caveat: we have assumed throughout that the sidebands are symmetric around the carrier, i.e. our transfer function is not of type (4) as described in the previous section. Asymmetric transfer functions can occur. For example, it is possible to produce single-sideband modulation. The state produced in such a situation is |𝜓 ′ ⟩ = |𝛽⟩Ωmod |0⟩Ω≠Ωmod

(4.119)

Such states are not fully characterized by homodyne quadrature measurements. 4.5.6 Another Example: A Coherent State Pulse Through a Frequency Filter So far we have only considered CW or quasi-CW situations where our beam can be approximated as many independent single frequency modes (or mode pairs as in the case of the sidebands). Now we wish to illustrate how the quantum transfer functions can also be used to describe the propagation of pulses through optical elements with non-uniform spectral response. The example we will discuss is a pulse of light in a coherent state transmitted through a frequency ﬁlter. We consider a plane wave Gaussian mode as described by Eq. (4.68). Written in the nomenclature of this section, we have â =

∫

̃ dΩF(Ω)A(Ω)

(4.120)

Suppose we ask: what is the expectation value of an amplitude quadrature measurement that projects onto this mode when the mode is prepared in a coherent state of amplitude 𝛼? Using the techniques of Section 4.4 and Eq. (4.24), we ﬁnd ̂ † (𝛼)(̂a + â † )D ̂ â (𝛼)|0⟩ ⟨𝛼|̂a + â † |𝛼⟩â = ⟨0|D â = ⟨0|̂a + â † + (𝛼 + 𝛼 ∗ )

dΩF(Ω)F(Ω)∗ |0⟩ ∫ = ⟨0|̂a + â † + 𝛼 + 𝛼 ∗ |0⟩ = 𝛼 + 𝛼 ∗

(4.121)

As expected we obtain 2 e 𝛼. The intermediate step of the calculation shown in the second line is included to emphasize that when generalizing from the displacement of a single frequency mode (Eq. (4.24)) to that of a pulse, the overlap between the mode that is displaced and that which is detected appears. In this case the detected mode is identical to the displaced mode, and so the integral evaluates to unity via normalization (Eq. (4.64)). Now suppose that a frequency ﬁlter is placed in front of the detector. The detector mode is now â o =

∫

̃ o (Ω) dΩF(Ω)A

(4.122)

̃ o is determined by the transfer function of the frequency ﬁlter. where A A frequency ﬁlter can be modelled as the transmission through an optical cavity,

129

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4 Quantum Models of Light

as described in Section 5.3. Here we just quote its transfer functions: 𝜅 iΩ ̃ ̃ B(Ω) (4.123) A(Ω) + 𝜅 − iΩ 𝜅 − iΩ ̃ where 𝜅 is the linewidth of the ﬁlter and B(Ω) is a second mode, initially in the vacuum state, coupled in by the ﬁlter. Now the expectation value of the amplitude quadrature becomes ̃ o (Ω) = A

̂ † (𝛼)(̂ao + â †o )D ̂ â (𝛼)|0⟩ ⟨𝛼|̂ao + â †o |𝛼⟩â = ⟨0|D â ) ( 𝜅 𝜅 F(Ω)F(Ω)∗ |0⟩ = ⟨0|(̂ao + â †o ) + dΩ 𝛼 + 𝛼∗ ∫ 𝜅 − iΩ 𝜅 + iΩ ( ) 𝜅2 Ω𝜅 ∗ = dΩ|F(Ω)|2 (𝛼 + 𝛼 ∗ ) 2 − i(𝛼 − 𝛼 ) ∫ 𝜅 + Ω2 𝜅 2 + Ω2 (4.124) In general the ﬁlter produces two eﬀects: a rotation of the quadratures such that a component of the imaginary part of 𝛼 now appears in the expectation value and attenuation by a Lorentzian factor. Notice that if |F(Ω)| = |F(−Ω)|, then the m 𝛼 term goes to zero as it is an odd function of Ω. Hence, if the signal is symmetric and the centre frequency is on resonance with the ﬁlter, there will be no rotation. Restricting to this case we can get some insight √about the attenuation by 2

−(Ω) 1 √ e 2𝑣 (c.f. Eq. 𝑣 π ⟨𝛼|̂ao + â †o |𝛼⟩â ≈ 𝛼 +

considering two limits. Suppose F(Ω) is the Gaussian

(4.68).

𝛼 ∗ , i.e. If 𝑣 ≪ 𝜅, then we can approximate F(Ω) ≈ 𝛿(Ω) and if the frequency spread of the pulse is narrow compared to the linewidth of the ﬁlter, almost all the power of the pulse is transmitted to the detector. On√the other 1 √ and hand, in the opposite limit where 𝑣 ≫ 𝜅, we can approximate F(Ω) ≈ 𝑣 π

⟨𝛼|̂ao + â †o |𝛼⟩â ≈ 0, i.e. if the frequency spread of the pulse is broad compared to the linewidth of the ﬁlter, almost none of the power of the pulse is transmitted to the detector. 4.5.7

Transformation of the Covariance Matrix

Throughout this section we have been considering linear transfer functions arising from linear, or linearized, optical interactions. Such transfer functions always map one set of Gaussian states to another set of Gaussian states and as such are known as Gaussian operations. Thus, if our system is initially in a Gaussian state and it is transformed according to linear transfer functions, then the ﬁnal state will also be Gaussian. This means that in such a situation all the information required to completely describe the output state can be obtained by ﬁnding the transfer functions of the elements of the input displacement vector and covariance matrix (see Section 4.3.4). For transfer function cases (1)–(3), a covariance matrix initially in standard form, Eq. (4.61), will remain in standard form, and it is only necessary to calculate how the variances transform. See also Section 2.1.5. However, for case (4), a covariance matrix initially in standard form will transform to one not in standard form, and hence the transformation of all matrix elements must be calculated.

4.6 Quantum Correlations

4.6 Quantum Correlations A key diﬀerence between classical optical ﬁelds and their quantum counterparts is that the correlations that can exist between spatially separated beams in quantum mechanics can be stronger than is allowed by classical theory. We now look at examples of how this diﬀerence can be quantiﬁed both in the single-photon and linearized domains. 4.6.1

Photon Correlations

In Section 2.3.3 we saw that the intensity correlations between diﬀerent beams could be characterized by the second-order correlation function, g (2) , which was deﬁned by g (2) (𝜏) =

⟨I(t)I(t + 𝜏)⟩ ⟨I(t)2 ⟩

(4.125)

We also saw that for classical ﬁelds g (2) (0) ≥ 1. In contrast we argued in Section 3.6 that g (2) (0) < 1 should be possible for quantum mechanical ﬁelds. We will now treat this question with the full quantum formalism. The calculation that led to the restriction on g (2) in the classical case relied on the fact that ⟨A2 ⟩ ≥ ⟨A⟩2 . This relationship still holds if A is an operator, so how is it that we can have g (2) (0) < 1 for a quantum mechanical ﬁeld? To answer this we must look more carefully at what the actual measurement of g (2) involves. As outlined previously we need to divide the ﬁeld in half at a beamsplitter and then separately measure the intensities of the two halves with a variable time delay, 𝜏, introduced. For a classical ﬁeld the action of the beamsplitter is I(t) → 1∕2I(t). Thus by ﬁnding the average of the product of the intensities of the two beams and normalizing by their individual averages, we arrive at the above expression for g (2) . Now consider the quantum mechanical case. As we have seen in Section 4.4.4 (see Chapter 5 for more details), the correct quantum mechanical expressions for the outputs of a 50: 50 beamsplitter are √ √ â ′ = 0.5 â + 0.5 b̂ √ √ (4.126) b̂ ′ = 0.5 â − 0.5 b̂ where â represents the beam whose intensity we are measuring, whilst b̂ is initially in the vacuum state. If we now form the ratio of the average of the product of the intensities of the two beams normalized to their individual averages, we obtain g (2) (𝜏) =

̂ ̂ + 𝜏))⟩ ⟨(̂a† (t) + b̂ † (t))(̂a(t) + b(t))(̂ a† (t + 𝜏) − b̂ † (t + 𝜏))(̂a(t + 𝜏) − b(t ̂ ̂ + 𝜏))⟩ ⟨(̂a† (t) + b̂ † (t))(̂a(t) + b(t))⟩⟨(̂ a† (t + 𝜏) + b̂ † (t + 𝜏))(̂a(t + 𝜏) − b(t (4.127)

Using the ﬁeld commutation relations and the fact that the b̂ mode is initially in the vacuum state (see Section 4.2.1), this expression can be signiﬁcantly

131

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4 Quantum Models of Light

simpliﬁed to g (2) (𝜏) =

⟨̂a† (t)̂a† (t + 𝜏)̂a(t)̂a(t + 𝜏)⟩ ⟨̂a† (t)̂a(t)⟩2

(4.128)

where we have also assumed the individual beam averages are not a function of time (the system is ergodic). Equation (4.128) is then the correct quantum mechanical expression for the second-order correlation function. Notice that the ordering of the operators under the averages is such that all creation operators stand to the left whilst all annihilation operators stand to the right. This is referred to as normal ordering and is often a convenient way of expressing results. We can now consider two simple examples. First, suppose the mode â was initially in a coherent state. Using Eq. (4.128) we immediately ﬁnd that for a coherent state, g (2) (0) = 1, as expected for an ideal classical ﬁeld. However, if the mode â is in a number state, then the numerator is immediately zero, as two applications of the annihilation operator to the one photon state produces zero. Thus, for a single photon number state g (2) (0) = 0, in stark contrast to the classical limit. In Chapters 12 and 13, we will see other examples of quantum intensity correlations as functions of diﬀerent degrees of freedom such as polarization. 4.6.2

Quadrature Correlations

We can also consider correlations between the quadrature amplitude ﬂuctuations of spatially separated beams, recorded by separate detectors. These correlations can be quantiﬁed by measuring the quadrature diﬀerence and sum variances deﬁned by 1 ̃ 2 )2 ⟩ ̃ 1 + 𝜹X𝟏 V 1s = ⟨(𝜹X𝟏 2 1 ̃ 2 )2 ⟩ ̃ 1 + 𝜹X𝟐 V 2s = ⟨(𝜹X𝟐 2 for the sum variances of the two quadratures and

(4.129)

1 ̃ 2 )2 ⟩ ̃ 1 − 𝜹X𝟏 V 1d = ⟨(𝜹X𝟏 2 1 ̃ 2 )2 ⟩ ̃ 1 − 𝜹X𝟐 V 2d = ⟨(𝜹X𝟐 (4.130) 2 for the diﬀerence quadrature variances. Here the subscripts label the beams 1 and 2. If the beams are correlated, say, in the amplitude quadrature, we would expect V 1d < V 1s . Conversely, if the beams were anti-correlated, we would ﬁnd V 1d > V 1s . Two coherent beams are found to be uncorrelated, with their noise at the QNL; that is, for two coherent beams, we have V 1d = V 1s = V 2d = V 2s = 1. Beams are said to be quantum correlated if they have sum or diﬀerence quadrature variances that fall below 1. Such correlations cannot be consistently accommodated in a classical theory. In particular, if, for example, V 1d < 1 and V 2s < 1, then the quantum state of the beams is necessarily said to be entangled (see Section 12.3) and no consistent local realistic description of the correlations is possible.

4.6 Quantum Correlations

More generally we might consider the conditional variance of, say, beam 1’s amplitude quadrature, given that we have measured beam 2’s amplitude quadrature. This tells us how accurately we can infer the value of beam 1’s amplitude quadrature given the measurement of beam 2 and is deﬁned as ̃ 1 + g 𝜹X𝟏 ̃ 2 )2 ⟩ V 11|2 = ⟨(𝜹X𝟏

(4.131)

Here g is a gain that is chosen such as to minimize the value of V 11|2 . By calculus we can ﬁnd that the optimum value of g is g=−

̃ 2 𝜹X𝟏 ̃ 1 𝜹X𝟏 ̃ 2 + 𝜹X𝟏 ̃ 1⟩ ⟨𝜹X𝟏 2 V 12

(4.132)

Substituting back into Eq. (4.131), we obtain V 11|2 = V 11 −

̃ 2 𝜹X𝟏 ̃ 2 + 𝜹X𝟏 ̃ 1 ⟩2 ̃ 1 𝜹X𝟏 ⟨𝜹X𝟏 2 V 12

(4.133)

Similarly for the phase quadrature conditional variance, we have V 21|2 = V 21 −

̃ 2 𝜹X𝟐 ̃ 2 + 𝜹X𝟐 ̃ 1 ⟩2 ̃ 1 𝜹X𝟐 ⟨𝜹X𝟐 2 V 22

(4.134)

For coherent states we ﬁnd V 11|2 = V 21|2 = 1, that is, the conditional variance is no better than the single beam variance indicating no correlation. Quantum correlation occurs when a conditional variance falls below 1. A special situation, which can only happen when the beams are entangled, is when both V 11|2 and V 21|2 fall below 1, and this will be discussed in Chapter 13. It is convenient to deﬁne the expectation value of the product of the quadratures of the two beams as the amplitude quadrature correlation coeﬃcient, C1, i.e. 1 ̃ ̃ 2 𝜹X𝟏 ̃ 2 + 𝜹X𝟏 ̃ 1⟩ (4.135) ⟨𝜹X𝟏1 𝜹X𝟏 2 Experimentally we can obtain C1 by measuring the sum and diﬀerence quadrature variances, i.e. 1 C1 = (V 1s − V 1d ) (4.136) 2 In terms of C1 we then have that uncorrelated beams have C1 = 0, correlated beams have C1 > 0, and anti-correlated beams have C1 < 0. Similarly for the phase quadrature correlation coeﬃcient, we have C1 =

C2 = 4.6.3

1 (V 2s − V 2d ) 2

(4.137)

Two-Mode Covariance Matrix

If the state of our two-mode system is Gaussian, then a complete description is provided by the covariance matrix and displacement vector. The covariance matrix of a two-mode system can be written in the following way: ( ) V1 C V1,2 = (4.138) CT V 2

133

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4 Quantum Models of Light

where Vi is the covariance matrix of the ith mode as deﬁned in Eq. (4.59). The matrix C is deﬁned as ) (1 ̂ 1 ΔX𝟏 ̂ 2 + ΔX𝟏 ̂ 2 ΔX𝟏 ̂ 1 ⟩ 1 ⟨ΔX𝟏 ̂ 1 ΔX𝟐 ̂ 2 + ΔX𝟐 ̂ 2 ΔX𝟏 ̂ 1⟩ ⟨ΔX𝟏 2 2 C= 1 ̂ 2 ΔX𝟐 ̂ 1 + ΔX𝟐 ̂ 1 ΔX𝟏 ̂ 2 ⟩ 1 ⟨ΔX𝟐 ̂ 1 ΔX𝟐 ̂ 2 + ΔX𝟐 ̂ 2 ΔX𝟐 ̂ 1⟩ ⟨ΔX𝟏 2 2 (4.139) and describes the correlations between the modes. The matrix CT is the transpose of C. In standard form the two-mode covariance matrix reduces to

V1,2

⎛ V 11 ⎜ ⎜ 0 =⎜ ⎜ C1 ⎜ ⎝ 0

0 V 21 0 C2

C1 0 V 12 0

0 ⎞ ⎟ C2 ⎟ ⎟ 0 ⎟ ⎟ V 22 ⎠

(4.140)

where the subscripts on the variances indicate the mode and C1 and C2 are deﬁned in the previous section. The displacement vector of the two-mode state is ̃ 1⟩ ⎞ ⎛ ⟨𝜹X𝟏 ⎜ ⎟ ̃ 1⟩ ⎟ ⎜ ⟨𝜹X𝟐 d=⎜ (4.141) ̃ 2 ⟩ ⎟⎟ ⎜ ⟨𝜹X𝟏 ⎜ ̃ ⎟ ⎝ ⟨𝜹X𝟐2 ⟩ ⎠ The covariance matrix approach will turn out to be very useful when analysing quantum information protocols with Gaussian states.

4.7 Summary We have introduced quantum optical theory and have discussed various quantum models of light. As is generally true in quantum mechanics, there are diﬀerent pictures and/or representations that can be used in calculations that seem very diﬀerent but lead to the same solutions. Which one to use can depend on the application. Here we have focused mainly on single photon states and coherent states as examples of the most quantum and most classical states, respectively; however the approaches described will be applied more generally in the coming chapters. 4.7.1

The Photon Number Basis

The photon picture is intuitive and obvious for the explanation of emission and detection processes. However, care has to be taken since photons do not behave like classical particles. Their quantum properties have to be taken into account. The description of state evolution in the Schrödinger picture using the photon number basis is optimized for situations with low photon numbers and photon

4.7 Summary

counting. Technology for generating and detecting single photons and photon coincidences has been developed that matches this representation well. The special quantum properties of particles are directly accessible in this regime, and they are used extensively for quantum applications such as cryptography and quantum information processing, described in Chapter 13. We will ﬁnd elsewhere in this guide, Chapter 9, that the photon number representation is very useful in explaining the properties of non-linear processes of CW laser beams, such as second harmonic generation (SHG) or OPO. Mathematically the discrete form of the number basis can have calculational advantages and is particularly useful for numerical solutions. 4.7.2

Quadrature Representations

We can also expand our quantum optical state in terms of the eigenstates of the quadrature observables in various ways, leading to continuous probability and quasi-probability distributions in phase space. The most popular probability distribution is the Q-function, which can be interpreted as the probability density of measuring the state to be in the coherent state |𝛼⟩. The most popular quasi-probability distribution is the Wigner function whose marginals are the probability densities for measuring particular quadrature outcomes, but which can take on negative values for non-Gaussian states. For the special but ubiquitous case of Gaussian states, the ﬁrst and second quadrature moments are suﬃcient to completely characterize the state. It is useful then to tabulate these moments into a displacement vector and covariance matrix that can be used to calculate various properties of the state. This is a compact representation particularly when one is considering multimode states. 4.7.3

Quantum Operators

A useful quantum optical approach is to calculate how the quantum mode operators evolve in the Heisenberg picture. Starting from some ﬁducial initial state, often the vacuum state, the transformation of the annihilation operators of the various modes involved completely characterizes the ﬁnal quantum state of the system. We have described how propagation and detection are modelled in the Heisenberg and Schrödinger pictures and highlighted the diﬀering physical interpretations associated with each. For many practical applications a simpliﬁed linearized model is fully suﬃcient. In this case we can completely describe the beam by the evolution of the operators ̃ ̃ 𝜹X𝟏(Ω) and 𝜹X𝟐(Ω). This model can be used to derive the noise transfer functions for V 1(Ω) and V 2(Ω), describing how the quantum (and classical) noise propa2 ̃ gates through diﬀerent devices, where V (Ω) = ⟨|𝜹X(Ω)| ⟩. In this way we can analyse complete optical systems from source to detector by evaluating the operators at the various locations. This model of transfer functions will be extremely useful to derive the properties of light beams that have deeper quantum properties, such as squeezed light (Chapter 9), entanglement between the quadratures, and CW quantum processing such as optical teleportation (Chapter 13).

135

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If all the interactions are linear or linearizable and our initial state is the vacuum, then all output states will be Gaussian, and so the transfer function approach can conveniently be used to calculate the covariance matrix elements and their evolution. 4.7.4

The Quantum Noise Limit

We have found that the QNL deﬁnes the quietest light that can be produced with a conventional source, such as thermal light from a lamp, or coherent light from a laser. Quantum noise is the way the quantum nature of light manifests itself in CW laser beams and in individual pulses of laser light. However, there is more to come, and other types of light states are possible. Within the quantum model it is possible to deﬁne light that has reduced ﬂuctuations in one of its quadratures. Such squeezed states of light and their generation are the topic of Chapter 9. We ﬁnd that for all devices that are sensitive enough to detect quantum noise, and these are getting more and more common due to the advances in technology, it is important to understand the quantum properties of the light. They will inﬂuence the performance of the devices. In many cases the quantum eﬀects will set a fundamental limit to the performance, the QNL. We will ﬁnd tricks to avoid these limitations. In other cases the performance will be quite diﬀerent near the QNL compared with the classical regime well above the QNL. Finally we will show applications and devices that are only possible because of the existence of these quantum eﬀects.

References 1 Goldstein, H. (1980). Classical Mechanics. Singapore: Addison-Wesley. 2 Power, E.A. (1964). Introductory Quantum Electrodynamics. Longmans. 3 Vogel, W. and Welsch, D.-G. (1994). Lectures on Quantum Optics. Berlin:

Akademie Verlag. 4 Casimir, H.B.G. and Polder, D. (1948). The inﬂuence of retardation on the

London - van der Waals forces. Phys. Rev. 73: 360. 5 Kitchener, J.A. and Prosser, A.P. (1957). Direct measurement of the

long-range van der Waals forces. Proc. R. Soc. A 242: 403. 6 Glauber, R.J. (1962). Coherent and incoherent states of the radiation ﬁeld.

Phys. Rev. 131: 2766. 7 Walls, D.F. and Milburn, G.J. (1994). Quantum Optics. Springer-Verlag. 8 Pegg, D.T. and Barnett, S.M. (1988). Unitary phase operator in quantum

mechanics. Europhys. Lett. 6: 483. 9 Freyberger, M., Heni, M., and Schleich, W.P. (1995). Two mode quantum

phase. Quantum Semiclass. Opt. 7: 187. 10 Caves, C.M. (1981). Quantum-mechanical noise in an interferometer. Phys.

Rev. D 23: 1693. https:doi.org/10.1103/PhysRevD.23.1693. 11 Weedbrook, C., Pirandola, S., Garca-Patron, R. et al. (2012). Gaussian quan-

tum information. Rev. Mod. Phys. 84: 621. 12 Fabre, C. (1992). Squeezed states of light. Phys. Rep. 219: 215.

Further Reading

13 Scully, M.O. and Zubairy, M.S. (1997). Quantum Optics. Cambridge Univer-

sity Press. 14 Gardiner, C.W. and Zoller, P. (2000). Quantum Noise, 2e. Berlin:

Springer-Verlag.

Further Reading Louisell, W.H. (1973). Quantum Statistical Properties of Radiation. New York: Wiley. Mandel, L. and Wolf, E. (1996). Optical Coherence and Quantum Optics. Cambridge University Press. Schleich, W.P. (2001). Quantum Optics in Phase Space. Wiley-VCH. Grynberg, G., Aspect, A., and Fabre, C. (2010). Introduction to Quantum Optics. Cambridge University Press.

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5 Basic Optical Components Real experiments consist of many components – at ﬁrst sight, any quantum optics experiment looks like a bewildering array of lenses, mirrors, etc. ﬁlling an optical table in a more or less random fashion. However, they all serve a purpose, and their individual characteristics are described in this chapter. In quantum optics experiments we consider light as laser beams or modes of radiation. Using optical components, we want to change the parameters of the mode, such as beam size, direction, polarization, or frequency. This means we wish to transform one mode into another mode. This can be achieved by a whole range of optical components, all of which are linear, which means that their output is proportional to their input, independent of the intensity of the beam. One of the simplest optical processes is attenuation by a beamsplitter, and we will see how it aﬀects both the intensity as well as the ﬂuctuations of the light. Whilst the ﬁrst is simple, that is, the intensity is reduced, the latter requires more detailed modelling. The complementary process, gain, is similar but requires much more elaborate equipment and theory, which will be covered in Chapter 6, describing lasers. Combining several mirrors allows us to build interferometers and cavities. Several other optical components are used to manipulate the light: spatial parameters can be controlled with lenses and curved mirrors; polarization can be inﬂuenced with various crystals and surface eﬀects; and frequency can be shifted using electro-optic modulators (EOM). In this chapter we will ﬁrst describe the eﬀect of the various components on a single spatial and temporal mode. The quantum description in Chapter 4 will be used to demonstrate and model the distinctive quantum eﬀects. The concept of linear systems, beamsplitters, interferometers, and resonators is immensely useful and can be used to describe many devices. Any machine that is based on the propagation of waves and modes will contain these components. This could be as simple as the unavoidable losses that take power out of the mode, any coherent splitting and recombination that leads to interference, and any resonators that have tuned transmission functions. Finally, we summarize the many practical ways of building beamsplitters, interferometers, and resonators that have been developed to create devices that use the principles of quantum physics. These go well beyond optics.

A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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5.1 Beamsplitters One of the most common and simplest optical components is a mirror. This is made either of thin layers of metal or multilayer dielectric ﬁlms deposited on glass. A mirror has two output beams for every input beam, one transmitted and the other reﬂected. Thus it is referred to as a beamsplitter. Let us analyse the properties of a beamsplitter. We can use the diﬀerent models described in Chapter 4; they will obviously lead to the same conclusion but will provide diﬀerent interpretations of the eﬀect of a beamsplitter. 5.1.1

Classical Description of a Beamsplitter

Consider a beamsplitter as shown in Figure 5.1 with an incoming beam described classically by the complex amplitude 𝛼in exp(i𝜙in ). There are two waves leaving the beamsplitter: the transmitted beam, 𝛼t exp(i𝜙t ), and the reﬂected beam, 𝛼r exp(i𝜙r ). We describe the properties of the beamsplitter by the intensity reﬂection coeﬃcient 𝜖: |𝛼r |2 = 𝜖|𝛼in |2 |𝛼t |2 = (1 − 𝜖)|𝛼in |2

(5.1)

The special case 𝜖 = 12 describes a balanced 50/50 beamsplitter. However, there is also a second input axis deﬁned such that a wave 𝛼u exp(i𝜙u ) would produce at the output a wave in the same place and propagating in the same direction as 𝛼t and 𝛼r . For a simple beamsplitter only 𝛼in exists and 𝛼u = 0. In other applications two waves 𝛼in and 𝛼u are superimposed and, in this way, are made to interfere. The optical frequencies of all input and output beams are identical. All the frequencies, including all the sidebands, are preserved. Assuming it has no losses, the mirror will preserve energy. This means |𝛼in |2 + |𝛼u |2 = |𝛼r |2 + |𝛼t |2

(5.2)

The intensity response to the other input beam can be written similarly to Eq. (5.1). The response for amplitude transmission and reﬂection is not that simple, since the eﬀect of the beamsplitter on the phase of the light has to be considered. Let us arbitrarily set the phase of the two incoming waves to zero, which means 𝜙in = 𝜙u = 0. The relative phase between the two output beams is of interest: √ √ 𝛼t = (1 − 𝜖) 𝛼in exp(i𝜙in,t ) + 𝜖 𝛼u exp(i𝜙u,t ) √ √ 𝛼r = 𝜖 𝛼in exp(i𝜙in,r ) + (1 − 𝜖) 𝛼u exp(i𝜙u,r ) Φu

αu

ε

αin

Figure 5.1 The classical beamsplitter.

αt

Φt

Φin Φr

αr

5.1 Beamsplitters

There is not suﬃcient information to determine all four relative phase terms 𝜙in , t; 𝜙n , r; 𝜙u,t ; and 𝜙u,r directly. This could be done by solving the electromagnetic boundary conditions explicitly. The solutions, known as Fresnel equations, can be found in textbooks on physical optics. These equations are diﬀerent for metal and dielectric mirrors, and they depend on the polarization of the input wave. However, we can apply energy conservation and ﬁnd a general condition for the four waves: |𝛼in |2 + |𝛼u |2 = |𝛼t |2 + |𝛼r |2 = (1 − 𝜖)|𝛼in |2 + 𝜖|𝛼u |2 + 𝜖|𝛼in |2 + (1 − 𝜖)|𝛼u |2 √ + 𝜖(1 − 𝜖) 𝛼u∗ 𝛼in exp(i(𝜙in,t − 𝜙u,t )) + c.c. √ + 𝜖(1 − 𝜖) 𝛼in 𝛼u∗ exp(i(𝜙in,r − 𝜙u,r )) + c.c.

(5.3)

which leads to exp(i(𝜙in,t − 𝜙u,t )) + exp(i(𝜙in,r − 𝜙u,r )) = 0

(5.4)

The usual choice is to set three of the relative phases to zero: 𝜙in,t = 𝜙in,r = 𝜙u,r = 0, and 𝜙u,t = π, which means √ √ 𝛼r = 𝜖𝛼in + 1 − 𝜖𝛼u √ √ 𝛼t = 1 − 𝜖𝛼in − 𝜖𝛼u (5.5) The normal explanation is that one of the reﬂected waves has a phase shift of 180∘ in respect to all other waves. For a simple dielectric layer, this would be the reﬂection of the layer with the higher refractive index. But this is only one possible choice; various mirrors have diﬀerent properties. All mirrors can be represented by Eq. (5.5). The above analysis had been discussed as early as 1852 by Stokes [1, 2] and leads to a more general matrix format: ( ) ( ) ( )( ) 𝛼r 𝛼in Mr,in Mr,u 𝛼in =M = (5.6) 𝛼t 𝛼u Mt,in Mt,u 𝛼u If there is no loss, the matrix M has to be a unitary transformation, which means the following conditions apply: |Mr,in | = |Mt,u | and |Mt,in | = |Mr,u | |Mr,in |2 + |Mt,in |2 = 1 Mr,in M∗t,in + Mr,u M∗t,u = 0 and our choice of the beamsplitter matrix is ( √ ) √ 𝜖 1 − 𝜖 √ √ 1−𝜖 − 𝜖

(5.7)

We should note that any real beamsplitter will introduce losses due to absorption and internal scattering. This can be described by a loss term (1 − A) such that |𝛼t |2 + |𝛼r |2 = (1 − A)(|𝛼in |2 + |𝛼u |2 )

141

142

5 Basic Optical Components

In most cases the transmitted and the reﬂected waves are aﬀected by the same amount. For metal mirrors typical values are A = 0.05; for dielectric mirrors, A of 10−2 –10−4 is common, but losses as low as 5 × 10−6 have been achieved. 5.1.1.1

Polarization Properties of Beamsplitters

In some cases we might want a beamsplitter that acts equally on all polarizations. In the other extreme case one could imagine a polarizing beamsplitter (PBS) that reﬂects only one polarization and transmits the orthogonal polarization. Both these cases can be approximately achieved with diﬀerent types of beamsplitters. Another possibility would be a partially polarizing beamsplitter (PPBS) that reﬂects all of one polarization but only partially transmits the orthogonal polarization. In general, the polarization response is not trivial, and the reﬂectivity 𝜖 is polarization dependent. For this purpose we deﬁne two orthogonal polarization states: S and P linear polarization. For P polarization the electric ﬁeld vector is parallel to the surface of the mirror. This is out of the plane of the drawing in Figure 5.1. For S polarization, the electric ﬁeld is perpendicular (or senkrecht in German) to the P polarization. This is in the plane of Figure 5.1. The S polarization contains a component that is perpendicular to the surface of the mirror, and the boundary condition on the mirror surface for the parallel and perpendicular ﬁelds is diﬀerent. Consequently the S and P components of the light are reﬂected diﬀerently, both in amplitude and in phase, and in general 𝜖P and 𝜖S differ from each other. They depend on the way the mirror is manufactured and changes with the input angle and wavelength. For a simple dielectric material, such as uncoated glass, the problem can be solved easily, resulting in the Fresnel reﬂection coeﬃcients [3]. For real multilayer mirrors such calculations are rather involved. One special solution is 𝜖S = 0 for a dielectric surface under Brewster’s angle. Stacks of tilted glass plates, near Brewster’s angle, are used as polarizers. As long as we consider only light linearly polarized in the S or P direction, the polarization is maintained, and the relative phase shift between the S and the P wave is of no concern. For any other light the polarization is modiﬁed by the mirror. The mirror acts like a combination of a non-PBS and a wave plate. In summary, to describe a mirror it is not suﬃcient to state just the reﬂectivity 𝜖; the following properties have to be speciﬁed as well: input angle, wavelength, polarization, and losses. Finally, the following practical hints are useful for the selection of mirrors: there is a choice of materials for mirrors. For non-PBS, that means ideally 𝜖P = 𝜖S . Metal surfaces are very useful, and 𝜖S = 𝜖P = 0.48 can be readily achieved, but the losses are considerable. With dielectric mirrors the diﬀerence between 𝜖P and 𝜖S is usually signiﬁcant, unless they are speciﬁcally designed for the speciﬁc application. Dielectric mirrors should preferably be used with linearly polarized light aligned in the S or P direction. The losses can be small ( 1. The second beamsplitter recombines the beams. Figure 5.5 The components of a classical interferometer.

M1

ε1

α|

αin

M3

M2

M4

α||

∆ϕ

α1

ε4 α2

153

154

5 Basic Optical Components

5.2.1

Classical Description of an Interferometer

In the classical wave model, a balanced interferometer with 50/50 beamsplitters can be described by the following evolution of the complex amplitudes: √ 𝛼I (t) = 1∕2 𝛼in (t) √ 𝛼II (t) = 1∕2 𝛼in (t) √ √ 𝛼1 (t) = 1∕2 𝛼I (t) exp(iΔ𝜙) − 1∕2 𝛼II (t) √ √ (5.25) 𝛼2 (t) = 1∕2 𝛼I (t) exp(iΔ𝜙) + 1∕2 𝛼II (t) √ √ where the amplitude reﬂectivity 𝜖 has been set to 1∕2. The calculation can easily be generalized to other values of 𝜖. Here the convention of Eq. (5.7) is used. The minus sign in the third equation represents the 180∘ phase shift imposed by one, but not the other, of the reﬂections. Strictly speaking, this set of equations applies only to plane waves, but it can also be applied to spatial Gaussian beams. The output intensity of the interferometer is given by I1 (Δ𝜙) = Iin cos2 (Δ𝜙∕2) I2 (Δ𝜙) = Iin sin2 (Δ𝜙∕2)

(5.26)

and I1 + I2 = Iin . There are many applications for interferometers: they can be used to measure physical processes that aﬀect the relative phase diﬀerence Δ𝜙. One should think about an interferometer as a device that transforms changes in the optical phase into changes of intensity. Equation (5.26) holds for any interferometer that is based on the interference of two waves – independent on the layout and the details of the apparatus. This includes Michelson interferometers, Lloyd’s mirrors, ﬁbre-optic interferometers, polarimeters, etc. The main task in describing a speciﬁc conﬁguration is to evaluate Δ𝜙; once this is done the intensity distribution can be directly evaluated from Eq. (5.26). Under practical conditions the interference will not be perfect. The overlap of the two wave fronts will not be perfect, or the two waves might not be completely coherent. The spatial or temporal coherence between the two waves, which means the correlation between 𝛼I and 𝛼II , will determine the strength of the interference. In the case of perfect correlation, the result in Eq. (5.26) is valid. In the case of no correlation between 𝛼II and 𝛼I , or if they are incoherent, the resulting intensity will be I1 (Δ𝜙) = (𝜖4 |𝛼II |2 + t4 |𝛼I |2 )hΩ = constant I2 (Δ𝜙) = (t4 |𝛼II |2 + 𝜖4 |𝛼I |2 )hΩ = constant I1 (Δ𝜙) + I2 (Δ𝜙) = Iin

(5.27)

where we are now considering an interferometer that is not necessarily balanced. Thus the visibility, or contrast, of the interference system can be used to measure the degree of coherence. We can measure the visibility VIS by comparing the extrema Imax and Imin of the intensity in one output beam and use the deﬁnition VIS =

Imax − Imin Imax + Imin

(5.28)

5.2 Interferometers

We can then compare the measured value of VIS with the best possible value VISperfect that we would have achieved for completely correlated beams, where 𝛼II and 𝛼I completely interfere: √ 2 III II (𝛼II + 𝛼I )2 − (𝛼II − 𝛼I )2 VISperfect = = (5.29) (𝛼II + 𝛼I )2 + (𝛼II − 𝛼I )2 III + II VISperfect can be determined separately by individually blocking the beams and by recording the intensity of the beams independently. Combining these four measurements of Imax , Imin , III , and II in Eq. (5.29) yields the value =

VIS VISperfect

(5.30)

This is the value for the coherence of the beams, assuming a perfectly aligned interferometer, which is also the value of the correlation between the beams. This measurement cannot produce easily the sign of the correlation since correlated and anti-correlated lights produce fringe systems of identical visibility – only shifted with respect to each other. A knowledge of the relative phase between 𝛼II and 𝛼I at the maximum interference would allow us to determine the sign. Please note that this technique can be used for any situation; the intensity of the interfering beams do not have to be balanced. However, a simple, direct evaluation is only possible for the balanced situation |III | = |II | since in this case VISperfect = 1 and = VIS. In all cases we have = |C(𝛼II , 𝛼I )|. One application of this procedure is to determine the degree of spatial coherence of a beam of light using a double slit experiment. In this particular case 𝛼II and 𝛼I are parts of the same wave but from spatially diﬀerent sections of the beam. The phase diﬀerence Δ𝜙 is a purely geometrical term that is given by Δ𝜙 = d sin(Θ) where d is the separation of the two slits and Θ the angle of detection measured from the axis normal to the plane of the two slits. If the detection occurs at a distance much larger than the slit separation, the fringes are intensity minima and maxima that are parallel and equally spaced. The visibility of these fringes is a quantitative measure of the spatial coherence of the light. Similarly, a measurement of the visibility can be used to ascertain the correlation between two electronic signals, for example, the currents from two photon detectors. By adding and subtracting the signals, or by adding them with a variable time delay, interference signals can be generated. The visibility, and thus the correlation coefﬁcient, can be determined. In Section 7.3.4 this application is discussed in detail. 5.2.2

Quantum Model of the Interferometer

We now consider a quantum model of the interferometer. As in the classical case, we will assume that the instrument is perfectly aligned and that the spatial overlap of the two beams at the recombining beamsplitter is perfect. The eﬀect of spatial and temporal mode mismatch will be discussed in Section 5.2.6 (Figure 5.6). The interferometer is illuminated by modes with the operators â in and â u ; inside we have the operators â I and â II . After recombination we have the two output operators â 1 and â 2 , which are detected individually. As with the classical model, the case of a Mach–Zehnder interferometer with a phase diﬀerence Δ𝜙 is treated

155

156

5 Basic Optical Components

Avac Ain

ε1

Figure 5.6 Description of an interferometer in the quantum model.

A|

A||

A1

∆ϕ

ε4 A2

ﬁrst. The results can be generalized to any other interferometer and to applications where Δ𝜙 is a function of time or space. By canonical quantization the internal operators are linked to the input operators via √ √ â I = 1 − 𝜖1 â in − 𝜖1 â u √ √ â II = 𝜖1 â in + 1 − 𝜖1 â u (5.31) The output operators are linked to the internal states via √ √ â 1 = exp(iΔ𝜙) 1 − 𝜖4 â II − 𝜖4 â I √ √ â 2 = exp(iΔ𝜙) 𝜖4 â II + 1 − 𝜖4 â I

(5.32)

For the balanced interferometer, we can set 𝜖1 = 𝜖4 = 12 , and the equations can be combined to give â 1 = exp(iΔ𝜙∕2)(cos(Δ𝜙∕2) â in + i sin(Δ𝜙∕2) â u ) â 2 = exp(iΔ𝜙∕2)(i sin(Δ𝜙∕2) â in + cos(Δ𝜙∕2) â u ) 5.2.3

(5.33)

The Single-Photon Interferometer

The output state of a single-photon incident on a balanced interferometer can be calculated by inverting Eq. (5.33). For the input state |1⟩in |0⟩u = â †in |0⟩in |0⟩u , we obtain the output state cos(Δ𝜙∕2) |1⟩1 |0⟩2 + i sin(Δ𝜙∕2) |0⟩1 |1⟩2

(5.34)

Depending on the phase shift in the interferometer, the photons can be made to exit from one or other output ports with certainty or be in a superposition of both ports. This eﬀect is clearly a wave eﬀect and cannot be understood by thinking of the beamsplitters as random photon partitioners. The behaviour of the photons depends on path length diﬀerences between the arms of the interferometer on the order of fractions of the optical wavelength. 5.2.4

Transfer of Intensity Noise Through the Interferometer

We now consider the intensity noise observed at the interferometer output when a bright beam is incident. In order to determine the photon ﬂux, for example, at

5.2 Interferometers

output A2 , we have to evaluate the operator n̂ 2 = â †2 â 2 . However, for a bright beam we can consider this as a classical part 𝛼 for the average values and the operator 𝛿 â for the ﬂuctuations, e.g. â 2 = 𝛼2 + 𝛿 â 2 . To calculate the average intensities, we only have to consider the terms in 𝛼; all terms with 𝛿 â make no contribution, and we obtain N 1 = N in cos2 (Δ𝜙∕2) N 2 = N in sin2 (Δ𝜙∕2)

(5.35)

Not surprisingly, this is the same result as for the classical calculation. The interferometer is like a variable beamsplitter that distributes the energy between the two output ports. The ratio of the distribution can be adjusted by the diﬀerential phase Δ𝜙. Now we come to the transfer function for the ﬂuctuations. The direct calculation of the ﬂuctuations would require the evaluation of V N1 = ⟨𝛿 â †1 𝛿 â 1 𝛿 â †1 𝛿 â 1 ⟩ − ⟨𝛿 â †1 𝛿 â 1 ⟩2 . This would be very laborious, but for bright beams we can linearize (see ̃ 1 (Ω)|2 ⟩. Consequently, turning Section 4.5) and use V N1 = 𝛼12 V 11 (Ω) = 𝛼12 ⟨|𝜹X1 Eq. (5.33) into a transfer function for the amplitude quadratures, the result is V 11 (Ω) = cos2 (Δ𝜙∕2) V 1in (Ω) + sin2 (Δ𝜙∕2) V 2u (Ω) V 12 (Ω) = sin2 (Δ𝜙∕2) V 1in (Ω) + cos2 (Δ𝜙∕2) V 2u (Ω)

(5.36)

Again, this is not a surprising result. The ﬂuctuations of the two outputs are proportional to the average photon ﬂuxes. The size of the ﬂuctuations reﬂects the division of the ﬂux between the two outputs. However, it is important to note that the ﬂuctuations of the phase quadrature of the unused port enter these calculations, not the phase ﬂuctuations of the input laser beam, which cancel completely. It was ﬁrst pointed out by C.M. Caves [7] that the noise in the output can be interpreted as a beat between the coherent input beam and the vacuum state with V 2u = 1. In this balanced interferometer the phase noise of the input beam cancels; it does not contribute. If we have only one input beam, we can make the assumptions that (i) the second port is not used, but the other input beam is the vacuum mode and V 1u = 1, and (ii) the photon ﬂux of the input beam is dominant (N in ≫ 1). Now we obtain V 11 (Ω) = cos2 (Δ𝜙∕2) V 1in (Ω) + sin2 (Δ𝜙∕2) V 12 (Ω) = sin2 (Δ𝜙∕2) V 1in (Ω) + cos2 (Δ𝜙∕2)

(5.37)

This is identical to the transfer function of a single beam with a beamsplitter of variable reﬂectivity, determined by the phase shift diﬀerence Δ𝜙 in the interferometer. Again, the spectrum of the ﬂuctuations remains unchanged.

5.2.5

Sensitivity Limit of an Interferometer

How should we operate the interferometer to get the best sensitivity? What is the limit in sensitivity? It is necessary to determine the signal that can be created by modulating the phase diﬀerence. For a small change 𝛿𝜙 of the phase

157

5 Basic Optical Components

shift (𝛿𝜙 ≪ π), the change in the optical ﬂux at the detector is equal to the ﬁrst derivative of the output photon ﬂux in regard to the phase multiplied with 𝛿𝜙: 𝛿N2 = 𝛿𝜙

d(N2 ) 𝛿𝜙 1 = 𝛿𝜙2 cos(Δ𝜙∕2) sin(Δ𝜙∕2) Nin ≈ sin(Δ𝜙)Nin d𝜙 2 2 (5.38)

An AC detector measures the signal power psig that is proportional to the square of the change in optical power (see Section 2.1) and includes 𝜏, the inverse of the detection bandwidth: ( )2 𝛿𝜙 (5.39) sin(Δ𝜙) Nin 𝜏 psig = 2 As illustrated in Figure 5.7, the signal is largest when the change in power is most rapid, that is, at Δ𝜙 = π∕2, in the middle between the maximum and minimum of a fringe. The signal can be compared with the variance of the photon number or the quantum noise on the detector: pqnoise = V N2 = sin2 (Δ𝜙∕2) Nin 𝜏 and we can evaluate the SNR: sin2 (Δ𝜙) 𝛿𝜙2 SNR = Nin 𝜏 2 4 sin (Δ𝜙∕2)

(5.40)

(5.41)

which has the largest value when Δ𝜙 = 0, 2π, where the output optical power is at a minimum. In the jargon of the trade, we operate the interferometer at a dark fringe. This is a very diﬀerent operating point than one would use for a classical interferometer with intensity-independent noise. The reason for this result is that whilst both signal and noise disappear at Δ𝜙 = 0, the quantum noise disappears more rapidly than the signal and the ratio of the two reaches a maximum (see the SNR trace (c) in Figure 5.7). Note that this is a somewhat academic result. In practice the perfect dark fringe is not an optimum since there are always some additional intensity-independent noise terms such as electronic noise. The optimum will be at a point where the quantum noise component in the current ﬂuctuations is larger than the additional 20

Linear units (QNL)

158

15

(b) Quantum noise

(c) SNR

10

(a) Signal

5

0 0

π/2

π

3π/2

Phase offset (∆ϕ)

2π

Figure 5.7 Results for an interferometer that is limited by quantum noise. Shown as a function of the detuning Δ𝜙 are (a) the optical power that also corresponds to a signal with ﬁxed modulation depth, (b) the quantum noise, and (c) the signal-to-noise ratio.

5.2 Interferometers

noise. Details are given in Section 10.3. We deﬁne a QNL instrument as a device where the quantum noise is the dominant noise source. The minimum detectable phase change 𝛿𝜙QNL in a QNL interferometer corresponds to an SNR of 1. Near the optimum dark fringe, we can approximate sin2 (Δ𝜙) (Δ𝜙)2 =4 ≈ sin2 (Δ𝜙∕2) (Δ𝜙∕2)2 1 𝛿𝜙QNL = √ Nin 𝜏

(5.42)

for a coherent input state. This is the standard QNL for interferometry. It is the smallest achievable measurement of a change in phase using coherent light. A comment on units is that 𝛿𝜙QNL is measured in rad. Nin is the number of photons travelling from the interferometer to the front face of a detector in one measurement interval. Thus Nin is a number with no dimensions. The value of 𝛿𝜙QNL is not ﬁxed; it depends on the integration time 𝜏. If we spend more time on the measurement, integrate for longer, the value would improve. But the integration time cannot be longer than the period of the signal. As an example the QNL for an interferometer operated with 1 (mW) of light at 𝜆 = 1(μm) and a detection interval of 1 (ms) is 4.4 × 10−7 (rad). This quantum description is not complete. Equation (5.42) shows that the eﬀect of the quantum noise decreases with more optical power. It seems that it would be possible to reach any arbitrary precision; this must be impossible. It is one of the deep truths of quantum mechanics that we need to complement (using an expression coined by Niels Bohr) the concept of quantum noise of the light with a description of the measurement process. We have to consider both the system we investigate and the process of measurement. In straightforward analogy to the Gedanken experiment of Heisenberg’s microscope, there is a back eﬀect of the measurement process onto the object we measure. In an interferometer we investigate the position of the mirrors, which deﬁnes the optical path length of the interferometer. The measurement process is the interference of the two reﬂected modes of light. The reﬂection of a beam of light with power Preﬂ exerts a force on the mirrors: 2Preﬂ c The ﬂuctuations of this force are due to the quantum noise ﬂuctuations in the photon ﬂux, as discussed above, and the amplitude spectral density of the force will be 2 ℏ Preﬂ 2 (Ω) = (5.43) 𝛿FRP c𝜆 which is independent of the detection frequency. Such a stochastic force will produce ﬂuctuations in the mirror position and results in a position noise spectrum [8]: √ ℏPreﬂ 1 1 2 2 𝛿F (Ω) = (5.44) 𝛿x (Ω) = m(2πΩ)2 RP mΩ2 8π3 c𝜆 FRP =

159

160

5 Basic Optical Components

where m is the mass of the mirror. This displacement occurs in both arms of the interferometer but with opposite sign. The total eﬀect gives a spectrum of the angular ﬂuctuations: ℏ √ Nin 𝜏 𝛿𝜙RP (Ω) = mΩ2 𝜆2 This noise contribution increases with the input power. It also rises at lower detection frequencies. The total noise will be 𝛿𝜙2QM = 𝛿𝜙2QNL + 𝛿𝜙2RP

(5.45)

and the balance between the two terms depends on detection frequency and the power level chosen. Only at low frequencies and high powers does the radiation pressure dominate. At higher frequencies and lower powers, the system is limited by intensity quantum noise alone (QNL). Very special arrangements, such as high-power lasers and very lightweight mirrors, are required to make the light pressure visible. An example for this balance is shown in Figure 10.10 in Chapter 10. For each frequency there is a minimum noise level, also called the standard quantum limit (SQL). This limit can be reached for an optimum power of Popt = πc𝜆mΩ2 , which is typically quite large. For example, with m = 1 kg, 𝜆 = 1 μm, and Ω = 100 Hz, one obtains Popt = 9.4 MW. At the optimum condition the system reaches its limit in sensitivity as dictated by the quantum physics of the measurement process. The QNL described in Eq. (5.45) cannot be avoided with any measurement process based on normal coherent light, but it can be surpassed by using correlated photon states and quantum non-demolition techniques that are described in Chapters 9–11.

5.2.6

Eﬀect of Mode Mismatch on an Interferometer

So far we have assumed that the modes aI and aII that meet at the second beamsplitter in the interferometer are perfectly mode-matched. In general this will not be true, leading to the visibility of the interferometer being reduced. There are many ways in which mode mismatch can occur, for example, the transverse modes might not perfectly overlap, either because they are displaced from each other or because their spot sizes are unequal; their polarizations might not perfectly match due to some birefringence in one arm; or their longitudinal mode functions might not perfectly overlap. Qualitatively, all these diﬀerent possibilities lead to similar eﬀects. As a particular example we will consider the latter case for a single photon through an interferometer. We consider a single-photon input state as in Section 5.2.3 but now show its mode decomposition explicitly using the techniques of Section 4.4.1: â †in (ti ) |0⟩in |0⟩u =

∫

̃ † |0⟩in |0⟩u d𝜔e−i𝜔ti F(𝜔)A in,𝜔

(5.46)

where ti is some initial time. We model propagation through an interferometer as shown in Figure 5.6 but now allow for an arbitrary time delay, td , in the lower arm.

5.2 Interferometers

Transforming the frequency operators and assuming a balanced interferometer, we obtain the output state |𝜙⟩1,2 =

∫

1 ̃ † + ((e−i𝜔(t+td ) + e−i𝜔t )A ̃ † ) |0⟩1 |0⟩2 d𝜔F(𝜔) ((e−i𝜔(t+td ) − e−i𝜔t )A 1,𝜔 2,𝜔 2 (5.47)

This is a spectrally entangled state. To evaluate the visibility of the interferometer, we need to introduce a particular detection model. We model broadband photon counting of the output modes of the interferometer by the operator n̂ i =

∫

̃† A ̃ d𝜔 A i,𝜔 i,𝜔

(5.48)

̃ represent ̃† A Here it is assumed that the although the individual operators A i,𝜔 i,𝜔 number detection with strong frequency resolution, the output photocurrent of the detector averages over all frequencies. The photon counting rates are then given by 𝜔t 1 d𝜔|F(𝜔)|2 |1 − ei𝜔td |2 = d𝜔|F(𝜔)|2 sin2 d ∫ ∫ 2 2 𝜔 t 1 d𝜔|F(𝜔)|2 |1 + ei𝜔td |2 = d𝜔|F(𝜔)|2 cos2 d ⟨n̂ 2 ⟩ = ∫ ∫ 2 2 ⟨n̂ 1 ⟩ =

(5.49)

We can identify two limits to this expression. If 𝜔 td is suﬃciently small, the sine and cosine terms will be approximately constant over the width of the mode function and so can be taken outside the integrals giving 𝜔o td 𝜔t d𝜔|F(𝜔)|2 = sin2 o d 2 ∫ 2 𝜔 t 𝜔 t 2 o d ⟨n̂ 2 ⟩𝜔tdlim d𝜔|F(𝜔)|2 = cos2 o d →0 = cos 2 ∫ 2

2 ⟨n̂ 1 ⟩𝜔tdlim →0 = sin

(5.50)

where 𝜔o is the central frequency of the single-photon pulse. In this case the time delay is so small that it only imparts a phase shift. The result agrees with those of Section 5.2.3. In this limit there is perfect mode matching and the visibility is unity. Alternatively, if 𝜔td is suﬃciently large, the sine and cosine terms will oscillate very rapidly. As a result, over intervals for which the mode function is approximately constant, both the sine and cosine terms will average to one-half. We obtain 1 1 d𝜔|F(𝜔)|2 = 2∫ 2 1 1 ⟨n̂ 2 ⟩𝜔tdlim d𝜔|F(𝜔)|2 = →∞ = 2∫ 2 ⟨n̂ 1 ⟩𝜔tdlim →∞ =

(5.51)

Now the time delay in the lower arm is so long that there is no overlap at the ﬁnal beamsplitter between pulses travelling along the upper arm and the lower arm. In this limit there is complete mode mismatch, and the visibility is zero. For delay times in between these limits, there will be partial mode match, and the visibility will be between zero and one.

161

162

5 Basic Optical Components

From the point of view of the broadband detection model (Eq. (5.48)), it is clear that the visibility is reduced because, although individual frequency components still display interference, the fringes shift their position with frequency and hence are washed out when we average over many frequencies. Interestingly, Eq. (5.48) can also arise from a completely diﬀerent detection model in which our individual detection operators have excellent temporal resolution, but we then integrate the photocurrent over time. Speciﬁcally 1 ̂† A ̂ d𝜏 A i,𝜏 i,𝜏 2π ∫ ′ 1 ̃† ̃ i,𝜔′ d𝜔′ ei𝜔 𝜏 A = d𝜏 d𝜔e−i𝜔𝜏 A i,𝜔 ∫ ∫ 2π ∫

n̂ i =

=

∫

̃† A ̃ d𝜔 A i,𝜔 i,𝜔

(5.52)

̂† A ̂ represent number Now it is assumed that the individual operators A i,𝜏 i,𝜏 detection with strong temporal resolution. The expansion in moving from lines one to two uses the Fourier transform relation between the time and frequency domains. The simpliﬁcation in moving from the second to third line results from recognizing that the integral over the detection time, 𝜏, creates the Dirac delta function 𝛿(𝜔′ − 𝜔). We arrive at the same measurement operator as for the broadband case, and so the visibility also decreases with increasing time delay for this detection model. However, now the reduction of visibility can be attributed to which-path information in the interferometer. In principle the time-resolved detectors can resolve the arrival times of the photons at the two outputs with high precision. When the time delay is very small, the arrival time of the photons reveals no information about which path they took through the interferometer. However, when the time delay is large, photons that take the lower arm will arrive noticeably later than those that take the upper arm, thus revealing the path taken through the interferometer and erasing the interference. Even though we integrate over diﬀerent times and hence do not reveal the arrival times to the experimenter, the in-principle existence of which-path information erases the interference.

5.3 Optical Cavities One component of widespread use in quantum optics experiments is the optical cavity. In the simplest case this consists of two mirrors that are aligned on one optical axis such that they exactly retro-reﬂect the incident light beam. In the more general case, it is a group of mirrors aligned in such a way that the beam of light is reﬂected in a closed path, such that after one round trip, it interferes perfectly with the incident wave. The curvature of the mirrors is chosen such that they match the curvature of the internal wave fronts at all reﬂection points. The optical path length for one round trip through the cavity is the perimeter p, which takes into account all the refractive indices inside the cavity. The light is reﬂected

5.3 Optical Cavities

(a)

(b)

Figure 5.8 Schematic diagrams of linear and ring cavities.

many times around the cavity. Each round trip generates one partial wave; all these waves will interfere (Figure 5.8). For the case that p is an integer multiple of the wavelength of the incident light, a strong ﬁeld will be built up inside the cavity due to constructive interference; the cavity is on resonance. The resonance frequencies satisfy the condition 𝜈cav =

jc ; p

j = integer

(5.53)

Note that for a linear cavity the optical length is p = 2nd12 , where d12 is the physical separation between the two mirrors and n is the refractive index. If either the cavity length or the wavelength of the light is scanned, the cavity will go in and out of resonance. The cavity resonance frequencies are equally spaced, and the diﬀerence between two adjacent resonance frequencies is called the free spectral range (FSR): FSR = c∕p

(5.54)

The magnitude of the enhancement of the ﬁeld and the width of the resonance depend on the reﬂectivities of the mirror and losses inside the cavity. Cavities are used for a number of purposes: the laser itself is a cavity with an active medium inside that provides the gain. The cavity provides the optical feedback required to achieve oscillation, hence the name laser resonator. The cavity determines the spatial properties of the laser beam. In addition, it selects the frequencies of the laser modes. A cavity can be used to investigate the frequency distribution of the incident light. For this purpose, the length of the cavity is scanned, and the transmission is plotted as a function of the length. In this mode of operation, a cavity is called an optical spectrum analyser. On the other hand, a cavity can be used as a very stable reference for the frequency of the light. The detuning of the incident light from the resonance frequency can be measured, and the frequency of the laser light can be controlled via electronic feedback. In this way the laser is locked to the cavity, and the frequency of the laser light is stabilized. For details see Section 8.4. The light reﬂected from the cavity provides optical feedback. This is very undesirable for many laser systems and sometimes even destructive. However, feedback can be used with some lasers, such as diode lasers, to form an external cavity and thereby control the laser frequency. The high optical ﬁeld inside the cavity can be used to enhance non-linear optical eﬀects. Such build-up cavities are used in many of the CW squeezing experiments described in Chapter 9.

163

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5 Basic Optical Components

On resonance, the cavity will set up internal spatial modes. In the case of a laser resonator, this determines the shape of the laser mode and its waist position and waist size. Only certain combinations of mirror curvatures will lead to a stable resonance condition [9]. A passive cavity will only be resonant with that component of the incident ﬁeld that corresponds to its internal spatial mode. All other light will be directly reﬂected. In other words, the incident light has to be mode-matched to the cavity in order to get into the cavity. In return, the transmitted light has a very well-deﬁned, clean spatial mode. For this reason cavities can be used as spatial mode cleaners. There are several applications of cavities for inﬂuencing the ﬂuctuations of the light. The build-up of the internal ﬁeld is not instantaneous. It requires the interference of many waves that have to travel through the entire cavity. A steady state can be achieved after a time tcav . This is also the time it takes for the internal ﬁeld to decay in the absence of an incident light ﬁeld. Thus the cavity acts like an integrator on the ﬂuctuations with a time constant tcav . The high frequency components of the incident light, at frequencies larger than 𝛾 = 1∕tcav , will not enter the cavity; they will be reﬂected. These frequency components will be strongly attenuated in the transmitted light. On resonance, the cavity acts like a low-pass ﬁlter on the intensity ﬂuctuations. Finally, there are interesting applications of a cavity that is close to, but not exactly on, resonance with the input ﬁeld. The phase of the reﬂected amplitude, compared with the phase of the incident light, varies strongly, by about π, when scanned across one cavity linewidth. Thus the sidebands of the light, which have diﬀerent cavity detunings to the carrier frequency, are reﬂected with diﬀerent optical phases. As a consequence, the quadrature of the reﬂected light is rotated in respect to that of the incident light. We will ﬁnd (in Section 9.3) that cavities can be used to analyse the quadratures of light, coherent, or squeezed and as a device to change the squeezing quadrature. 5.3.1

Classical Description of a Linear Cavity

The intensity of the internal, reﬂected, and transmitted light and the relative phase of these beams can be evaluated with a purely classical model. Let us consider a simple cavity with input mirror M1 and output mirror M2 . This could be a standing wave cavity with two mirrors only or a ring cavity where all other mirrors are near-perfect reﬂectors. The cavity can be described by the parameters and the corresponding total cavity properties in Figure 5.9 and Table 5.1. These properties can be derived using the classical equations for multiple interference of waves. Here we will use the standard textbook approach of adding partial waves and combine it with the more elegant approach of solving d12

Figure 5.9 Properties of a linear cavity.

Loss: e–α2d R1 T1

R2 T2

5.3 Optical Cavities

Table 5.1 Parameters and properties of a linear cavity. Parameters of a linear cavity

T1

= intensity transmission of the input mirror

T2

= intensity transmission of the output mirror

R1

= intensity reﬂectivity of the input mirror

R2

= intensity reﬂectivity of the output mirror

p

= optical path length for one round-trip

d12

= distance from input to output mirror

𝛽

= the loss coeﬃcient for internal absorption or scattering

exp(−𝛽p)

= round-trip intensity loss of the cavity

gm

= loss parameter

Pin

= power of the incident beam

Pout

= power of the transmitted beam

Preﬂ

= power of the reﬂected beam

𝛾

= cavity linewidth

Δ

= normalized cavity detuning = (𝜈L − 𝜈cav )∕𝛾

Properties of a linear cavity

FSR

= free spectral range = c∕p

F

= ﬁnesse = FSR∕𝛾

self-consistency equations at the boundaries of the cavity. The amplitude 𝛼in of the incident ﬁeld will be partially transmitted through the √ ﬁrst mirror, and the internal ﬁeld just past the ﬁrst mirror ﬁeld will be 𝛼0 = T1 𝛼in . It will circulate inside the cavity, and during each round trip it will be reduced in amplitude by a factor [9]: √ (5.55) gm = R1 R2 exp(−𝛽p) which includes the amplitude reﬂectivities at the mirrors and the internal losses. The wave will also pick up a phase diﬀerence 𝛿𝜙 = 2π(𝜈L − 𝜈cav )p∕c for every round trip (Figure 5.10). T1 R1 A1 Ein

Pin

T2 R2 A2

E0 e–αp ei δϕ E1 E2 E3 E4

Erefl

Prefl

Ecav

Pcav

Eout

Pout

Figure 5.10 Multiple amplitude interference inside a linear cavity.

165

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5 Basic Optical Components

Let us describe the ﬁeld amplitude 𝛼ca𝑣 inside the cavity as a sum of all circulating terms: 𝛼cav = 𝛼0 + 𝛼1 + 𝛼2 + 𝛼3 + 𝛼4 + · · · =

∞ ∑

𝛼j

j=0

where 𝛼j+1 = g(𝜈)𝛼j and g(𝜈) = gm exp(−i𝛿𝜙(𝜈))

(5.56)

is a complex number that describes the total eﬀect of the cavity during one round trip. The amplitude of 𝛼2 is related to the amplitude of 𝛼1 by the same factor. The total ﬁeld is described by 𝛼0 (5.57) 𝛼cav = 𝛼0 (1 + g(𝜈) + g(𝜈)2 + g(𝜈)3 + g(𝜈)4 + · · · ) = 1 − g(𝜈) Note that this is equivalent to the eﬀect of a closed feedback system that adds a component g(𝜈)𝛼0 inside the feedback loop. The intensity, and thus the power of the light inside the cavity, is built up to |𝛼0 |2 |1 − g(𝜈)|2 |𝛼0 |2 = |1 − gm exp(−i𝜙)|2 |𝛼0 |2 = (1 − gm )2 + 4gm sin2 (𝛿𝜙∕2)

Pcav ∝ |𝛼cav |2 =

(5.58)

Thus we get 1 1 + (2F∕π)2 sin2 (𝛿𝜙∕2) )2 ( 𝛼 with Pmax = 1−g0 and m √ π gm F = Finesse = (1 − gm ) Pcav = Pmax

(5.59)

The ﬁnesse is generally used as a measure for the quality of cavity. It is the number most frequently quoted in experimental publications. We can now link the amplitudes of the various ﬁelds through the following boundary conditions, where we take 𝛼cav as the internal ﬁeld just on the inside of mirror M1 : √ 𝛼0 = T1 𝛼in √ 𝛼out = T2 exp(−𝛽d12 + i𝛿𝜙∕2)𝛼cav √ √ − R1 𝛼in + T1 𝛼reﬂ = (5.60) √ R1 𝛼cav

5.3 Optical Cavities

The second condition takes into account the internal losses on the path d12 from mirror M1 to mirror M2 and the transmission through mirror M2 . The last condition takes into account that the wave at the outside of mirror M1 is out of phase (minus sign) with the wave transmitted through this mirror from the inside. Combining Eq. (5.60) with Eq. (5.57) leads to the link between the transmitted and reﬂected powers with the input power Pout exp(−2𝛽d12 ) = T1 T2 Pin |1 − g(𝜈)|2 Preﬂ |R − (R1 + T1 )g(𝜈)|2 = 1 Pin R1 |1 − g(𝜈)|2 Pcav T1 = Pin |1 − g(𝜈)|2

(5.61)

Consider the special case of a linear cavity with no losses in the mirrors, which means p = 2d12 and R1 + T1 = 1, R2 + T2 = 1. This is commonly discussed in textbooks as the case for an optical spectrum analyser or a Fabry–Perot etalon. In this case we obtain the result Pout T1 T2 |g(𝜈)|2 = Pin R1 R2 |1 − g(𝜈)|2 Preﬂ |R1 − g(𝜈)|2 = Pin R1 |1 − g(𝜈)|2 Pcav T1 = Pin |1 − g(𝜈)|2

(5.62)

The cavity build-up can be dramatic, and the transmission of a cavity can be much larger than the individual transmissions of the mirrors. Note that these equations hold for all detunings; on resonance, simply replace g(𝜈) by gm . For illustration it is worthwhile to consider some special cases: (i) A symmetric, lossless cavity. That means T1 = T2 = T and 𝛽 = 0. The loss parameter for this case is gm = R = (1 − T), and we obtain from Eq. (5.61) Pout ∕Pin = T 2 R∕R(1 − R)2 = 1 and Preﬂ ∕Pin = (1 − R)2 ∕R(1 − R)2 = 0. Such a cavity is perfectly transmitting and has no reﬂected power at all. This clearly illustrates the dramatic eﬀect of the constructive interference. The internal power is Pcav = Pin T∕(1 − R)2 = Pin ∕T, which means that mirrors of transmission T = 0.01, corresponding to R = 0.99, are required for a power build-up by a factor of 100. The ﬁnesse of such a cavity can be approximated as F = π∕T = 300. (ii) A cavity with loss. All cavities will have some losses. An interesting case is a cavity where the input transmission T1 just balances the sum of all losses. That means R1 = gm , which is equivalent to R1 = R2 exp(−𝛽p). For this case we get Preﬂ = 0; there is no reﬂection. The leakage of the intra-cavity power out through the input mirror is equal to the directly reﬂected amplitude. They cancel each other. This case is known as the impedance matched case,

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in analogy to the matching of electronic transmission lines: Under-coupled:

R1 < R2 exp(−𝛽p)

Impedance matched:

R1 = R2 exp(−𝛽p)

Over-coupled:

R1 > R2 exp(−𝛽p)

The two other cases, under-coupled and over-coupled, lead to some reﬂection, and the transmission is not optimum. Whilst both cases result in similar reﬂected power, the sign of the reﬂected wave changes from the under-coupled to the over-coupled case. However, in most experiments this sign cannot be detected. All these cases are illustrated in Figure 5.11 by the traces (i), (ii), and (iii). 1 (iii) A single ported cavity (T2 = 0). This obviously has no transmission. However, it is still useful. It shows internal power build-up, and all the leakage back out is through the input mirror. The cavity will have a partial power reﬂection. This light will contain information about the internal ﬁeld. The reﬂected beam can be separated and detected by using a linear cavity and an optical isolator or alternatively a ring cavity where the light is reﬂected into a diﬀerent direction. The loss parameter gm describes the reduction in the intensity of the internal ﬁeld on resonance with the cavity after one round trip. The fraction (1 − gm ) leaves the cavity during each round trip. One round trip takes the time p∕c. Thus the energy loss from the cavity is (1 − gm )Pcav c∕p measured in J/s. Since this loss is proportional to the energy stored in the cavity, the power Pcav will decay exponentially in time in the absence of an incident ﬁeld: Pcav (t) = Pcav (0) exp(−t∕tcav ) c 1 = (1 − gm ) tcav p The amplitude of the ﬁeld decays with twice the time constant, and we can deﬁne a coupling rate 𝜅 for the amplitude 𝛼cav (t) = 𝛼cav (t = 0) exp(−t∕2tcav ) = 𝛼cav (t = 0) exp(−𝜅t) with 𝜅 = (c∕p)(1 − gm ). This decay constant is the appropriate term for the temporal description of the cavity response. It can be linked to the cavity linewidth 𝛾. The Fourier transform of the intensity decay gives the spectral response function of the cavity. It results in a function proportional to 1∕(1 + i4πtcav ). The resonance of the spectral response has a full width half maximum (FWHM) of 𝛾 = 1∕(2πtcav ) and is measured in units of Hz. The same result could have been obtained by determining the FWHM of the cavity response function given in Eq. (5.59). 1 Note that the deﬁnition of under-coupled and over-coupled used here is the opposite of the conventional deﬁnition used, for example, in the book Lasers by A.E. Siegman [9].

5.3 Optical Cavities 0.4 π/2

(ii)

Pout

Φout (Rad)

0.3 0.2

Pin

0.1

(i)

0 0.064

(a)

(iii) 0

0

−π/2 0.064

0.064

0

0.064

0

0.064

0 Detuning (∆ν)

0.064

40 π/2

Φrefl (Rad)

30

Pcav 20

Pin

0

10 −π/2

0 0.064

(b)

0

0.064

0.064

1

π

Φcav (Rad)

0.75

Prefl ___ 0.5 Pin

0

0.25

−π 0 0.64

(c)

0

0.64

0.064

Detuning (∆ν)

Figure 5.11 Illustration of the cavity response. On the left side is the power response, and on the right side the phase response. Shown are the three results: (a) transmitted light, (b) the internal ﬁeld, and (c) the reﬂected light. For each situation three diﬀerent cases are shown. (i) The largely under-coupled case (long dashes), (ii) the impedance matched case (solid line), and (iii) the over-coupled case (dotted line).

5.3.2

The Special Case of High Reﬂectivities

It is useful to treat the transmission through the mirrors and the loss inside the cavity in a similar way. They are all processes for the leakage of the ﬁeld out of the cavity. For this purpose one can describe the properties of the mirrors and the losses with coupling constants 𝛿 [9]. These replace the exponential terms that are required for cavities with large losses. We convert the mirror reﬂectivity into

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a mirror coupling constant via Ri = exp(−𝛿mi )

or 𝛿mi = ln(1∕Ri ) = − ln(Ri )

and deﬁne an equivalent coupling constant for the intra-cavity losses as 𝛿lo = 𝛽p. In this way the loss parameter gm , deﬁned in Eq. (5.55) for a cavity with two partially reﬂecting mirrors, turns into gm = exp(−(𝛿m1 + 𝛿m2 + 𝛿lo )) = exp(−𝛿) An important case in quantum optics is the case of a high ﬁnesse cavity. Here the round-trip losses are small. That means 𝛿mi ≪ 1, 𝛿lo ≪ 1, and thus 𝛿 ≪ 1. This eﬀectively means that the amplitude is changing so little per round trip that spatial eﬀects along the propagation axis can be neglected. We assume a uniform internal amplitude. We also assume that losses inside the mirrors can be neglected, which means (Ri + Ti ) = 1. This leads to the following approximations: gm = exp(−𝛿) ≈ 1 − 𝛿 𝛿mi = − ln(Ri ) ≈ 1 − Ri = Ti (1 − gm )c (𝛿 + 𝛿m2 + 𝛿lo )c 𝛿c 𝜅= ≈ = m1 2p 2p 2p

(5.63)

We can deﬁne the loss rates for the total cavity and for the individual processes as 𝜅 = 𝜅m1 + 𝜅m2 + 𝜅lo

with 𝜅mi =

Ti c 2p

and 𝜅lo =

𝛽c 2

(5.64)

These deﬁnitions are commonly used in describing high ﬁnesse cavities. We ﬁnd them, for example, in laser models (Chapter 6) and many papers on non-linear processes inside cavities. They are very useful for most quantitative models. However, the limitations of this approximation should not be forgotten. There are many experiments where intra-cavity losses as large as several percent are unavoidable. Or we might have a strong non-linear conversion out of the internal mode into another mode. One example would be eﬃcient second harmonic generation (SHG). In these cases the approximations of the high ﬁnesse limit might not hold, and the exact equations should be used.

5.3.3

The Phase Response

The cavity introduces a phase shift between the incoming, the intra-cavity, the reﬂected, and the transmitted ﬁelds. The boundary conditions for the electric ﬁeld, given in Eq. (5.60), can be used to evaluate these phase shifts. Let us deﬁne all phase shifts in respect to the phase of the incoming amplitude. That means 𝜙cav = arg(𝛼cav ) − arg(𝛼in ) = arg(𝛼cav ∕𝛼in ), where the function arg(z) determines the phase angle 𝜙 of the complex number z. Using the results of Eq. (5.60) and

5.3 Optical Cavities

some algebraic manipulation, the following results are obtained: ( ) gm sin(𝛿𝜙) 𝜙cav = arctan 1 − gm cos(𝛿𝜙) ( ) − sin(𝛿𝜙) 𝜙out = arctan g − cos(𝛿𝜙) ) ( m √ −T1 R1 sin(𝛿𝜙) 𝜙reﬂ = arctan √ √ R1 + T1 R1 (1 − gm cos(𝛿𝜙))

(5.65)

A graphical evaluation of these equations for three special cases of under-coupled, over-coupled, and impedance matched is shown in Figure 5.11 (i)–(iii) alongside the results for the transmitted powers. On resonance, which means 𝛿𝜙 = 0, all three phase terms are equal to zero, modulus π. The electric ﬁelds of the incoming, intra-cavity, and transmitted light are all in phase. The reﬂected light is either in phase or out of phase. Around the resonance the phase terms show a dispersive behaviour. The width of the dispersion curve (right-hand side) is comparable with the cavity linewidth 𝛾 (left-hand side). There is a 2π phase change right across the cavity resonance. The intra-cavity ﬁeld and the transmitted ﬁeld have a phase advance over the incoming ﬁeld that increases with detuning. For the reﬂected light the phase can be advanced (+ sign) or retarded (− sign), depending on the parameters of the cavity. If the directly reﬂected light dominates (under-coupled case), the phase diﬀerence is 180∘ on resonance, and the phase is retarded. If the light from the cavity dominates (over-coupled case), the phase is advanced. Note that sidebands of the incoming wave can experience diﬀerent attenuation and phase shifts. For the limiting case where the modulation frequency Ωmod of the sidebands is less than the cavity linewidth (Ωmod ≪ 𝛾), this eﬀect is negligible. In this case the sidebands remain in phase with the cavity. For modulation frequencies much larger than the cavity linewidth (Ωmod ≫ 𝛾), the two sidebands are shifted by +π∕2 and −π∕2, respectively, and again they appear to be in phase with the central component. If, however, the modulation frequency is of the order of the cavity linewidth, the two sidebands at ±Ωmod will experience very diﬀerent phase shifts. This eﬀect can be exploited to measure the detuning of the cavity. The incoming light is modulated, and the transmitted or reﬂected light intensity is detected. At Ωmod we obtain a signal that is the sum of two beat signals between the sidebands and the central laser frequency. Commonly, phase modulation is used with the consequence that on resonance these two beat signals are out of phase and cancel each other. On resonance, there is no detectable intensity modulation on either the incoming, the transmitted, or the reﬂected light. However, for a detuned cavity the diﬀerence in the phase shifts for the two sidebands will prevent this cancellation. A modulation signal at frequency Ωmod appears, and phase modulation is partially turned into intensity modulation. The signal strength can be measured by demodulating the photocurrent, for example, with a mixer driven at Ωmod . The shape of this response as a function of detuning is closely linked to the dispersion curve for the individual phase terms. It can be derived analytically, and details are given in Section 8.4.

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5.3.4

Spatial Properties of Cavities

So far we have treated the cavity and the light in the cavity as one-dimensional systems, and we have lost all spatial information. However, not all cavities with the same mirror reﬂectivity and length are the same. The curvature of the mirrors and the shape of the wave front have to be taken into account. There are certain combinations of mirror separation and mirror curvature where a resonance is possible. In this case the wave front, returning after one round trip, is keeping its size and is not expanding signiﬁcantly. A spatial resonance will build up, and as a consequence the curvature of the wave front of the resulting resonant mode will be identical to the curvature of the mirror. As the simplest example let us consider a linear two-mirror cavity. Here the resonator is deﬁned by the separation d12 of the mirrors and the mirror curvatures 1 and 2 . The focal length of the mirrors is given by f1 = 1 ∕2 and f2 = 2 ∕2. In the standard notation we deﬁne the mirror parameters [9] g1 = 1 −

d12 2f1

and g2 = 1 −

d12 2f2

(5.66)

The condition for a stable resonator, which means a resonator that builds up a resonant wave, can be derived [9] and results in the stability criterion 0 ≤ g1 g 2 ≤ 1

(5.67)

Based on this criterion we ﬁnd that for a given set of mirror curvatures (focal lengths), resonance only occurs for a restricted range of cavity lengths. We have to choose the length and the curvatures together. Some examples of stable resonators are: • The (near) concentric cavity: d12 = 2 1 = 2 2 , which means g1 = g2 = −1 and g1 g2 = 1. • The (near) planar cavity: 1 = 2 = ∞, which means g1 = g2 = 1 and g1 g2 = 1. • The confocal cavity: d12 = 1 = 2 , which means g1 = g2 = 0 and g1 g2 = 0. The concentric and planar cavities are right at the limit of the stability criterion. They can be stable, but even minor changes in length or small misalignments tend to aﬀect the performance. These cases are not recommended. It is better to satisfy the stability criterion with some margin. The confocal cavity is in the middle of the stability region and thus very useful as a robust design for a resonator. Cavities can support higher-order TEMi,j spatial modes, and if the cavity is not confocal, this will occur at cavity lengths diﬀerent to the resonance length for the TEM0,0 mode. This feature can be used to devise techniques for locking a cavity to the frequency of a laser beam [10] by analysing the spatial properties of the transmitted and/or reﬂected beam (see Section 8.4). 5.3.4.1

Mode Matching

The cavity has its own internal geometric mode, determined by the curvature of the mirrors. The wave front of the mode of light illuminating the cavity might have a diﬀerent wave front than the internal mode. The incoming wave can only fully interact with the cavity if the incoming wave front and the internal wave front are

5.3 Optical Cavities

matched. This is known as the mode-matched case. The spatial properties of the incoming wave are matched, the intensity distribution (mode size) is identical to the internal mode, and the wave front curvature is identical to the curvature of the mirror, which in turn is the curvature of the internal mode. In this particular case, which is highly desirable but in practice requires extensive alignment work, the description of the cavity and the coupled light is simple, and the one-dimensional model in the previous section is correct. All the light interacts with the cavity. Most theoretical models assume mode matching. What will happen to light incident on a cavity that has a diﬀerent geometric mode? The cavity will support one fundamental mode and a whole series of higher-order spatial modes. In general, these modes will not be resonant simultaneously – diﬀerent detunings, which means small shifts in the mirror separations, will be required for the resonance condition of the higher-order modes. For any given mirror separation, the cavity will only support one mode. The one special case, where all spatial modes are resonant simultaneously, is the confocal cavity. For all other cases the resonators are non-degenerate – the resonances occur at diﬀerent cavity detunings. On resonance, the cavity will select the part of the external ﬁeld that corresponds to the resonant internal mode, an eigenmode of the cavity. In order to calculate the reﬂection and transmission of a cavity, it is useful to describe the external ﬁeld in terms of all the internal eigenmodes of the cavity. These spatial eigenmodes are a useful basis set for a mathematical expansion of the external ﬁeld. Let us classify the spatial modes as Hermite–Gaussian modes (see Section 2.1) with the classiﬁcation TEMmn describing the possible modes. We can calculate the fraction Pmn of the total power that corresponds to the resonant internal mode TEMmn . In most cases we align the cavity to resonate in the fundamental mode TEM00 . Only this light will enter the cavity and contribute to the build-up of the internal power. The remaining power will be reﬂected, assuming that the front mirror has a high reﬂectivity. The equations for the transmitted and reﬂected power (Eq. (5.61)) have to be modiﬁed to include the fraction Pmm . For example, the transmission will then be Pmn × Ptrans ∕Pin , and the reﬂectivity will be (1 − Pmn ) + Pmn × Preﬂ ∕Pin . Obviously, P00 = 1 corresponds to the perfectly mode-matched situation. To achieve P00 > 0.95 requires considerable patience and good-quality optical components. Once aligned to the fundamental mode, both the internal and the transmitted ﬁelds will have the spatial intensity distribution of a Gaussian beam. The cavity selects the Gaussian component from the incident ﬁeld; it acts like a spatial ﬁlter. Such a cavity is therefore also referred to as a spatial mode cleaner. How can the mode-matched condition be achieved, and how can the mode mismatch be quantiﬁed? By detuning the cavity, which means by changing the mirror separation, a small fraction of a wavelength, the cavity can be set on resonance with the higher-order spatial modes. The exact resonance frequencies can be evaluated by taking into account the phase diﬀerences between the various higher-order modes and the fundamental mode, known as Gouy phase shifts. These phase √ shifts depend on the order numbers m and n and can be expanded as cos−1 (± g1 g2 ) where g1 and g2 are the mirror parameters deﬁned in Eq. (5.66). The changes in the resonance frequencies Δ𝜈mn for the modes TEMmn away from

173

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5 Basic Optical Components

the resonance of the fundamental mode are given by [9] √ (n + m + 1)cos−1 (± g1 g2 ) c Δ𝜈mn = π2d12

(5.68)

with the following special cases: =0

for confocal cavity

1 FSR for near planar cavity 2 = FSR for near concentric cavity

Δ𝜈mn =

If the cavity is not confocal, the higher-order spatial modes map out a whole series of resonances as shown in Figure 5.12b. These can be made visible by scanning the cavity length or the frequency of the laser. At each side peak the cavity is in resonance with one of the higher-order spatial modes, and the transmitted light will show the corresponding spatial patterns of this mode. In most practical situations we want to achieve the mode-matched case. That means we want to minimize the higher-order modes, as shown in Figure 5.12. By using lenses to change the wave front curvature and by optimizing the position of the lenses systematically, mode matching can be achieved. In the same way the alignment of the beam to the axis of the cavity is improved. This is largely an iterative process, since in most situations, there are six degrees of freedom that have to be optimized: two degrees for the tilt of the input beam in regard to the cavity, two degrees for a lateral shift, and two degrees for the wave front curvature. Measuring the size of the high-order peaks Pmn is far more sensitive than measuring the reﬂected power Preﬂ . Clever strategies have been devised to detect misalignment and mode matching separately. There have been several approaches how alignment can be automized [11–13], but the problem of automatic mode matching is still largely unresolved. 5.3.4.2

Polarization

Polarization was not included in the above discussion. For a completely isotropic cavity, the polarization does not matter; the description using a single, arbitrary polarization state of light would be correct. However, in most practical cases at least one of the components of the cavity, or inside the cavity, is anisotropic and induces some birefringence or dichroism. In that case one has to ﬁnd the polarization eigenstates of the cavity. Light in these states will keep its polarization whilst P00 Pmn

FSR

(a)

νcav

(b)

νcav

Figure 5.12 The frequency response of a scanning cavity that is longer than confocal to (a) a mode-matched input beam and (b) a mismatched beam. Improved alignment will get us from case (b) to the optimum case (a).

5.3 Optical Cavities

propagating. It will return after one complete round trip with the same polarization. Note that at any individual point inside the cavity, these eigenstates can be quite diﬀerent. There are always two such orthogonal states. Losses, detuning, and thus the ﬁnesse can be diﬀerent for the two states, and the response of the cavity is described by two independent sets of equations, one for each polarization eigenstate. The best-known example is the unidirectional device inside a ring laser that uses polarization and losses to suppress laser oscillation in one of the two counter-propagating directions. 5.3.4.3

Tunable Mirrors

In many practical situations it would be very desirable to be able to tune the reﬂectivity of a mirror in a cavity. This can be achieved by making one of the mirrors a cavity itself or by using a trick called frustrated internal reﬂection. Here the two uncoated surfaces are used with a small gap to form a mirror. The light is aligned such that each surface would totally reﬂect the light. However, when the surfaces have a separation of a wavelength or less, the evanescent ﬁeld, on the outside of the totally reﬂecting surfaces, reaches to the next surface, and light is transmitted. The magnitude of this eﬀect can be controlled by changing the separation, thus a mirror of tunable reﬂectivity can be constructed [3, 14]. It should be noted that such mirrors are polarization dependent. 5.3.5

Equations of Motion for the Cavity Mode

For the rest of our discussion of cavities, we will assume that the technical requirements for obtaining high ﬁnesse operation have been satisﬁed. Under such conditions the equations of motion of a single cavity mode can be obtained in a straightforward way. Conceptually it is easiest to consider the case of a ring cavity as depicted in Figure 5.13. The cavity has two partially transmitting mirrors with transmissions T1 and T2 and internal loss with a transmission of Tl . External ﬁelds 𝛼in1 and 𝛼in2 enter at the ﬁrst and second mirrors, respectively. We calculate the change in the circulating mode amplitude, 𝛼c , incurred in making one round trip of the cavity under the assumption that that change is small. We have √ √ √ 𝛼c (t + 𝜏) = 𝛼c (t)ei𝜙 1 − T1 1 − T2 1 − Tl √ √ √ + 𝛼in1 (t)ei𝜙 T1 1 − T2 + 𝛼in2 (t)ei𝜙 T2 (5.69) where 𝜏 is the cavity round-trip time and 𝜙 is the diﬀerential phase shift acquired by the ﬁelds in that time. If T1 , T2 , 𝜙 ≪ 1, then the change in the ﬁeld in one round trip will be small. As such we expand to ﬁrst order the LHS of Eq. (5.69) in a Taylor Figure 5.13 A cavity described in the quantum model.

Aout1

Ain2 acav

Ain1

κ1

κloss

Aout2

κ2

175

176

5 Basic Optical Components

series and make small T approximations on the RHS. We obtain ) ( ( √ √ ) 1 1 1 𝛼c (t) + 𝜏 𝛼̇ c (t) = 𝛼c (t) 1 − T1 − T2 − Tl + 𝛼in1 T1 + 𝛼in2 T2 ei𝜙 2 2 2 Making a small 𝜙 (modulo 2π) approximation, i.e. assuming we are close to a cavity resonance, and rearranging, we obtain ( ) √ √ i𝜙 − 12 T1 − 12 T2 − 12 Tl √ √ T1 T2 𝜏 𝛼̇ c = 𝜏𝛼c + 𝛼in1 + 𝛼in2 (5.70) 𝜏 𝜏 𝜏 √ Equation (5.70) is normally presented with the following substitutions: 𝛼 = 𝜏𝛼c is the amplitude of the standing mode of the cavity; Δ = 𝜙∕𝜏 is the detuning of the cavity; and 𝜅i = Ti ∕(2𝜏) are the decay rates of the cavity. With these substitutions we obtain the following equation of motion for the cavity mode: √ √ (5.71) 𝛼̇ = 𝛼(iΔ − 𝜅1 − 𝜅2 − 𝜅l ) + 𝛼in1 2𝜅1 + 𝛼in2 2𝜅2 This equation of motion will be the starting point for our quantum description. The ﬁeld exiting from the ﬁrst mirror is √ √ 𝛼out1 = T1 𝛼c − 1 − T1 𝛼in1 √ ≈ 2𝜅1 𝛼 − 𝛼in1 (5.72) and similarly the ﬁeld exiting from the second mirror is √ √ 𝛼out2 = T2 𝛼c − 1 − T2 𝛼in2 √ ≈ 2𝜅2 𝛼 − 𝛼in2

(5.73)

Equations (5.72) and (5.73) are referred to as the boundary conditions for the cavity. 5.3.6

The Quantum Equations of Motion for a Cavity

The fundamental object of the quantum model of light is the cavity mode. The cavity is the experimental equivalent to the abstract concept of a single frequency mode and its subsequent identiﬁcation with the quantum model of a harmonic oscillator. In general, a description of the quantum evolution of a cavity requires the speciﬁcation of the Hamiltonian describing the interactions within the cavity as well as those between the cavity mode and the continuum of ﬁeld modes outside the cavity. Equations of motion for the ﬁeld can be obtained either in the Heisenberg picture (operator Langevin equations) or in the Schrödinger picture (master equation) [15]. Combining this with quantum boundary conditions [16] allows calculations of the properties of ﬁelds leaving the cavity. It was techniques such as these that were used to rigorously establish the behaviour of quantum ﬁelds that we have described in this and the previous chapters. In later chapters we will use such approaches. However, here we proceed via the simpler approach of canonical quantization of the classical equations of motion (Eq. (5.71)). This leads directly to √ √ √ ̂ in1 + 2𝜅2 A ̂ in2 + 2𝜅l A ̂l (5.74) â̇ = −(𝜅 − iΔ)̂a + 2𝜅1 A

5.3 Optical Cavities

̂ j are travelling modes where â is the standing cavity mode, whilst the A and 𝜅 = 𝜅1 + 𝜅2 + 𝜅l is the total decay rate of the cavity. As for the beamsplitter and the interferometer, the essential diﬀerences between the quantum and classical equations are the transformation to operators and the explicit inclusion of an operator representing the vacuum mode introduced by loss. Equation (5.74) is an example of an operator quantum Langevin equation. The boundary condition for the transmitted ﬁeld √ ̂ in2 ̂ out2 = 2𝜅2 â − A (5.75) A and the reﬂected ﬁeld √ ̂ out1 = 2𝜅1 â − A ̂ in1 A

(5.76)

can also be obtained directly from the classical boundary conditions (see Eqs. (5.72) and (5.73)). 5.3.7

The Propagation of Fluctuations Through the Cavity

In order to analyse the eﬀect of a cavity on the noise of an incident propagating mode of light, we consider the transfer functions for the cavity. We use the model described in Chapter 4. We approach the problem step by step: 1. The ﬂuctuations of the input ﬁeld are described as the spectrum of the amplitude variance V 1Ain (Ω). 2. The cavity is described through the equation of motion. The resulting ﬂuctuations are analysed. We ﬁrst consider the amplitude quadrature of the ﬂuctuations. 3. The variances V 1out of the various output ﬁelds are determined. They are given by a linear combination, the transfer function, of the variances of all the input ﬁelds. The variances of the ﬂuctuations of the internal ﬁeld are of academic interest only. They cannot be measured by conventional means. We ﬁnd that, for a cavity on resonance, the output amplitude ﬂuctuations are linked directly to the input amplitude ﬂuctuations. For a cavity with detuning, both quadrature amplitudes of the input contribute to the output amplitude ﬂuctuations. The output noise spectrum can be written as a linear function of the variance of the input mode and of contributions from the various unused input ports, which is the major diﬀerence to the classical model. Whilst the unused ports do not contribute to the spectrum of the output power, they contribute to the noise spectra of the light. In a way, the eﬀect of the cavity on the ﬂuctuations becomes similar to that of an electronic ﬁlter. The response is frequency dependent; it is linear, and it can be described by a transfer function. We consider the system shown in Figure 5.13 and include the ﬂuctuations. As usual each operator is written as a classical complex value for the amplitude and ̂ in1 , etc. The classical ̂ in1 = 𝛼in1 + 𝜹A ̂ A an operator for the ﬂuctuations: â = 𝛼 + 𝜹a, parts obey Eqs. (5.71)–(5.73), whilst the operator parts obey Eqs. (5.74)–(5.76). Considering just the quantum ﬂuctuations, we ﬁnd √ √ √ ̂̇ = −𝜅 𝜹a ̂ + 2𝜅 𝜹A ̂ + 2𝜅 𝜹A ̂ ̂ + 2𝜅 𝜹A (5.77) 𝜹a 1

in1

2

in2

l

l

177

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5 Basic Optical Components

By adding the complex conjugate, an equation for the amplitude quadrature can be formed: √ √ √ ̂̇ = −𝜅 𝜹X𝟏 ̂ + 2𝜅1 𝜹X𝟏 ̂ in1 + 2𝜅2 𝜹X𝟏 ̂ in2 + 2𝜅l 𝜹X𝟏 ̂ l 𝜹X𝟏 (5.78) The boundary conditions of Eqs. (5.75) and (5.76) are rewritten for the ﬂuctuations √ ̂ out2 = 2𝜅2 𝜹X𝟏 ̂ − 𝜹X𝟏 ̂ in2 𝜹X𝟏 √ ̂ out1 = 2𝜅1 𝜹X𝟏 ̂ − 𝜹X𝟏 ̂ in1 (5.79) 𝜹X𝟏 The next step is to consider the Fourier components of the ﬂuctuation. Using the property of Fourier transforms (df (t)∕dt) = −i2πΩ (f (t)), we can combine Eq. (5.78) with Eq. (5.79) and obtain √ √ ̃ in1 + (2𝜅2 − 𝜅 + i2πΩ) 𝜹X𝟏 ̃ l ̃ in2 + 4𝜅1 𝜅l 𝜹X𝟏 4𝜅1 𝜅2 𝜹X𝟏 ̃ out2 = 𝜹X𝟏 𝜅 − i(2πΩ) (5.80) In order to obtain the spectrum of the ﬂuctuations transmitted through the cavity, we determine the variance. This is the desired set of transfer functions. It shows how the diﬀerent input variances are linked and how the ﬂuctuations propagate through the system. We can repeat the above calculations and obtain for the reﬂected beam V 1out1 (Ω) = {((2𝜅1 − 𝜅)2 + (2πΩ)2 ) V 1in1 (Ω) + 4𝜅2 𝜅1 V 1in2 (Ω) + 4𝜅1 𝜅l V 1l (Ω)} 1 × 2 (5.81) 𝜅 + (2πΩ)2 and for the transmitted beam 4𝜅 𝜅 V 1in1 + ((2𝜅2 − 𝜅)2 + (2πΩ)2 ) V 1in2 (Ω) + 4𝜅2 𝜅l V 1l (Ω) V 1out2 (Ω) = 2 1 𝜅 2 + (2πΩ)2 (5.82) The ﬁrst useful check is to consider the special case where all inputs are QNL, which means they are not either used or illuminated by a coherent state. We ﬁnd V 1out2 (Ω) = V 1out1 (Ω) = 1 As expected, the two output noise spectra for this special case are also QNL. This is an important result that can be applied to all passive systems no matter how complicated the instrument is: if all the input beams are at the QNL, then also all the output states are at the QNL. But note that even when all output variances have the variances V 1out = 1, the origin of this noise can still vary with the frequency since the components of the transfer function itself are frequency dependent. This becomes important when at least one of the input beams has ﬂuctuations above or below the QNL, and either noise or squeezing is propagating through the cavity. In general, the transfer function is frequency dependent.

5.3 Optical Cavities

Consider the special case of a perfect ring cavity, with 𝜅1 = 𝜅2 = 𝜅∕2. Then the set of transfer functions will become 𝜅 2 V 1in1 + (2πΩ)2 V 1in2 𝜅 2 + (2πΩ)2 2 (2πΩ) V 1in1 + 𝜅 2 V 1in2 V 1out1 (Ω) = 𝜅 2 + (2πΩ)2 V 1out2 (Ω) =

We can now examine the low and high frequency performances of the cavity, which are distinctly diﬀerent. For low frequencies, we see that in the limit Ω → 0 the spectra approach V 1out2 (0) = V 1in1 V 1out1 (0) = V 1in2 In other words, the input noise passes directly through to the output, and the noise of the unused port, normally vacuum noise, passes straight through into the reﬂected beam. The cavity is transparent to noise in either direction. Interestingly, at high frequencies, we see that in the limit of Ω → ∞, these spectra approach V 1out2 (∞) = V 1in2

(5.83)

V 1out1 (∞) = V 1in1

(5.84)

This is the opposite situation to that above. The cavity is completely opaque to noise at high frequencies. The input noise is reﬂected into the reﬂected ﬁeld, and the noise of the unused port is reﬂected into the output ﬁeld. This result is true for all cavities, as described by Eqs. (5.82) and (5.81). For high frequencies ((2πΩ)2 ≫ 𝜅 2 ), the noise from the input source will not be able to propagate through the cavity. Only low frequency noise, within the linewidth of the cavity, is transmitted. The cavity acts like an integrator and smoothes out fast ﬂuctuations. However, these ﬂuctuations are not lost; they appear in the reﬂected light. For low frequencies, (2πΩ)2 ≪ 𝜅 2 , the noise is transmitted. For frequencies around (2πΩ) = 𝜅, there is a smooth transition. This response is shown in Figure 5.14 for the two cases of a noisy input (V 1in1 = 2) and a squeezed input (V 1in1 = 0.1). 2 Vrefl = Vout2

Vin = 2

1.5 Vtrans = Vout1 V

Figure 5.14 Noise response of a cavity as a function of the detection frequency. Both the transmitted and the reﬂected noise are shown. Two cases: (i) a cavity with noisy input (V1in1 = 2) and (ii) a cavity with an input noise below the QNL (V1in1 = 0.1).

1 Vtrans = Vout1 Vin = 0.1

0.5 Vrefl = Vout2

0 0

2

4

Ω/∆ν

6

8

179

180

5 Basic Optical Components

In practice, it means that for a laser beam with modulation at Ωmod incident on a cavity, the transmission function for the laser carrier frequency 𝜈L and for the modulation sidebands is not the same (compare Eqs. (5.61) and (5.82)). In particular, for high frequencies the modulation sidebands cannot be transmitted; they are reﬂected. A cavity can be used as a ﬁlter for noise. The cavity is placed after the output coupler of the laser, all high frequency classical noise is rejected, and the output beam is QNL at these frequencies. Please note that all of the above equations have treated the resonant case. The transfer functions for the phase quadrature, V 2, or indeed any arbitrary quadrature angle, are equivalent to those we have calculated for the amplitude quadrature, V 1, for this case. It is straightforward to include detunings from cavity resonance, but this generates signiﬁcantly more complex equations. The main diﬀerence is that in the detuned case both quadrature amplitudes, X1 and X2, have to be taken into account. A detuned cavity couples the ﬂuctuations of the two quadratures – it links amplitude and phase ﬂuctuations. This eﬀect can be exploited for very sensitive measurements of phase ﬂuctuations, for cavity locking (see Section 8.4), and for the rotation of the squeezing axis (see Section 9.3).

5.3.8

Single Photons Through a Cavity

We now look at the eﬀect of the cavity on a single-photon ﬁeld. As for previous elements we will use the techniques of Section 4.4 to generate the state evolution. However, in contrast to the beamsplitter and interferometer, here we need to explicitly work in the frequency picture. Let us consider the case of a resonant, lossless, two-sided cavity with the input and output mirrors of equal reﬂectivity. Using Eqs. (5.74), (5.75), and (5.76) with 𝜅1 = 𝜅2 = 𝜅∕2 and 𝜅l , Δ = 0, we obtain the following operator evolutions through the cavity in frequency space: ̃ ̃ ̃ out2 = 𝜅 Ain1 + i𝜔 Ain2 A 𝜅 − i𝜔

(5.85)

̃ ̃ ̃ out1 = 𝜅 Ain2 + i𝜔 Ain1 A 𝜅 − i𝜔

(5.86)

and

where 𝜔 = 𝜔o − 𝜔cav is the diﬀerence between the optical frequency of the mode and the cavity resonant frequency in radians per second. Let us consider a single-photon input at the ﬁrst input port and vacuum at the other. We can write the input state as ∞

|1⟩f |0⟩ =

∫−∞

̃ † |0⟩in1 |0⟩in2 d𝜔f (𝜔)A in1,𝜔

̃ † is the photon creation operator at frequency 𝜔 in the input mode to where A in1,𝜔 ̃ † |0⟩in1 = |1⟩in1,𝜔 . Also |f (𝜔)|2 d𝜔 is the probability to the ﬁrst port, such that A in1,𝜔 ﬁnd the photon in the small frequency interval d𝜔 around 𝜔.

5.3 Optical Cavities

Using the inverted forms of Eqs. (5.85) and (5.86), we can evolve the state through the cavity giving ̃ U(𝜔)|1⟩ f |0⟩ =

∞

∫−∞

d𝜔f (𝜔)

̃† ̃ † + i𝜔 A 𝜅A out2 out1 𝜅 + i𝜔

|0⟩out1 |0⟩out2

(5.87)

As expected, the eﬀect of the cavity in frequency space is to act like a frequency-dependent beamsplitter, transmitting the low frequency components of the single-photon state but reﬂecting the high frequency components. For example suppose the input photon has a very narrow frequency range centred at 𝜔 = 0. We can write approximately f (𝜔) = 𝛿(𝜔), and so the input state is just |1⟩in1,0 |0⟩in2 , whilst the output state is |0⟩out1 |1⟩out2,0 , that is, the photon is deﬁnitely transmitted through the cavity. On the other hand if the photon has a narrow frequency range centred on some frequency 𝜔 ≫ 𝜅, then the output state will be |1⟩out1,𝜔 |0⟩out2 , that is, the photon will be completely reﬂected by the cavity. For an input state with a very broad frequency distribution, which we can approximate by supposing f (𝜔) = K, where K is a constant, the output state will be in a superposition of the two output modes: ̃ U(𝜔)|1⟩ f |0⟩ = K

∞

∫−∞

d𝜔

̃ † + i𝜔 A ̃† 𝜅A out2 out1 𝜅 + i𝜔

|0⟩out1 |0⟩out2

(5.88)

If we consider the reﬂected ﬁeld to be ‘lost’, then the transmitted ﬁeld will be a mixed state of the vacuum and a single photon state with the Lorentzian frequency distribution 𝜅2 (5.89) + 𝜔2 In this way cavities select speciﬁc components of the radiation ﬁeld, even in the limit of single photons. 𝜅2

5.3.9

Multimode Cavities

In our quantum treatment, so far we have focused on a single high ﬁnesse cavity mode. This occurred because we made several approximations that assumed high reﬂectivity mirrors and low round-trip loss and that we were close to a single cavity resonance, to obtain Eq. (5.71), before applying canonical quantization to obtain Eq. (5.74). The advantage of doing this was it gave a simple equation of motion, useful in many situations, which corresponds to the quantum Langevin equation one obtains from a quantum Hamiltonian approach. However, it is also possible to simply quantize the exact expression (Eq. (5.69)) in order to obtain the more general quantum expression √ √ √ ̂ c (t + 𝜏) = A ̂ c (t)ei𝜙 1 − T1 1 − T2 1 − Tl A √ √ √ ̂ in1 (t)ei𝜙 T1 1 − T2 + A ̂ in2 (t)ei𝜙 T2 +A (5.90) In principle this expression can be used to describe multiple resonances of a cavity of arbitrary ﬁnesse. Here we will still consider a high ﬁnesse cavity but

181

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5 Basic Optical Components

retain the multimode feature. Taking T1 , T2 ≪ 1 and using the deﬁnitions in Section 5.3.5, we obtain √ √ √ ̂ in1 (t) + 2𝜅2 A ̂ in2 (t) + 2𝜅l A ̂ l (t)) â (t + 𝜏) = ei𝜙 (𝜅 â (t) + 2𝜅1 A (5.91) i𝜏𝜔 ̃ Using the Fourier transform relationship â (t + 𝜏) = a(𝜔)e and the input–output relations (Eqs. (5.75) and (5.76)), we can solve for the output ﬁeld in Fourier space to obtain √ √ ̃ in2 + 4𝜅1 𝜅l A ̃ in1 + (2𝜅2 − 𝜅 + Γ(𝜔)) A ̃l 4𝜅1 𝜅2 A ̃ out2 = (5.92) A 𝜅 − Γ(𝜔)

where 1 − ei(𝜔−Δ)𝜏 (5.93) 𝜏 This now describes a multimode cavity in which cavity resonances explicitly occur whenever (𝜔 − Δ)𝜏 = 2πn (with n as an integer). Γ(𝜔) =

5.3.10

Engineering Beamsplitters, Interferometers, and Resonators

Whilst in Section 5.1 we focussed on one speciﬁc type of beamsplitter, the well-known device made of a plain piece of glass with speciﬁcally prepared surface, many other devices have the same eﬀect. Some of them transfer the power into one or several speciﬁc other modes; this could, for example, be done by a grating that reﬂects into diﬀerent orders. Or we can use surface or evanescent coupling when two modes approach each other by bringing two surfaces close to each other, or we can achieve this inside an optical ﬁbre coupler. We can set up periodic nanostructured materials that couple diﬀerent propagating modes. The list includes various forms of active modulators that produce gratings or other spatial modes that couple into other spatial modes and those that create diﬀerent frequency components. We can combine several beamsplitters into interferometers and cavities and resonators. This creates many technological options for coherent linear devices. This is illustrated in Figure 5.15, which shows the diverse range of optical resonators that are used today. All are using diﬀerent approaches and techniques, ranging from simple optical cavities made out of two mirrors to multi-mirror ring cavities, ﬁbre couplers, and ﬁbre rings, evanescent waves in discs, toroids and spheres, Bragg reﬂectors, planar optical devices, and photonics crystals. The ﬁrst generation of devices shown in the top row of Figure 5.15 was designed as resonators for lasers. One of the main design criteria is to make the resonator stable, which means a large build-up after many round trips is possible and minor mechanical perturbations have only a small eﬀect. A second design goal is to minimize losses such that the components can handle high internal ﬁelds. All of these resonators have several spatial eigenmodes, which are the diﬀerent stable solutions for the resonance. For example a long linear resonator can support the family of Hermite–Gaussian spatial modes, shown in Chapter 2 in Figure 2.2, and these modes can be generated in a laser. For some of these conﬁgurations, in particular the case of the confocal resonator, the spatial modes

Planar mirror

Spherical mirror

Microtoroid

Figure 5.15 Various optical resonator designs.

DBR

Microdisk

Microsphere

Rectangular cavity

Integrated optic ring

DBR

Fibre ring

Ring mirror

Micropillar

Photonic crystal

184

5 Basic Optical Components

exist at the same resonance frequency; they are degenerate and will appear at the same time. However, for most geometries the spatial modes have diﬀerent optical resonance frequencies, the modes are non-degenerate, and the laser either operates on several modes with diﬀerent frequencies or in one spatial mode at a time. There is a considerable knowledge about the engineering of such resonators to make them insensitive against mechanical vibration and also to make their resonance frequencies tunable. A good starting point for the design techniques can be found in the widely used book by A.E. Siegman [9]. The lower part of the diagram shows a variety of resonators that have emerged in the last few years and have been optimized to create resonators that are stable, have low losses, and more recently are designed to be very small and mass produced as building blocks for complex systems. Each technology has its own design limits, but in general they are improving quite dramatically to achieve almost perfect resonators. The ever-increasing quality of these devices is well described in a series of review articles and textbooks [17].

5.4 Other Optical Components 5.4.1

Lenses

We require optical components to change the spatial properties of a laser beam or mode of light. Actually, in most of optics imaging is the main concern. The ability of lenses and mirrors to form and reproduce images is the reason for using these components. The eﬀect of lenses is a change of the wave front curvature and consequently a transformation from one Gaussian input beam to another Gaussian beam with diﬀerent waist position and size. In practice, lenses are used to transform the spatial mode coming from the laser into the mode required in the experiment, to match it to any cavity that exists in the experiment, and ﬁnally, to focus the mode onto the detector (Figure 5.16). The quality of a lens or mirror is determined by the shape of the lens, its surface ﬁnish, and the uniformity of the material. Most imaging lenses are multi-element lenses, and the details of the lens design are of crucial importance to the quality of the image, the spatial resolution that can be achieved. For practical details consult one of the handbooks on optical instruments [18]. For most cases the paraxial transformation will hold, and we can use the results discussed in Section 2.1. It

Lens Gaussian beam

Figure 5.16 Spatial mode transformation by a lens.

Gaussian beam

5.4 Other Optical Components

assumes that the input is in a single spatial mode and that the lens is well represented by a parabolic phase term. Consequently, the output will be in a single spatial mode. Imperfections of the lens will result in deviations from a Gaussian beam, and aspherical aberrations are fairly common. In this case the light has no longer a parabolic wave front; it is distributed over a variety of spatial modes. This will make any interference experiment, including homodyne detection, very diﬃcult. It will reduce the visibility of the interference fringes and therefore make it an ineﬃcient process. The loss of light due to reﬂection on the lens surfaces is a major experimental problem. As an example, for glass lenses, n = 1.5, the intensity reﬂection per surface is typically 4% for every surface. This can quickly add up in a whole experiment. Thus most surfaces have an anti-reﬂection coating that can reduce the reﬂection to less than 0.5%. Scattering losses inside the components are usually even smaller than 0.1% and will only be of concern if the optical component is placed inside a cavity or is extremely sensitive to scattered light. Whilst losses are one concern in quantum experiments, the other is the need to maintain one single mode. Assume we have a mode of light with an interesting intensity noise statistics that is sub-Poissonian. It passes a lens with distortions and turns into a non-Gaussian mode. If we detect all of this light, we still ﬁnd a sub-Poissonian result. If, however, the detector is sensitive to only the Gaussian mode, because it contains a cavity or it is a homodyne detector that requires the interference and beating with an optical local oscillator, the results will be diﬀerent. If the partition between the modes is random, then the other modes will appear like a power loss, and the statistics will be closer to Poissonian. If the partition is somehow deterministic, then additional noise could be introduced. Fortunately all passive components, such as mirrors and lenses, result in a random partition. The other modes can be treated as loss terms. In contrast, active systems such as lasers are diﬀerent and can introduce additional partition noise, as shown in Chapter 6. A problem that exists with high intensity experiments, and in particular pulsed experiments, is thermal lensing. Glasses will absorb some small fraction (typically 0.1 [%/cm]) of the light, and the absorbed power can lead to local heating of the glass that results in local changes of the refractive index. At such high intensities, light becomes non-linear. In this case self-focussing can occur since the intensity-dependent refractive index eﬀectively forms the proﬁle equivalent to a lens, formed by the beam itself. Self-focussing leads to high higher intensities at the core of the beam increasing the eﬀect event more. This results in positive feedback, which can have disastrous eﬀects. Many lenses have been destroyed with holes and channels burned into the glass. With clever design, and the right materials and pulse length, this can be used to create a localized bubble in the glass, which can be seen from the outside and can serve as on voxel for a 3D structure, very popular with artists and tourists. 5.4.2

Holograms and Metasurfaces

The concept of using structured surfaces to alter the spatial wave front and to manipulate the shape of beams of light has a very long tradition. Fresnel lenses,

185

186

5 Basic Optical Components

which use circular phase shifts or even absorption, were a very early invention that could take on the task handled by lenses. Early versions, such as those used in light houses, had structures that were much larger than the wavelength of light, frequently several millimetres, and can be understood in the same way as lenses. Later it was possible to make small structures using photographic processes, and the understanding of coherent light with known phase distributions allowed the concept of holograms by D. Gabor [19] even lasers were available and in many versions since. Holograms can be used to record and play back complex waveforms that result, for example, in the reconstruction of 3D images. Fourier optics, which means the spatial frequencies of the light ﬁeld, created a whole new ﬁeld of understanding imaging systems, designing instruments, and shaping light. Holograms have been reﬁned, and it is not possible to create computergenerated hologram using the high-resolution spatial light modulators (SLM) available today. That means we can now design optical components that can be manipulated from the outside. This allows the shaping of light for many purposes, such as displays, microscopy, optical traps, and optical tweezers for atoms and particles. Practical examples are devices for the investigation of biological processes or the functional imaging of brain cells. The external control also allows the use of the modulators, and phase structures inside feedback loops, which allows for intelligent alignment and imaging processes. Recent development in the design and manufacture of structured surfaces allow the direct control of the wave fronts by very thin components. New nanoscale building techniques can create the required three dimensional structures and have resulted in a completely new class of materials and of passive and active components for the manipulation of light. The properties of metamaterials were explored in 1990s, ﬁrst as a curiosity, and initially tested at longer wavelengths, such as microwaves. Now we have optically thin structured materials made from sub-wavelength nanostructures, sometimes referred to as meta-atoms, which allow us to inﬂuence the properties of the light interacting with them. Metasurfaces enable the manipulation of the wave front at the surface. They can be used as optical elements for beam shaping and holography. At the same time, metasurfaces enable the control of light in the intermediate and near-ﬁeld zones. By tuning the phase properties of waves in the intermediate regime, also known as the Fresnel zone, it is possible, for example, to focus beams or engineer-desired point spread functions. At shorter distances, and straight after the surface where the behaviour of light is governed by its near-ﬁeld characteristics, carefully crafted components allow the concentration of the local intensity as required for the non-linear generation of new frequencies, surface-enhanced Raman scattering, or enhancement of the interaction with atoms via the Purcell eﬀect. For an overview of the design and use of metasurfaces, see, for example, S. Keren-Zu et al. [20]. As long as the metasurfaces, and thicker strictures made of metamaterials, behave in a linear way, they can be seen as the equivalent of the bulkier conventional optical components, such as lenses, interferometers, and optical cavities. They will not inﬂuence the photon statistics or add noise. Increasingly these will be the technique of choice for designing optical instruments, for focussing, and for the generation of multiple spatial modes.

5.4 Other Optical Components

5.4.3

Crystals and Polarizers

Glass and crystals vary in two very important points: crystals are non-isotropic, and they can show an electro-optics response and they can show non-linear eﬀects. The anisotropy of the crystals results in properties such as birefringence, optical activity, and dichroism. The details obviously vary with diﬀerent types of crystals and their composition and geometrical structure. It also varies with the alignment of the light beam with the crystal axis. All these properties can be used to control the polarization properties of the light. The idea here is to introduce a relative phase change between two diﬀerent polarization components of the ﬁeld. For example, introduce a π∕2 phase change of one linear polarization and not the other, and thereby turn linearly polarized light, aligned with the axis of the crystal, into circularly polarized light. This would be called a quarter wave plate. The results of any such component can be analysed using the rules of classical optics, in particular the formalism of Jones matrixes [3]. Birefringent crystals can be manufactured into wave plates and polarizers. In this regard quantum optics experiments are no diﬀerent to classical optics experiments. All tricks available to control and inﬂuence the polarization will be used in the various experiments. One particularly useful component is the optical isolator. This device is equivalent to a diode in electronics; it provides a diﬀerent eﬀect in the forward and backward direction. The light propagating in the reverse direction is redirected and does not go back to the source. This avoids interference between the incoming beam and the retro-reﬂected light. This eﬀect is described as optical feedback that can lead to very noisy or even unstable performance of a laser. At the same time an optical isolator will give access to the retro-reﬂected beam, which is important in experiments where all of the light has to be detected. It is also useful for situations where locking of a cavity to a laser is important. In this case the redirected light contains the information about the cavity detuning. Two types of optical isolators are commonly used: the Faraday isolator and wave plates (see Figure 5.17). The Faraday isolator uses a material, usually a special glass or crystal, with a high Verdet constant. This means a large rotation of the plane of polarization for linearly polarized light can be induced by a static magnetic ﬁeld along

Laser

PBS

B Mirror Faraday isolator

(a)

Laser

PBS

4 Mirror Quarter wave plate

(b)

Figure 5.17 Optical isolators using either a Faraday rotator (a) or a quarter wave plate (b).

187

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5 Basic Optical Components

the crystal. The isolator is tuned to the wavelength used in the particular experiment by adjusting the magnetic ﬁeld strength to give a 45∘ rotation. The return beam is rotated again by 45∘ in the same direction, which means both beams are linearly polarized at 90∘ to each other and can be separated by a PBS. The second design uses wave plates (see Figure 5.17b). The source generates linearly polarized light. A quarter wave plate is used to turn this into left-hand circularly polarized light (𝜎+ light). Upon reﬂection in the experiment, the returning light is right-hand circularly polarized (𝜎− light), and the wave plate turns this into a counter-propagating beam with orthogonal linear polarization. This light can be separated with a PBS. Crystals can also contribute entirely new eﬀects. They can have a non-linear response that allows us to generate new frequencies, such as twice the frequency (SHG), half the frequency (optical parametric gain), or sums and diﬀerent frequencies (four-wave mixing (4WM)). Crystals can show a Kerr eﬀect, which means an intensity-dependent refractive index, or non-linear absorption. These eﬀects will inﬂuence not only the intensities but also the ﬂuctuations of the light, and consequently they are discussed in Chapter 9 on squeezing experiments. 5.4.4

Optical Fibres and Waveguides

Optical ﬁbres are a convenient component to deliver light from one place to another. The idea is simple: the ﬁbres have a refractive index structure that guides the light. For suﬃciently small core sizes, a few times the optical wavelength, the ﬁeld inside the ﬁbre has one single mode with low loss, and all other modes are highly attenuated over length. In addition, larger diameter ﬁbres show many modes. The mode properties of ﬁbres are complex and are discussed in great detail in the specialized literature [21]. It is possible to design a lens that will couple very eﬃciently a free-space Gaussian laser beam into a single-mode ﬁbre. This requires a lens of short focal length, such as a microscope objective, which has a numerical aperture matched to the ﬁbre geometry. When the beam parameters are correctly chosen, at least 80% of the power can be coupled into the ﬁbre. The losses along the ﬁbre itself are so exceedingly small that the ﬁbre length is normally of no consequence. Fibres can be spliced together with low losses, and eﬃciencies of 95% are feasible. In this way whole ﬁbre networks, including couplers, can be built. In recent years it has also been possible to create other optical waveguides, frequently through direct writing optical circuits inside blocks of glass and other transmitting materials. Materials that are structured on nanometre scale have been used as so-called photonics crystals that are used as waveguides and optical circuits. All of this has extended the options for guiding and distributing light tremendously. However one should note that in communication applications an overall loss of 3 dB or 50% is fully acceptable. This is also the case for some single-photon experiments, where the overall quantum eﬃciency may not be critical. However, in CW applications these devices are not yet suitable since the high losses reduce the precious quantum properties. It should be noted that simple symmetric ﬁbres completely scramble the polarization of the light. Any polarization dependence of the optics after the ﬁbre

5.4 Other Optical Components

can lead to enormous intensity ﬂuctuations introduced by the ﬁbre. There are special non-symmetric polarization-preserving ﬁbres, which have to be carefully aligned. And even these are very sensitive to mechanical movements of the ﬁbre. The output mode shape from a single-mode ﬁbre is very close to a Gaussian beam. As a result ﬁbres can be used as spatial mode cleaners, and they will transform an input beam with a complex wave front into a very nice Gaussian beam. The light that is not matched will be lost by scattering. The higher losses of a ﬁbre mode cleaner are frequently compensated by its simplicity and the fact that, unlike a mode cleaner cavity, it requires no active locking. Modern technology now allows the writing of waveguides inside solid materials, and the advantages of ﬁbres can be extended to more complex structures that can contain splitters and combiners and other components all integrated into one chip. One of the main attractions is the high internal intensity inside the ﬁbre or waveguide, which make it very attractive for non-linear interactions. Some of the ﬁrst and some of the best squeezing experiments have been carried out in ﬁbres (see Section 9.4.4). Apart from producing quantum eﬀects, the non-linearities in waveguides can be exploited for future active control and the operation of quantum logic operations, all ﬁrmly integrated into solid components [22].

5.4.5

Modulators

Modulators are an essential component of most electro-optical systems. They allow us to encode information onto laser beams. The most common application is to modulate a single beam, which means one single optical mode, without noticeably changing the mode but simply adding frequency sidebands that contain the information. The classical description of modulation (see Section 2.1.7) can be transferred into the quantum mechanical description (see Section 4.5.5) by introducing a time-dependent component to the model of the beamsplitter. The properties of the classical transfer equation remain the same. However, a quantum mechanical vacuum operator appears that represents the coupling of vacuum ﬂuctuations back into the beam. This term is responsible for the special quantum noise properties that are so important in any system operating close to the QNL. For example an intensity modulator is essentially a time-dependent beamsplitter, where an external modulation function controls the strength of the beam splitter. In general a time-dependent beamsplitter that mixes two input modes ̂ transforms them such that ̂ and b(t) represented by the operators a(t) √ √ ̂ â o (t) = 𝜂(t)̂a(t) + 1 − 𝜂(t)b(t) √ √ ̂ − 1 − 𝜂(t)̂a(t) (5.94) b̂ o (t) = 𝜂(t)b(t) where 𝜂(t) is the time-dependent transmitivity of the beamsplitter. Transforming into the frequency domain, we obtain ̃ ̃ o (Ω) = 𝜈(Ω) ̃ ̃ ∗ A(Ω) + 𝜇(Ω) ̃ ∗ B(Ω) A ̃ ̃ ̃ ∗ B(Ω) − 𝜇(Ω) ̃ ∗ A(Ω) B̃ o (Ω) = 𝜈(Ω)

(5.95)

189

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5 Basic Optical Components

√ where 𝜈(Ω) ̃ is the Fourier transform of 𝜂(t) and 𝜇(Ω) ̃ is the Fourier transform √ of 1 − 𝜂(t). The ∗ symbol indicates a convolution, that is, M(Ω) ∗ L(Ω) =

∫

dΩ′ M(Ω′ )L(Ω − Ω′ )

(5.96)

A good model for the modulator is a beamsplitter that takes oﬀ a small fraction of the light 𝜖o and then adds an even smaller modulation term with amplitude 𝜖(t). Hence we can write 𝜂(t) = 1 − 𝜖o + 𝜖(t) cos(2πΩmod t)

(5.97)

where Ωmod is the modulation frequency and 𝜖o > 𝜖(t). If we assume the modulation amplitude is very small such that 1 − 𝜖o > 𝜖o ≫ 𝜖(t), then we can make the approximations √ √ 𝜖(t) 𝜂 ≈ 1 − 𝜖o + √ cos(2πΩmod t) 2 1 − 𝜖o √ √ 𝜖(t) 1 − 𝜂 ≈ 𝜖o − √ cos(2πΩmod t) (5.98) 2 𝜖o and hence we can write √ 1 𝜖(Ω) ̃ ∗ (𝛿(Ω + Ωmod ) + 𝛿(Ω − Ωmod )) 𝜈(Ω) ̃ = 1 − 𝜖o 𝛿(Ω) + √ 4 1 − 𝜖o √ 1 𝜇(Ω) ̃ = 𝜖o 𝛿(Ω) + √ 𝜖(Ω) ̃ ∗ (𝛿(Ω + Ωmod ) + 𝛿(Ω − Ωmod )) (5.99) 4 𝜖o where 𝜖(Ω) ̃ is the Fourier transform of 𝜖(t). Let us assume that the modulation is turned on at some time −T∕2 and turned oﬀ at T∕2 and has the value 𝜖 when on. Hence 𝜖(Ω) ̃ = 𝜖T sinc(πΩT) where sinc(x) = sin(x)∕x. Further we assume that ̃ ̃ the {b(t) mode is initially in the vacuum state and write B(Ω) ≡ V(Ω). As a result, in the expectation values, non-zero modulated terms coming from the vacuum input will only appear to second order or higher in 𝜖 and so can be neglected. Thus, inserting the convolutions, we obtain the following expression for the output mode from the intensity modulator: √ √ ̃ o (Ω) = 1 − 𝜖o A(Ω) ̃ ̃ A + 𝜖o V(Ω) 𝜖T ̃ ′ + Ωmod ) + A(Ω ̃ ′ − Ωmod ))sinc(π(Ω − Ω′ )T) dΩ′ (A(Ω + √ 4 1 − 𝜖o ∫ (5.100) In the limit that the modulation is on for a long time (T ≫ Δtmod ), the sinc function approaches a 𝛿 function, and the sidebands become very sharp (as assumed in Section 4.5.5). In contrast, if the modulation is only on for a short time, as a burst, the sidebands are spread out over a range of frequencies around the sideband frequencies. Similarly the information about a slowly varying time evolution of the modulation will be contained in the spectral components close to the pair of sidebands.

5.4 Other Optical Components

5.4.5.1

Phase and Amplitude Modulators

1

1

FM

n

atio

cal

ti Ver (a)

riz pola

Fast axis

Modulators normally operate across a frequency bandwidth: some are broadband, and others are more resonant. By feeding a signal at a frequency Ωmod into the modulator, we create the speciﬁc sidebands, which contain the information we want to impose. Within the bandwidth complete modulation spectra can be used, for example, in communication systems. The modulation can be either a phase or an amplitude modulation, or even a combination of both, depending on the design of the modulator. An intensity modulator controls only the magnitude of the transmitted beam and should leave all other properties unchanged. A phase modulator introduces a time-dependent phase shift and should leave the overall power constant. However, we should note that many instruments, for example, interferometers, convert phase modulation into amplitude modulation, and this can make it diﬃcult to build a pure phase modulator. Reducing crosstalk between amplitude and phase modulation is a technical challenge (Figure 5.18). Many crystals have signiﬁcant electro-optic coeﬃcients. That means the refractive index can be manipulated with an external electric ﬁeld. The direction of the electric ﬁeld that aﬀects the refractive index along a particular optical axis depends on the type of crystal used. In some cases the transverse eﬀect is large, which means the electric ﬁeld is applied orthogonal to the optical beam. In other crystals a longitudinal ﬁeld, parallel to the optical beam, is required. The refractive index will follow the ﬁeld up to very high frequencies; more than 20 GHz are possible. Thus these devices can be used as EOM. Large changes in refractive index will require electric ﬁelds of the order of 100–1000 V/cm. The limit in speed is normally set by the electric power requirements. Fast response and large degrees of modulation can be achieved by using resonant electronic circuits that provide a large electric ﬁeld in a very narrow frequency range. This eﬀect can be used to make a phase modulator. Either a crystal with isotropic response is chosen, or the axis of linear polarization is carefully aligned

ted dula mo ization s t ron polar se f n Pha ange i PBS ch o N

n

atio

ariz

du mo

m n A latio u d mo

tion

riza

ola ar p

l

u Circ (b)

n

latio

de

Pol

axis

1

plitu

1

AM

Slow

Figure 5.18 Electro-optical modulators for phase or amplitude.

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5 Basic Optical Components

parallel to the crystal axis. For a single modulation frequency Ωmod , the output light will get one set of sidebands, Ωmod and its higher harmonics, as described in Eq. (2.21). In the ideal case phase-modulated light does not produce any modulation of the photocurrent. In practice, considerable care has to be taken to avoid amplitude modulation. FM sidebands are frequently used to probe the detuning of cavities and to lock lasers to cavities. Alternatively, a birefringent crystal can be used and aligned such that the linear input polarization is at 45∘ to the optical axis of the crystal. In this case the two orthogonal components of the linear polarization are aﬀected diﬀerently. The voltage induces a diﬀerence in the change of the refractive index, and the modulator is acting like the AC equivalent of a wave plate. The polarization is rotated synchronously with the applied electric ﬁeld. After transmission through a polarizer, the light has a mixture of amplitude modulation, from the polarization rotation, and phase modulation. This is a common conﬁguration for intensity modulation. Normally a small reduction in intensity is chosen, and the intensity can be increased and decreased around this oﬀset point, as per the example in Eq. (5.97). An entirely diﬀerent eﬀect is used in acousto-optical modulators (AOM). These utilize the fact that an acoustic standing wave can be generated inside crystals through an RF electric ﬁeld. Light transmitted through such a standing wave sees a periodic density grating and is diﬀracted. The diﬀracted beam is shifted in frequency by ΩRF and is also shifted in its output direction, proportional to ΩRF (Figure 5.19). Since the diﬀraction angle is linked to the modulation frequency, we can redirect a beam by simply changing the frequency ΩRF of the RF driving voltage, which is typically in the range of 10–1000 MHz as a function of time. We can change the intensity of the diﬀracted beam by changing the RF power. Since both RF and power can be controlled rapidly, the output beam is thus scanned in one dimension. One common application of an AOM is as fast scanners, such as laser printers and displays. The AOM can be used as an intensity modulator by keeping the RF ﬁxed and modulating the RF intensity at frequencies Ωmod ≪ ΩRF . In quantum optics experiments the main application is the shift in optical frequency. For an ideal AOM the diﬀracted beam has the same mode properties as the input beam, apart from a change in frequency and direction. All the ﬂuctuations of the light are simply transmitted, and when carefully aligned there should be no additional noise. There are many applications, such as tuning to Crystal with internal acoustic standing wave vL

Figure 5.19 Acousto-optical modulator. vL + Ωmod vL

~

Ωmod

Modulation frequency

5.4 Other Optical Components

an atomic or cavity resonance or the probing of atoms with a frequency oﬀset. AO modulators can also be used as isolators. The return beam from the experiment, for example, a cavity, will have twice the frequency shift 2ΩRF imposed by the modulator and is less likely to interfere with the laser. This retroﬂection trick can also be used to produce a beam with high frequency modulation and no directional shift and intensity modulation. 5.4.6

Spatial Light Modulators

SLMs allow the modiﬁcation and control of the optical wave front through many pixels by electronic means. They are a fairly new technology that is steadily improving in quality and are now almost omnipresent through their applications in laser scanners, data projectors, and adaptive optics. They are part of much of the optics that is being developed. SLMs allow completely new applications, such as laser movie projection [23], in the form of spatial intensity modulators, or they can modify the shape of the optical wave front in the form of spatial phase modulators, for example, to produce dynamically controlled optical tweezers and traps [24] or neurostimulators [25]. At the same time they can be used to optimize the shape of optical beams, which means they can be used as an alternative to optical phase plates to create diﬀerent spatial modes, for example, beams with orbital angular momentum, or they can dynamically improve a wave front that has been distorted by the atmosphere in the case of astronomy or by propagating through biological samples or tissue in the case of medical applications. In quantum optics laboratories SLMs are used to optimize mode matching and have been used for quantum logic and entanglement measurements. The main advantage of the SLM is the high speed and the ability to control through suitable software very complex wave fronts and intensity distributions. The commonly quoted characteristics are the number of pixels controlled, now up to several megapixels; the contrast ratio, which is important for intensity modulators and can now reach up to 100 000 or higher; and the dynamic range, in multiples of full wavelength in the case of phase modulators. The other factor is the speed, which is linked to the ability to switch one pixel, typically on the timescale of 1–100 μs. The second aspect of speed is the ability to control so many pixels simultaneously, which means transmission rates of 10 Mbit/s to 1 Gbit/s and is rapidly progressing through the use of dedicated graphics cards and optimized software. Soon speed and resolution will not be a limiting factor in the applications. There are several diﬀerent technologies: the SLMs can be used in transmission and in reﬂection. The transmission mode with very small mirrors, known as digital light projection (DLP) is widely used in display technology such as video and cinema projectors. However, it does not have the characteristics required in quantum optics due to the considerable losses. More useful are phase modulators that work either in transmission mode or deformable mirrors that operate in reﬂective mode. The liquid crystal display (LCD)-type phase modulator is the most common; it combines large pixel numbers with good access speed. Such a phase modulator can be eﬀectively programmed as a hologram and can be used

193

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5 Basic Optical Components

as variable lens, changing the focal length, compensating for beam distortions, or transforming the mode, for example, from one TEM mode to another TEM mode or from one Gauss Leguerre LG mode to another LG mode, or to create completely diﬀerent type of modes, for example, special modes for optical tweezers. This can be done with reasonable eﬃciency, and in particular when used as a grating, where the transmission into ﬁrst order or higher order is used, the contrast can be very high. For high eﬃciency applications, transformable mirrors perform far better. In particular those mirrors that have a continuous mirror surface can reach eﬃciencies above 90%. These SLMs have a low number of pixels. For reasonable cost one can get about 1000 pixels, which is suﬃcient for a number of mode matching and entanglement applications. The question arises: do SLMs add technical noise that will reduce their usefulness in quantum optics? The LCD-type modulators have considerable phase jitter that is commonly very noticeable due to the strong phase modulations created by the electronic drivers. In many normal display applications, the image on the SLM is refreshed frequently, for example, every millisecond, which means a rapid reorientation of the liquid crystals at that rate. The crystals then tend to oscillate around their equilibrium position, and this produces a strong driving of the phase setting for each pixel, leading to a large phase noise. This eﬀect can be avoided either by using very diﬀerent electronic drivers that have a very fast refresh rate that cuts the oscillation out of the frequency spectrum that is of interest or by using the SLM as a grating. The phase noise will be visible in the zero order beam. However, by observing the ﬁrst-order transmission or reﬂection, the phase noise will appear as translation noise only and thus not be detectable with the normal optical detectors. This is the reason why transmission SLMs have been used so successfully in combination with single-photon detection where optical losses are not so important [26]. For CW applications with squeezed light, the deformable mirror appears to be promising. A systematic test has been performed to show that no noticeable phase noise is present. This experiment shows that the mirror can be used to transform the mode shape of a beam of squeezed light with an eﬃciency of larger than 95% and that no detectable noise is present that would add to the quantum noise and destroy the quantum correlations [27]. That means no fundamental limits have to be considered – the right type of SLM will exist for every experiment. The future generation of SLMs will have further improved speciﬁcations, and this opens up the ﬁeld of quantum optics experiments to electronic mode matching that enhances the reliability of experiment. The controller of the SLM can use learning algorithms to iterate the many pixel settings with the single goal to optimize a chosen global parameter. An example would be to optimize fringe visibility between two interfering beams to get the best possible mode matching. In this way the combination of SLM and intelligent multidimensional controller will tune itself, and it can be an alternative to the manual alignment process performed now. Future students will have at least one SLM in each experiment and will not need to worry about frequent mirror realignment. The SLMs will also allow more complex experiments with multiple beams. They will allow the design

5.4 Other Optical Components

of reconﬁgurable layouts, for example, for quantum logic, and alternative modulation techniques for quantum communication. 5.4.7

Optical Noise Sources

Optical materials can contribute noise. This is equivalent to saying that additional light can be scattered into the mode under investigation. One example is atoms that are excited by one of the laser beams in the experiment and that spontaneously emit at the frequency and into the spatial mode that is detected. In a similar way stimulated processes can add light to the mode. These could be Raman transitions or other non-linear processes, such as 4WM or SHG. We have to be very careful in experiments that contain strong optical ﬁelds; basically any process that can generate light at the frequency of the mode we wish to detect is potentially a source of competing noise that could swamp an eﬀect as small as quantum noise. These types of noise add to the intensity ﬂuctuations in the mode. In addition, the refractive index of the optical material can be aﬀected by ﬂuctuations that will generate noise in the phase quadrature of the transmitted light. For example, a common source of noise in optical ﬁbres is guided acoustic wave Brillouin scattering, abbreviated as GAWBS. This noise occurs in any system where the wave is guided by geometrical structures such as a ﬁbre or planar waveguide. The noise spectrum shows very pronounced maxima, which can be related to the resonances of acoustic waves that are reﬂected by the boundaries of the ﬁbre. It covers such a wide spectrum that the detection of QNL light becomes very diﬃcult. Non-linear crystals can be the source of additional noise [29]. In all experiments a careful characterization of the noise spectrum has to be carried out, before we can assume that quantum noise is actually observed. Chapter 8 describes these techniques in detail. 5.4.8

Non-linear Processes

In summary, all linear components have a very simple eﬀect on the ﬂuctuations: a coherent state of light will remain a coherent state. Even for light that is not a coherent state, the eﬀects are simple; we will ﬁnd that any losses make the ﬂuctuations look more like those of a coherent state. In the limit of large losses, any input light will approach a coherent state (Figure 5.20). Quantum optics would not be much fun if there were only linear processes, but fortunately there is more. Gain combined with saturation allows us to make a laser, the all-important light source (see Chapter 6). One of the main features of non-linear processes is that they can generate additional new frequencies. In particular, SHG and degenerate optical parametric oscillation (OPO) are used to generate beams at twice or half the original frequency. The non-linear Kerr eﬀect transfers changes in the intensity to the refractive index. 4WM links modulation sidebands. The non-linear processes are too numerous to list here explicitly, and instead the extensive literature should be consulted, for example, the textbooks by M.D. Levenson and S.S. Kano [30], and R.W. Boyd [31] as the current reference. For a recent review of non-linear ﬁbre eﬀect, see the review by

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5 Basic Optical Components

40 Noise power (rel. units)

196

Figure 5.20 An example of optical noise. The noise introduced in a single-mode optical ﬁbre by guided acoustic wave Brillouin scattering (GAWBS) measured with a CW laser. Source: Courtesy of Shelby and Levenson [28].

Hitachi fibre GAWBS

20 10 8 4 2

Quantum noise

1 0

500 MHz

1000

S.M. Hendrickson et al. [22], and for non-linear metamaterials, see the colloquium review by M. Lapine and the team lead by Y.S. Kivshar and coworkers [32]. These non-linear eﬀects allow us to change the wave front of the light, leading to self-focussing or defocussing; they can also turn a single spatial mode, like a TEM modes, into a high-order mode or generate even multimode situations, all as a consequence of the interplay between high local intensities and non-linear materials. Important for quantum optics is the fact that non-linear eﬀects can change the modulation and information carried by the beam. And it can change the statistics of the light and can be used to manipulate quantum noise and thus allows to create light for measurements with better SNR than allowed by the QNL. Non-linear media are essential for the generation of photon pair, sub-Poissonian light, and squeezed and entangled light, and such experiments will be discussed, one by one, in Chapter 9.

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Cambridge University Press. 3 Born, M. and Wolf, E. (1999). The Principles of Optics. Pergamon Press. 4 Luis, A. and Sanchez-Soto, L.L. (1995). A quantum description of the beam-

splitter. Quantum Semiclass. Opt. 7: 153. 5 Edwards, P.J. (1993). Reduction of optical shot noise from light emitting

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optics in quantum Hall edge channels. Mod. Phys. Lett. B 25: 1053–1073. 7 Caves, C.M. (1981). Quantum-mechanical noise in an interferometer. Phys. Rev. D 23: 1693.

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Detection. World Scientiﬁc. 9 Siegman, A.E. (1986). Lasers. University Science Books. 10 Shaddock, D.A., Buchler, B.C., Bowen, W.P. et al. (2000). Modulation free

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metasurfaces. Rev. Mod. Phys. 86: 1093.

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6 Lasers and Ampliﬁers 6.1 The Laser Concept The laser is the key component in any quantum optics experiment. It was the invention of the laser (see Figure 6.1) that gave us access to the world of coherent optics, non-linear optics, and the quantum features of light. Without the laser we would be restricted to thermal light – missing out on most modern applications and on almost all the experiments described in this guide. In quantum optics we are concerned with the characteristics of the light generated by a laser. In particular, we will investigate the ﬂuctuations in the intensity of the light. In theoretical terms the laser is the system of choice for the realization of many quantum optics ideas: the laser light itself is close to a pure coherent state. In many cases, the laser can be regarded as the source for a single mode of radiation as described in the theory. If we consider a continuous-wave (CW) laser, the mode will have constant properties, and even pulsed lasers can be considered as constant in the sense that we describe them as a steady stream of pulses – each pulse like any other. Many books have been written about the laser while the technology constantly evolves and makes the devices more powerful and more accessible. One of the most comprehensive introductions remains the masterpiece by A.E. Siegman [1] in which he provides all the technical details required to understand and operate lasers. In simple technical terms, the laser is an optical oscillator (see Figure 6.2) that relies on feedback, much like an electronic circuit. It consists of a narrow band optical ampliﬁer, which generates a well-deﬁned output beam of light and an optical arrangement that feeds part of the light, in phase, back to the input. The ampliﬁcation is provided by a gain medium, atoms or molecules that are excited by an external source of energy, the pump. The realization that such a situation is possible was one of the early results of atomic quantization [2]. The feedback is provided by a cavity surrounding the active medium. The mirrors of the cavity generate a large internal optical ﬁeld, which is ampliﬁed during each round trip. Of course there are also losses inside the cavity, and in order to achieve suﬃcient feedback for oscillation to occur, the gain has to be larger than the losses. Without any input there would be no output, no optical feedback signal, and no laser beam. However, the presence of even a small amount of light, some small optical amplitude within the bandwidth of the resonant ampliﬁer system, A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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6 Lasers and Ampliﬁers

Figure 6.1 The original laser demonstrated by Theodore Maiman in 1960. The ruby crystal with coatings is the gain medium, the ﬂash lamp surrounding it the pump source. The pulses wore out the metal coating of this version of the laser. Source: Courtesy Welling (2015).

Pump Output beam

Feedback Output Amplifier Power (a)

Cavity with gain medium

(b)

Figure 6.2 Comparison between an (a) electronic ampliﬁer/oscillator and (b) a laser.

will initiate the process and lead to a rapid growth of the output. Spontaneous emission from atoms in their excited state can provide such a seed. The amplitude increases until the gain of the ampliﬁer saturates at a level that just balances the losses due to the mirrors and other sources. The output power will stabilize at a steady-state value, and the laser achieves constant CW operation. The conditions for lasing are twofold. Firstly, the ampliﬁer gain must be greater than the loss in the feedback system so that a net gain is achieved in each round trip. All lasers show a threshold, which means they require a minimum pump power Pthr to start steady oscillation and generate an output beam. Secondly, the total phase accumulated in a single round trip must be an integer multiple of 2π to provide phase matching to the feedback signal, i.e. the laser frequency must correspond to a cavity mode. Ignoring small frequency pulling eﬀects, which can be somewhat larger in diode lasers, the laser mode(s) will correspond with the cavity mode(s) closest to the peak of the gain proﬁle. In the simplest case this is a single ﬁxed frequency.

6.1 The Laser Concept

Above threshold the laser produces a steady output beam. The geometrical properties of the output beam are those of the spatial modes of the cavity, in the simplest case the fundamental Gaussian TEM00 beam (see Eq. (2.8)). The combination of these two cases is known as single-mode operation, both transversely (spatially) as well as longitudinally (spectrally). This is the ideal case for the study of the quantum properties of light. The intra-cavity ﬁeld se well described as one quantum mechanical mode of the radiation ﬁeld that extends throughout the cavity. The output beam can be described by a propagating mode or coherent state as described in Section 4.5. As we will see, the present technology gets us extremely close to the realization of such a perfect laser. 6.1.1

Technical Speciﬁcations of a Laser

For CW lasers we can use a wide range of lasing materials, excitation mechanisms, and cavity designs. CW lasers include gas lasers, such as the once widely used He–Ne lasers and the powerful Ar ion and CO2 lasers, to the complex tunable dye lasers. We do not have space here to describe all the technical details and numerous engineering tricks that are used in the design of these lasers. A range of specialized literature is available to provide this information [1, 3]. Nowadays, the majority are the more robust, technically almost simplistic solid-state diode lasers. The trend is to ﬁbre lasers, and various other solid-state systems all pumped by diode lasers [4]. These lasers are starting to dominate the technological applications. In the area of atomic spectroscopy, where tunability and narrow bandwidth is required, diode lasers and Ti/sapphire rapidly replace the traditional dye lasers. For short pulses solid-state lasers have allowed much shorter pulses. For complex experiments and for applications, the compactness and robustness of the diode laser are the main advantage. As a consequence most of the practical examples of laser performance in this book refer to solid-state lasers. The performance of a laser can be described by the following parameters: optical power, spatial mode structure, output wavelength, frequency spectrum, and noise spectrum. The power is quoted as the total optical power Pout of the output beam. Of practical interest is the total eﬃciency of the laser 𝜂total = Pout ∕Ppump , where Ppump is the pump power required to achieve an output power Pout . Well above threshold the power scales linearly, and the slope eﬃciency 𝜂slope = dPout ∕dPpump is constant. This is shown in Figure 6.3. Note that operation near threshold is not reliable. At that point the power is not stable; it tends to be noisy, and Pthr may not be easy to measure. The spatial properties of the laser beams are determined by the properties of the optical resonator. In most cases a stable resonator is used, and the mode analysis in terms of Hermite–Gaussian TEMm,n modes, as discussed in Chapter 2, can be used. There are a few exceptional cases, particularly for the design of high-power lasers where more complicated unstable resonators are used. However, many lasers operate in a single spatial mode, the lowest order TEM0,0 mode, which is particularly simple to describe. In practice the resonator is never a perfect optical cavity, there will always be some spatial distortions in the output beam, and a parameter Mm,n is normally quoted that describes how closely the beam is

201

6 Lasers and Ampliﬁers

Figure 6.3 Optical output power versus pump power. The laser has a threshold above which the output increases linearly with the pump power.

Pout Output power

202

Pthr

Ppump Pump power

approximated by a TEMm,n mode. The perfect match corresponds to Mm,n = 1. Some lasers produce beams that have elliptical rather than circular beam proﬁles. For these lasers it is important to know the aspect ratio of the ellipse and whether the spatial mode is simply a stretched version of a Gaussian beam or a more complex wave front. The output wavelength is determined by the properties of the gain medium and can vary from the UV (excimer laser 169 nm) to the far-infrared NH3 laser (440 μm). Many laser materials have gain for several output wavelengths. For example, the Ar ion laser has several green/blue laser transitions. The desired wavelength can be selected by optimizing the mirror coatings and by using selective intra-cavity components, such as a prism or a grating. Some lasers are tunable, that is, a range of laser wavelengths can be chosen. Some materials with a tuning range have a broad gain proﬁle – prominent examples are dye lasers and the Ti/sapphire laser; in other materials the gain proﬁle is temperature dependent. For example, in a diode laser the wavelength can be tuned by changing the temperature of the device, controlling the current, and using an extended cavity for precise control of the laser frequency. The frequency distribution of the laser output is determined by the cavity properties and the gain proﬁle. In addition, it is important whether the gain proﬁle is an inhomogeneous or homogeneous proﬁle. In the ﬁrst case diﬀerent cavity modes obtain their gain from diﬀerent active atoms, and the various modes can oscillate independently. The result is a series of spectral output modes, separated by the free spectral range of the laser resonator. In the latter case all active atoms in the laser contribute to the laser in the same way. There will be competition between the spectral modes, and normally only one will win and the output spectrum will be dominated by one single spectral mode. Such a single-mode laser still has a frequency bandwidth. The absolute frequency can change with time. The bandwidth is given as the range of frequencies that are present in a set observation time. The bandwidth is a measure of the size of the frequency ﬂuctuations. For example, suppose only frequencies within a 1 MHz interval are measured within one second interval. This information tells us that the laser could be used to resolve a detailed feature of a given spectrum with a resolution of 1 MHz and no less. However, short-term laser bandwidth

6.1 The Laser Concept

and long-term frequency drift can vary widely. The same laser might drift by 100 MHZ in 10 minutes. We would have to carry out the experiment very quickly. A more complete description of the frequency properties is given by the Allan variance, a statistical analysis of the frequency changes for a range of diﬀerent observation times [5]. We will ﬁnd that the frequency ﬂuctuations are commonly a consequence of technical imperfections, such as vibrations and electrical and acoustical noise. The quantum mechanical ﬂuctuations play hardly a role in the frequency noise. The phase of a laser is unconstrained and will drift. This is known as phase diﬀusion. This means that the variance of the phase quadrature (see Chapter 4) will grow unlimited over long time periods. On the other hand for short times, or high frequencies, the phase can be quite stable and will tend to the coherent state level at suﬃciently high frequencies. Finally, there are ﬂuctuations of the intensity of the output beam. These are described as intensity noise spectra, as the relative changes of the laser power, 𝛿Pout ∕Pout , measured with a bandwidth RBW at a frequency Ω. The measured variance of the optical power ﬂuctuations can be directly compared with the modelled variance of the amplitude quadrature V 1las (Ω) (see Chapter 4), which is normalized by the quantum noise associated with a coherent state of the same optical power. In summary, lasers are available that produce output beams, which are very close to the theoretical limit. Their light can be considered as the real-world analogue of a coherent state. But not all lasers do this. Some lasers are not built well enough and produce beams that are distorted or noisy. Some other lasers have noise that is intrinsic to the lasing process. Finally, we will ﬁnd some lasers that actually perform better than a coherent state. 6.1.2

Rate Equations

Although diﬀerent laser systems diﬀer widely in their details, it is possible to model many aspects of laser behaviour from a quite general rate equation approach. The main assumption in such an approach is that coherent atomic eﬀects are negligible. Coherent eﬀects can be neglected in the presence of strong dephasing eﬀects, such as interatomic collisions in a gas or lattice vibrations in a solid. This is an excellent approximation for all but a few lasers, such as far-infrared gas lasers, which we will not consider in this discussion. In a rate equation model, we consider the gain medium to be made up of M atoms. These atoms have internal states of excitation, with energies E1 , E2 , and E3 as shown in Figure 6.4. We assume lasing occurs between the states with energies E3 and E2 . The frequency of the laser light is centred around 𝜈L = (E3 − E2 )∕h. The actual laser frequency will be determined by the cavity resonance, taking into account all the non-linear eﬀects inside the laser resonator. The excitation of the medium is described by the populations M1 , M2 , M3 of the individual states. These populations should really be interpreted as the probabilities J1 , J2 , J3 of ﬁnding any individual atom in one of the three states. The total population is preserved, thus M = M1 + M2 + M3 = M(J1 + J2 + J3 )

(6.1)

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6 Lasers and Ampliﬁers

Figure 6.4 The semi-classical model of a three-level laser system.

E3 , J3 γ32

G Energy

204

E2 , J2 Γ

γ31 γ21

E1 , J1

A number of processes link the states. Pumping is modelled as a uni-directional transfer of population from state |E1 ⟩ to state |E3 ⟩. Normally any pumping process will also produce de-excitation of the upper pump level at a rate proportional to its population. The uni-directional pumping is to be understood as the limit of pumping to a higher state, which then decays very rapidly to state |E3 ⟩. The population of all excited states decay through spontaneous emission. Stimulated emission and absorption of photons in the laser ﬁeld couples the states |E3 ⟩ and |E2 ⟩. All these processes are described by rates. As an example, the spontaneous emission from state |E2 ⟩ to state |E1 ⟩ results in M2 𝛾21 = MJ2 𝛾21 photons emitted per second, which are leaving the laser in random directions, and a corresponding reduction in the population of state |E2 ⟩ and increase of population of state |E1 ⟩. Similarly the rate of spontaneous emission from state |E3 ⟩ is described by rate constant 𝛾32 . The pumping rate depends on the pump power, the excitation cross section, and the pumping eﬃciency. It is described by one rate M1 Γ. For simplicity, all we neglect spatial eﬀects and consider one spatial mode only. Similarly only one spectral mode, at frequency 𝜈L , is considered. All optical eﬀects are described by the gain and loss of light from this single mode. For a high ﬁnesse cavity, the total loss of the resonator is just the sum of the individual losses (see Section 5.1.7). We write the total loss 𝜅 as the sum of losses due to the output mirror 𝜅m and those due to all other mechanisms 𝜅l . If the mode inside the laser contains nt photons, then the number of photons leaving the cavity per second will be 2𝜅nt . Of these only the fraction 𝜅m ∕𝜅 exits in the output beam; the remainder are dissipated internally. The energy diﬀerence between the atomic states |E3 ⟩ and |E2 ⟩ corresponds to a frequency that is resonant with the mode. Stimulated emission occurs at a rate G32 J3 nt , and stimulated absorption at a rate G32 J2 nt . The rate constant G32 is deﬁned as G32 = 𝜎s 𝜌c∕nr

(6.2)

where 𝜎s is the stimulated emission cross section for the |E3 ⟩ to |E2 ⟩ transition, 𝜌 is the density of lasing atoms in the medium, c is the speed of light, and nr is

6.1 The Laser Concept

the refractive index. The two processes produce an overall change of the photon number of G32 (J3 − J2 )nt (photons per second). The inversion (J3 − J2 ) drives this ampliﬁcation process. At this point we ignore the small contribution to the photon number due to spontaneous emission into the lasing mode. Combining all these individual rates, we can write a system of four coupled rate equations that describe the dynamics of the laser: dJ1 = −ΓJ1 + 𝛾21 J2 dt dJ2 = G32 (J3 − J2 )n + 𝛾32 J3 − 𝛾21 J2 dt dJ3 = −G32 (J3 − J2 )n − 𝛾32 J3 + ΓJ1 dt dn (6.3) = G32 (J3 − J2 )n − 2𝜅n dt where n is the photon number per atom such that nM = nt . We can also write the equation of motion of the cavity ﬁeld (per atom) as d𝛼c (6.4) = G32 (J3 − J2 )𝛼c − 𝜅𝛼c dt where 𝛼c∗ 𝛼c = n. The steady state solution for the photon number per atom, n, can be obtained by setting the time derivatives to zero. We ﬁnd either n = 0 or n=

Γ(𝛾21 − 𝛾32 ) 𝜅 Γ𝛾21 + 𝛾21 𝛾32 + Γ𝛾32 − 2𝜅(2Γ + 𝛾21 ) G32 2𝜅(2Γ + 𝛾21 )

Provided we have 𝛾21 > 𝛾32 , there is a threshold pump rate, Γt , above which the non-zero solution for n will be positive and stable. This represents laser oscillation while the zero solution is unstable. For pump rates below the threshold, the non-zero solution is negative, and it is the zero solution that is stable. That is, the laser does not oscillate. Physically that means the losses in the resonator, per round trip, are larger than the gain so that any light introduced into the cavity is quickly damped away. Generally the lifetime of the lower lasing level will be very short so we can set 𝛾21 ≫ 𝛾32 , Γ. Under this condition the population of the second level can be neglected, and the ground state can be considered undepleted (J2 = 0, J1 = 1). Our equations of motion reduce to dJ3 = −G32 J3 𝛼c∗ 𝛼c − 𝛾32 J3 + Γ dt d𝛼c (6.5) = G32 J3 𝛼c − 𝜅𝛼c dt and we obtain the following simpler solution for the output photon ﬂux per atom of the laser, nout : ( ) 𝜅m 2𝜅 nout = 2𝜅m n = Γ − 𝛾32 (6.6) G32 𝜅

205

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6 Lasers and Ampliﬁers

The pump threshold is Γt = steady state, 2𝜅 J3 = G32

2𝜅 𝛾 . G32 32

We also have that above threshold, in the (6.7)

and the following alternative form of Eq. (6.6) can be written 𝜅 (6.8) nout = (Γ − 𝛾32 J3 ) m 𝜅 That is, the rate at which laser photons leave the cavity is simply the rate at which atoms are placed in the upper lasing level minus the rate at which photons are lost through spontaneous emission. Threshold occurs when pumping exceeds the loss due to spontaneous emission. The useful output is scaled by the fraction of photons that leave via the cavity mirror and hence depends on the internal losses. The total eﬃciency of the laser, 𝜂total , depends on this fraction, the relative pump power Ppump /Pthr , as well as the pumping eﬃciency. Very close to threshold the performance is obviously very dependent on ﬂuctuations. The laser can switch on and oﬀ, and our simple theoretical model would have to be extended to include such eﬀects. For reliable operation the actual pump rate Γ should be considerably higher than Γt , which means Ppump > Pthr . Above the threshold the output power increases linearly with the pump rate (see Figure 6.3). Hence the slope eﬃciency, 𝜂slope , is constant above threshold, depending only on the internal losses and pump eﬃciency. At even higher pump powers, the output power can level oﬀ. Theoretically this occurs when the pump rate begins to deplete the ground state. In practice this tends to happen at lower pump powers than a simple estimate would suggest, since a number of additional non-linear eﬀects might occur that can have power limiting back eﬀects. For example, laser materials show thermal expansion, due to absorption, which can lead to a non-uniform refractive index and thus thermal lensing. This and other power-induced wave front distortions introduce additional losses in the resonator, which limit the internal power. An alternative possible conﬁguration of the laser is where the lasing transition occurs between state |E2 ⟩ and the ground state, state |E1 ⟩, with a stimulated emission cross section proportional to G21 . Important examples of this type of laser are the solid-state Ruby and Erbium lasers. The analysis proceeds as before. Making the assumption now that state |E3 ⟩ decays very rapidly, we get the following solution for the output photon ﬂux per atom: ) ( 𝜅m 1 2𝜅 (Γ − 𝛾21 ) nout = 2𝜅m n = (6.9) Γ − 𝛾21 − 2 G21 𝜅 The threshold pump rate is given by Γt =

1+

2𝜅 G21

1−

2𝜅 G21

𝛾21

(6.10)

The major diﬀerence between this laser system and the previous one is the threshold behaviours. Whilst in the previous case, the pump threshold could in principle be made arbitrarily small by reducing the ratio G2𝜅 , that is, increasing the 32

6.1 The Laser Concept

cavity ﬁnesse or active atom density, in this case the threshold cannot become less than the spontaneous emission rate between the laser levels, 𝛾21 . Physically this is because we have to at least half empty the ground level before we can create an inversion. 6.1.3

Quantum Model of a Laser

In order to determine the dynamics of the ﬂuctuations of the laser System, a quantum model is required. In the quantum description the laser consists of a laser medium of M atoms with internal states |E1 ⟩, |E2 ⟩, |E3 ⟩, a pump source with pump rate Γ and noise spectrum V 1P (Ω), and an internal optical mode â with an expectation value of 𝛼c . Operators Ĵ1 , Ĵ2 , Ĵ3 describe the collective excitation of the atoms. As in the semi-classical model, given in the previous section, the expectation values of these operators, J1 , J2 , J3 , respectively, give the probability of ﬁnding any one atom in the respective state. Figure 6.5 shows the laser system in this notation. Making the simplifying assumptions that led to Eq. (6.5) (i.e. negligible second-level population and an undepleted ground state), the quantum operator Langevin equations of motion can be written as √ √ √ dĴ3 ̂ p − Γ 𝜹X ̂ 𝛾 + G32 â † â 𝜹X ̂P = −G32 Ĵ3 â † â − 𝛾32 Ĵ3 + Γ + 𝛾32 𝜹X dt √ √ √ dâ G32 ̂ ̂ l + 2𝜅m 𝜹A ̂ m − G32 Ĵ3 𝜹A ̂ †p (6.11) = J3 â − 𝜅 â + 2𝜅l 𝜹A dt 2 where all terms with ‘𝛿’ preﬁxes are vacuum ﬁelds, with zero expectation values. The equations of motion of the expectation values correspond to the semi-classical rate equations (Eq. (6.5)). All the processes are described in a similar way – now by using operators to describe the states of the atoms and the single mode of the light. In addition, the full quantum model has to contain operators describing all possible noise sources. These include the pump ̂ P , and with an intensity noise spectrum noise, characterized by the operator 𝜹X 2 ̂ m , and ̃ V 1P (Ω) = ⟨|𝜹XP | ⟩, the vacuum input through the cavity mirror, 𝜹A ̂ the vacuum input due to other intra-cavity losses, 𝜹Al . The rapid phase decay

|3〉, σ3

Energy

G

V1las |2〉, σ2

Γ

V1vac

V1pump

γ32

γ31 γ21

|1〉, σ1

Figure 6.5 The quantum model of a three-level laser.

V1spont V1dipole V1losses

207

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6 Lasers and Ampliﬁers

of the atomic coherence due to collisions with atoms or, in the case of solids, lattice-induced disturbances introduces dipole ﬂuctuations that can be charaĉ p + 𝜹A ̂ †p . Finally, noise in the populations due ̂ p = 𝜹A terized by the operator 𝜹X ̂ 𝛾 . The eﬀects of to spontaneous emission has to be included via the operator 𝜹X all these noise sources appear explicitly in the ﬂuctuations of the output operator ̃ las . The derivation of Eq. (6.11) is quite involved and beyond the scope of this A book. Details of the derivation and the identiﬁcation of the noise operators with their various physical origins can be found in Ref. [6]. The steady-state solutions for the probabilities J3 and photon number per atom, n, in the quantum model, obtained when all the time derivatives are set to zero, are identical to those obtained from the semi-classical Eq. (6.5). We proceed by ̂ linearizing around these stable solutions (see Section 4.5.1). Writing â = 𝛼c + 𝜹a and Ĵ3 = J3 + 𝜹Ĵ3 and only keeping terms to ﬁrst order in the vacuum ﬁelds, we obtain the following linearized equations for the quantum ﬂuctuations: √ d𝜹Ĵ3 ̂ a − (𝛾32 + G32 𝛼c2 )𝜹Ĵ3 + 𝛾32 𝜹X ̂𝛾 = −G32 J3 𝛼c 𝜹X dt √ √ ̂ p − Γ 𝜹X ̂P + G32 𝛼c2 𝜹X √ √ √ ̂ G d𝜹a ̂ l + 2𝜅m 𝜹A ̂ m − G32 J3 𝜹A ̂ †p (6.12) = 32 𝛼c 𝜹Ĵ3 + 2𝜅l 𝜹A dt 2 √ where we have taken 𝛼c = n real without loss of generality (see Section 6.1.5). These equations can be solved in Fourier space. The internal mode is coupled to the external √ ﬁeld (see Section 5.3 – resulting in the frequency space ̃ las = 2𝜅m ã − 𝜹A ̃ m , describing the ﬂuctuations of the output ﬁeld. operator, A The result of interest is the noise spectrum V 1las (Ω) of the photon number, or intensity, of this mode. Under our linearization condition this is given by the ̃ A |2 ⟩ where amplitude quadrature spectrum of the output mode: V 1las (Ω) = ⟨|𝜹X † ̃ ̃ ̃ 𝜹XA = 𝜹Alas + 𝜹Alas . We obtain the following solution for the system of transfer functions: ) ( 4𝜅m2 ((2πΩ)2 + 𝛾L2 ) − 8𝜅m 𝜅G32 n𝛾L Vm V 1las (Ω) = 1 + ((2πΩRRO )2 − (2πΩ)2 )2 + (2πΩ)2 𝛾L2 + + + +

2 2𝜅m G32 nΓ

((2πΩRRO )2 − (2πΩ)2 )2 + (2πΩ)2 𝛾L2 2 (2𝜅m G32 n𝛾32 J3 )

((2πΩRRO )2 − (2πΩ)2 )2 + (2πΩ)2 𝛾L2 2𝜅m G32 ((𝛾32 + Γ)2 + (2πΩ)2 ) ((2πΩRRO )2 − (2πΩ)2 )2 + (2πΩ)2 𝛾L2 4𝜅m 𝜅l (𝛾L2 + (2πΩ)2 ) ((2πΩRRO )2 − (2πΩ)2 )2 + (2πΩ)2 𝛾L2

VP (Ω) V𝛾 Vp Vl

(6.13)

where 𝛾L = G32 n + 𝛾32 + Γ, (2π ΩRRO )2 = 2G32 n𝜅, and the solutions for n and J3 are given by Eqs. (6.6) and (6.7). It should be noted that the parameters have to be chosen separately for each laser. The material constants 𝜎s , 𝜌, nr , 𝛾32 , and

6.1 The Laser Concept

𝛾21 are speciﬁc to the type of laser and reasonably well known. In particular the rapid decays from the highly excited states and the lower lasing level for most lasers justify the use of this simpliﬁed three-level model. Other parameters such as the resonator parameters 𝜅l and 𝜅m and the total number of lasing atoms vary with the individual laser. How close to a coherent state with V 1las (Ω) = 1 can this solution be? The general answer is: many lasers approach this limit if they are operated well above threshold. However, there are important diﬀerences for the various types of lasers. We will now discuss a number of typical cases focusing especially on solid-state and diode lasers. 6.1.4 6.1.4.1

Examples of Lasers Classes of Lasers

The ﬁnesse of the laser resonator, determined by the magnitude of 𝜅, is generally chosen to achieve optimum laser eﬃciency. The ﬁnesse is controlled by changing the transmission of the output mirror (𝜅m ). Reducing the mirror transmission lowers the laser threshold and hence acts to increase eﬃciency. However, it also 𝜅 reduces the ratio 𝜅m due to the presence of internal losses (𝜅l ), which decreases the eﬃciency. Thus for a particular laser material, there will be an optimum value of 𝜅. Three classes of lasers with distinctly diﬀerent dynamical behaviours can be identiﬁed on the basis of the relationship between the sizes of 𝜅 and the decay rate of the upper lasing level (𝛾32 ). They are: • Class 1: 𝛾32 ≫ 𝜅 – e.g. dye lasers. • Class 2: 𝛾32 ≈ 𝜅 – e.g. visible gas lasers. • Class 3: 𝛾32 ≪ 𝜅 – e.g. solid-state and semiconductor lasers. We will now examine the predicted behaviour of examples from each of these classes, focusing particularly on the class 3 solid-state and semiconductor lasers. A fourth class of lasers, mentioned earlier, in which the photon decay rate is more rapid than the coherence dephasing rate cannot be described via rate equations and will not be discussed here. 6.1.4.2

Dye Lasers and Argon Ion Lasers

There is a wide range of possible lasers that are used in quantum optics experiments. In particular when the non-linearity is a resonant atomic or molecular medium, it is important to have a laser that can be tuned on or arbitrarily close to the resonance. While even that can be achieved nowadays with diode lasers, using external cavities, the traditional work horse in these situations is the argon ion laser pumped dye laser. In these devices the active medium is a dye solution, made up of large, complex dye molecules that can provide gain continuously over a wide range of optical frequencies. The actual laser frequency is selected with a number of frequency-dependent components inside the laser resonator. Typically a dye laser will have parameters of 𝜅 ≈ 1.5 × 107 1/s and 𝛾32 ≈ 3 × 108 1/s and hence is a class 1 laser. Provided we are operating more than about ﬁve times above threshold, we can approximate the noise spectrum (6.13) by V 1las (Ω) = 1 +

(2𝜅)2 (V 1P − 1) (2𝜅)2 + (2πΩ)2

(6.14)

209

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This is a Lorentzian spectrum with a half width equal to the cavity decay rate. At low frequencies the spectrum will be dominated by the noise of the pump laser. At high frequencies (2πΩ ≫ 2𝜅), the spectrum will be quantum noise limited (QNL). In order to obtain QNL operation from a dye laser at frequencies in the tens of MHz, a QNL pump laser is needed. A dye laser is pumped by an Ar ion laser, providing large optical powers (more than 10 W) at wavelengths in the blue end of the visible spectrum. Typically an argon laser will have 𝜅 ≈ 1.0 × 106 1/s and 𝛾32 ≈ 2 × 106 1/s and hence is a class 2 laser. Suﬃciently far above threshold a class 2 laser has a noise spectrum very similar to that of a class 1 laser (see Eq. (6.14)). Due to the slower cavity decay rate and hence narrower pump (electric discharge) noise region, we would expect the noise spectrum of the argon laser to be QNL at frequencies of tens of MHz. However, due to the small gain that the Ar plasma provides, these lasers have long cavities and thus in the presence of inhomogeneous broadening frequently run on several longitudinal modes. The output spectrum of such a pump laser would hence be dominated by strong spectral features with a spacing of typically 100 MHz. This noise from an Ar ion laser will appear on the intensity noise spectrum of the dye laser. It is possible to suppress the higher-order modes using intra-cavity etalons but only with a large power penalty. For quantum optics experiments, which require a laser beam that is as close as possible to the quantum limit, the noise properties of the Ar ion and dye lasers have posed considerable problems. Most experiments were carried out at speciﬁc detection frequencies, in between the large noise spikes, where the noise spectrum happened to be QNL. In recent years the progress in the development of high-power, low noise diode lasers and diode laser pumped solid-state lasers has been so remarkable that now dye lasers are avoided as much as possible. 6.1.4.3

The CW Nd:YAG Laser

We will now discuss the most popular solid-state laser, the Nd:YAG laser. Gain is provided by excited neodymium ions in a yttrium aluminium garnet glass substrate. Traditionally the neodymium ions were excited by a ﬂash lamp, but it is becoming more common these days to use diode laser arrays as the pump source. In a miniature monolithic ring conﬁguration, these lasers can be made to perform very close to the ideal single-mode laser described by our rate equation (6.3). Thus a careful quantitative comparison between theory and experimental results is warranted here. Typically solid-state lasers have 𝜅 ≈ 107 − 108 1/s depending on their size and 𝛾32 ≈ 102 − 103 1/s and hence are class 3 lasers. For a class 3 laser, all the terms in the noise spectrum (Eq. (6.13)) remain important even when operating up to 10 times above threshold and the spectrum is quite complex. It contains contributions from the pump noise V 1P (Ω), spontaneous emission, dipole ﬂuctuations, and vacuum ﬂuctuations. The mixture of these contributions varies with the detection frequency. It is instructive to plot these various contributions separately. This is done in Figure 6.6a, where the various contributions are described in the ﬁgure caption. There are some very clear features, and the spectrum is dominated by a resonant feature, called the resonant relaxation oscillation that occurs at the frequency √ 2π ΩRRO = 2G23 n𝜅 (6.15)

Noise power relative to QNL (dB)

6.1 The Laser Concept

80 60 (v)

40 (iv) 20

(vi)

(i)

QNL

0

0.01

0.1

(a)

Noise power relative to QNL (dB)

(ii)

(iii) –20 0.001

1

10

100

500

10

30

Frequency (MHz)

75 (i) 55

(ii)

35

15

(iv)

(iii) QNL

–5 0.001 (b)

0.01

0.1

1

Frequency (MHz)

Figure 6.6 Noise spectrum of a free-running Nd:YAG laser. (a) The diﬀerent contributions to Eq. (6.3) are drawn separately. Trace (i) is the contribution of the vacuum noise, (ii) is the eﬀect of the pump noise, (iii) is the noise due to spontaneous emission, (iv) is the noise associated with the dipole ﬂuctuations, (v) is the noise due to intra-cavity losses, and (vi) is the total noise for a QNL pump. Part (b) shows experimental results for such a Nd:YAG laser. In this realistic case (i) the pump noise level was about 40 dB above the QNL. For comparison the predicted noise performance is shown in (ii). Trace (iii) was obtained with a pump noise of 20 dB. Trace (iv) is the corresponding theoretical prediction. Source: Adapted from Harb et al. [7].

In this particular case ΩRRO = 400 kHz. This oscillation is due to the interaction of the atomic state inversion (J3 − J2 ) ≈ J3 with the excitation of the internal optical mode. In physical terms, the energy in the laser is moving back and forth between the laser medium and the cavity mode, resulting in a strong modulation of the output power. This oscillation is a prominent feature and corresponds to an underdamped oscillation that is continually excited by the noise sources within the laser [8]. Note that this oscillation does not occur in class 1 and 2 lasers because for them 𝛾L ≫ ΩRRO , making the oscillation overdamped, as can

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be seen from the denominator in Eq. (6.13). An alternative expression for ΩRRO in terms of the pump rate is √ √ Γ − Γt (6.16) 2π ΩRRO = 2𝜅𝛾32 Γt That means the harder the laser is pumped, increasing Γ, the higher is the frequency of the relaxation oscillation and the less pronounced it is. This can be seen by comparing the traces in Figure 6.7 that are plotted for the same laser but with diﬀerent pump powers. In some situations ﬁtting the measured frequencies ΩRRO for a range of pump powers is a more reliable way to determine Γt than an attempt to measure the power at which the laser turns on. Above ΩRRO the laser has a long, drawn-out noise tail. For very high frequencies the laser approaches the quantum noise limit. The origin of the quantum noise can be traced back to the various vacuum inputs in the model. It should be noted that the noise of the pump source has no inﬂuence on the laser noise for frequencies Ω ≫ ΩRRO . A noisy pump will not aﬀect the laser at such high frequencies. However, it also means that in this frequency regime the laser output power cannot be controlled by modulating the pump power. This is certainly a disadvantage for communication systems, since pump power modulation is normally the simplest and cheapest modulation scheme. At low frequencies, Ω ≪ ΩRRO , the noise spectrum of the laser is dominated by the noise of the pump source. There is a strong transfer from the pump to the laser output. By carefully choosing the noise properties of the pump, the laser output can be quietened down. Figure 6.6b shows two such cases. The ﬁrst case, traces (i) and (ii), is realistic for a laser with large pump noise. Here V 1p (Ω) is a factor 10 000, or 40 dB, above the standard quantum limit. The second case, –20

129

170

205 266

–40 Power (dBm)

318 360 413

78 kHz

–60 455 –80

1

0.1

–100 0.01

212

Frequency (MHz)

Figure 6.7 Experimental noise trace for a Nd:YAG laser with various pump powers. The relaxation oscillation frequency ΩR is stated for each case in units of kHz. It increases with pump power as stated in Eq. (6.16). Source: Adapted from Harb et al. [7].

6.1 The Laser Concept

traces (iii) and (iv), was measured with a quieter pump, V 1p (Ω) = 20 dB. Note that even for the case of a quantum-limited pump (V 1p (Ω) = 0 dB), the resonant relaxation oscillation would survive. Only for the impossible case of a completely noise-free pump source (V 1P (Ω) = 0) and no cavity losses or dipole ﬂuctuations would the relaxation oscillation completely disappear. 6.1.4.4

Diode Lasers

Perhaps the simplest and most eﬃcient lasers are semiconductor or diode lasers. Diode lasers use the band gap in doped semiconductor materials to produce optical gain directly from an electrical current. Under suitable conditions diode lasers can be described by our rate equation (6.3) as class 3 lasers. Major diﬀerences with solid-state lasers arise from their small size and high gain. Typically a diode laser will have 𝜅 ≈ 1011 1/s and will be run 5–10 times above threshold. Under these conditions the RRO will be pushed out to frequencies of tens of GHz and will be quite small. Most detection schemes are only sensitive up to frequencies of a few GHz; thus for all practical purposes the spectrum can be written as 𝜅 (6.17) V 1las (Ω) = m [1 + 𝜁 (V 1P (Ω) − 1)] 𝜅 where it is understood that 2πΩ ≪ 𝜅 and 𝜁 is the pump eﬃciency. Moreover, unlike the situation of the dye laser where the pump source is a potentially very noisy Ar ion laser, the pump source of the diode laser is an electric current, which can be made very quiet. Indeed we can have V 1las (Ω) = 1 if the pump current is given by V 1P (Ω) = 1. This means that for a quiet Poissonian noise-limited pump current, a diode laser has a QNL output. Equation (6.17) also means the laser output responds directly to modulation of the pump source. This is important for communication applications where we want to use the current driving the diode laser as the input of modulation. In fact, there is no fundamental reason why we cannot have V 1P (Ω) < 1, that is, a sub-Poissonian current. Unlike light, electric currents can easily be made sub-Poissonian. This would imply V 1las (Ω) < 1; the laser output would be sub-Poissonian. Diode lasers have successfully been made to produce squeezed light using this method, and technical details are discussed in Chapter 9. From Eq. (6.17) we see that, at least in theory, the squeezing is limited only by the internal losses and the pump eﬃciency 𝜁 . 6.1.4.5

Limits of the Single-Mode Approximation in Diode Lasers

From the previous discussion of diode lasers, it would appear that, provided sufﬁcient care was taken to suppress noise in the pump current, these lasers would always run at or below the QNL. For a number of reasons, in practice this is not the case. In particular, multimode behaviour can lead to increased noise. Normally one expects single-mode operation from a homogeneously broadened medium, such as that of a diode laser, due to gain competition. However, this is not the whole story; if there is strong spatial dependence in the laser mode, some regions of the gain medium can become gain depleted, whilst others do not, allowing modes with diﬀerent spatial dependence to access the undepleted gain and get above the lasing threshold. Competition between these modes can lead to elevated noise levels. An example is the high-power diode array, consisting

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of a strip of diode gain medium sharing a common cavity. The cavity can support a number of transverse modes, many of which lase simultaneously due to spatial hole burning, thus introducing excess noise. Further, the far-ﬁeld spatial patterns of the diﬀerent modes are diﬀerent, such that diﬀerent spatial regions of the beam can have diﬀerent noise properties. Even in a purely homogeneously broadened medium, spontaneous emission into the lasing mode, which is neglected in our rate equation (6.3), can allow for the existence of weakly lasing longitudinal side modes [9]. The homogeneity of the medium means that the production of a photon in the main mode implies no photon is concurrently emitted in the side modes and vice versa. Hence there is a strong anti-correlation between the side modes and the main mode. If just the main mode is observed, it is found to be very noisy due to the deletion of photons to the side modes [10]. However, the anti-correlation means that the noise tends to cancel when all the light is observed. Unfortunately, diﬀerences in the losses experienced by the main mode and the side modes can lead to a small amount of excess noise, which does not cancel. This can be suﬃcient to destroy squeezing. This problem has been solved using injection locking or external cavities to suppress the side modes. In this way reliable squeezing has been obtained – the details of these experiments are given in Chapter 9. In summary we ﬁnd that lasers themselves are interesting but in many cases complex light sources. Only the single-mode laser operating well above threshold will be QNL at all detection frequencies. For all other cases we have to explicitly derive the intensity noise spectrum or measure it directly. 6.1.5

Laser Phase Noise

In Section 6.1.2 we solved for the steady-state values of the intra-cavity photon number, n, and the upper laser level population, J3 . We also introduced the intra-cavity complex amplitude 𝛼c . Notice however that it is not possible to solve for the phase of 𝛼c . Physically this indicates that the laser phase is metastable and will tend to drift in time. This eﬀect is known as phase diﬀusion. When we linearized the quantum equations of motion to obtain the equations of motion for the ﬂuctuation operators (Eq. (6.12)), we took the phase of 𝛼c to be constant and real. This is of no consequence for the intensity spectrum because ̂ are phase insensitive. However, we need to be ̂ † A) intensity measurements (A more careful if considering phase-sensitive measurements. We can calculate the phase ﬂuctuation spectrum of the laser output from ̃ las |2 ⟩ (6.18) V 2las (Ω) = ⟨|𝜹X𝟐 ̃ las = i(𝜹A ̃ las − 𝜹A ̃ † ). The phase ﬂuctuation spectrum can be where as usual 𝜹X𝟐 las obtained by making a phase-sensitive measurement of the ﬁeld with respect to a stable phase reference, which means using homodyne detection described in Section 8.1.4. Our assumption of constant classical phase (𝛼c real) will only be valid for the phase spectrum if the ﬂuctuations around that constant phase are small. The solution is given by V 2las (Ω) =

(2πΩ)2 + 2(2𝜅)2 (2πΩ)2

(6.19)

6.2 Ampliﬁcation of Optical Signals

At very high frequencies the phase spectrum tends to the quantum noise level, V 2las (Ω) = 1. As we move to lower and lower frequencies, the phase noise becomes larger and larger, eventually diverging at zero frequency. The divergence of the phase noise spectrum at zero frequency indicates that our linearization breaks down for suﬃciently small Fourier frequencies. On long enough timescales, or low enough frequencies, the phase drifts signiﬁcantly, thus invalidating the linear approximation. This is the signature of laser phase diﬀusion. In the absence of technical noise, this phase diﬀusion determines the laser linewidth. In practice phase diﬀusion does not present a major problem as generally the same laser is used as a phase reference as is used to produce the signal. The phase drift is then common to both modes and so is not observed. Thus provided we stay within the coherence length of our laser (see Section 2.3.3 and work at frequencies for which the laser’s intensity noise is at the QNL, a laser beam can represent an excellent realization of an ideal coherent state. 6.1.6

Pulsed Lasers

The focus in this section is primarily CW lasers since they are very common in quantum optics experiments, in precision metrology. The steady optical power is necessary to avoid unwanted non-linear eﬀects and the related laser noise. They are also easy to explain and comprehend. However, the ﬁrst laser demonstrated was pulse, and much of the research work and many applications use pulsed lasers, ranging from nanosecond (ns) down to femtosecond (fs) pulses for commercial systems and attosecond (as) pulse lengths for the highness optical intensities required in research applications. This advance of laser technology uses non-linear processes and the elegant concept of beam pulse compression and expansion, as ﬁrst shown in the late 1980s [11]. These techniques have revolutionized the use of lasers and were honoured by the award of the Nobel prize to Donna Strickland and Gerard Mourou in 2018. In all of these systems, the optical ampliﬁcation is similar, and the optical resonator shares some commonalities. Most of the pulsed lasers have a high repetition frequency Ωrep , frequently, around 100 MHz for femtosecond lasers, and thus a typical applications uses many of these pulses to detect the desired eﬀect. Each pulse has a high peak power, which is useful to drive non-linear eﬀects, and this is key reason to use them in applications such as two- and three-photon imaging, materials processing, ionization, etc. The signal-to-noise ratio however of a measurement is normally measured at signal frequencies that are much lower than the repetition frequency Ωsig < Ωrep , and the inﬂuence of the noise is at low frequencies as well and basically identical to the case of a CW laser. Clear examples of this for the use of pulsed lasers in quantum optics are shown in Section 9.5.

6.2 Ampliﬁcation of Optical Signals It is possible to use an optical medium to amplify the power of a laser beam. For example, we can imagine an input power of 1 mW and an output power

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of 1 W, which means a power gain G of a factor of 1000. A medium that can achieve this purpose is called an active optical medium. Normally it is just a laser medium without optical feedback from a cavity. Such a device would obviously be very useful for practical applications where high power is required. Some of the biggest and most expensive optical systems are the Nd:glass ampliﬁers that have been built to generate powerful laser pulses for fusion studies using inertial conﬁnement [12]. Another application is in optical communications where the beam is attenuated over distance, for example, by losses inside an optical ﬁbre. In this case it is necessary to amplify not only the power but also the modulation sidebands, which contain the information that is communicated. In a ﬁbre system an in-line ampliﬁer, such as an erbium-doped ﬁbre ampliﬁer, is a very useful and attractive device. The gain bandwidth should extend to the largest modulation sideband that is used. Typically this is a few GHz, which is not diﬃcult to achieve. The next question is: how does an optical ampliﬁer aﬀect both the signal and the noise? Does the signal-to-noise ratio vary? It might be easiest to understand the properties of optical ampliﬁers by ﬁrst describing the simpler and more obvious case of an electronic ampliﬁer. Consider a circuit with a linear ampliﬁer that has a power gain Gel (Ω). The gain is frequency dependent; usually Gel (Ω) drops to less than unity at high frequencies. The input into the ampliﬁer is a current with modulation at Ωm . In linear systems we can consider each Fourier component, each modulation frequency, separately and independently of all the others. Ideally the input and output are related by √ 𝛿iout (Ω) = Gel (Ω)𝛿iin (Ω) The variances of modulations, or ﬂuctuations, of the input and the output are linked by Var(iout ) = Gel

Var(iin )

(6.20)

This ampliﬁer aﬀects both the signal and the noise in the same way. In principle, the signal-to-noise ratio will be maintained. In practice a small additional noise component will be added by the ampliﬁer. However there is no fundamental limit to how small this added noise can be. As long as the added noise is much smaller than the noise ﬂoor of the input, the signal-to-noise ratio is maintained. The performance of this system could be improved further if the ampliﬁer is given a reference signal, which is at the same frequency and phase locked to the modulation signal. In this case a signiﬁcant part of the noise, which has random phase, can be suppressed by this apparatus, normally called a lock-in ampliﬁer. The performance of an optical ampliﬁer is quite diﬀerent [13–15]. The input and the output modes can be described in the Heisenberg picture by the operators â in and â out . Suppose our input state is a coherent state, |𝛼⟩. We expect the coherent amplitude to be ampliﬁed such that √ 𝛼out = ⟨̂aout ⟩𝛼 = G 𝛼in This suggests the following operator description: √ â out = G â in

6.2 Ampliﬁcation of Optical Signals

However, this violates the commutation rule [̂aout , â †out ] = G [̂ain , â †in ] = G ≠ 1 This problem is similar to the quantum theory of damping, where a careful consideration of the role of quantum noise has to be made. The vacuum state entering through any unused port has to be taken into account. See the description of a beamsplitter in Section 5.1 for details. In a similar way we have to include the vacuum state in this ampliﬁcation process. The correct description is [16] √ √ â out = G â in + G − 1 v̂ † (6.21) where v̂ is an operator describing an input mode originally in the vacuum state that is mixed in by the ampliﬁer. It is completely independent of â in . The result is that the light, after ampliﬁcation, is considerably noisier than before. The ampliﬁer is increasing not only the power but also the noise level. The variance of the intensity of the ampliﬁed light is V 1out = G V 1in + (G − 1) V 10

(6.22)

Here V 10 = 1 is at the QNL. For large gains (G ≫ 1), the second term results in an increase of an extra 3 dB compared to the classical case (Eq. (6.20)). As shown in Figure 6.8, both the average amplitude and the ﬂuctuations increase. To see that this is indeed the eﬀect of ampliﬁcation by a laser medium, consider again the quantum mechanical equations of motion for the laser (Eq. (6.11)). Suppose that we are below threshold such that 𝜅 > G32 J3 ∕2 but that instead of ̂ in , is injected. Further let us assume vacuum entering the output mirror, a ﬁeld, A that we are suﬃciently below threshold, that there is very little depletion of the upper lasing level, and that we can therefore replace the operator Ĵ3 with its expectation value J3 . We obtain the following equation of motion: √ √ √ G dâ ̂ l + 2𝜅m A ̂ †p ̂ in − G32 J3 𝜹A (6.23) = 32 J3 â − 𝜅 â + 2𝜅l 𝜹A dt 2 Equation (6.23) is now linear and can be solved exactly in Fourier space. Neglecting internal losses (𝜅l = 0) leads to the following solution for the output from the ampliﬁer: √ 2𝜅G32 J3 𝜅 + G32 J3 ∕2 + i2πΩ ̃ in − ̃ †p ̃ out = (6.24) A 𝜹A A 𝜅 − G32 J3 ∕2 + i2πΩ 𝜅 − G32 J3 ∕2 + i2πΩ Figure 6.8 Eﬀect of optical gain on a quantum state.

X2

X1

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The spectrum for either quadrature is then V iout (Ω) =

(𝜅 + G32 J3 ∕2)2 + (2πΩ)2 2𝜅G32 J3 V iin + (𝜅 − G32 J3 ∕2)2 + (2πΩ)2 (𝜅 − G32 J3 ∕2)2 + (2πΩ)2

which is of the same form as Eq. (6.22) with G(Ω) =

(𝜅 + G32 J3 ∕2)2 + (2πΩ)2 (𝜅 − G32 J3 ∕2)2 + (2πΩ)2

We see that in this case the additional quantum noise is introduced via the dipole ﬂuctuations of the gain medium. The 3 dB penalty is imposed by the optical ampliﬁer when it acts equally on all quadratures. Such an ampliﬁer is called phase insensitive. This penalty can be avoided by using an ampliﬁer that is phase sensitive, as will be discussed in the next section.

6.3 Parametric Ampliﬁers and Oscillators Phase-insensitive ampliﬁcation is of limited utility when dealing with quantum ﬁelds because of the excess noise it introduces. We would really like to be able to amplify our quantum-limited light beam without adding additional noise. Let us consider the fundamental commutation problem again. If we write the annihilation operator in terms of the amplitude and phase quadratures, we see the problem clearly: 1 ̂ ̂ + iX𝟐) (6.25) â = (X𝟏 2 If both quadratures are ampliﬁed equally, then â is just multiplied by some gain factor, and via the arguments of the last section, it is clear that an additional ﬁeld, inevitably introducing noise, must be mixed in by the ampliﬁer. However, consider √ the situation in which the amplitude quadrature is ampliﬁed by some amount, K, but the phase quadrature is de-ampliﬁed by the same amount. The output operator would now look like ( ) i ̂ 1 √ ̂ K X𝟏 + √ X𝟐 (6.26) â out = 2 K It is straightforward to conﬁrm that the commutation relations still hold in this case and that this is a legitimate transformation and no additional noise operators are required. Thus, provided that the product of the gains of the two quadratures is unity, noiseless ampliﬁcation of one of the quadratures can be achieved. This is a very important quantum optical interaction. In analogy with the linear ampliﬁer, we can also write Eq. (6.26) in the form √ √ (6.27) â out = G â in + G − 1 â †in where the gain is G=

(K + 1)2 4K

6.3 Parametric Ampliﬁers and Oscillators

For historical reasons a phase-sensitive ampliﬁer is referred to as a parametric ampliﬁer. 6.3.1

The Second-Order Non-linearity

Classically the response of an atomic medium to an incident optical beam can be described by the induced dipole polarization P(E) caused by the beam’s electric ﬁeld E: P(E) = 𝜒E + 𝜒 (2) E2 + 𝜒 (3) E3 + · · ·

(6.28)

A non-linear optical material is one in which, at the available ﬁeld strength, higher than linear terms in the expansion become signiﬁcant. With the invention of the laser, incident ﬁeld strengths became suﬃciently high that many materials could be made to show non-linear behaviour. We will ﬁnd that phase-sensitive ampliﬁcation and other interesting quantum optical phenomena can be produced by materials exhibiting a second-order 𝜒 (2) non-linearity. The key feature of the second-order non-linearity is its ability to couple a fundamental ﬁeld (oscillating at 𝜈) to a harmonic ﬁeld (oscillating at 2𝜈). The Hamiltonian, or energy function, for such an interaction can be written as H = iℏ𝜒 (2) (𝛽 ∗ 𝛼 2 − 𝛼 ∗ 2 𝛽) where 𝛼 is the fundamental whilst 𝛽 is the harmonic ﬁeld. Canonical quantization then leads to the quantum mechanical Hamiltonian ̂ ̂ = iℏ𝜒 (2) (b̂ † â 2 − â †2 b) H This Hamiltonian has an elegant interpretation in terms of photons. The ﬁrst term represents the annihilation of two photons at the fundamental frequency and the creation of one at the harmonic, whilst the second represents the reverse process of a single harmonic photon being annihilated and two fundamental photons being created. The Heisenberg equations of motion (see Section 4.4) are then given by 1 ̂ ̂ H] = 𝜒 (2) â 2 b̂̇ = [b, iℏ â̇ = −2 𝜒 (2) â † b̂

(6.29)

Notice that because the process is energy conserving (and coherent), i.e. two photons of energy h𝜈 equals one of energy h 2𝜈, then no additional vacuum modes are required to be coupled into the ﬁelds. In order that either the up-conversion (fundamental to harmonic) or the down-conversion (harmonic to fundamental) process takes place with some reasonable eﬃciency, we require not only energy conservation but also that momentum is conserved for the converted beam in some particular direction(s). In this way the emission from many dipoles can coherently add to produce a single output beam. This condition is known as phase matching. Let us consider the situation shown schematically in Figure 6.9. A non-linear crystal is placed inside a pair of cavities, one resonant at the fundamental and the other resonant at the harmonic, which are oriented such that there is good phase

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M1 Ain

vA

M2 ζ

αA

Detunings

αB

Aout

κA κB

Δa

Δb

vB =

1 v 2 A

κℓa κℓb

κa = total loss at vA

κb = total loss at vB

Figure 6.9 A parametric medium inside a cavity. The various input, output, and intra-cavity modes are labelled. This system has a threshold and can operate below threshold as a phase-sensitive ampliﬁer of the vacuum modes and above threshold as an oscillator. The subscript A refers to the fundamental frequency, and the subscript B refers to the subharmonic frequency.

matching between the two spatial modes deﬁned by the cavities. By combining the cavity equations of motion derived in Section 5.3.6 with those of Eq. (6.29), we obtain the equations of motion for the internal cavity modes as √ √ (6.30) b̂̇ = 𝜒 (2) â 2 − 𝜅b b̂ + 2 𝜅B B̂ in + 2𝜅lb 𝜹B̂ l â̇ = −2 𝜒 (2) â † b̂ − 𝜅a â +

√ √ ̂l ̂ in + 2𝜅la 𝜹A 2𝜅A A

(6.31)

where 𝜅A and 𝜅B are the loss rates of the input/output mirrors for the â and b̂ modes, respectively. Similarly 𝜅la and 𝜅lb are the respective internal loss rates for the two modes and 𝜅a = 𝜅A + 𝜅la , 𝜅b = 𝜅B + 𝜅lb . We also have the respective input ̂ in and B̂ in and the respective vacuum modes that couple in due to the modes A ̂ l and 𝜹B̂ l . Equations (6.30) and (6.31) can describe a large range internal losses 𝜹A of important quantum optical experiments. 6.3.2

Parametric Ampliﬁcation

Let us now consider parametric ampliﬁcation. We will assume that the harmonic ﬁeld is an intense ﬁeld, which is virtually undepleted by its interaction with the non-linear crystal, whilst the fundamental ﬁeld is ‘small’. We expect down-conversion under these conditions with energy being transferred from the harmonic to the fundamental ﬁeld. We shall also assume that 𝜅a ≪ 𝜅b such that the dynamics of the harmonic mode can be neglected. Under these conditions we can replace the operator b̂ with the c-number 𝛽. We are then left with just a single, linear equation for the â mode: √ √ ̂l ̂ in + 2𝜅la 𝜹A (6.32) â̇ = −𝜒 â † − 𝜅a â + 2𝜅A A where

√ 𝜒 = 2𝜒 (2) 𝛽 = 2𝜒 (2)

2 𝛽 𝜅b in

(6.33)

6.3 Parametric Ampliﬁers and Oscillators

and 𝛽in = ⟨B̂ in ⟩. The parameter 𝜒 is intensity dependent. Equation (6.32) can be solved in Fourier space to give the output mode at the fundamental frequency ̃ out = A

𝜒 2 + 𝜅a2 + (2πΩ)2 2𝜒𝜅a ̃ in + ̃† A A 2 2 (𝜅a + i2πΩ) − 𝜒 (𝜅a + i2πΩ)2 − 𝜒 2 in

(6.34)

Here we have neglected internal losses to the fundamental mode (𝜅la = 0) for simplicity. Equation (6.34) is formally equivalent to Eq. (6.27). The variance of the amplitude quadrature is given by V 1out (Ω) =

(𝜅a2 + 2𝜒𝜅a + 𝜒 2 + (2πΩ)2 )2 𝜒 4 + (𝜅a2 + (2πΩ)2 )2 − 2𝜒 2 (𝜅a2 − (2πΩ)2 )

V 1in (Ω)

(6.35)

where V 1in is the amplitude quadrature variance of the input at the fundamental. The output is an ampliﬁed version of the input amplitude quadrature with no added noise as hoped. This can be seen more clearly at zero frequency where the expression reduces to V 1out (0) =

(𝜅a + 𝜒)2 V 1in (0) (𝜅a − 𝜒)2

(6.36)

As the harmonic intensity is increased 𝜒 becomes larger, the denominator of Eq. (6.36) becomes smaller, and hence the ampliﬁcation increases. On the other hand the phase quadrature (at zero frequency) is given by V 2out (0) =

(𝜅a − 𝜒)2 V 2in (0) (𝜅a + 𝜒)2

(6.37)

which suﬀers increasing de-ampliﬁcation as 𝜒 increases. Notice that if V 2in is at the QNL, then V 2out will be below the QNL, which means V 2out < 1. This is termed squeezing and is discussed in detail in Chapter 9. 6.3.3

Optical Parametric Oscillator

Examining Eq. (6.36) we note that the spectrum diverges as the value of 𝜒 approaches that of 𝜅a . This indicates that an approximation is failing, in this case that of negligible pump depletion. As the ampliﬁcation of the harmonic ﬁeld becomes larger and larger, eventually it starts to signiﬁcantly deplete the harmonic. To see what happens in this case, we must go back to Eqs. (6.30) and (6.31) and linearize. We continue to consider the experimental situation where the cavity for the harmonic mode is either only weakly resonant or even non-existent, that is, we eﬀectively have 𝜅a ≪ 𝜅b . As a result the evolution of the mode b̂ occurs on a much faster timescale than that of the â mode, and we can adiabatically eliminate it by setting its time derivative to zero. We also continue to assume that 𝜅lb ≪ 𝜅b . That means the losses for the mode b̂ can be neglected and Eqs. (6.30) and (6.31) reduce to √ (2)2 √ √ 2 ̂ †̂ ̂l ̂ in + 2𝜅la 𝜹A ̂ † â 2 − 2𝜒 (2) ̂ + 2𝜅A A ̂̇ = − 2𝜒 𝜹a 𝛿a 𝜹a Bin − 𝜅a 𝜹a 𝜅b 𝜅b

221

222

6 Lasers and Ampliﬁers

To proceed we ﬁrst consider the semi-classical steady-state solution of this equation. By taking expectation values, and making the approximation that they may be factored, we obtain √ 2 √ 2 𝜒 (2) ∗ 2 2 ∗ 𝛼̇ = − 𝛼 𝛼 − 2𝜒 (2) 𝛼 𝛽in − 𝜅a 𝛼 + 2𝜅A 𝛼in (6.38) 𝜅b 𝜅b The steady-state solution is obtained by setting the time derivative to zero. If we assume that the ﬁrst term on the RHS of Eq. (6.38) is small and can be neglected, then we obtain the parametric ampliﬁcation considered in the previous section. Instead we now retain that term but consider the situation in which there is no input ﬁeld at the fundamental (𝛼in = 0). The equation that needs to be solved is √ 2 2𝜒 (2) ∗ 2 2 ∗ (2) − 𝛼 𝛼 − 2𝜒 𝛼 𝛽in − 𝜅a 𝛼 (6.39) 𝜅b 𝜅b Equation (6.39) has two solutions: either 𝛼 = 0 or 𝛼 ≠ 0. We recognize this as threshold behaviour similar to that of the laser. At a suﬃciently high value of the harmonic pump ﬁeld, the system will start to oscillate and produce coherent output at the fundamental. Under these conditions the device is called an optical parametric oscillator. If 𝛼 ≠ 0 we have √ 2𝜅b 𝜅 𝜅 ∗ 𝛼 𝛼 = − (2) 𝛽in e−i2𝜙𝛼 − a (2)b 2 (6.40) 𝜒 2𝜒 where 𝜙𝛼 is the phase of 𝛼. To be a valid stable solution, 𝛼 ∗ 𝛼 must be real and positive. This is only possible if 𝜙𝛽 − 2𝜙𝛼 = π. For simplicity we take 𝛽in real and negative (𝜙𝛽 = π); then 𝛼 will be real, and the solution for the output photon number is 𝜅 𝜅 nout = 2𝜅a 𝛼 2 = a(2)b2 (𝜒 − 𝜅a ) (6.41) 𝜒 where we have used the deﬁnition of 𝜒 from Eq. (6.33). When 𝜒 < 𝜅a the 𝛼 = 0 solution will be stable; when 𝜒 > 𝜅a then Eq. (6.41) gives the photon ﬂux of the output ﬁeld at the fundamental. Notice that the threshold condition is the same as the point of divergence of the noise spectrum in the previous section. 6.3.3.1

Noise Spectrum of the Parametric Oscillator

We can now investigate the ﬂuctuations of the above threshold fundamental output by linearizing around the semi-classical solution. By linearizing Eq. (6.34) we obtain the following equation of motion for the ﬂuctuations: √ (2)2 2 ̂ † ̇ = − 2 𝜒 𝛼 2 (𝜹a ̂ † + 2𝜹a) ̂ − 2𝜒 (2) ̂𝛿a (𝜹a 𝛽in + 𝛼 𝜹B̂ in ) 𝜅b 𝜅b √ √ ̂ in + 2𝜅la 𝜹A ̂l ̂ + 2𝜅A 𝜹A (6.42) −𝜅a 𝜹a This equation can be solved in Fourier space to give the quadrature ﬂuctuations of the output ﬁeld: √ 2 2𝜅a (𝜒 − 𝜅a ) 4𝜅a − 2𝜒 − i2πΩ ̃ ̃ ̃ Bin 𝜹X𝟏out = 𝜹X𝟏Ain + 𝜹X𝟏 2𝜒 − 2𝜅a + i2πΩ 2𝜒 − 2𝜅a + i2πΩ √ 2 2𝜅a (𝜒 − 𝜅a ) 2𝜅a − 2𝜒 − i2πΩ ̃ out = ̃ Ain + ̃ Bin 𝜹X𝟐 (6.43) 𝜹X𝟐 𝜹X𝟐 2𝜒 + i2πΩ 2𝜒 + i2πΩ

6.3 Parametric Ampliﬁers and Oscillators

where we have neglected internal losses of the fundamental for simplicity ̃ Ain and 𝜹X𝟐 ̃ Ain are the amplitude and phase quadrature ﬂuctuations and 𝜹X𝟏 entering at the fundamental input/output mirrors, respectively. Notice also that non-negligible pump depletion now couples ﬂuctuations from the pump ̃ Bin and 𝜹X𝟐 ̃ Bin , into the fundamental ﬁeld. The quadrature harmonic ﬁeld, 𝜹X𝟏 variances at zero frequency are V 1(0) = 1 +

𝜅a2 (𝜒 − 𝜅a )2

V 2(0) = 1 −

𝜅a2 𝜒2

(6.44)

where we have assumed that the harmonic pump ﬂuctuations are at the QNL. The coupling of the harmonic ﬂuctuations means that the ampliﬁcation of the amplitude quadrature is no longer noiseless above threshold. Strong noise reduction (or squeezing), that is, V 2 ≪ 1, is still present on the phase quadrature close to threshold. However, it becomes negligible far above threshold where the output tends towards a coherent state.

6.3.4

Pair Production

Finally let us return to the parametric ampliﬁer operating far below threshold for another important application of the 𝜒2 interaction. Far below threshold, with no input at the fundamental, we may form the following physical picture: the harmonic beam is mostly transmitted without change through the non-linear medium; however every now and then a harmonic photon is converted into a pair of fundamental photons. This picture suggests that if we counted photons at the fundamental output under these conditions, we should only ever see them arrive as pairs. To test this proposition, we consider the Schrödinger evolution of the fundamental far below threshold where its initial state is the vacuum. The evolution of the fundamental mode is not passive (energy conserving) as energy is being added; however it is unitary (reversible) as no noise is being added. This means we cannot directly use the techniques described in Chapter 4 but we can use a modiﬁed version. Let us work in the Fourier domain and use the fact that we can ̃A ̃ † |0⟩ that is clearly still the vacwrite the vacuum state in the following way: A uum state as the action of creation and then annihilation nullify each other. This motivates us to write the output state as ̃A ̃ † |0⟩ ̃ ̃A |𝜙⟩ = U|0⟩ =U ̃ †U ̃U ̃A ̃ ̃A ̃ † )(U ̃ † )(U|0⟩) = (U ̃ ′A ̃ ′† |𝜙⟩ =A

(6.45)

Equation (6.45) shows that the output state is the eigenstate of the operator ̃ ′A ̃ ′† with eigenvalue ‘one’. Here the dashes indicate reverse time HeisenỸ = A berg evolution (see Section 4.4). Solving this eigenvalue equation leads to (at worst) a second-order recurrence relation for linear or linearized systems.

223

224

6 Lasers and Ampliﬁers

Inverting Eq. (6.34) and its corresponding conjugate equation, we obtain † ̃ ̃ ̃ ̃ ̃ − pq∗ A(Ω) Y(Ω) = |p|2 A(Ω) A(Ω) A(−Ω) ∗ ̃ †̃ † 2̃ ̃ −p qA(−Ω) A(Ω) + |q| A(−Ω)† A(−Ω)

(6.46)

where 𝜒 2 + 𝜅a2 + (2πΩ)2 (𝜅a + i2πΩ)2 − 𝜒 2 2𝜒𝜅a q= (𝜅a + i2πΩ)2 − 𝜒 2

p=

(6.47)

Note the presence of creation and annihilation operators for both upper and lower sidebands in the result (see Section 4.5.5). Solving the eigenvalue equation under the assumption that 𝜒 ≪ 𝜅a leads to the following output state for a particular frequency component: |𝜙⟩Ω = |0⟩ +

2𝜒𝜅a 𝜅a2

+ (2πΩ)2

|1⟩Ω |1⟩−Ω

(6.48)

As expected, most of the time ‘vacuum in’ results in ‘vacuum out’. However, occasionally a pair of photons is produced: one in the upper sideband and one in the lower sideband, as required for energy conservation. Given that the input state is a superposition of all frequencies, we may write the total output state as |𝜙⟩T = |0⟩ +

∫

dΩ

2𝜒𝜅a 𝜅a2

+ (2πΩ)2

|1⟩Ω |1⟩−Ω

(6.49)

Operating in this conﬁguration the system is often referred to as a down-converter or photon pair generator. The technical speciﬁcations have been improved steadily via a variety of new non-linear media, engineering of the non-linear crystals and novel optical arrangements. The main emphasis is on improving the brightness of the source, measured as the number of pairs per second, the suppression of individual photons and other noise sources, and the quality of the optical modes of the pairs. Optical down-converters and related systems are presently the workhorses of single-photon experiments, as described in Chapters 12 and 13. The quality of the entanglement that has been measured is a direct indication of the quality of the source of photon pairs.

6.4 Measurement-Based Ampliﬁers Ampliﬁcation of quantum states can also be implemented via techniques that employ partial measurement of the state followed by feedforward of the measurement results. We will refer to such devices as measurement-based ampliﬁers. These can be divided into two distinct classes: (i) deterministic and (ii) heralded. Deterministic measurement-based ampliﬁers implement ampliﬁcation techniques like those discussed in the previous section. However, they avoid the need for in-line, non-linear crystals or other gain media. This can be quite an advantage as coupling into and out of non-linear media can be ineﬃcient and imperfections in the media can lead to unwanted technical noise. In contrast,

6.4 Measurement-Based Ampliﬁers

heralded measurement-based ampliﬁers can beat the quantum limits discussed in the previous section. This is allowed because these devices are probabilistic in their operation. Nevertheless successful operation is heralded by particular measurement results. Hence such devices can be useful for certain applications.

6.4.1

Deterministic Measurement-Based Ampliﬁers

We consider ﬁrst a noiseless, phase-sensitive ampliﬁer. The idea here is to amplify one particular quadrature whilst maintaining the signal-to-noise ratio of any signals on that quadrature (i.e. not adding additional noise). Initially we are not interested in what happens to the other quadrature. A brute force measurement-based technique is the complete detection and re-emission of the light using high eﬃciency detectors and LEDs [17, 18]. The disadvantage here is that no phase reference or coherence exists between the input and output beams. An alternative, which does not require any non-linear process whilst its output still retains optical coherence, was developed and demonstrated by Lam et al. using electro-optic feedforward [19]. This scheme is based on partial detection of the light with a beamsplitter and detector (see Figure 6.10A). The light reﬂected from the beamsplitter is detected, and the photocurrent is ampliﬁed and fed forward to the transmitted beam via an electro-optic modulator (EOM). The electronic gain and phase can be chosen such that the intensity modulation signals carried by the input light are ampliﬁed, whilst the vacuum ﬂuctuations that enter through the empty port of the beamsplitter are cancelled, thus making the ampliﬁcation noiseless. Since not all of the input light is destroyed, the output is still coherent with the input beam. Here a half-wave plate controlled the transmittivity, 𝜀1 , of the polarizing beamsplitter. On the in-loop beam, a high eﬃciency (𝜀2 = 0.92 ± 0.02) photodetector was used. The photocurrent is then passed through multiple stages of RF ampliﬁcation and ﬁltering to ensure suﬃcient gain. An amplitude modulator is formed by using the EOM in conjunction with a polarizer, and ﬁnally the ﬂuctuation spectrum of the output is measured using detector Dout and a spectrum analyser. This system can be modelled with the linearized operator approach as used for the electro-optic noise eater (see Section 8.3). We obtain the following transfer functions for the spectrum of the output beam normalized to the QNL: √ |√ |2 Vout (Ω) = 𝜀3 | 𝜀1 + 𝜆 (1 − 𝜀1 )𝜀2 | Vin (Ω) | | √ |√ |2 + 𝜀3 | (1 − 𝜀1 ) − 𝜆 𝜀1 𝜀2 | V1 | | |2 | √ + 𝜀3 |𝜆 (1 − 𝜀2 )| V2 | | + (1 − 𝜀3 )V3 (6.50) where the electronic gain 𝜆(𝜔) is in general a complex number. Vin (Ω) is the amplitude noise spectrum of the input ﬁeld. The vacuum noise spectra due to the beamsplitter V1 , the in-loop detector eﬃciency V2 , and the out-of-loop losses V3 are shown separately to emphasize their origins. All vacuum inputs are QNL, i.e.

225

6 Lasers and Ampliﬁers

ε2

Dil

Filter

Amplifier Dout

PBS Signal

ε1 λ/2

EOM AM

δv1

(A)

ε3

–73 Input signal

2.8 dB (i)

–78 Measured noise power (dBm)

226

(ii)

–83 –75 Output signal

(c)

2.6 dB

–80 (a)

–85 (b)

–90 5 (B)

10

15

20

25

Frequency, f (MHz)

Figure 6.10 (A) Schematic of the feedforward experiment. Dil , in-loop detector; Dout , out-of-loop detector; PBS, polarizing beamsplitter; 𝜆∕2, half-wave plate; AM, amplitude modulator. (B) Top: noise spectra of the squeezed input beam. Bottom: noise spectra of the output beam. (a) Direct detection of the input light after lossy (86%) transmission. (b) Output noise spectrum without feedforward (eﬀectively 98.6% loss). (c) With optimum feedforward gain the signal and noise are ampliﬁed with little loss of SNR. Source: After Lam et al. [19].

V1 = V2 = V3 = 1. Due to the opposite signs accompanying the feedback parameter 𝜆 in Eq. (6.50), it is possible to amplify the input noise (ﬁrst term) while cancelling the vacuum noise from the feedforward beamsplitter (second term). The third and fourth terms of Eq. (6.50) unavoidable experimental √ represent √ losses. In particular if we choose 𝜆 = 1 − 𝜀1 ∕ 𝜀1 𝜀2 , the vacuum ﬂuctuations from the beamsplitter, V1 , are exactly cancelled. Then, under the optimum condition of unit eﬃciency detection and negligible out-of-loop losses (𝜀2 = 𝜀3 = 1),

6.4 Measurement-Based Ampliﬁers

we ﬁnd Vout (Ω) =

1 V (Ω) 𝜀1 in

(6.51)

That is, the ﬂuctuations are noiselessly ampliﬁed by the inverse of the beamsplitter transmittivity. Hence the system ideally can retain the SNR for a signal gain of G = 1∕𝜀1 . This system was used to demonstrate low noise ampliﬁcation of amplitude squeezed light (see Chapter 9). The second harmonic output from a singly resonant frequency doubling crystal (Section 9.4.3) provided the input beam. The top half of Figure (6.10B) shows the input noise spectra. Trace (i) shows the QNL, and trace (ii) is the noise spectrum of the input light. Regions where (ii) is below (i) are amplitude squeezed. The maximum measured squeezing of 1.6 dB is observed in the region of 8–10 MHz on a 26 mW beam. A small input modulation signal is introduced at 10 MHz that has SNRin = 1.10 ± 0.03. Other features of the spectra include the residual 17.5 MHz locking signals of the frequency doubling system (see Section 9.4.3 and the low-frequency roll-oﬀ of the photodetector, introduced to avoid saturation due to the relaxation oscillation of the laser. The bottom half of Figure 6.10B shows the noise spectra obtained from the single output detector. The input beam is made to experience 86% downstream loss. As trace (a) shows, the SNR is greatly reduced by the attenuation. Signal ampliﬁcation can be performed by ﬁrst introducing a beamsplitter of reﬂectivity 90%. This further attenuates the output beam to 𝜀tot = 𝜀1 𝜀3 = 0.014. With no feedforward gain, as trace (b) shows, the modulation signal is completely lost, and the trace itself is QNL to within 0.1 dB over most of the spectrum. Finally, by choosing the optimum feedforward gain, trace (c) shows the ampliﬁed input signal with SNRout = 0.82 ± 0.03 and G = 9.3 ±0.2 dB. Note that both (b) and (c) are of the same intensity; hence the output signal is now signiﬁcantly above the QNL. This means the ampliﬁed output is far more robust to losses than the input. The results are best described by the transfer coeﬃcient Ts = SNRout ∕SNRin . The spectra in Figure 6.10B corresponds to a value of Ts = 0.75 ± 0.02, in good agreement with the theoretical result of Ts = 0.77. This is to be compared with the performance of an ideal phase-insensitive ampliﬁer, with similarly squeezed light, of Ts ≈ 0.4. Signal ampliﬁcation via feedforward is a versatile technique that can be adapted to various diﬀerent applications. For example, by replacing the homodyne detector in the feedforward loop with a dual homodyne detection set-up, one can implement phase-insensitive linear ampliﬁcation. In particular, now the light reﬂected oﬀ the beamsplitter is split again on a 50 : 50 beamsplitter and sent to two homodyne detectors, one of which measures the amplitude quadrature and the other the phase quadrature. The signals are fed forward to amplitude and phase modulators, respectively, beam. By choosing the √ in the transmitted √ feedforward gains to both be 𝜆 = 1 − 𝜀1 ∕ 𝜀1 𝜀2 , one can obtain (in the unit eﬃciency limit) the input/output relation √ â out =

1 â + 𝜖1 in

√

1 − 1 v̂ † 𝜖1

(6.52)

227

228

6 Lasers and Ampliﬁers

Given the substitution G = 𝜖1 , Eq. (6.52) becomes formally equivalent to Eq. 1 (6.21). Hence quantum-limited phase-insensitive ampliﬁcation can be implemented in this way without needing a laser gain medium (and avoiding various technical noise sources associated with gain media). The vacuum mode, v̂ , leading to the quantum ampliﬁcation noise, is coupled in via the dual homodyne beamsplitter in this case. An experimental demonstration of this technique was carried out by Josse et al. at the University of Erlangen [20]. Coherent state sideband states at 14.3 MHz on a 1064 nm Nd:YAG laser were ampliﬁed using this method, achieving transfer coeﬃcients within about 10% of the theoretical limit (see Eq. (6.22)) and in good agreement with the theory when ﬁnite detection eﬃciency is included. For many applications we are only interested in the ampliﬁcation of the quantum state carried on the sidebands (see Section 4.5.5), i.e. the signal. However, if we are considering an optical beam with a bright carrier, an unavoidable consequence of this signal ampliﬁcation technique is the reduction in intensity of the carrier beam. It turns out that injection locking can be used to amplify the intensity of the output beam without aﬀecting the signal or noise (see Section 8.5). When the signal frequency is much larger than the linewidth of the laser cavity, the output spectrum of the injection-locked ﬁeld is the same as the input spectrum (Vout = Vin ). Because the injection-locked output is of a higher optical intensity, we can thus regain the optical intensity lost by the feedforward ampliﬁcation scheme. This was demonstrated by E. H. Huntington et al. [21] using the output of the feedforward set-up to injection lock a Nd:YAG nonplanar ring oscillator. A transfer coeﬃcient of Ts = 0.88 ± 0.05 with signal gain of G = 9.3 ± 0.2 dB was obtained for a range of output powers from 14 to 350 mW. Hence with this set-up it is possible to independently vary the amount of signal and carrier ampliﬁcation with minimal loss of signal-to-noise ratio. 6.4.2

Heralded Measurement-Based Ampliﬁers

So far we have seen how measurement-based ampliﬁers can mimic the performance of quantum-limited ampliﬁers but without the technical diﬃculties associated with non-linear optical media. We now consider a situation in which the measurement result obtained heralds whether the ampliﬁcation has been successful or not. Because such an ampliﬁer is nondeterministic, i.e. only certain results herald success, it is able to exceed the quantum limits that apply to deterministic ampliﬁers. Consider a device that performs the transformation |𝛼⟩⟨𝛼| → 𝜌(𝛼) = P|g𝛼⟩⟨g𝛼| + (1 − P)|0⟩⟨0|

(6.53)

where g is a real number obeying |g| > 1, |𝛼⟩ is an optical coherent state with complex amplitude 𝛼, and P is a probability. The RHS of Eq. (6.53) is a mixed state, but we assume a heralding signal identiﬁes which term in the output density operator has been produced by any particular run of the device: the ﬁrst term being produced by a successful run and the second term by an unsuccessful run. Thus with probability P the coherent amplitude of the input is ampliﬁed

6.4 Measurement-Based Ampliﬁers

whilst keeping the heralded state a pure coherent state. The quantum limits discussed in the previous sections would clearly be violated if this occurred with P = 1. What probability of success is allowed? The linearity of quantum mechanics requires that the distinguishability of two quantum states cannot be increased by any transformation. This condition is equivalent to requiring that the magnitude of their overlap does not decrease under any transformation. Consider the input states |0⟩ and |𝛼⟩. We require |⟨0|𝛼⟩|2 ≤ ⟨0|𝜌(𝛼)|0⟩ = P|⟨0|g𝛼⟩|2 + 1 − P. ′ 2 Inserting the values of the overlaps using |⟨0|𝛼 ′ ⟩|2 = e−|𝛼 | and taking the limit of small 𝛼, we ﬁnd 1 (6.54) P≤ 2 g We conclude that provided P is bounded according to Eq. (6.54), then the transformation of Eq. (6.53) is physically allowed for small coherent states. The transformation of Eq. (6.53) was discussed by Ralph and Lund who referred to it as heralded noiseless linear ampliﬁcation (NLA) [22]. They proposed a method for realising NLA based on single-photon sources, single-photon detection, and linear optics. The simplest NLA circuit is shown schematically in Figure 6.11a. The optical mode to be ampliﬁed is interacted with a single-photon ancilla as shown. This interaction is a generalization of the quantum scissors introduced by D. Pegg et al. [23]. The interaction is successful if one and only one photon is counted at one or other of the indicated ports. |ψ〉 |0〉

|0〉

Ns p l i t t e r

Tˆ |ψ〉

Ns p l i t t e r

A A

A

〈0|

〈0|

(a)

|in〉 A

=

a″

a′

〈1|〈0| or 〈0|〈1|

a′″

|0〉

|out〉

η |1〉 (b)

Figure 6.11 Schematic of the noiseless linear ampliﬁer. (a) The interaction labelled ‘A’ interacts a single-photon ancilla with an input beam as shown. The upper beamsplitter is 50 : 50, whilst the lower beamsplitter has transmission 𝜂 as shown. The interaction succeeds if a single count is recorded at a′ and no count at a′′ , or vice versa. (b) The N-splitter is an array of beamsplitters that evenly divides the input beam. The second N-splitter coherently recombines the beams and is considered to have succeeded if no light exits through the other ports, as determined by photon counters. Each arm contains the interaction shown in (a).

229

230

6 Lasers and Ampliﬁers

Consider the eﬀect of this device on a small input coherent state, i.e. |𝜓⟩ = |𝛼⟩, where we assume 𝛼 ≪ 1. The action of the generalized quantum scissor is to truncate the coherent state to ﬁrst order and simultaneously amplify it. Speciﬁcally, detection of a single photon at output port a′ and zero photons at output port a′′ , or detection of a single photon at output port a′′ and zero photons at output port a′ , produces the transformation ) √ √ ( 2 𝜂 1−𝜂 † − |𝛼|2 |𝛼⟩a′ → e (6.55) â 𝛼 |0⟩a′′′ 1± 2 𝜂 where the plus sign corresponds to the former case and the minus sign to the latter case. If the latter case occurs, the phase ﬂip can be corrected by feeding forward to a phase shifter. Equation (6.55) is approximately equivalent to 1

(1−g 2 )|𝛼|2

(6.56) |𝛼⟩ → 𝜂 2 e− 2 |g 𝛼⟩ √ where g = (1 − 𝜂)∕𝜂, provided that g𝛼 ≪ 1 holds. Thus for 𝜂 < 1∕2 we have g > 1, and hence the device in Figure 6.11a achieves NLA as per Eq. (6.53). The probability of success is given by the norm of the state P = 𝜂e−(1−g

2

)|𝛼|2

≈

1 g2 + 1

(6.57)

in agreement with the condition of Eq. (6.54). This device can be scaled up such that it can amplify larger coherent states as depicted in Figure 6.11b; however the complexity of the set-up increases rapidly, and the probability of success decreases rapidly [22]. The basic operation of the NLA in the low photon number regime was ﬁrst demonstrated by G.Y. Xiang et al. [24], F. Ferreyrol et al. [25], and A. Zavatta et al. [26]. The ﬁrst two publications implemented the quantum scissor approach described above. However, Zavatta et al. used a diﬀerent approach based on single-photon subtraction and addition that was able to noiselessly amplify somewhat larger photon number states with high ﬁdelity. The NLA has been proposed as a tool for improving various quantum information protocols, and such demonstrations are discussed in Chapter 13.

6.5 Summary In this chapter we have examined the theory of laser oscillators, the conditions under which real lasers behave like the simpliﬁed theory, and the characteristics of the laser light under those conditions. We conclude that well-built lasers can produce output beams that behave approximately like coherent states. We call these QNL lasers or beams. We considered the theoretical limits to ampliﬁcation of quantum limited light and discussed the theory of the non-linear 𝜒2 interaction. We found a general description of this process and considered the speciﬁc examples of parametric ampliﬁcation, parametric oscillation, and pair production. We also discussed measurement-based quantum ampliﬁers based on electro-optic feedforward and heralded devices based on photon counting. The latter type could circumvent

References

the quantum limits to ampliﬁcation because they are nondeterministic. Experiments based on examples discussed in this chapter and many variations on these themes will feature strongly in the following experiments in quantum optics.

References 1 2 3 4 5

6 7

8 9

10 11 12

13 14 15 16 17

18

19

Siegman, A.E. (1986). Lasers. University Science Books, and many reprints. Einstein, A. (1917). Zur Quantentheorie der Strahlung. Phys. Z. 18: 121. Saleh, B.E.A. and Teich, M.C. (2007). Fundamentals of Photonics. Wiley. Koechner, W. (1999). Solid State Laser Engineering, 5e. Springer-Verlag. Allan, D.W. (1987). Time and frequency (time-domain) characterization, estimation, and prediction of precision clocks and oscillators. ICEE Trans. Ultrason. Ferroelectr. Freq. Control 6: 647. Ralph, T.C. (2003). Squeezing from lasers. In: Quantum Squeezing (ed. P. Drummond and Z. Ficek), 141. Springer-Verlag. Harb, C.C., Ralph, T.C., Huntington, E.H. et al. (1997). Intensity-noise dependence of Nd:YAG lasers on their diode-laser pump source. J. Opt. Soc. Am. B 14: 2936. Yariv, A. (1989). Quantum Electronics. Wiley. Inoue, S., Ohzu, H., Machida, S., and Yamamoto, Y. (1992). Quantum Correlations between longitudinal-mode intensities in a multi-mode squeezed semi-conductor laser. Phys. Rev. A 46: 2757. Marin, F., Bramati, A., Giacobino, E. et al. (1995). Squeezing and inter-mode correlations in laser diodes. Phys. Rev. Lett. 75: 4606. Strickland, D. and Mourou, G. (1985). Compression of ampliﬁed chirped optical pulses. Opt. Commun. 56: 219. Di Nicola, J.M., Bond, T., Bowers, M. et al. (2018). The national ignition facility: laser performance status and performance quad results at elevated energy. Nucl. Fusion 59, 3. Haus, H.A. and Mullen, J.A. (1962). Quantum noise in linear ampliﬁers. Phys. Rev. 128: 2407. Caves, C. (1982). Quantum limits on noise in linear ampliﬁer. Phys. Rev. D 26: 1817. Haus, H.A. (2000). Electromagnetic Noise and Quantum Optical Measurements. Springer-Verlag. Vogel, W. and Welsch, D.-G. (1994). Lectures on quantum optics. Berlin: Akademie Verlag. Goobar, E., Karlsson, A., and Björk, G. (1993). Experimental realization of a semiconductor photon number ampliﬁer and a quantum optical tap. Phys. Rev. Lett. 71: 2002. Roch, J.-F., Poizat, J.-Ph., and Grangier, P. (1993). Sub-shot-noise manipulation of light using semiconductor emitters and receivers. Phys. Rev. Lett. 71: 2006. Lam, P.K., Ralph, T.C., Huntington, E.H., and Bachor, H.-A. (1997). Noiseless signal ampliﬁcation using positive electro-optic feedforward. Phys. Rev. Lett. 79: 1471.

231

232

6 Lasers and Ampliﬁers

20 Josse, V., Sabuncu, M., Cerf, N.J. et al. (2006). Universal optical ampliﬁcation

without nonlinearity. Phys. Rev. Lett. 96: 163602. 21 Huntington, E.H., Lam, P.K., Ralph, T.C. et al. (1998). Independent noiseless

signal and power ampliﬁcation. Opt. Lett. 23: 540. 22 Ralph, T.C. and Lund, A.P. (2009). Nondeterministic noiseless linear ampliﬁ-

cation of quantum systems. AIP Conf. Proc. 1110: 155. 23 Pegg, D.T., Phillips, L.S., and Barnett, S.M. (1998). Optical state truncation by

projection synthesis. Phys. Rev. Lett. 81: 1604. 24 Xiang, G.Y., Ralph, T.C., Lund, A.P. et al. (2010). Noiseless linear ampliﬁcation

and distillation of entanglement. Nat. Photon. 4: 316. 25 Ferreyrol, F., Barbieri, M., Blandino, R. et al. (2010). Implementation of a non-

deterministic optical noiseless ampliﬁer. Phys. Rev. Lett. 104: 123603. 26 Zavatta, A., Fiurášek, J., and Bellini, M. (2011). A high-ﬁdelity noiseless ampli-

ﬁer for quantum light states. Nat. Photon. 5: 52.

233

7 Photon Generation and Detection What are photons? In the simplest language, they are what make a photon counter click. A practical working deﬁnition of a photon links directly with the process of detection. Photons are not directly detected in the laboratory: they are nearly always converted to electrical signals, and it is these events that are detected and analysed. In this way we can recover the information that is encoded on the light and is carried by the photons. Light propagates as a high frequency electromagnetic wave, and, as with all such ﬁelds, there are two methods of detection: ﬁeld detection, where both amplitude and phase information can be measured, and intensity detection, which yields no phase information. In the radio, microwave, and mid-infrared (mid-IR) regions of the spectrum, ﬁeld detection can be done directly. In the near-infrared (near-IR), visible, and ultraviolet (UV) regions, ﬁeld detection is achieved indirectly via homodyne or heterodyne techniques: the light ﬁeld is interfered with a reference ﬁeld of known amplitude and phase of either the same frequency (homodyne) or shifted frequency (heterodyne), and the resultant ﬁeld is detected with intensity detectors, as described in Section 8.1.4. Thus in the optical region of the spectrum, intensity detectors are used for both ﬁeld and intensity detection. Broadly speaking there are two diﬀerent ways of recording light intensity: we can detect individual photons or we can measure currents generated by a stream of photons. The time response of the photodetector and the magnitude of the photon ﬂux determine which mode of operation is suitable. Both technologies exist in parallel and are used for diﬀerent purposes.

Figure 7.1 The origin of quantum noise. A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

234

7 Photon Generation and Detection

Table 7.1 Comparison of diﬀerent detection modes, the type of detection, the output results, the processing required, and the applications made possible with this type of detection. Type

Detection

Result

CW weak

Fast, single photon

|0⟩, |1⟩, …

Many samples Pulsed

Fast, synchronized

weak

Single photons

|1⟩ on demand

Processing

Application

Store

Q-info, Q-logic

each event

QKD

Store

Q-info, Q-logic

each event

Many samples CW strong

Slow

Modulation

Cont. homodyne

Sensors, anologue +

averages

Noise variance

Record + FFT

dig. communication

Wigner function

Tomography

QKD

Pulsed

Slow

Modulation

Cont. homodyne

Sensors, analogue +

strong

averages

Noise variance

Record + FFT

dig. communication

Δtmeas ≫ Δtrep

Wigner function

Tomography

Hybrid

Fast, CW, and

Negative

Cont. homodyne

single photon

Wigner function

and post-selection

Many samples

Cat states

Tomography

Q-info, Q-logic

Table 7.1 provides an overview of the types of detection that are required for the diﬀerent applications. All of these are now possible with the diﬀerent photodetectors and processing electronics that are described in this chapter (Figure 7.1). If the photon ﬂux is low enough and the detector is fast enough, with a recording time Δtmeas shorter than the time intervals between the photons, we can record the individual events, resulting in a sequence of numbers, typically 0 for no photon and 1 for at least one photon in each of the counting intervals. This would be the most fundamental way of sending information using this elementary quantum state of light. In addition photons can carry more complex information that is encoded in the quantum state using additional degrees of freedom such as polarization, momentum, or frequency. In these cases we use photons in beams that have several independent modes, for example, a beam with orthogonal polarizations, with multiple spatial modes or optical frequencies. Photons are ideal for these types of applications; since they travel at the speed of light (ﬂying qubits), they interact only weakly with the environment, even over long distances, and photons can be manipulated very eﬃciently and with linear and non-linear optics. By directing the modes to separate detectors, we can record the state of the qubit. Many quantum communication protocols make use and demand exactly one single-photon travelling in a mode, or channel, and more than one photon recorded in one click would compromise the results. For this reason increasing eﬀort is made to both create single photons on demand and to have detectors that can alert to the presence of additional photons that could compromise the security of the communication. Ideally, we would have detectors

Photon Generation and Detection

that can distinguish the number of photons in every counting interval and could then work with multiphoton states. As an alternative to photon counting, we can integrate the individual events and continuously record a photocurrent i(t) that varies in time. We average during the measurement interval Δtmeas over a large number of photons and analyse the optical power P(t) that contains information in the form of a slowly varying modulation of the light, up to a maximum modulation frequency Ωmod < 1∕2Δtmeas . There will always be some fast ﬂuctuations, or noise, and the more accurate the information measured in the photocurrent will get, the more photons have been detected. We can characterize the statistics of the ﬂuctuations by recording and analysing large sets of these short measurement intervals. Depending on the nature of the light source, the photons arrive with quite diﬀerent temporal distributions: for example, thermal (a natural light source), Poissonian (a coherent light source such as a laser), or regularly spaced in time (a Fock state or single-photon source). The information of the recorded power in a particular beam of light is frequently not dependent on the underlying statistics of the light source. Indeed, many quantum optics phenomena based on self-interference are insensitive to the photon arrival statistics with the general caveat that the light is suﬃciently attenuated such that the chance of a multiple photon count in any given measurement interval is very low. Examples include quantum key distribution (QKD) and ‘interaction-free’ measurements, which are discussed in Chapters 12 and 13. In principle, photons are detected as discrete events, and the experiment is about registering the place and time of arrival of these events. In arguably the ﬁrst quantum optics experiment, G.I. Taylor attenuated a thermal light source down to the single-photon level, passed the light through a double slit, and recorded the photon arrival positions as an image on photographic ﬁlm, as discussed in Chapter 3. It was the race to develop television that led to the development of the photomultiplier in the 1930s, which allowed the multiplication of the charge created by a single photon into a useful electronic pulse that could be made audible as a click or as a dot on a screen [1, 2]. In the 1960s the Geiger-mode avalanche photodiode (APD) made solid-state optical single-photon detection possible [3]. Many more innovations, some in the last decade, have extended our detector technology. We can now combine speciﬁcally designed light sources with the corresponding detectors to cater for the various applications. This includes imaging, sensing with strong laser beams or a few photons, and reading optical memories. It could be classical communication with either analogue or digital signals or quantum information applications such as secure communication with QKD, or it could be quantum logic, quantum memories, or potentially quantum computation. For each case there are now sources and detectors available. Many sophisticated applications involve the interference of several modes, which are all individually in a single photon state. These include non-classical interference, teleportation, and quantum computation as they are discussed in Chapter 13 on single-photon experiments and quantum information. In these experiments it is critical that the various modes are indistinguishable. That is, they have identical distributions in energy, which means photon number and frequency, momentum (spatial modes), and time. In general the modes must

235

236

7 Photon Generation and Detection

also have states of deﬁnite photon number, i.e. Fock states. In the case of pulsed modes, this requires pulses that are identical from pulse to pulse. Finally we want to know when such a pulse arrives, providing us with a source for photons on demand, which would be the ultimate quantum light source [4].

7.1 Photon Sources In practical terms, it is not easy to build a source that meets all these requirements. Approaches tried to date have been numerous and include single atoms [5], colour centres [6], molecules [7], atomic ensembles [8], quantum wells, and quantum dots [9]. We can describe and classify photon sources by a number of critical parameters and compare them with the requirements for the ideal single-photon source. Are they deterministic or probabilistic? Probabilistic sources, such as spontaneous down-converters, produce pairs of photons that are entangled but appear at random times. They can be used as heralded sources, where one photon is used to determine the time when the other photon can be observed. Many of the more recent approaches hold the promise of a deterministic photon source where the photons arrive at predetermined times or can be triggered from the outside [4], allowing the use of a range of advanced quantum information protocols. To date in 2018 we are approaching the goal of building such an idealized photon gun [10, 11]. At what temperature do they operate? Some materials work well close to room temperature (300 K), and some require precise temperature control at an elevated temperature in order to optimize the non-linear processes through phase matching. Others, in particular those designed for photon number resolution, work with ultra-cold atomic samples; individual atoms or ions trapped in a vacuum might require cryogenic conditions. In which wavelength region can they be used, and are they broadband or restricted to an atomic transition? Whilst the non-linear crystalline materials have broadband characteristics and work in the visible regime, some require near-IR wavelengths or near-UV wavelengths. Some sources are tunable and can be used in systems that include optical materials with speciﬁc spectral properties, in particular atomic samples or speciﬁc organic materials such as dyes or even biological samples. What is the eﬃciency of extracting the photon from the source and sending it into the detector, including any spectral ﬁltering that would be necessary for a typical quantum information application? This property determines the choice of protocol to be used. A low eﬃciency makes the apparatus slow. Whilst we can only process a small subset of the pulses, the results otherwise do not change. In deterministic sources we know that we will have only one photon at a time, but we do not know exactly when it works. The ideal photon on-demand source has close to 100% eﬃciency. Do they produce a good single spatial mode? This is an important ability because if the probability of ﬁnding the single photon is spread across incoherently related modes, the photon state will be mixed. In many situations, we want

7.1 Photon Sources

to overlap the diﬀerent modes at the output to create a quantum interference at the output of the apparatus. The interference will be maximized if each source operates in a single spatial and frequency mode. One measure of the quality of the single-photon source is the g (2) function. Ideally the value should be zero, which indicates that no more than one photon is contained in each of the outputs from the source and the apparatus is perfect. Values as low as 4 × 10−4 have been achieved, which is extremely useful in practical terms and almost closes potential loopholes for quantum demonstrations. A faint laser would be the simplest approach. It allows to direct all the light on to the detector and can create a single mode. However, photon arrival times are random and thus g (2) = 1. Many sophisticated sources are now available, as the review written by Eisaman et al. shows very clearly [12]. Table 7.2 lists the most common and advanced sources and presents their typical speciﬁcation. However, note that g (2) says nothing about the eﬃciency or modal purity of the source. The most common technology used in quantum optics experiments to date is still spontaneous parametric down-conversion (spontaneous PDC) and is probabilistic. PDC sources are bright with high photon count rates, can produce clean optical beams with well-deﬁned frequency and spatial characteristics [13–16], and can produce tunable entangled modes [17, 18]. In a spontaneous PDC, high frequency photons are passed through a non-linear crystal, where they are converted down into two lower frequency daughter photons, born within 10s of Table 7.2 Comparison of diﬀerent photon sources and their speciﬁcations.

Source

Type

Temperature (K)

Wavelength

Eﬃciency

Mode

g(2)

Faint laser

Prob

300

vis-IR

1

Single

1

PDC bulk

Prob

300

vis-IR

0.6–0.8

Multi

0.0014

PDC pp

Prob

350 ctr

vis-IR

0.85

Multi

—

PDC waveguide pp

Prob

350 ctr

vis-IR

0.07

Single

0.0007

FWM in ﬁbre

Prob

300

vis-IR

0.02

Single

0.02

PDC multiplexed

Prob/Det

300

vis-IR

0.1

Single

0.08

Q-dot (GaN)

Det

300

340–370 nm

10 dark counts per second with moderate Peltier cooling), and have a high spectral bandwidth (detecting light from 185 to 900 nm). Unfortunately, they are not very eﬃcient at turning photons into electrons: 𝜂 ranges from fractions of a percent at high wavelengths (600–900 nm) to a few percent at low wavelengths (185–300 nm) to tens of percent near the peak wavelengths. Currently the best commercial PMT achieves 𝜂 ∼ 40% at 550 nm. A photon counter with higher eﬃciencies than the PMT is the APD used in a regime where it operates as single-photon avalanche diode (SPAD). An APD works via the internal photoelectric eﬀect, where light incident on a semiconductor surface promotes an electron from the valence to the conduction band,

241

7 Photon Generation and Detection

Electrons

Energy

242

Ebg

hν

Figure 7.2 Semiconductor and band gap.

Photon

Holes

as illustrated in Figure 7.2. Of course, this requires a material with a band gap smaller than the photon energy, h𝜈 – a diﬃcult task in certain regions of the spectrum. As with a PMT, a single electron is typically liberated by a single photon, so SPADs are also run in Geiger mode to amplify the single electrons to macroscopic current pulses. Commercially available APDs are often packaged with an ampliﬁer, a discriminator, and a transistor–transistor logic (TTL) output driver, which turns the current pulses into pulses with well-deﬁned height and rise time, so-called TTL voltage pulses (deﬁned as Logic 1 = 2–5 V and Logic 0 = 0–0.8 V), which make them very user-friendly. These detectors are also reasonably fast, with maximum counting rates of ∼10 × 106 photons/s, linear to ∼ 106 photons/s, and suﬀer moderate dark noise (>100 dark counts per second). The wavelength range is smaller than PMTs but decidedly more eﬃcient, depending on the way they are designed. Examples of the possible peak eﬃciencies are for silicon SPADs are 50% at 450 nm, 70% at 600 nm, and 50% at 830 nm and for InGaAs SPADs, up to 33% at 1060 nm. An even more eﬃcient device is the solid-state photomultiplier (SSPM). This is a semiconductor material doped with neutral atoms where the energy levels of the dopant atoms are very close to the conduction band of the semiconductor (e.g. Si doped with As has a 54 meV gap). Brieﬂy, at an operating temperature of 6 ∼ 7 K, the electrons are frozen onto the impurity atoms, as they lack enough thermal energy to be excited into the conduction band. When an incoming photon is absorbed, it generates an electron–hole pair: the electron is accelerated to the anode, impact ionizing other impurities as atoms on the way and so generating an electron cascade within the device. Since little energy is needed to photoexcite an impurity atom so close to the conduction band, SSPMs are sensitive well into the infrared (IR): from 30 μm down to the visible, except for wavelengths in the silicon absorption band, which unfortunately includes the common telecom wavelengths. The broad IR sensitivity means that SSPMs must be carefully screened so that only light at the desired wavelength is coupled to the detector – or else ambient IR will quickly lead to detector saturation. So screened, and corrected for optical losses, the quantum eﬃciency of an SSPM in the visible has been measured to be 96±3% [17]. In an SSPM, photoabsorption and gain occur throughout the bulk of the detector: in a variant known as the visible light photon counter (VLPC), the absorption and gain regions are separated. This reduces

7.2 Photon Detection

the sensitivity to 1–30 μm radiation to ∼2% whilst maintaining the visible spectral response. At 694 nm, uncorrected quantum eﬃciencies of 88±5% have been measured [34], corresponding to corrected eﬃciencies of 95±5%. Unlike an APD or a PMT, the SSPM and VLPC can distinguish between one and several photons hitting the detector at the same time: an SSPM has been shown to distinguish two photons with 47% quantum eﬃciency [35]. Apart from requiring cryogenic operation, typically at 7 K, the other disadvantages to these systems are their cost and relatively high dark noise, typically 2 × 104 dark counts per second. They also have a remarkable wide wavelength range and thus require shielding from IR photons. 7.2.1.3

Photo-thermal Detectors

The superconducting transition-edge sensor (TES) detector operates as a bolometer where the energy of the absorbed photon is detected as a rise in temperature. In order to achieve the sensitivity required to detect a single photon, the heat capacity of the absorber has to be very small, and the sensor must create a large response to a very small temperature change. This is achieved by using a thin layer of superconducting material, deposited onto an insulating material. The device is operated at a temperature between superconducting and normal resistances, such that a minimal change in temperature yields a large change in resistance. The device is kept at the operating temperature by electrothermal control, which also produces the output current. The current sensitivity can be enhanced by using an array of superconducting quantum interference devices (SQIDs). Single-photon sensitivity was ﬁrst demonstrated in the visible and IR [36]. To date TES detectors have recorded the highest eﬃciency so far [37, 38] and a very low dark count rate. However, they suﬀer so far from several drawbacks: the need to operate at about 100 mK and, until recently, a slow response of typically 100 ns, resulting in low maximum count rates of typically 100 kHz. Major technical advances can be expected soon. There are several further approaches that are being followed. They include heat-to-voltage conversion with digital readout (QVD), superconducting nanowire single-photon detectors, quantum dot ﬁeld-eﬀect transistor-based detectors, superconducting tunnel junction (STJ)-based detectors, up-conversion single-photon detectors, and the use of atomic vapours with lambda three-level systems as reviewed by Eisaman et al. [12]. Even quantum non-demolition sensing of photon number using materials with giant Kerr non-linearities [39] is being pursued. Research into all these devices is very active, and it seems rather likely that the quality of detectors, their applicability, and price will improve steadily, and thus we should assume that in the future, almost perfect single number receiving detection with small pixel size and imaging capability will be achievable and practical (Table 7.3). 7.2.1.4

Multipixel and Imaging Devices

Spatial images can be obtained photoelectrically, although typically the response time is orders of magnitude slower than that possible with a single detection element device like an APD. There are several technologies for achieving

243

244

7 Photon Generation and Detection

Table 7.3 Comparison and typical speciﬁcations of diﬀerent single-photon detector technologies: PMT, SPAD, SSPM, VLPC, and TES. Detector

Eﬃciency (%)

Temperature (K)

Dark counts (1/s)

Max counts (1/s)

PNR

PMT

40 at 500 nm

300

100

10 × 106

Some

SPAD

76 at 600 nm

300

200

16 × 106

None

SSPM

76 at 700 nm

50

7 000

10 × 106

Full

VLPC

88 at 694 nm

7

20 000

10

Some

TES

95 at 1556 nm

0.1

3

0.1

Full

Source: From Eisaman et al. [12].

this, perhaps the most common being the photon counting charge-coupled device (CCD). A traditional conﬁguration is the intensiﬁed charge-coupled device (ICCD), where an image intensiﬁer (e.g. a photoelectric photocathode, followed by an array of multichannel plates for electronic ampliﬁcation, followed by a phosphor screen) is imaged, via free space or ﬁbre bundle, to a low noise CCD camera. More recent devices dispense with the image intensiﬁer, utilizing high voltage to produce an on-chip electron cascade, and thus gain. Current CCDs have a quantum eﬃciency in excess of 92% (with resolutions up to 1024 × 1024 pixels and an associated readout rate of 30 frames per second) and will doubtlessly improve. Figure 7.3 shows the output of a spontaneous down- conversion photon source imaged with an image- intensiﬁed photon counting CCD. The photoabsorbing surface in most photoelectric devices is a metal or semiconductor: high refractive index materials that intrinsically reﬂect a significant proportion of incident light (e.g. see Table 7.4). The quantum eﬃciency of photoelectric detectors can be substantially improved via the application of anti-reﬂection (AR) coatings that allow more light to be coupled into the bulk of the material, where the intrinsic quantum eﬃciency may be quite high, e.g. 𝜂 = 99% for Si [41]. In commercial single element devices, the AR coatings are relatively simple (e.g. a single layer of TiO2 ), and the improvement in quantum eﬃciency is modest. In commercial CCDs the situation is somewhat better: mid-band visible AR coatings are available that improve the eﬃciency from

Figure 7.3 Images acquired with a photon counting CCD camera: output of a type II spontaneous down-converter. Source: After [40].

7.2 Photon Detection

Table 7.4 Properties of photodiode materials. Si

Ge

InGaAs

Band-gap Ebg [eV]

1.11

0.66

≈ 1.0

Cut-oﬀ wavelength [nm]

1150

1880

≈ 1300

Peak detection wavelength 𝜆peak [nm]

800

1600

≈ 1000

Material quantum eﬃciency

0.99

0.88

0.98

𝜆min and 𝜆max for 𝜂mat = 0.5

600–900

900–1600

≈ 900–1200

Refractive index n at 𝜆peak

3.5

4.0

3.7

Reﬂectivity at normal incidence (%) Brewster’s angle (∘ )

31

36

33

74

75

76

𝜂mat at peak wavelength [nm]

∼61% to ∼92%. It is worth remembering that for a speciﬁc wavelength application, a custom coating can lead to a signiﬁcant improvement, up to the intrinsic eﬃciency of the device. 7.2.2

Recording Electrical Signals from Individual Photons

The improvements of electronics and computing in the last decade have resulted in a shift of approach to photodetection. The availability of very fast detectors and digitizing boards, large processing capacities, and data storage has led to a paradigm shift in the way experiments are designed and data are analysed. The technique that most people now consider is to store the entire output from the photodetectors directly and to process these digitally, either in real time or in post-processing. This has many advantages: it allows the implementation of a much wider range of processes; it has capabilities for both single-photon experiments and continuous-wave experiments; the data are stored and can be analysed again and again; and the processing tools can be reﬁned after the data were taken. This allows the experimentalist to concentrate on the experiment for the purpose of getting the best data and separately, at other times, on the analysis. In practice this is very conducive to daytime experimentation and automated repeats of the experiments. But it is still followed by long evenings, in front of computer displays. The ﬁrst and very common approach is the direct counting of photons within regular time intervals. The arrival of one or more photons will result in an electric pulse or click. Only PNR detectors will allow us to classify the pulses according to the number of photons. We store such a pulse above a preset threshold as binary 1, whilst we ignore all small events below the threshold and record them as 0, meaning no photon. The processing is now a simple matter of storing the string of 1s and 0s, which can then be used to plot histograms and analyse trends as a function of various parameters. The average value can be associated with the average photon number ⟨N⟩ in this mode. The next level will be to look for correlations of the output of two photodetectors. The most direct way is to set the system

245

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7 Photon Generation and Detection

such that it is only triggered by coincidences, that is, when both detectors have produced a pulse above the threshold and thus show a 1. There are many important experiments, such as entanglement, tests of Bell’s inequality, etc., which rely directly on coincidence counts. The results can be used to determine a two-mode correlation function g (2) quantitatively. The alternative to coincidence counting is to store the binary ﬁles and evaluate the sequences during or after the recording. This would yield the same results for coincidences but allows other types of protocols. We can consider conditional recording via post-selection where the pulse in one channel is recorded only when a pulse was present in the other channel. This is used in many protocols where one channel is designed to produce photons that herald a particular event in another channel. It has been used to demonstrate logic gates and describe their properties. Or it can be used to evaluate the conditional variance between the two channels. By recording the sequences of events from several detectors, more complex situations have been investigated, for example, the properties of a system with more than two modes. This could be systems that have entanglement between several modes, or degrees of freedom, or even systems where the modes are linked together to form a logic system. That means a large number of correlation functions between diﬀerent modes can be measured, the result leading to a full analysis of the covariance matrix (see Chapter 4). All of this is possible with the straightforward recording of binary sequences, typically with a sampling time of nanosecond, resulting in datasets with a few megabits. Analysis in these experiments is relatively straightforward, since the important parameter is the rate of photons detected at a given output port, measured over some integration time. If ﬁlm is used, the integration time is set by the length of the exposure. For photodetectors, the integration time is set by the counting time, which in turn is set by the electronics, including capacitors as low-pass ﬁlters, and the frequency of the trigger pulses. The electric current is digitized, e.g. TTL pulses, converted into binary values and counted with a commercial pulse counter; otherwise it can be digitized using a commercial discriminator, fast oscilloscope, or similar device and then counted. In some protocols, such as quantum cryptography, it is important that the photons arrive in a well-deﬁned time period. Even in these cases, any source, thermal, coherent, or Fock state source, can be used, as long as some combination of the source, the optical path, or the photodetector detection time is appropriately modulated. Such pulses will have regular time intervals. Alternatively, photon time-of-arrival information is a useful diagnostic in these experiments and may be obtained by using a time-to-amplitude converter (TAC) to measure the diﬀerence between the expected time, as provided by the modulator, and the actual time, as provided by the photodetector. The modulator provides a periodic start signal, for a measurement, which is ended by a photodetection event if one occurs. The time diﬀerence between the start and stop pulses then sets the amplitude of a regular pulse. The amplitude distribution of these pulses can be measured with a multichannel analyser (MCA), providing information on the distribution of photon arrivals.

7.3 Generating, Detecting, and Analysing Photocurrents

7.3 Generating, Detecting, and Analysing Photocurrents 7.3.1

Properties of Photocurrents

The technology used with CW laser beams is quite diﬀerent from that used with single photons, as described in Section 7.2. Here we analyse the full-time history of the photocurrent i(t), and the results we obtain for a laser beam are a value for the average optical power, which is obtained with a power meter, and in parallel a spectrum of the variance of the optical power, such as shown in Figure 7.9, which is obtained with an electronic spectrum analyser (ESA). For CW detection the electronic signal from the photodetector is integrated over many detection events for a time interval Δt, and this generates a current i(t), which is proportional to the power P(t), measured in Watt, of the light beam on the detector or, in other words, the rate at which photons reach the detector: i(t) =

ne (t)e P(t) e 𝜂det = Δt h𝜈

(7.1)

where ne (t) is the number of electrons released by the detector, e is the charge of a single electron, and 𝜂det is the eﬃciency of the conversion process (see for comparison Section 4.4.3). We will be interested in the average value of the current i = ⟨i(t)⟩. This value is proportional to the average optical power Popt ; thus a device that measures i is generally referred to as a power meter. Each such apparatus will have to be calibrated, and care should be taken that the response is linear, since the device will saturate if the optical power is too high.

7.3.1.1

Beat Measurements

The photodetector provides not only an average current i, the DC current, but also a whole spectrum of time-varying signals, the AC. Any AC component at detection frequency Ω can be traced back to a beat between two components of the optical spectrum with a separation Ω. These beat signals are very useful to identify the mode structure of the light source. Multimode laser operation will lead to strong beat signals, which can be detected and processed with high accuracy and sensitivity using conventional electronic RF equipment. The measurement of beat signals between two lasers, for example, will establish the frequency diﬀerence and the convolution of the linewidth of both lasers. Such observations are extremely useful for the establishment of optical frequency standards and also the measurement of laser linewidth, as discussed in Section 8.1.5. In addition, the AC signal also contains information about the noise and the signal carried by the laser beam, since the ﬂuctuations and modulation of the light intensity corresponds directly to the size of the amplitude sidebands. The presence of these sidebands leads to beat signals, which are similar but much weaker and noisier than the beat signals from two lasers.

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7.3.1.2

Intensity Noise and the Shot Noise Level

For the quantitative measurement of the ﬂuctuations, we are interested in the variance Δi(t)2 of the photocurrent. This is linked to the variance of the number of single-photon detection events via ( ) ne (t)e 2 Δ(ne (t))2 e2 2 Δi(t) = Δ = (7.2) Δt Δt 2 If we consider the generation of all the photoelectrons as independent processes, we can apply the statistics for independent particles. The counting statistics for the electrons should be a Poissonian distribution with variance Δ(ne (t))2 = ne = iΔt∕e. For large numbers of electrons in each counting interval, this should be a Gaussian distribution, resulting in ie Δt We will be investigating the spectrum of the current ﬂuctuations. In this case the detection interval Δt is related to the bandwidth B of the apparatus via B = 1∕2Δt, which leads to Δi(t)2 =

Δi(t)2 = 2 i e B

(7.3)

assuming that i does not vary during the measurement. This is the well-known shot noise formula in electronics that also applies to photodetection. It should be noted that Eq. (7.3) can be derived for any stream of electrons that are generated independently. However, this is a rather special case for a current, and generally a current can be noisier or quieter. There are many practical situations, such as the use of electronic current controllers, which can generate currents with less noise than the shot noise limit. It is important to note that shot noise is not a property of the detector. The noise reﬂects the ﬂuctuations of the light intensity, and we can use detectors to probe the statistics of the light ﬂuctuations. This will be correct as long as the quantum eﬃciency of the detector is high. For 𝜂det = 1 the statistics of the photons and electrons will be identical, the current ﬂuctuations are directly linked to those of the light, and we can investigate the statistical properties of the light by studying the current ﬂuctuations. It should be noted that after the detection of the light, any quantum properties of the ﬂuctuations no longer exist. Current noise can be treated classically, as already discussed in Section 5.1. For any practical purpose there is no quantum limit for the noise of a current. However, at the same time the current ﬂuctuations are a reliable measure of the quantum properties of the light. This distinction is important in the context of measurement theory, the generation of non-classical light (Chapter 9), and the applications of electro-optic feedback described in Section 8.3. How closely the current noise reﬂects the light ﬂuctuations depends on the eﬃciency 𝜂det . A quantum eﬃciency much less than unity corresponds to a random selection of a fraction 𝜂det of the light. This is fully equivalent to having a perfect detector behind a beamsplitter, which selects a small fraction of the photons. This results in a variance of the current that is closer to the shot noise limit than to the variance of the light intensity. To describe this quantitatively, it is useful to

7.3 Generating, Detecting, and Analysing Photocurrents

consider normalized variances for individual Fourier components of the current ﬂuctuations: ) ( Δi(t)2 Vi (Ω) = (7.4) 2ieB The corresponding statistics of the light is described by the normalized spectrum V 1out (Ω) of the photon ﬂux, as was shown in Chapter 4. The eﬀect of the detection eﬃciency 𝜂det is given by Vi (Ω) = 1 + 𝜂det (V 1out (Ω) − 1) This is identical to treating the ineﬃcient detector as a perfect detector that sees only the light reﬂected by a beamsplitter with reﬂectivity 𝜖 = 𝜂det as calculated in Section 5.1. The measured spectrum Vi (Ω) varies with the fraction of the light detected. For a very ineﬃcient detector, the normalized variance of the current approaches the shot noise limit Vi (Ω) = 1. 7.3.1.3

Quantum Eﬃciency

The detection eﬃciency 𝜂det contains all processes that reduce the conversion from light into a current. In practice, there are a number of such processes, as shown in Figure 7.4: only a fraction 𝜂r of the light reaches the photodetector material, and the rest is reﬂected or scattered at the surface of the detector. The detector absorbs a fraction 𝜂a of the light, and the rest is scattered or transmitted through the detector. The absorption scales exponentially with the thickness of the material and depends on the doping of the material. The material has a quantum eﬃciency 𝜂mat , which depends strongly on the wavelength of the light. This eﬃciency can be very close to unity as long as the wavelength of the light is close to but shorter than the cut-oﬀ wavelength set by the band-gap energy of the material. Finally, there is a chance for the electron–hole pair to combine before it can leave the detector. This recombination is more likely at the surface of the material than inside. At shorter wavelengths the light tends to be absorbed more strongly; it does not penetrate as deeply. Thus more detection processes occur close to a

N(t)

(1 – ηcom) recombination

(1 – ηr) reflection

ηdet =

N(t) i(t)

(1 – ηa) transmission

ηmat conversion

i(t)

Figure 7.4 The diﬀerent loss mechanisms present in a photodetector.

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surface, and the losses due to recombination are larger. Only the fraction 𝜂com will actually contribute to the photocurrent. All these losses are independent and can be combined to the total detector eﬃciency 𝜂det : 𝜂det = 𝜂mat × 𝜂a × 𝜂r × 𝜂com The eﬃciency 𝜂det is generally independent of the intensity and the detection frequency Ω. However, it has been noted that 𝜂com will get worse at higher intensities and that some photodetectors get slower at higher intensities when either the detector material is getting hot or there are insuﬃcient fast charge carriers to drive the AC. The eﬃciency 𝜂det has to be determined for the parameters of the light used in the speciﬁc experiment, in particular its wavelength, total power, and focusing geometry. Techniques for increasing 𝜂det in order to approach the ideal case of 𝜂det = 1 are very important in all experiments with non-classical light. 7.3.1.4

Photodetector Materials

Photodetection relies on the interaction of light with materials, which results in the absorption of photons and the generation of electron–hole pairs. These pairs change the conductance of the material, and they contribute to the photocurrent. The material for the photodetector has to be selected for a given frequency of the light. The energy of the photons h𝜈 has to be matched to the band-gap separation Ebg of the material. Semiconducting solids, such as germanium (Ge), silicon (Si), or indium–gallium–arsenide (InGaAs) are well suited for this application. These materials work in the visible and near IR. The properties of these materials are listed in Table 7.4, and the wavelength dependence of the eﬃciency is shown in Figure 7.5. Note that the properties of primary semiconductors (Si, Ge) are ﬁxed but that the properties of tertiary materials (such as InGaAs) depend on the relative composition, which means the concentration of the individual elements. The material quantum eﬃciency 𝜂mat of a very pure and selected material can be very high, up to 𝜂mat = 0.98. This is the limit of the eﬃciency of the detector and can be achieved in practice provided the sample of material used is thick enough to absorb all the light and reﬂection losses are avoided. Since most photodetector 1 Si

Ge

InGaAs η

250

0.5 InSb InGaAsP

Ag–ZnS 0 0.1

0.2

0.4

0.6 1 2 Wavelength λ (μm)

4

Figure 7.5 Quantum eﬃciencies of various photodiode materials.

6

7.3 Generating, Detecting, and Analysing Photocurrents

materials have very high refractive indices, their reﬂectivity is also very high; for example, for Si with a refractive index of 3.5 at the peak detection wavelength of 800 nm, the reﬂectivity under normal incidence is 31% of the intensity. To this the reﬂection loss at the cover glass (typically 8% from both surfaces of the glass) has to be added. In order to increase the quantum eﬃciency, it is common to remove the cover glass and to either tilt the detector closer to Brewster’s angle, which requires large detectors, or use an AR coating on the detector. In addition, the light reﬂected at the surface can be retro-reﬂected on to the same detector. Using these techniques typical quantum eﬃciencies that were achieved by 2005 were 65% at 553 nm (EG&G FND 100 Si photodiode) and 95% at 1064 nm (Epitax 500 InGaAs photodiode). Similarly high eﬃciencies around 1600 nm have also been achieved, which is important due to the interest in optical communication applications and the optimal performance of silicon ﬁbres in this wavelength range. In most cases one would expect technology to simply improve. However, there have been situations where certain optimized components are no longer produced, for example, the Epitax diodes, and it can be costly and time consuming to ﬁnd replacements. Alternatively, several detectors can be cascaded, each subsequent detector receiving the light reﬂected by the previous detector, and all the individual photocurrents are combined. Here electronic delays of the diﬀerent currents have to be carefully avoided in order to keep the signals synchronized with an accuracy better than the inverse of the detection frequency. Using cascaded detectors eﬃciencies close to 100% have been reported for detection frequencies of up to 100 kHz using Si detectors [42] and up to 10 MHz using InGaAs detectors. 7.3.2 7.3.2.1

Generating Photocurrents Photodiodes and Detector Circuit

The photodetection process leads to a change in the conductivity of the semiconductor material. A simple photoresistor could be used to produce a photocurrent. However, the most common conﬁguration is a PIN photodiode, which is a composite structure made out of P and N semiconductors with an intermediate layer. In this device an external voltage, the bias voltage, Ubias , creates an electrical ﬁeld in the region between the P layer and the N layer. Here electron–hole pairs can be separated, and they contribute to the photocurrent. There are three modes of operation for a photodiode. The diode can be operated with an open circuit. This will produce a voltage proportional to the photon ﬂux; no current will ﬂow. Alternatively, a low resistance is switched across the diode, and the circuit is eﬀectively a short circuit. In this conﬁguration a DC current is generated and the photodiode operates like a solar cell. Both these modes have slow responses. It is advantageous to apply a voltage, the reverse bias voltage Ubias , to the detector that accelerates the charge carriers. In this mode the photodetector is used like a diode to generate a voltage signal across an external resistor. The diode current is proportional to the photon ﬂux N(t). For a suﬃciently large bias voltage, the photocurrent is proportional to N(t). A schematic circuit and a typical response curve are shown in Figure 7.6. It is common to

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7 Photon Generation and Detection

i N(t)

DC Ubias

–Ubias

R i(t)

U

N(t) AC –Ubias / R

Figure 7.6 Circuit of a photodiode with reverse bias voltage Ubias and the typical diagram that shows the photocurrent i(t) as a function of the bias voltage for various values of the photon ﬂux N(t).

use the highest possible bias voltage (Ubias = 5–80 V depending on the diode) since this will increase the drift velocity of the charge carriers and thereby the speed of the detector. The photocurrent contains both a DC component and an AC component that can be directed to diﬀerent receivers using simple capacitors or more complicated electronic networks. The AC components contain all the information about modulation and the ﬂuctuations. 7.3.2.2

Ampliﬁers and Electronic Noise

The typical AC noise component is of the order of 1 part in 107 of the DC current. It requires ampliﬁcation. This can be achieved by various types of ampliﬁers, as described in [43]. The most obvious solution is to terminate the photodiode into a 50 Ω resistor and to use a commercial RF ampliﬁer. Such an ampliﬁer can be obtained with wide bandwidth (1–200 MHz), large gain (10–60 dB), and low noise performance. Please note that the gain is normally stated as the power ampliﬁcation in dB, where dB is 10 log Pout ∕Pin (see Eq. (7.9)). The ampliﬁer will increase both signal and noise at the input. Figure 7.7 shows the combination of noise and signal that is generated. The diﬀerent noise contributions are: • Dark noise of the detector Δi(t)2dark . Note that this is not the shot noise; it is only the noise due to false detection events. For most of our measurements with optical powers of a few milliwatt, this can be neglected. • Thermal, or Johnson, noise of the circuit given by Δi(t)2th ≈ 4kB TB∕R where T is the temperature, B is the bandwidth, and R is the impedance of the circuit. • Ampliﬁer noise Δi(t)2amp ≈ 2eGBF. Here G is the voltage gain and B the bandwidth. F is the excess noise factor, a composite of diﬀerent contributions inside the ampliﬁer. The excess noise factor is normally quoted in the technical speciﬁcations. Note that a large bandwidth detector contributes more noise than a narrowband ampliﬁer. All these noise contributions are independent; thus their variances can be added. Note that the noise Δi(t)2optical created by the light is the actual information

7.3 Generating, Detecting, and Analysing Photocurrents

Light

Photodiode

Gain

Output Thermal noise Gain noise

Noise

(a)

Dark noise Quantum noise

Signal

(b)

Modulation and classical noise

Figure 7.7 Signal and noise contributions in a photodetector. Noise terms (a) and signals (b) propagate diﬀerently through the stages of the detectors circuit.

we wish to detect. It corresponds to the signal in a communication application, and in order to achieve a clear detection of the signal, the total noise has to be less than the optical noise: Δi(t)2el = Δi(t)2dark + Δi(t)2amp + Δi(t)2th ≪ Δi(t)2optical

(7.5)

One of the largest contributions is normally the thermal noise. Consider the following example: for a DC current of 1 [mA], equivalent to 1 mW of power at 𝜆 = 1000 nm, the thermal noise at room temperature for an impedance 2 2 of 50 Ω and a detection bandwidth of 100 kHz is Δi(t)2thermal ∕i = 4kB TB∕Ri = 4 ∗ 1.38 × 10−23 ∗ 300 × 105 ∕(50 × 10−6 ) = 3.3 × 10−11 . The relative noise that corresponds to the shot noise for this bandwidth and average power is 2 Δi(t)2 ∕i = 2eB∕i = 2 ∗ 1.6 × 10−19 ∗ 105 ∕10−3 = 3.2 × 10−11 . This means that for the given conditions both quantum noise and electrical noise produce comparable relative ﬂuctuation Δi(t) of about 5 × 10−6 . In this example, the thermal noise is as large as the shot noise at this optical power, and such a detector could only show quantum noise at much higher powers. The importance of this so-called front end of the circuit cannot be overestimated; it will determine the noise characteristics of the detector circuit. Several technical tricks exist to optimize, which is summarized, for example, by P.C.D. Hobbs [44]. Most custom-made photodetectors contain a so-called transimpedance ampliﬁer, a circuit that converts the AC photocurrent into an AC voltage, which is fast and still has a high impedance. In such a circuit the thermal noise will be much lower, about 10−8 at 1 mA DC current, and we can clearly resolve quantum noise. State-of-the-art experiments provide a combined electronic noise of typically −10 dB and in some cases −30 dB below the quantum noise limit (QNL). This is the reason why photodetectors for quantum optics experiments are frequently custom made. For illustration, an example of a complete circuit diagram of such an ampliﬁer is shown in Figure 7.8. It shows the photodiode,

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7 Photon Generation and Detection

3

C1 22u

C2 100n

C3 100n

D1 ETX 500

7

I2 10u

2 – 3 +

5

1N4007 D2

+15

~4PF R1 2K 6

IC1 R2 CLC420 200R

U2 79L05 3 GND 2 IN –5V 1

–15

1N4007 D3

I3 10u C8 22u

C6 100n

C4 100n C10

GND 2

7

U1 78L05 Vin +5V

3 + 2 –

8

1

IC2 CLC430 6

R5

Vout

51R

4

I1 10u

4

254

R3 750R R4 300R

C5 100n

C7 100n

C9 100n

Figure 7.8 Example of an actual circuit diagram of a photodetector. For other examples, see M.B. Gray et al. [43].

the transimpedance ampliﬁer (IC1), the driver ampliﬁer (IC2), and all the ﬁlter components required for a practical low noise circuit. Whilst the development of electronics has continued constantly, the design of photodetectors still follows the same pattern. A good reference in 2017 is still the article by P.C.D. Hobbs [44], which summarizes the concept of the electronic design and the references given in the textbook by the same author [45]. 7.3.2.3

Detector Saturation

In many cases the light has strong modulations at speciﬁc detection frequencies. This could be due to laser noise, such as beats with other laser modes, or due to the sidebands introduced for the purpose of locking the laser to resonators, as discussed in Section 8.4. We expect the photodetector to handle both the very small ﬂuctuations and the modulation without distortions, and consequently we require a very large dynamic range for some applications as large as 60 dB. If the dynamic range of the ampliﬁers is not chosen large enough, the circuit will be saturated. This frequently leads to a decrease of all AC components, even those at other than the modulation frequency that originally caused the saturation. This saturation can be confusing and can mimic false results. It is particularly worrisome if several ampliﬁers are used, a ﬁrst ampliﬁcation stage with wide bandwidth and a later stage with narrow bandwidth. In this situation the ﬁrst stage can saturate from a modulation at frequencies outside the bandwidth of the ﬁnal stage. Thus the modulation is not visible at the output, and the noise level decreases when the signal is turned on – mimicking a noise suppression. In more than one case, this malfunctioning of the detector has fooled researchers and was taken as the demonstration of some quantum optical eﬀect. As a precaution one should always record spectra for the entire bandwidth of the detector, and any

7.3 Generating, Detecting, and Analysing Photocurrents

ampliﬁer and low-pass ﬁlter should best be placed directly after the photodiode. In addition, the scaling of the noise traces with the input optical power should be checked, as outlined in Section 8.1.1, to verify that no saturation exists. 7.3.3

Recording of Photocurrents

In most situations experiments are conducted by recording the photocurrents directly via fast analogue-to-digital (A/D) converters storing the entire information with a large bandwidth and analysing the physical properties, such as histograms and correlation functions. Historically, this is a form of post-processing that was typical for the analysis of single-photon events. However, due to the vastly increased speed of the processors, this can also be used now for all types of data analysis of continuous photon currents. For example, the modulation spectrum of the photocurrent can be determined with a numerical fast Fourier transform (FFT) providing similar results as the direct analogue ESA that were used in the beginning of the quantum experiments. Provided that the speed of recording, the dynamic range of the AD converter, and the size of the memory provides no limitation, we can assume that the Fourier transform will produce a true spectrum. In 2017, A/D converters with 18–20 bit conversion and sample speeds of 100 megasamples/s are available. In addition the size of the memory is now almost no issue, say, a few terabits, and the size is only noticeable in the time it takes to process a trace. However, there are limitations in the digital data recording that have to be considered and tested thoroughly to avoid misinterpretations: the main constraints are digitization speed or sampling frequency (samples per second), the dynamic range (bit per sample), storage capacity (total number of bits), and sensitivity to aliasing, which means the interference between the ﬁxed sample frequency and higher harmonics present in the input, leading to spurious low frequency eﬀects. For a simple practical example, we might consider a technical test where we record a communication channel that transmits signals with a carrier frequency of 100 MHz and modulation sidebands of around 100 kHz. We want to record an experiment for about 10 seconds. And we want to see a signal-to-noise ratio of 40 db. All this seems quite reasonable. Using the Nyquist theorem as a guide, we would set the sampling frequency to about 200 megasamples/s. To achieve a dynamic range, or contrast, of 10 000, which is necessary in order to see the ﬂuctuations and signal in reasonable detail, we would have to use a 16-bit A/D converter. This would generate 2 × 108 ∗ 65 536 ∗ 10 = 1.31 × 1014 = 130 terabits. That is, in excess of 2017 technology and such a brute force approach would simply create an unmanageable amount of data. However, the real information content is actually much smaller. We actually have only a maximum of 1 million possible signal values per second, we can average out the noise to a slowly varying value with an accuracy of one part in 10 000 of the biggest signal. So with little distortion a 12-bit A/D converter would be suﬃcient for simple communication applications. After mixing down the signal from 100 MHz and recording at a speed of 0.1 megasamples/s, only about 106 ∗ 4000 ∗ 10 = 4 gigabits of recording would be generated. This volume of data is comparable with the recoding of a

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high-deﬁnition video signal for that length of time and would be suﬃcient for most applications. In comparison, this could have been an experiment recorded with analogue instrumentation, such as an ESA that creates a printout of a spectrum centred at 100 MHz and a width of 200 kHz with a 50 dB vertical scale with a resolution of 0.5. Whilst such a spectrum contains only an average of the information over the entire 10 seconds scan, it would give us excellent information about the signal-to-noise ratio, and any potential problems or non-linearities within the system bandwidth would be recognized. The printout contains only a hundred values between 0 and 120 db, but this represents the complete spectrum. In order to describe the technical speciﬁcations of the apparatus, such a simple graph or even just a few numbers are suﬃcient in most cases. This example shows that the recording of an arbitrary signal with very ﬁne detail requires extraordinary number of samples. This not only exceeds our storage capacity, but it also makes processing practically impossible, and it is only necessary if we have no predictor of the signal that we wish to ﬁnd. That means that we should tailor our system to exactly what we need; there is no universal solution and just digitizing everything is not eﬃcient. In most technical applications we have a speciﬁc goal in mind. Let us consider two extreme examples. We can store and show with a smartphone or tablet any imaginable sequence of video images. These could potentially change completely from frame to frame, say, 30 times per second. However, in most cases the change from frame to frame might be small. In order to make this practical, the information has to be cleverly compressed, for example, in one of the current standards as MP4. This allows us to store only a fraction of the total information contained in all the pixels at all times. Clever algorithms project the images unto mathematical base functions and select a small amount of data that represents the video. The spatial resolution can be selected. The information can be stored and can be analysed quite readily. This technique requires online compression and storage of a medium number of data points. At the other end of the range of applications is the investigation of something unknown, for example, of gravitational waves (GW) detected as the correlation between the outputs of several optical interferometers placed at diﬀerent locations. Since late 2015 this has been dramatically demonstrated by the global LIGO collaboration. The GW have frequencies from megahertz to kilohertz. The detectors have good sensitivity above about 10 Hz and are being sampled at 40 kilosample/s continuously for 24 hours per day for months on end, since we do not know when the next event will happen and what it will look like. This leads to many terabits of data. Such large datasets have been recorded by LIGO since 2011 and have been constantly analysed by many groups around the world. In quantum optics experiments we typically lie between these extremes, and we have to design our experiments carefully to make use of the existing digital recording and processing technology. We can now record 100 megasamples/s with 16-bit resolution. In 2017 we can handle 1–100 gigabit ﬁles quite readily. However, we should also consider any analogue steps as part of the recording process, such as mixing the RF signal before recording, including low-pass ﬁlters or band ﬁlter to isolate the frequencies we are interested in and to protect

7.3 Generating, Detecting, and Analysing Photocurrents

us from aliasing. Thus the optimum system will be a combination of analogue preprocessing and digital recording and post-processing. 7.3.4 7.3.4.1

Spectral Analysis of Photocurrents Digital Fourier Transform

The time spectrum of the photon current contains a wealth of information that describes the properties of the apparatus. It is thus a widely used test procedure and the key to demonstrating many optical techniques. Figure 7.9 shows a typical spectrum and all the settings of the analyser being used. The most direct approach for creating a spectrum from digitized data is to use an FFT. This commonly available software routine is frequently directly built into the data acquisition system, for example, LabVIEW . The obvious requirements are as follows: a suﬃciently large number of samples, according to the Nyquist limit; the dynamical range, which is limited by the sampling resolution; and protection against interference with other periodic events, for example, at higher frequencies. The latter is an unavoidable consequence of recording with strictly periodic samples and can only be avoided by optimizing the recording electronics. Since FFT deals with stored data, the central RF, resolution bandwidth (RBW), and the averaging, or in traditional terms, video bandwidth (VBW) can all be selected after the recording. The recording should satisfy the following

Figure 7.9 A typical intensity spectrum as displayed by an electronic spectrum analyser. The modulation is about 8.9 MHz. The smaller signal at about 5.2 MHz is a radio interference from another source, aﬀecting both the signal and the dark trace. Here the settings of the instrument are shown together with the data.

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requirements: a sampling frequency must be at least twice the central RF; the number of samples should be at least larger than the ratio of RF to RBW; and the VBW is set by the inverse of the recording length. For slow long-term recordings, the choice is to either process all frequencies simultaneously and display long-term average values or alternatively analyse every section of the spectrum independently one at a time, which provides less averaging but allows to catch transient phenomena. It is worth noting that some of the commercially available ESA are actually a combination of A/D converter, storage, and FFT, rather than the full analogue version of using RF mixers. In these devices all the sample settings are automatically adjusted by selecting the required frequency and bandwidth settings. These machines are limited in their speciﬁcations but avoid the complication of having to individually optimize the digitization and sampling settings. 7.3.4.2

Analogue Fourier Transform

An ESA is used in some experiments to record the Fourier spectrum of the photocurrent. A typical spectrum displayed by such an instrument is shown in Figure 7.9. The principle function of an ESA is to determine and display the power spectrum of the electrical input signal. We consider in detail the transfer function from the spectrum of the input signal to the display. It is important that we understand the many options and settings of the analyser in order to obtain reliable data that can be interpreted as the properties of the light beam. A state-of-the-art analyser is a very powerful instrument with many options, menus, and knobs. An ESA contains a combination of analogue and digital electronics that produce a string of data that will be recorded alongside its actual settings. These machines are designed for the recording of signals, such as radio signals. In this section we will discuss the technical details of the ESA, such as the various bandwidth settings, sensitivity scales, sweep speeds, etc. and the tricks for using the analyser for the recording of noise and signals. Finally, we will discuss alternatives such as electronic mixers, which are useful for more advanced experiments such as tomography. 7.3.4.3

From Optical Sidebands to the Current Spectrum

We are interested in the sidebands of the optical ﬁeld that contain all the information about the ﬂuctuations and the statistics as well as all the information that is transmitted with the light in the form of modulation. Typical sideband frequencies range from kilohertz to gigahertz. These frequency intervals are too small for ordinary spectroscopic instruments, such as grating spectrometers or optical spectrum analysers (OSA), and are just about accessible with optical cavities. However, the most reliable and best separation of the sidebands is by the Fourier analysis of the photocurrents. The information is transferred from the optical domain (𝜈 ≈ 1015 Hz) into the domain of electronic instruments, which means detection frequencies Ω < 10 GHz. For this purpose several light beams are combined on a photodetector. The resulting beat signals contain information about the quadrature of the light. These techniques, homodyne and heterodyne detection, are described in

7.3 Generating, Detecting, and Analysing Photocurrents

Section 8.1.4. In all cases the information is contained in the temporal development of the current i(t) generated by the photodetector. The purpose of an ESA is to measure and display the spectral composition pi (Ω) of the electrical power of an electrical signal i(t). Mathematically this corresponds to a determination of the power spectrum, or the square of the Fourier transform, of the input voltage. The instrument measures the power pi (Ω) of the oscillating ﬁeld in a spectral interval around the detection frequency Ω. The spectral response of the analyser is given by the function res(Ω), which has a width given by the RBW. For a ﬁxed input resistance R, typically 50 Ω, the measured power is ( )2 pi (Ω) = time average res(Ω) i(Ω) R dΩ (7.6) ∫ where i(Ω) is the Fourier transform of the current i(t). We are measuring the square of the AC photocurrent integrated over the detection interval. The ESA is not sensitive to the phase of the electrical input signal. It will display only positive frequencies; there is no diﬀerence in the result of a measurement of signals with positive or negative frequencies, of sine or cosine modulations, of the current. 7.3.4.4

The Operation of an Electronic Spectrum Analyser

Technically, the ESA consists of a mixer that selects a certain frequency; an envelope detector, in most cases a logarithmic detector; and circuitry for averaging the signal, as shown in Figure 7.10. In most modern analysers the signal is digitized and displayed with a wide choice of scales and other options. In the following we analyse the transfer function between the electronic input signal and the display. The ESA is built around the concept of electronic heterodyne detection. A reference signal with ﬁxed amplitude and variable frequency, the local oscillator, is generated internally. The frequency Ωdet of the local oscillator determines the frequency under detection. The input signal is ampliﬁed and mixed with the local oscillator. The output from the mixer is ﬁltered by a narrow band pass ﬁlter with bandwidth RBW. The result of the measurement, at a given Ω, can still ﬂuctuate in time, and we can describe the output of the ESA by a distribution function with the mean value 𝜇disp , and it is this value from which we infer the spectral density pi (Ω) of the photocurrent. In our optical experiments we make relative measurements, which means we are not interested in the absolute values of 𝜇disp ,

Mixer Voltagein

Filter RBW filter

Envelope detector

Log amplifier

Filter VBW filter

Local oscillator

Display

Scale dB power

Figure 7.10 Block diagram of a typical spectrum analyser.

AD sample

259

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7 Photon Generation and Detection

but in relative values, such as the ratio of the 𝜇disp to a calibrated value that represents quantum noise. The output signal is integrated using a low-pass ﬁlter with a cut-oﬀ frequency known as the VBW. The VBW is the inverse of the time constant Δtave with which the measured signals is averaged, Δtave = 1∕VBW. In the limit of small VBW, large integration times, the output is one single, steady value 𝜇disp (Ω) for each detection frequency. In many practical situations we want to see the spectrum rapidly, and a lower value for the VBW is chosen; this does not vary 𝜇disp (Ω) but produces a more noisy trace. Modern RF spectrum analysers cover a wide range of input frequencies, from about 1 kHz to 100 GHz. They can achieve this wide span through a sophisticated combination of several internal mixing processes. Nevertheless, the idea outlined above of a narrow band scanning ﬁlter followed by an integrating power detector is appropriate. The instrument covers a total frequency interval, or span, SPA, and scans it with a set scan speed, measured in Hz/s. The instrument requires at least a time of 1/RBW for each measurement; after that it moves on by a frequency interval RBW. Consequently the maximum scan speed is RBW2 . For typical values of RBW of 104 Hz, the fastest possible speed is 108 Hz/s = 100 MHz/s. Faster scans would lose detailed information. The highest possible VBW is equal to the RBW, but usually the VBW is chosen lower. In particular for noise measurements, it is necessary to select a low VBW in order to obtain a well-deﬁned result. Frequently, measurements with ﬁxed detection frequency Ωdet are carried out. In this case the scan represents a time history of the power of the single Fourier components. 7.3.4.5

Detecting Signal and Noise Independently

It is not possible to distinguish between a pure noise source and a mixture of signal and noise by just looking at a trace recorded at one frequency. In this case we could not say if we have a laser beam with quantum noise alone, a laser beam with quantum and classical noise, or laser beam with noise and modulation. We need additional information. For example, a scan of frequencies would reveal a peak if the signal is narrowband and the noise is broadband. Or a measurement of the orthogonal quadrature, using a homodyne detector, would show the characteristics of a signal and of noise or squeezing and anti-squeezing. We can isolate signal and noise, as measured by the traces on the spectrum, and we can determine the SNR as SNR =

2 2 𝜎(signal+ − 𝜎noise noise) 2 𝜎noise

(7.7)

What would happen if we had more than one noise source and more than one signal present? As long as the noise terms have Gaussian distributions with 2 2 2 widths 𝜎n1 and 𝜎n2 , we can combine them into one value 𝜎noise = 𝜎n1 + 𝜎n2 . In a similar way we can combine two signals S1 and S2 with a random phase relationship. However, if the two signals have a ﬁxed phase relationship, the amplitudes 2 will have to be added coherently, before the variance 𝜎signal is formed. In either

7.3 Generating, Detecting, and Analysing Photocurrents

case, one combined signal intensity Ssignal can be found. Consequently, the calculation in Eq. (7.7) for one noise and one signal term is generally suﬃcient for all cases that we will encounter. 2 for a given noise source by decreasing Note that it is possible to reduce 𝜎noise the RBW, eﬀectively detecting less noise and the same amount of signal. This will improve the balance in Eq. (7.7) and allow smaller signals to be measured. However, this will make the detection process slower. We can also determine the measured value better by averaging the ﬂuctuations for a suﬃciently long time, which means using a low VBW. In the limit of VBW → 0, or extremely long integration times, the response of the analyser to a noise input with constant RMS will be a constant reading. The entire noise distribution would now be reduced to one piece of information. In most practical situations the integration time will be limited to about 0.1 s (VBW = 10 Hz) since systematic drifts inﬂuence the performance of the experiment and do not allow excessively long scans. When the noise is white, which means the ﬂuctuations are independent of the frequency, the spectrum will be a ﬂat horizontal line on the display. 7.3.4.6

The Decibel Scale

Some examples of actual measurements of laser intensity noise are given in Chapter 8. The display of the spectrum analyser is normally logarithmic. The units used are logarithmic power units, called dBm. The conversion of Watt into dBm is given by ( ) P[W] (7.8) pi [dBm] = 10 log 1[mW] The unit dBm is an absolute unit of power, for example, the RF power of 1 mW corresponds to 0 dBm, 0.001 mW = 1 μW corresponds to −30 dBm, etc. Whilst these numbers appear to be rather small, it should be noted that they refer to the power of the AC components only, not the optical power of the beam that generated the photocurrent. It would require a strong modulation of the intensity of the beam to get a signiﬁcant fraction of the energy into the sidebands. For example, a beam with average optical power Popt and single frequency harmonic modulation of the intensity by 1% has both a positive and a negative sideband at the modulation frequency, ±Ωmod . Each contains 1% of the amplitude of the optical ﬁeld, which corresponds to i(Ωmod ) = 0.02 of i and pi (Ωmod ) = 4 × 10−4 Popt . In most situations the relative change of the signal, or the noise, is of interest, and a relative unit of power, the decibel or dB, is used. It describes the ratio of two power levels. On a logarithmic scale this corresponds to a diﬀerence ΔP(Ω) given by ( ) P1 (Ω) (7.9) ΔP1,2 (Ω)[dB] = 10 log P2 (Ω) An expression such as ‘the signal was ampliﬁed by 3 dB’ means that the power of the signal increased by a factor of 103∕10 = 1.995 ≈ 2. Note √ that for such an increase the actual AC current would grow by only a factor 2. Similarly 6 dB corresponds to a factor 4 increase in power. The ratios in Eq. (7.9) are independent of the units; we record them from the display as ratios between values of 𝜇disp , and

261

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we think of them as the ratio of two noise powers P and interpret them as the ratio of quadrature variances V 1. All results of noise increase or noise suppression are quoted in dB, which gives directly the factor of change in the quadrature variance V 1 of the light. 7.3.4.7

Adding Electronic AC Signals

In many situations more than one signal or one noise component are present at the same detection frequency. Only the total power of all the contributions will be displayed. It is important to be able to determine the contributions from the individual terms, and we need rules for the adding of several AC. These rules depend on the nature and the correlation of the individual currents. Clearly, there is a distinction between currents that are coming from the same source or from different independent sources. First consider the extreme case of two currents, i1 (Ω) and i2 (Ω), which are generated by the same source. They are perfectly correlated, and their cross-correlation coeﬃcient as deﬁned in Chapter 2 is equal to unity: C(i1 , i2 ) = 1. In practical terms that means that the two signals have identical time histories. For single harmonic functions, this means that they have the same frequency and are locked in phase to each other. For these signals the combined power is simply the sum of the amplitudes: Pi1 +i2 (Ω) =

∫

res(Ω)[i1 (Ω′ ) + i2 (Ω′ )]2 R dΩ′

(7.10)

If the ratio of the amplitudes of the two currents is given by the number c = i2 (Ω)∕i1 (Ω), the measured power will change by a factor Pi1 +i2 = (1 + c)2 (7.11) Pi1 or in dB by ΔPi1 +i2 [dB] = 10 log((1 + c)2 )

(7.12)

For the case of two identical signals, i1 (Ω) = i2 (Ω) or c = 1, the power increase is 6 dB. The other extreme case is that of two signals that have a random phase diﬀerence to each other, which means they are uncorrelated. In this case the two signals add in quadrature, which means ΔPi1 +i2 [dB] = 10 log(1 + c2 )

(7.13)

and for the case of two signals with equal amplitude, the power increase is 3 dB. The same technique can be applied to noise terms, which are only diﬀerent from signals insofar as the power of the noise at detection frequency Ω varies in time whilst the power of a signal remains ﬁxed. In particular, signal and noise terms are always uncorrelated to each other. Two noise terms are uncorrelated when they come from diﬀerent sources. They are correlated when they can be traced back to the same source. Correspondingly, these noise signals are added in the respective way, using either Eq. (7.12) or Eq. (7.13). In the ﬁrst case, uncorrelated terms, the variances of the two noise terms are added in quadrature. Note that this procedure is strictly speaking only correct if both noise signals have the same statistical distribution functions, e.g. Gaussian distributions, which we can assume for many practical situations.

7.4 Imaging with Photons

7.4 Imaging with Photons All of the images shown in Figure 2.14 can be described by classical optics. At the same time, they use low intensities and thus photon ﬂux, and the limiting eﬀects of photon statistics are clearly visible. The ﬁrst step is to consider the eﬀects of photons in the generation of light in the light source. This could be due to the various processes that generate the light, such as thermal processes for an incandescent light, excitation of atoms in a plasma in a discharge lamp, excitation of atoms in a semiconductor that operates as a LED, or ampliﬁcation of light in a laser resonator. In each of these situations, the clearest and most elegant description of the process is on the basis of photons. There are well-described probabilities for an atom to emit a photon, and the wavelength of the light is determined by the energy of each photon via Ephoton = hc∕𝜆, and this is controlled by the energy level structure of the atoms or the semiconductor being employed. The ampliﬁcation of light inside a laser resonator is best described in terms of stimulated emission. Starting with the object that is being illuminated and imaged, the major processes are the transmission and absorption of the light from the illumination, where some of the photons are absorbed and the ﬂux is reduced in proportion. In addition photons can be generated in the object plane with new propagation directions through elastic scattering. These photons have the same energy per photon, which means they carry the same wavelength information. There could also be inelastic scattering, or ﬂuorescence, where the emitted photon has less energy than the photon required for the excitation of the atom involved in the process. This leads to new wavelengths being created in the object. There can be time delays between the excitation and ﬂuorescence, which can be used through precise time gating to distinguish between the photons from the illumination and the sample and enhance contrast and sensitivity. The improvement in detectors and gating has allowed imaging and spectroscopy to reach the level of detecting single molecules. This in turn has led to a major expansion of imaging techniques in biological and medical applications [46]. Another option is the use of multiphoton excitation where the presence of several low energy photons is required to created one high energy photon. For example, Figure 2.14d was created using 2-photon excitation of a dye in the brain sample. This has the advantage that the excitation has a much larger probability in the focus of a laser beam since the probability scales with the square to the local photon ﬂux. Therefore superior spatial resolution is achieved in the x, y plane and to some degree in the z direction. It is possible to reach with high precision individual or many points simultaneously for photostimulation and to image light from multiple spots. Higher-order three-photon excitation is possible. Such non-linear processes, which use multiphoton processes, are also at the core of so-called super-resolution techniques, including stimulated emission depletion (STED) microscopy proposed by Stefan Hell in 1994 and demonstrated in 1998 [47, 48], ﬂuorescence photoactivation localization microscopy (FPALM), and other techniques in microscopy that bypass the conventional diﬀraction limits. For a summary of the techniques and some of the wide impact they have had, see, for example, [49].

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As we have seen in Chapter 4, the time-averaged photon ﬂux of the quantized ﬁeld is proportional to the intensity of the classical ﬁeld, N(X, Y ) ∝ I(X, Y ), and the losses, redistribution of light, and change in ﬂux can be described by one standard model. The quantum eﬃciency 𝜂 is the one single parameter that is used to model the change in photon ﬂux. This includes all the ﬁner details of Fourier optics and any of the imaging techniques. As a consequence we can easily translate the intensity distribution to determine the photon ﬂux captured by each of the pixels of the photon detector. This applies independently to each pixel. For all these linear processes, the photon statistics will be either a Poissonian distribution or approach this distribution for lower and lower values of 𝜂. Consequently, we have a universal description of the noise in each part of the imaging system and for each pixel. Moreover, as with multiple beamsplitters, there are no correlations between the ﬂuctuations in each of the pixels; the photon counting noise in diﬀerent pixels is independent and only scales with the photon ﬂux. It is easy to visualize an image that has been taken in very dim light by just a few photons. It is grainy and noisy and is obviously not of the best quality. With modern technology, and cooled CCDs, it is now quite common to work with just a few photons/pixel and time interval. More photons, more intensity, would obviously help until we have a beautiful image at high intensities. Many situations where this is not possible come to mind, in night viewing, astronomy, etc. The human eye is close to the limit of being able to detect individual photons. Whether this is possible is still a topic of active research, and it seems that this is possible for some situations and some people but not for all, as discussed in Section 3.1. Detectors have steadily improved, with improvements to the sensitivity and reduction of other noise sources, as listed in Section 7.2. Thus it is now more common to image with a few photons per pixel and to be photon counting and shot noise limited. This means imaging has in many cases reached the limits set by the quantization of the light. Examples are given in Figure 2.14 where parts (a) and (d) were made using specialized CMOS detectors. In addition, there has been rapid progress with the development of arrays of single-photon detectors based on SPADs as used in part (d) where both fast timing and high sensitivity are important. All these types of single-photon detectors are now employed in more and more low light imaging situations in astronomical, medical, biological, and engineering applications.

References 1 Iams, H.E. and Salzberg, B. (1935). The secondary emission phototube. Proc.

IRE 23: 55–64. 2 Kubetsky, L.A. (1937). Multiple ampliﬁer. Proc. Inst. Radio Eng. 254: 421. 3 McIntyre, R.J. (1961). Theory of Microsplasma instability in silicon. J. Appl.

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23 Collins, M.J., Xiong, C., Rey, I.H. et al. (2013). Integrated spatial multiplexing

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of heralded single-photon sources. Nat. Commun.. 4. https://doi.org/10.1038/ ncomms3582. Kim, J., Benson, O., Kan, H., and Yamamoto, Y. (1999). A single-photon turnstile device. Nature 397: 500. McKeever, J., Boca, A., Boozer, A.D. et al. (2004). Deterministic generation of single photons from one atom trapped in a cavity. Science 303: 1992. Riebe, M., Monz, T., Kim, K. et al. (2008). Deterministic entanglement swapping with an ion-trap quantum computer. Nat. Phys. 4: 839. Home, J.P., Hanneke, D., Jost, J.D. et al. (2009). Complete methods set for scalable ion trap quantum information processing. Science 325: 1227. Zhang, J., Pagano, G., Hess, P.W. et al. (2017). Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator. Nature 551: 601. Brouri, R., Beveratos, A., Poizat, J.-Ph., and Grangier, P. (2000). Photon antibunching in the ﬂuorescence of individual color centres in diamond. Opt. Lett. 25: 1294. Beveratos, A., Kuhn, S., Brouri, R. et al. (2002). Room temperature stable single-photon source. Eur. Phys. J. D 18: 191. James, D.F.V. and Kwiat, P.G. (2002). Atomic vapor-based high eﬃciency optical detectors with photon number resolution. Phys. Rev. Lett. 89: 183601. Taylor, G.I. (1909). Interference fringes with feeble light. Proc. Cambridge Philos. Soc. 15: 114. Kwiat, P.G., Mattle, K., Weinfurter, H. et al. (1995). New high-intensity source of polarization-entangled photons. Phys. Rev. Lett. 75: 4337. Takeuchi, S., Kim, J., Yamamoto, Y., and Hogue, H.H. (1999). Development of a high-quantum-eﬃciency single-photon counting system. Appl. Phys. Lett. 74: 1063. Kim, J., Takeuchi, S., Yamamoto, Y., and Hogue, H.H. (1999). Multiphoton detection using visible light photon counter. Appl. Phys. Lett. 74: 902. Cabrera, B., Clarke, R.M., Colling, P. et al. (1998). Detection of single infrared, optical, and ultraviolet photons using superconducting transition edge sensors. Appl. Phys. Lett. 73: 735. Lita, A.E., Miller, A.J., and Nam, S.W. (2008). Counting near-infrared single-photons with 95% eﬃciency. Opt. Express 16: 3032. Fukuda, D., Fujii, G., Numata, T. et al. (2009). Ti/Au TES to discriminate single photons. Metrologica 46: S288. Munro, W.J., Nemoto, K., Beausoleil, R.G., and Spiller, T.P. (2005). High-eﬃciency quantum-nondemoltion single-photon-number resolving detector. Phys. Rev. A 71: 033819. Takeuchi, S. (2001). Beamlike twin-photon generation by use of type II parametric downconversion. Opt. Lett. 26: 843. Zalewski, E.P. and Geist, J. (1980). Silicon photodiode absolute spectral response self calibration. Appl. Opt. 19: 1214. Gardner, J.M. (1994). Transmission trap detectors. Appl. Opt. 33: 5914.

Further Reading

43 Gray, M.B., Shaddock, D.A., Harb, C.C., and Bachor, H.-A. (1998). Photode-

44 45 46 47

48 49

tector designs for low-noise, broadband and high power applications. Rev. Sci. Instrum. 69: 3755. Hobbs, P.C.D. (2001). Photodiode front needs: the real story. Opt. Photon. News, OSA, April. 42. Hobbs, P.C.D. (2001). Building Electro-Optical Systems: Making it All Work. Wiley. Michalet, x., Siegmund, O.H.W., Vallerga, J.V. et al. (2007). Detectors for single-molecule ﬂuorescence imaging and spectroscopy. J. Mod. Opt. 54: 239. Hell, S.W. and Wichmann, J. (1994). Breaking the diﬀraction resolution limit by stimulated emission: stimulated emission depletion microscopy. Opt. Lett. 19: 780. Hell, S.W. (2009). Microscopy and its focal switch. Nat. Methods 6: 24. Chi, K.R. (2009). Super-resolution microscopy: breaking the limits. Nat. Methods 6: 15. https://doi.org/10.1038/nmeth.f.234.

Further Reading Rieke, G. (2002). Detection of Light: From the Ultraviolet to the Submillimeter. Cambridge University Press. Saleh, B.E.A. and Teich, M.C. (2007). Fundamentals of Photonics, 2e. Wiley.

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8 Quantum Noise: Basic Measurements and Techniques We now turn our attention to the measurement of quantum noise – that is the fundamental noise we observe on the photocurrent when we detect optical beams with a large coherent amplitude (|𝛼|2 ≫ 1) using photodetectors. Several theoretical models have already been described in Chapters 2 and 4, and the experimental components have been discussed in Chapters 5 and 7. Based on this let us build up, step by step, the experiments for measuring and manipulating quantum noise.

8.1 Detection and Calibration of Quantum Noise We will concentrate on CW laser beams and the noise associated with the intensity, amplitude, and phase of the light. We have to devise ways of interpreting the electrical noise spectra. How do we distinguish quantum noise from ordinary technical noise? How can we can calibrate our measurement? Three diﬀerent techniques are commonly used: direct detection (Section 8.1.1), balanced detection (Section 8.1.2), and homodyne detection (Section 8.1.4). The choice depends on the properties of laser beam. 8.1.1

Direct Detection and Calibration

The obvious way of detecting quantum noise is to directly detect the light with a single photodetector and to analyse the resulting photocurrent. As long as we can assume that the noise is Gaussian, we will have the full information in the form of the variance of the noise, and we can use an electronic spectrum analyser or we can perform a spectral decomposition of the digitized data using a Fourier transform. This is the same arrangement as one would use to measure an intensity modulation of the laser beam, and both of these eﬀects will be present in the resulting spectrum. In most applications we will be interested in the quantum noise as the ultimately limiting eﬀect in the signal-to-noise ratio (SNR) (Figure 8.1). Direct detection gives us an electrical noise power pi (Ω), which for a perfect laser mode and detection system should be entirely due to the quantum noise of the light. There are obviously some technical pitfalls that need to be avoided, and we need a reliable procedure that convinces us, and anybody else, that we A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

8 Quantum Noise: Basic Measurements and Techniques

V1las(Ω) Laser

Photodetector efficiency η

Amplifier gain gel Noise pel

Spectrum Analyser Vi (Ω) pi (Ω)

Figure 8.1 Schematic diagram of direct detection.

are really measuring quantum noise. The noise power pi (Ω) displayed by the spectrum analyser is a measure of ﬂuctuations of the photocurrent, the ampliﬁcation, and the noise contributions of the electronics. For a quantum noise limited source, we obtain the value pi,QNL (Ω). All measurements will be normalized to this value, resulting in the normalized variance Vi (Ω) =

pi (Ω) pi,QNL (Ω)

(8.1)

Once the recorded noise traces have been calibrated, it is obvious that a new unit should be used. A creative suggestion was to label the standard quantum noise limit (QNL) as 1 Hoover, in recognition of the role of the vacuum noise [1]. All noise diagrams in this guide with linear scales, or a logarithmic scale in decibels (dB) deﬁned on page 261, use this unit. We need to know the total quantum eﬃciency of the detection system to link the variance Vi (Ω) to the ﬂuctuations of the light V 1(Ω) (Eq. (4.57)). All values for the noise power are long-term averages, as shown on the display depicted in Figure 8.2. In this case a quantum noise limited light source with modulation at about 9 MHz illuminates the detector. The left-hand side traces show the display for a spectrum from 6 to 10 MHz with constant resolution bandwidth (RBW) of 100 kHz and variable video bandwidths (VBWs) of 30, 300, and 3000 Hz. Note that the average noise remains constant whilst the actual time ﬂuctuations of the noise increase with VBW. The signal remains unchanged, and the width of the signal is given by RBW. The right-hand side traces show the results for the RBW –86 Noise power (dbm)

270

–90

–94

–98

VBW: 10 Hz RBW: 100 kHz

VBW: 100 Hz RBW: 100 kHz

VBW: 300 Hz RBW: 100 kHz

VBW: 300 Hz RBW: 300 kHz

VBW: 300 Hz RBW: 1 MHz

Figure 8.2 Experimental noise and signal spectra for diﬀerent combinations of resolution and video bandwidth. These are four separate experimental traces, with settings as shown in Figure 7.9. The increases in the noise ﬂoor with larger resolution bandwidth are clearly visible.

8.1 Detection and Calibration of Quantum Noise

of 300 kHz. The average noise level increases with RBW. In contrast, the signal height remains the same, since the signal is much larger than the noise and the signal width increases with RBW. Clearly we gain by selecting a larger RBW and get further away from the electronic noise. However, there are limitations, in particular if we have not a ﬂat but a structured noise spectrum, and there are a number of technical complications: 1. Electronic noise. The photodetector generates independent electronic noise, pel (Ω), which adds to the optical noise spectrum. We can avoid this complication by designing special detectors with low electronic noise (Section 7.3.2) and by independently measuring the dark noise of the detector. This is usually done by simply blocking the detector. If the dark noise is signiﬁcant, we can correct the measured signal by subtracting the dark noise after the measurement. It can be assumed that the electronic noise and the optical noise are independent and are added in quadrature. It should be noted that some exotic ampliﬁers have the unpleasant property that the noise depends on the actual photon current. In these cases a normal dark current measurement is not sufﬁcient, and several measurements at various photocurrents, or intensities, are required to investigate the scaling of the electronic noise. The performance of a given detector is commonly described by the noise equivalent power (NEP), which is the optical power on the detector at which quantum noise and electronic noise are equal (pi,QNL (Ω) = pel (Ω)). That means the recombined trace is 3 dB above the dark noise trace. Values of NEP = 1 mW are typical. Values as low as 10 μW have been achieved. 2. AC response. Most detector circuits will have a frequency-dependent gain, given by gel (Ω). For all detectors there will be a high frequency roll-oﬀ, depending on the type of photodiode and ampliﬁer used. Frequently, there is also some electronic mismatch between the components leading to a sinusoidal frequency response curve. This is more a nuisance than a complication. It is taken into account by normalizing the traces (Eq. (8.1)), and gel (Ω) does not aﬀect Vi (Ω) directly. 3. Eﬃciencies. The detector eﬃciency is not perfect. A low eﬃciency has the same eﬀect as a beamsplitter that diverts only a fraction of the light into the detector. For measurements of the quantum noise alone, this is not crucial. For example, we can measure the QNL even from only a fraction of the beam and compensate with more electronic gain. However, knowledge of the eﬃciency is crucial to calculate the SNR of a signal or the degree of squeezing in the light, which means diﬀerences to the QNL. These eﬀects disappear with high quantum eﬃciency. Note that electronic gain and eﬃciency play diﬀerent roles. The golden rule for any squeezing experiment is that all of the light should be detected and that quantum eﬃciencies have to be close to unity. 4. Power saturation. Detectors can only handle a limited amount of optical power. The actual limit varies from mW for small fast photodiodes up to W for large detectors. Once the limit in power is exceeded, the detector can actually be damaged due to heating processes. The absorbed light will heat the diode material, and, more importantly, the bias current will lead to further ohmic heating. Even below this damage limit, the performance can become

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non-linear. The number of available carriers inside the detection material is limited, and the response drops if the local intensity is too high. Thus the detector has to be uniformly illuminated even at intensities below the recommended power limit. 5. Saturation of the ampliﬁers. Several cases have been reported where a detector showed a peculiar response and the reason was the presence of a strong modulation signal that saturated the entire circuit. Saturation at one radio frequency (RF) can aﬀect the performance of the detector at many other frequencies. Care should be taken that no such strong signal is present by recording large bandwidth spectra up to the roll-oﬀ frequency of the detector. A particular pitfall is to use a series of ampliﬁers where the bandwidth of the earlier stages is broader than that of later stages, for example, by using a low-pass ﬁlter at the output of an ampliﬁer. In this case the large high frequency signal would not be visible in the output, but it is saturating the ampliﬁers ahead of the ﬁlter. Saturation can change the noise spectrum over a wide bandwidth, and it can even lower the noise level, and in some experiments saturated ampliﬁers have generated noise traces that mimicked, and were mistaken for, squeezing. After all precautions have been taken, we can trust the detector. But in order to measure the QNL, we still need to calibrate it. One obvious technique is to calculate the expected noise power pi,QNL (Ω) from the formula for shot noise (Eq. (3.9)), and using the appropriate constants, we get pi,QNL (Ω) (dBm) = −138.0 + 10 log(i × RBW)(dBm) + pel (Ω)(dBm) + gel (Ω)(dB) where RBW is measured in Hz and i in A and an input resistance of 50 Ω is assumed. pi,QNL (Ω) can be determined once the average photocurrent iave is known. This in turn should be done best by a direct measurement of the DC from the photodiode. It also requires a knowledge of the detection bandwidth, which is normally the RBW of the spectrum analyser. The AC gain of the various amplifying stages has to be included. Finally, the actual response characteristics of the spectrum analyser to noise, rather than to a signal, have to be taken into account (Section 7.3.4), which could require corrections as large as 2 dB. After all these measurements and calculations, one could expect to get a value for the noise level with an accuracy of typically 20% (1 dB). This is good enough to test the apparatus for inconsistencies but insuﬃcient to calibrate the traces accurately. A useful check is to investigate the scaling of the detected electrical noise power with the optical power, as shown in Figure 8.3. Quantum noise, electronic noise, and classical photon noise, or modulations, all scale diﬀerently. Electronic or dark noise is usually independent of the optical power Popt . The noise power 0.5 . The noise due to classical optical due to quantum noise is proportional to Popt noise is proportional to Popt . Figure 8.3 shows four cases of diﬀerent mixtures of noise and their scaling. A reliable measurement of the quantum noise should only be attempted for situations that show a linear dependence of the measured noise level on Popt , in this case near Popt = 1.

8.1 Detection and Calibration of Quantum Noise

Log noise power, Pi (Ω)

100 (d) 10 (c) 1 (b) (a) 0.1 0.1

1 10 Log optical power, Popt

100

Figure 8.3 Scaling of diﬀerent types of noise for a beam of light with ﬁxed classical relative intensity noise (RIN = constant) and variable optical power Popt . The noise terms scale diﬀerently with Popt : (a) electronic noise, (b) quantum noise, (c) classical noise, and (d) total noise as seen in an experiment. For convenience all terms are normalized by the quantum noise present at Popt = 1.

8.1.1.1

White Light Calibration

An alternative is to calibrate the detector with a light source that we know is quantum noise limited. For example, we could use a thermal white light source, which has Poissonian photon statistics (Section 2.3). In principle all that is required is to illuminate the detector with white light that generates the same DC photocurrent and to record the RF noise spectrum. However, care has to be taken: the wavelength should be ﬁltered to the same wavelength (about ±20–50 nm) as used in the actual experiment since the response of photodiodes is known to be wavelength dependent. The absorption depth depends on the wavelength. In addition, the illumination pattern has to be similar since localized illumination can lead to charge carrier depletion. Finally, the diode, particularly the very small InGaAs diodes, can easily be geometrically overﬁlled. Once a large fraction of the light misses the photosensitive area, it can heat up the entire chip and distort the results. From experience, large photodetectors can be calibrated using white light to about ±0.1 dB, whilst small InGaAs detectors can be calibrated to an accuracy of about ±0.4 dB. 8.1.2

Balanced Detection

An alternative technique for calibration is to use a balanced detection system. Here the incoming light is divided into two beams of equal optical power, which are detected by two matched photodetectors (Figure 8.4). The two photocurrents are either added or subtracted. Two noise spectra are created by the electronic spectrum analyser and are recorded as the sum and diﬀerence spectrum. This technique can distinguish between optical quantum noise and classical optical noise. The beamsplitter sends classical noise onto both detectors – the classical optical noise in both beams, and thus the photocurrents are correlated to each other. That means they can be subtracted and cancelled. When added, the full

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Vvac (Ω)

η

Vlas (Ω)

η

+

– gel (Ω) pel (Ω)

Vi (Ω)

Figure 8.4 Balanced detector.

noise power is recovered. For quantum noise the eﬀect of the beamsplitter is diﬀerent – the two resulting photocurrents are not correlated. That means the two channels add in quadrature for both the sum and the diﬀerence. In the quantum noise limited case, the two spectra are identical. This argument is based purely on the correlation properties of the sidebands containing the information about the noise, as described in Section 4.5. A more formal derivation of these results is given in Appendix B, which describes the balanced detector in terms of the propagation of coherent states through a beamsplitter. Electronic noise from the two detectors is obviously not correlated and adds in quadrature. It cannot easily be distinguished from quantum noise. Consequently, the apparatus can suppress all classical technical noise on this mode of light and leaves only the optical quantum and the electronic noise. The description is p+ (Ω) = gel (Ω) pi (Ω) Vi (Ω) + pel (Ω) p− (Ω) = gel (Ω) pi (Ω) + pel (Ω) Vi (Ω) =

p+ (Ω) − pel (Ω) p− (Ω) − pel (Ω)

(8.2)

In the case of sub-Poissonian intensity noise, the plus trace displays the noise suppression, and the minus trace shows quantum noise only. The technical difﬁculties with this detection scheme is the balancing of the detectors. The two photodetectors have to have the same electronic RF response. We have eﬀectively built an electronic interferometer, and both the gain and the phase have to be matched at all RF that are used. This can be extraordinarily time consuming since the response frequently depends not only on the notional values of the components used but also on the actual physical layout of the circuit. Obviously cable lengths and delay times in the circuits have to be matched to achieve a proper balance. Passive splitters and combiners are used as adders and subtractors at high frequencies (≥ 50 MHz), whilst active circuits are useful for low and medium frequencies. The optical balancing can be tricky since it ideally requires a precise 50/50 beamsplitter. This can be achieved by careful adjustment since the splitting ratio for most dielectric beamsplitters depends on the input angle. Alternatively, a polarizing beamsplitter (PBS) can be used. For a linearly polarized output beam, the ratio can be continuously adjusted by rotating the polarization direction ahead of the beamsplitter with a half-wave plate.

8.1 Detection and Calibration of Quantum Noise

In all these arrangements optical losses have to be avoided if squeezing is to be detected since they would eﬀectively decrease the quantum eﬃciency. It is tempting to compensate a mismatch in the electronic gain with an adjustment in the optical power. But again, this would decrease the quantum eﬃciency. Thus many of the common tricks used in detecting ordinary coherent light are not allowed for squeezed light. 8.1.3

Detection of Intensity Modulation and SNR

The size of a modulation can be measured directly. After all, electronic spectrum analysers were designed for this purpose. The height of the spectral peak is proportional to the power of the Fourier component. The relative intensity noise is linked to the measured power by pi (Ω) = (i RIN(Ω))2 × RBW 1

RIN(Ω) = 10 2 pi (Ω) dBm

(8.3)

For example, a beam that generates an average photocurrent of 1 mA with a modulation of one part in 104 (RIN = 10−4 ) produces a noise peak of −10 dBm when measured with a RBW of 100 kHz. For large modulations, the photocurrent SNR, as deﬁned in Eq. (7.7), can be measured directly as the diﬀerence, in dB, between the signal peak and the noise ﬂoor. For small modulations we have to take into account that the signal peak contains both the signal and the noise and have to calculate the value SNR = (signal peak − noise)/noise. For very small SNR (< 2), the exact response of the spectrum analyser has to be considered (Appendix E). Electronic gain will not aﬀect the SNR. Finally, the total quantum eﬃciency, including any form of attenuation of the light, has to be taken into account when using Eq. (7.7) to evaluate the optical SNR from the measurements of the photon current. 8.1.4

Homodyne Detection

The direct detection schemes described in Section 8.1 are used to measure the intensity as well as the intensity noise of a laser beam. For a single-mode ﬁeld, the combination of two detectors with a beamsplitter can be used to evaluate the contribution of classical modulation and quantum noise. However, the direct detection cannot diﬀerentiate between the quadratures, eﬀectively only measuring the in-phase (or amplitude) quadrature. To measure diﬀerent quadrature angles, we need a reference beam, commonly called the local oscillator. This beam has to be phase-locked to the input; otherwise it cannot provide a phase reference to distinguish between the quadratures. If the local oscillator has the same frequency as the input, we have a homodyne detector. Alternatively, if the local oscillator is frequency shifted, the system is known as a heterodyne detector. 8.1.4.1

The Homodyne Detector for Classical Waves

For a homodyne detector we can assume that the local oscillator is created by the same source as the input beam we want to measure. In practice the link

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Unknown beam αin Laser

M1 M4 Local oscillator αlo

D1

ϕlo Phase shifter

D2

_

i_(t)

Figure 8.5 Homodyne detector.

could be much more indirect. The entire system can be represented by one large interferometer (Figure 8.5). The input beam 𝛼in and the local oscillator beam 𝛼lo are combined by the mirror M4 that has a balanced 50/50 reﬂectivity. The whole apparatus has to be aligned in the same way and with the same precision as a conventional interferometer. The two detectors D1 and D2 will receive the interference signal. Firstly, let us classically analyse the performance of the system. The two beams can be represented by the amplitudes 𝛼in (t) = 𝛼in + 𝛿X1in (t) + i𝛿X2in (t) 𝛼lo (t) = [𝛼lo + 𝛿X1lo (t) + i𝛿X2lo (t)] ei𝜙lo

(8.4)

i𝜙lo

Here e represents the relative phase diﬀerence between the input amplitude and the local oscillator amplitude. It can be controlled by changing the length of the one arm of the interferometer designated as the local oscillator. In practice, this involves the ﬁne control of a mirror position by a piezo or the rotation of a glass plate in the local oscillator beam. The local oscillator should dominate the output signal. We consider the case that the local oscillator beam is far more 2 . In this case we can assume that all the intense than the input beam: 𝛼lo2 ≫ 𝛼in intensity is from the local oscillator and that the intensities on both detectors are the same: 1 |𝛼D1 |2 = |𝛼D2 |2 = |𝛼lo |2 (8.5) 2 All terms proportional to 𝛼in can be dropped in comparison with those proportional to 𝛼lo . The signal at the two detectors can be described using the standard convention for the phase shift in a single beamsplitter (Eq. (5.1)), including a 180∘ phase shift for one of the reﬂected beams (Figure 8.6): √ √ 𝛼D1 = 1∕2 𝛼lo (t) + 1∕2 𝛼in (t) √ √ (8.6) 𝛼D2 = 1∕2 𝛼lo (t) − 1∕2 𝛼in (t) The intensities and the photocurrents from the two detectors are proportional to |𝛼D1 |2 and |𝛼D2 |2 , respectively, ∗ |𝛼D1 |2 = 1∕2 [ |𝛼lo (t)|2 + 𝛼lo (t)𝛼in (t) + 𝛼in (t)𝛼lo∗ (t) + |𝛼in (t)|2 ]

8.1 Detection and Calibration of Quantum Noise

Figure 8.6 Projection through the uncertainty area.

δX2

θ = ϕlo δX1

which can be approximated as |𝛼D1 |2 = 1∕2 [|𝛼lo |2 + 2𝛼lo 𝛿X1lo (t) + 2𝛼lo (𝛿X1in (t) cos(𝜙lo ) + i𝛿X2in (t) sin(𝜙lo )) ]

(8.7)

∗ compared with |𝛼lo |2 and 𝛼lo 𝛿X ∗ compared by neglecting terms such as 𝛼lo 𝛼in ∗ with 𝛼in 𝛿X . In addition all higher-order terms in 𝛿X have been left out. Similarly, the result for the other detector can be found. We can now deﬁne the diﬀerence current i− (t) from the two detectors:

i− (t) ≈ 2𝛼lo (𝛿X1in (t) cos(𝜙lo ) + i𝛿X2in (t) sin(𝜙lo ) )

(8.8)

This is a remarkable result. The current i− (t) scales with the amplitude of the local oscillator, but the ﬂuctuations, or noise, of the local oscillator are completely suppressed. Its noise level could be well above the QNL, and the result would not be aﬀected. On the other hand, the power of the input beam has no inﬂuence, as long as it is small compared to the power of the local oscillator. This is an extremely useful and somewhat magic device. The diﬀerence current i− (t) contains only higher frequency, or AC, terms. Obviously the DC cancel each other. In the experiment we could use a splitter/combiner or a diﬀerential AC ampliﬁer to generate this ﬂuctuating current. Finally, the variance of this current is evaluated. We should note that in this calculation all cross terms of the type 𝛿X1𝛿X2 cancel out in the averaging process. We obtain Δi2− ≈ 4𝛼lo2 (𝛿X12in cos2 (𝜙lo ) + 𝛿X22in sin2 (𝜙lo ))

(8.9)

This means the measured variance is a weighted combination of the variance of the ﬂuctuations in the two quadratures. In particular for the phase 𝜙lo = 0, the variance in the quadrature amplitude is measured, whilst for 𝜙lo = 90 = 𝜋∕2 the variance in the quadrature phase is measured. An alternative interpretation of Eq. (8.9) is that the homodyne detector measures the variance of the ﬂuctuating state under a variable projection angle 𝜃 = 𝜙lo . By scanning the local oscillator phase, the projection angle can be selected. The sideband interpretation says that the diﬀerent types of modulation diﬀer by the relative phase 𝜃 between the two beats between the two sidebands and the carrier. The homodyne detector selects one such phase. The modulation with the other phase (𝜃 + 90) is not detected at all. The homodyne detector ﬁlters out speciﬁc quadrature of the modulation sideband. With the ﬁxed phase shift 𝜙lo for the local oscillator, the detector will have best sensitivity for one particular type of modulation. For example, for 𝜙lo = 0, only amplitude modulation is detected, and

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8 Quantum Noise: Basic Measurements and Techniques

for 𝜙lo = 90, only phase modulation is detected. The sensitivity of the detector to this quadrature changes with cos2 (𝜙lo ). The homodyne detector is very popular in radio applications and is a key technology in our present communication systems. Signals are detected in phase with a local oscillator, in the case of radio waves an electronic oscillator inside the receiver, which can be controlled in frequency very well, for example, using quartz crystal oscillators, or could even be locked to the incoming RF (known as phase-locked loop (PLL)). Radio homodyne and all the later heterodyne detection techniques have allowed us to place diﬀerent carriers very close to each other in the radio spectrum and to detect very weak signals. This increases dramatically the carrying capacity compared to a simple intensity-coded system and allowed applications such as the mobile phone, satellite communications, etc. Quantum mechanical calculations are required to check the validity of the above classical results for the quantum properties of the light. The balanced homodyne detector is a special case of a Mach–Zehnder interferometer (Section 5.2) with a 50/50 output mirror. The two incident ﬁelds on the beamsplitter M4 can be described by the operators ̂ in (t) + i 𝜹X𝟐 ̂ in (t) â in (t) = 𝛼in + 𝜹X𝟏 ̂ lo (t) + i 𝜹X𝟐 ̂ lo (t)] ei𝜙lo â lo (t) = [𝛼lo + 𝜹X𝟏

(8.10)

where we are using the nomenclature of Section 4.5.1. In complete analogy to the classical derivation, we can now derive the equivalent to Eq. (8.9) but now for the full quantum operators: ) ( ̂ 2 ⟩sin2 (𝜙lo ) ̂ 2 ⟩cos2 (𝜙lo ) + ⟨𝜹X𝟐 Δi2− ≈ 4𝛼lo2 ⟨𝜹X𝟏 in in = 4𝛼lo2 (V 1in cos2 (𝜙lo ) + V 2in sin2 (𝜙lo ))

(8.11)

The derivation of Eq. (8.11) holds for the approximation that 𝛼lo ≫ 𝛼in . In this simple interferometer we have a beautiful device for the measurement of the variance for a single quadrature, including all quantum properties. We can select the quadrature by simply varying the phase angle of the local oscillator. For a coherent state, where all quadratures are identical, this technique has no particular advantage over direct detection. However, once the variances of the quadratures change, as in a squeezed state, the homodyne detector is the essential measuring system. It was ﬁrst proposed by H.P. Yuen and J.H. Shapiro [2] and D.F. Walls and P. Zoller [3] and has since been a key component in all experiments with quadrature squeezed states, as discussed in Chapter 9. Finally, a few practical hints are given: It should be noted that the homodyne detector can be operated even when the intensity of the input is small. As long as |𝛼in |2 is smaller than the intensity |𝛼lo |2 of the local oscillator, all the results given above are correct. Thus the detector can be used to measure squeezed vacuum states as well as bright squeezed light. If |𝛼in |2 is comparable with |𝛼lo |2 , the output intensity will change periodically with 𝜙lo , and the variance will be a mixture of variances from both orthogonal quadratures. Balancing the homodyne detector is the most diﬃcult part of this experiment. Both the amplitude and the wave front curvature have to be matched for both outputs. This generally means that

8.1 Detection and Calibration of Quantum Noise

both the local oscillator and the input beam should travel comparable distances and should experience similar optical components such as lenses and mirrors. In squeezing experiments we cannot aﬀord any losses; otherwise some vacuum noise would be added, and the noise suppression would deteriorate. That means some of the tricks quite common with normal classical interferometers cannot be used here. For example, we have to detect all of the beam, and using an aperture is not allowed. We cannot balance the two beamsplitter outputs by attenuating one of the beams. It is also important that all the electrical components are balanced and they should have the same gain and phase diﬀerence at all detection frequencies used. The quality of the alignment can be judged by measuring the fringe visibility VIS of the system (deﬁnition in Eq. (5.28)). This can be done in some experiments by turning up the power of the input beam to be the same as that of the local oscillator. As long as some input intensity is available, the normalized visibility = VIS∕VISperfect can be measured, where VIS is the directly measured value and VISperfect is the visibility predicted from the measurement of the intensities. For = 1 the homodyne detector records all the correlations accurately. The measurement of the extrema is reduced when < 1, and the correlations between the input and the local oscillator beams appear to be smaller than they really are. The quality of the homodyne detector is described by an overall eﬃciency: 𝜂hd = 𝜂det 2

(8.12)

where 𝜂det is the normal eﬃciency of the photodetector. In most situations 𝜂det is smaller than 2 . Values as good as 2 = 0.98 have been achieved. These details will be discussed in the description of the individual experiments. The homodyne detector is the most common technique to measure the noise and signal properties of both quadratures. It is more complex and requires more patience than direct detection, but it provides the full information. 8.1.5

Heterodyne Detection

Whilst in homodyne detection we are mixing two beams with the same optical frequency on one detector, with a beamsplitter we can also mix beams with diﬀerent optical frequencies 𝜈1 and 𝜈2 . This is known as heterodyne detection. As long as the diﬀerence frequency ΩHT = 𝜈1 − 𝜈2 is within the detection range of the photodetector, and in practice that is limited typically to about 1–2 GHz, we will record a strong, narrow signal at ΩHT , which is surrounded by beat signals created by all the sidebands of either laser beam beating with the respective carriers of the other beam. Note that a spectrum analyser cannot distinguish between positive and negative beat frequencies and these two will be folded together (Figure 8.7). Heterodyne detection is frequently used to determine the linewidth Δ𝜈2 of an unknown laser. The spectrum around the detection frequency ΩHT = 𝜈1 − 𝜈2 contains information about both beams. The lineshape of the heterodyne signal is a convolution of the lineshapes of the two lasers. If one of the laser beams has a narrow linewidth compared with the other, e.g. Δ𝜈1 ≪ Δ𝜈2 , the width ΔΩHT of

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8 Quantum Noise: Basic Measurements and Techniques

Spectrum analyser Laser 1

Laser 2 Mode matching optics

Figure 8.7 Example of heterodyne detection of the bandwidth of two lasers.

the heterodyne signal corresponds to the broader linewidth, in this case Δ𝜈2 . The exact link between ΔΩHT and Δ𝜈 depends on the lineshape. For example, if both lasers have Lorentzian lineshapes, which are quite common, and Δ𝜈1 ≈ Δ𝜈2 , the link is ΩHT = Δ𝜈1 + Δ𝜈2 . Furthermore, the heterodyne spectrum will show any additional laser modes in the input beams. These will appear as strong, narrow spectral features around ΩHT . In heterodyne detection we are measuring signals at high detection frequencies, and outside the prevailing technical noise at low frequencies, the heterodyne technique is very sensitive in the amplitude quadrature. Ideally it is only limited by the QNL of the stronger of the two input beams, and in this case the detection is at the QNL, or commonly called shot noise limited. This technique has been used to detect with very high sensitivity the ﬂuorescence spectrum of individual ions in traps or individual atoms and to determine such properties as the ion trap frequency (as low as hundreds of Hz) and the excitation temperature. In this regime heterodyne detection competes with single-photon detection in sensitivity. 8.1.5.1

Measuring Other Properties

The detection schemes discussed so far concentrate on the measurement of the intensity and the quadratures of the laser light. However, there are other properties of the light, in particular the position and the polarization of the light, that are worth a consideration. Many applications rely on the modulation and detection of these properties, as can be seen from Figure 8.8. Detector (c) measures the beam position by comparing the diﬀerence in the intensity falling onto the two halves of a split detector. This could also be done in both transverse directions using a quadrant detector. Detector (d) uses a PBS to analyse the degree of polarization, in this case horizontal and vertical linear polarization. Other states of polarization can be determined by including a wave plate ahead of the PBS. This comparison shows that in all cases we are using two photodetectors and we are processing the diﬀerence of the photocurrents. In all these cases we will ﬁnd that for a coherent and single-mode beam, the perfect laser beam, each detector produces a photocurrent that includes quantum noise and that simply adds to the quantum noise from the other detector. This means we have a quantum noise for the detection of all these properties: intensity, quadrature position, and

8.2 Intensity Noise

D1 Laser

Laser D2

D1

+/–

(a)

Phase shifter

(b)

D2

_

D1

Laser (c)

Laser (d)

PBS

D2

Figure 8.8 Comparison of detection schemes: (a) balanced detector for V1(Ω), (b) homodyne detector for the quadratures, (c) split detector for the position of the beam, and (d) polarization detector.

polarization. The last one is normalized diﬀerently, and the uncertainty can be expressed in terms of the Stokes parameters that have a cyclic commutation relationship [4, 5], but is still a QNL. We will ﬁnd in Chapters 9 and 10 that for all these properties we can generate non-classical states of light with reduced noise below the QNL.

8.2 Intensity Noise 8.2.1

Laser Noise

Lasers are the light sources in all quantum optics experiments, but they are far from ideal; most lasers have intensity noise well above the QNL as discussed in Chapter 6, page 210. With careful preparation some lasers can be QNL in restricted parts of the spectrum. Good examples are the diode-pumped monolithic Nd:YAG laser, which is one of the quietest and most reliable lasers available (https://www.coherent.com/lasers/main/mephisto-lasers) [6, 7]. These lasers are QNL for frequencies well above 5 MHz, as shown in Figure 6.6, and thus are rather useful light sources for quantum optics experiments. Diode lasers themselves can be excellent low noise light sources at frequencies up to several hundred MHz, as discussed on page 213. In particular the low-power ( 0, the noise is increased. This is not correct. The suppression depends on the value |1 − h(Ω)|2 . This is best explained with a Nyquist diagram (Figure 8.10b). Equation (8.16) means that the feedback loop decreases the variance at all those frequencies where |1 − h(Ω)| > 1, that is, where h(Ω) is outside a circle of radius 1 drawn around the point (1, 0) on the Nyquist diagram. The reduction in the variance is proportional to the square of the distance between h(Ω) and (1, 0). If the line that traces h(Ω) is inside the circle, the variance is actually increased, not reduced. If h(Ω) gets close to (1, 0), the resulting noise is extremely large. The stability of the systems can be described by one simple rule: if the line that traces h(Ω) in the Nyquist diagram encloses the point (1, 0), the system is unstable. It will oscillate madly. This includes the case where h(Ω) = 1. All other systems, where h(Ω) does not enclose (1, 0), are stable. But note that even in a stable system, there can still be frequencies where the noise is increased, as demonstrated in the example shown in Figure 8.11. Finally, we ﬁnd that the variance of the modulation of the beam leaving the classical system is scaled by the transmission 1 − 𝜖 of the beamsplitter and the quantum eﬃciency 𝜂 of the detector, in the same way as the intensity of the output beam, resulting in the transfer Figure 8.11 The action of the noise eater on a large classical signal using the feedback gain shown in the previous diagram. Input noise is V1las (Ω) = 1000, very large compared to the QNL. The resulting output V1out (Ω) remains well above the QNL of V1out (Ω) = 1.

10 000 Vin = 1000 1000 Vout

284

100

10

1

10

100

1000 Ω

10 000

100 000

8.3 The Intensity Noise Eater

function V 1out (Ω) = 𝜂2 (1 − 𝜖) V 1il (Ω) = 𝜂2 (1 − 𝜖)

V 1las (Ω) |1 − h(Ω)|2

(8.17)

where 𝜂2 is the quantum eﬃciency of the output detector. To illustrate Equation (8.17) consider the output noise spectrum shown in Figure 8.11, which is calculated for a large constant laser noise V 1las (Ω) = 1000, very much larger than the QNL. The feedback system has the gain h(Ω) shown in Figure 8.10. We see the eﬀect of a feedback ampliﬁer with a maximum gain at 300 kHz and unity gain points at 6 and 1000 kHz. The gain drops below 1 at frequencies below 6 kHz and above 1000 kHz. This feedback loop has an internal delay time of about 4 μs. The phase response is determined by the somewhat resonant nature of this circuit. In the Nyquist diagram this corresponds to a line that starts with straight negative feedback until h(Ω) changes sign at about 400 kHz. However, the noise is reduced even at higher frequencies up to 1000 kHz since the noise reduction is proportional to the square of the distance to the point (1, 0). The complete spectrum of the output variance is shown in Figure 8.11. The noise is suppressed from very low frequencies to 1000 kHz. The maximum gain results in the best noise suppression at 300 kHz. But when the line h(Ω) enters the circle around (1, 0), the noise increases. A clear hump is visible from 1000 kHz to 40 MHz. This feedback loop functions well at medium frequencies with high gain. But at higher frequencies the phase has increased too much, and excess noise is created. This result is not atypical for a realistic feedback system. In practice one has to work very hard to ﬁnd a design that reduces the gain quickly enough and simultaneously avoids too much electronic phase change. 8.3.2

Quantum Noise Control

The behaviour of the feedback system near the standard quantum limit (QNL) is quite diﬀerent. In this case we have to include the vacuum ﬂuctuations from the unused input port of the beamsplitter. We can also no longer assume that the variance inside and outside the feedback loop will remain the same. Figure 8.12 shows the apparatus including all vacuum ﬁelds. The input into Modulator

Avac

η2

V1las (Ω) Avac

Laser PID h(Ω)

η1

Avac V1out (Ω)

V1il (Ω)

Figure 8.12 Schematic diagram of a noise eater with all vacuum terms.

285

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8 Quantum Noise: Basic Measurements and Techniques

̃ las (Ω). We distinguish between the in-loop the noise eater is described by 𝜹A ̃ il in front of the beamsplitter, the ﬁeld in front of the in-loop detector ﬁeld 𝜹A ̃ f , the ﬁeld that is leaving the apparatus 𝜹A ̃ o and the two electrical signals, one 𝜹A ̃ el that is fed back to the modulator, and 𝜹A ̃ out from inside the feedback loop 𝜹A the detector that measures the beam leaving the apparatus. We recall that we are using nomenclature introduced in Section 4.5.1 whereby capitals and tildes indicate continuum operators in the frequency domain with commutators: ̃ ̃ † (Ω′ )] = 𝜹(Ω − Ω′ ). [𝜹A(Ω), 𝜹A It is now necessary to deﬁne a quantum description of the feedback. This problem was addressed as early as 1980 and was approached in various ways. The most obvious and direct approach was pioneered by Wiseman et al. [9]. The main feature is that the current signal at time t is not only related to the instantaneous value of the photon ﬂux Nil but is the convolution with a response function that takes the history of the signal into account and simulates the gain of the feedback loop. For frequencies that are smaller than the inverse of the round-trip time of the feedback loop (Ω < 1∕tloop ), the eﬀect is similar to the eﬀect in a classical feedback system, and we can describe the action of the modulator simply by √ ̃ il = 𝜹A ̃ las + g(Ω) 𝜹A ̃ el 𝜹A (8.18) which describes the action of the intensity modulator that combines the input ﬁeld and the electrical signal. The very small absorption of this device, which also corresponds to a vacuum input, has been ignored. The gain of the in-loop electrical ampliﬁer is given by the value g. The various ﬁelds are linked by √ √ ̃ il + 1 − 𝜖 𝜹A ̃ vac ̃ f = 𝜖𝜹A 𝜹A √ √ ̃ el = 𝜂𝜹A ̃ f − 1 − 𝜂 𝜹A ̃ vac 𝜹A √ √ ̃ o = 1 − 𝜖𝜹A ̃ il + 𝜖 𝜹A ̃ vac 𝜹A √ √ ̃ o − 1 − 𝜂 𝜹A ̃ vac ̃ out = 𝜂𝜹A 𝜹A (8.19) where we have assumed that the two detectors have the same quantum eﬃciency 𝜂 and recall that 𝜖 is the reﬂectivity of the feedback beamsplitter. We can now ̃ out , ̃ el and 𝜹A combine the equations in (8.19) to provide explicit expressions for 𝜹A ̃ ̃ convert these into the quadrature operators 𝜹X𝟏el and 𝜹X𝟏out , and use our standard recipe to calculate the noise spectra V 1el (Ω) for the current inside the loop and V 1out (Ω) for the current from the external detector. The details of the calculation are given in Appendix D, and the resulting transfer functions are V 1el (Ω) =

𝜂𝜖(V 1las (Ω) − 1) + 1 |1 − h(Ω)|2

(8.20)

and V 1out (Ω) = 1 +

2 1 − 𝜖 𝜂𝜖(V 1las (Ω) − 1) + |h(Ω)| 𝜖 |1 − h(Ω)|2

(8.21)

√ where h(Ω) = 𝜂𝜖 g(Ω) is the open-loop round-trip gain, which includes both the electronic gain g(Ω) and the optical attenuation. Note that our model

8.3 The Intensity Noise Eater

of the modulator requires the intensity attenuation to be combined with the voltage gain. These results show that the properties of the light inside the controller and the beam leaving the controller are entirely diﬀerent. Inside (Eq. (8.20)), the modulation and the noise are strongly suppressed, and for a large gain the variance V 1el (Ω) can be well below the QNL. The noise eater works well inside, which is possible, since the laser beam inside is a conditional ﬁeld. It only exists inside the feedback system, and the conditions for a freely propagating beam do not apply. The standard commutation rules do not apply and allow this violation of the normal quantum noise rule. Such light is sometimes referred to as squashed light [10]. For high gain, |h(Ω)| > 1, and a low level of laser noise, V 1las (Ω) ≈ 1, the variance V 1el (Ω) of the control current can be substantially less than the QNL. This was observed by A.V. Masalov et al. [11, 12] and others. It was initially concluded that the light inside the loop was sub-Poissonian and could be used for measurements with a better SNR than the QNL would allow. However, it was overlooked that any signal, for example, a modulated absorption, would be controlled by the feedback loop in a similar way as the classical noise. A detailed analysis has shown that even with sub-Poissonian control currents, it is not possible to improve the SNR. Any apparent advantage of the low noise of the control current cannot be used [13]. The properties of the freely propagating output beam (Eq. (8.21)) are quite different. This beam obeys the standard commutation rules and conditions for the quantum noise, and the spectrum V 1out (Ω) in most cases is above the QNL. We ﬁnd that this light has properties that are quite diﬀerent from the predictions of the classical model (see Eq. (8.17)) and contain unique quantum features. In many situations the eﬀect of the noise eater is actually counterproductive, and the machine increases the noise level. This is one of the striking examples where the properties of an optical instrument with small signals, near the QNL, are considerably diﬀerent to the conventional classical predictions, and standard engineering solutions will actually fail. In the area of laser noise control, this lesson had to be learned, and quantum optics rules are important for the technical progress. It is worthwhile to consider the following special cases: 1. For very low gain (h(Ω) → 0), this result shows simply the eﬀect of attenuation by the combination of beamsplitter transmission (1 − 𝜖) and quantum eﬃciency 𝜂 with V 1out (Ω) = 1 + (1 − 𝜖)𝜂 (V 1las (Ω) − 1). This is identical to a simple beamsplitter. Note that the eﬀect of the quantum noise contribution is included. 2. For a QNL input laser (V 1las (Ω) = 1), the feedback loop creates extra noise |h(Ω)|2 that increases with the gain h(Ω) with V 1out (Ω) = 1 + 1−𝜖 . This case 𝜖 |1−h(Ω)|2 is interesting for engineering applications since it means that in systems with very small signals, close to the QNL, the real feedback system shows a behaviour that is quite diﬀerent from the classical approximation. The higher the gain and the smaller the fraction 𝜖 of the light that is used for control, the more excess noise is generated. The physical explanation is that the vacuum noise at the beamsplitter is added and ampliﬁed. For example, if only

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8 Quantum Noise: Basic Measurements and Techniques

10 8 Vlas = 10 6 V1out

288

Vlas = 5 4 Vlas = 1 2 0

0

–1

–2

–3

–4

–5

Round-trip gain, h

Figure 8.13 The eﬀect of the feedback system with 𝜖 = 0.1, for diﬀerent modulations V1las = 10, 5, 1, as a function of the open-loop gain h. At small gains the larger signals are suppressed, and at larger gains (h = −5) all signals approach the same value, in this case V1out = 7.3.

10% of the light is used (𝜖 = 0.1) and the open-loop gain is h = −5 (negative feedback), the output variance will be 1 + ((1 − 0.1)∕0.1) 52 ∕(1 + 5)2 = 7.25. It could be signiﬁcantly larger if the feedback phase is not optimum. In order to avoid this noise penalty completely, all the light would have to be detected (𝜖 = 1), which is clearly impractical. One can minimize the excess noise by detecting most of the light, leaving only a small intensity at the output. In . the limit of very large gain, we obtain the limiting value V 1out (Ω) = 1 + 1−𝜖 𝜖 This value is independent of the gain, a kind of saturation eﬀect producing a ﬁxed noise level, as shown in Figure 8.13. ), the system shows the classical proper3. For very large signals (V 1out (Ω) > h(Ω) 𝜂𝜖 ties with the noise suppression given by V 1out (Ω) = no longer play any role.

𝜂𝜖V 1las . |1−h(Ω)|2

Quantum eﬀects

To illustrate this quantum eﬀect, the performance of the noise eater previously described in Figure 8.10b has been evaluated for a wide range of input laser noise levels. They range from V 1las = 1000 to V 1las = 1. All traces in Figure 8.14 are normalized to the QNL. At the highest frequencies where the gain is much less than one, the noise spectrum is unaﬀected. The top trace, with the large noise level, V 1las = 1000, clearly shows the same response as the classical feedback loop. Biggest suppression is achieved close to the maximum gain 300 kHz and some excess noise at higher frequencies, 2000 kHz. However, the lowest trace for a QNL laser clearly demonstrates the noise penalty. In this particular case the feedback system produces about 8 dB of excess noise. At laser noise levels of about 10 times the QNL, the feedback loop has very little eﬀect indeed since in this case noise suppression and noise penalty balance each other. One explanation for the noise penalty is the fact that the beamsplitter acts as a random selector of photons, not a deterministic junction (see the discussion in Section 5.1). The noise measured by the in-loop detector is a random selection

8.3 The Intensity Noise Eater

104 Vin = 1000

1000

300 100

Vout

100

30 10

10

1

3 1

0.1

1

10

100

1000

104

105

Ω

Figure 8.14 The eﬀect of a noise eater on a laser. Various input noise levels (V1las = 1, 3, 10, … , 1000) are shown. At low noise levels this device creates rather than eats noise.

of the laser noise that is not correlated to the ﬂuctuations transmitted to the output. This small fraction of the noise is detected, converted into a current, ampliﬁed, and added by the modulator to the remaining ﬂuctuations of the light from the laser. Obviously all this has to happen within a time less than 1∕Ω. Unlike in a classical feedback circuit, the noise that was detected cannot cancel the remaining noise since there is no correlation between them. Instead, they are added in quadrature, leading to the excess noise. The more gain we apply, the worse it gets. This phenomenon has been observed and described by a number of researchers. One particularly instructive description was given by Mertz and Heidmann who used a Monte Carlo technique to describe the properties of the beamsplitter [14]. 8.3.2.1

Practical Consequences

The consequence of this performance at the QNL is that we have to reduce the gain of the feedback loop to much less than unity at all detection frequencies where the laser should be quantum noise limited. Most lasers are QNL at large frequencies. Consequently, the task is to develop special electronic ampliﬁers that combine high gain at low frequencies, where classical noise dominates, with low gain (|h| < 10−2 ) at the frequencies where we need QNL performance. The trick to achieve this is a very steep roll-oﬀ in the gain without too much phase lag being incorporated [15], and this requires specialized electronic designs. There are diﬀerent options for intensity control. As discussed so far, an external intensity modulator can be added to the laser. It is also possible to control the pump power to the laser by feedback. In the case of diode lasers, the injection current can be directly controlled, which is a very elegant approach. The performance of this system can be predicted by incorporating the laser noise transfer function for pump noise into the description of the feedback loop. The solution is not trivial. The practical diﬃculty is that most current control units are not

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suitable for this since their external inputs include ﬁlters designed to avoid noise transients. This produces a phase lag that is far too large for a good feedback performance. In many cases it is better to leave the current source untouched and to add a small control current. This was demonstrated successfully by C.C. Harb et al. [16] for a diode laser-pumped CW Nd:YAG ring laser. The relaxation oscillation was suppressed by more than 40 dB, and the noise at low frequencies was largely eliminated. Similar circuits are available in commercial lasers [17]. In conclusion, a feedback intensity controller, or noise eater, is extremely useful to reduce technical noise in the intensity of a laser. With careful design the QNL can be achieved. However, it cannot be used to either generate an output beam that is sub-Poissonian (squeezed) or measure any eﬀect with an SNR better than the standard quantum limit that can already be reached with a coherent state. A complementary design, electro-optic feedforward, has quite diﬀerent properties that can be utilized for noiseless ampliﬁcation (Chapter 10) or quantum information processing (Chapter 13).

8.4 Frequency Stabilization and Locking of Cavities In many experiments it is very important to operate lasers and cavities that are locked to each other, which means they are tracking each other in the frequency domain. Active feedback techniques are required to achieve this aim. The length of either the cavity or the laser resonator is constantly adjusted to keep both of them in resonance at one frequency. Applications for such a system are build-up cavities for non-linear processes, mode cleaner cavities, and the stabilization of the laser frequency. In the latter case, a very quiet reference cavity is chosen, which is protected as much as possible from external inﬂuences, such as mechanical vibrations and temperature ﬂuctuations. The laser is locked to this reference and will have a much narrower linewidth than a free-running laser. The performance of lasers can best be described using the concept of the Allan variance and quoting the timescale on which the smallest variance is observed [18]. Frequency stabilization has been used since the very early days of laser design. Many of the milestones in this technology have been achieved by J. Hall and his colleagues in a continuous series of technical developments since the 1970s. Two current examples illustrate the degree of improvement that can be achieved. A free-running dye laser has a linewidth of many MHz, measured over an interval of 1 second. Using active control techniques, the linewidth can be reduced to about 1 kHz. A free-running diode laser-pumped CW Nd:YAG laser has already a narrow linewidth, 10 kHz measured over 100 seconds, due to its simple mechanical structure. Using active control techniques, linewidths of less than 1 Hz measured in 100 seconds can be achieved. And this is not a fundamental limit [19]. Many tricks are used in the design of the special reference cavities, such as low expansion materials, special mounting systems, and cryogenic control. Extraordinary precision has been achieved through painstaking engineering and some ingenious design. Here only the principles of the locking technique

8.4 Frequency Stabilization and Locking of Cavities

will be outlined, as an introduction to the techniques commonly used. In most CW quantum experiments, at least one cavity locking system is required. Frequently several are used simultaneously. In most cases the locking system can be described by the schematic block diagram shown in Figure 8.15a. The aim of Modulator PBS

Piezo element

Faraday isolator

Reference cavity

Laser

RF signal generator

~ Error signal Mixer

(a)

PBS

Piezo element

Faraday isolator

Reference cavity

Laser

Error signal (b)

Transmitted power (V)

2 TEM00

1.5

TEM10

TEM20

TEM30

(i)

1 0.5

PDH error signal (V)

0 0.05 (ii) 0

Tilt error signal (V)

–0.05 1 0.5

–0.5 –1

(c)

(ii)

0

0

1

2

3 4 5 Scanning time (a.u.)

6

7

Figure 8.15 Frequency locking system. (a) Schematic layout for PDH locking and phase response of the cavity. (b) Layout for tilt locking. (c) Experimental error signals as demonstrated by D.A. Shadock et al. (i) Transmitted power, (ii) error signal generated via PDH locking, and (iii) error signal generated via tilt locking technique. Source: Courtesy of D.A. Shadock et al. [20].

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the device is to dynamically control the detuning Δ𝜈, the diﬀerence between the cavity and the laser frequency, and to maintain resonance (Δ𝜈 = 0). The light from the laser is mode-matched, using a lens system, into the reference cavity. An error signal is generated, which is a measure of the detuning Δ𝜈. This signal is processed using an ampliﬁer of suitable gain, bandwidth, and phase response. Finally, an actuator is used, such as a piezo crystal mounted on a mirror, to control either the laser frequency or the cavity frequency. Let us discuss the most important features of these components. The cavity parameters are chosen to produce a suitable control range. Typically the control range is as wide as the cavity linewidth. For lasers with large frequency excursions, such as dye lasers, a low ﬁnesse cavity is required. For narrowband lasers, such as CW Nd:YAG lasers, a high ﬁnesse (up to 20 000 or more) cavity is chosen to provide a very steep frequency response. The cavities can be confocal, which reduces the diﬃculties of mode matching, or can have a conventional stable resonator conﬁguration.

8.4.1

Pound–Drever–Hall Locking

The error signal has to be responsive to very small changes in Δ𝜈. We require a technique that measures the detuning accurately and with low noise. The most popular technique is to measure the response of the cavity to frequency modulation sidebands. This technique was ﬁrst used by Pound for microwave systems and later reﬁned by several scientists for the optical domain. It is frequently known as the Pound–Drever–Hall (PDH) technique. In this case a phase modulator, driven by a frequency Ωpm , is placed between the laser and the cavity. The light reﬂected from the cavity is detected. Since optical feedback has to be avoided, an optical isolator (Section 5.4) is normally placed between the cavity and the laser, with the reﬂected beam directly accessible. The reﬂected light has modulation sidebands at frequency Ωpm . These are detected using a mixer and, as local oscillator, a part of the modulator driving signal. The amplitude of the resulting low frequency signal provides a suitable error signal. If the cavity is on resonance with the laser beam, which means Δ𝜈 = 0, the reﬂected light will be purely phase modulated. After the mixing process, no signal is available since the two sidebands balance each other exactly. The error signal for Δ𝜈 = 0 is zero. However, once the cavity is detuned, both PM sidebands will experience a diﬀerent phase shift, and the two beat signals will have a phase diﬀerence Δ𝜙out (Δ𝜈) with respect to each other. They will no longer cancel, and there will be some AM, a detectable error signal after the mixer. The shape of the error signal, as a function of cavity detuning, is easily explained for the case that the modulation frequency Ωpm is much larger than the cavity linewidth Γcav . The relative phase of the two beat signals is determined by the phase diﬀerence between the central component 𝜙c and the two sidebands 𝜙− , 𝜙+ . The three components have a distinctively diﬀerent phase, as discussed in Section 5.3. A typical phase response is shown in Figure 8.15c(ii). The corresponding power response, in this case for an impedance-matched cavity, is given in Figure 8.15c(i). We can arbitrarily deﬁne 𝜙(0) = 0 at resonance. That means

8.4 Frequency Stabilization and Locking of Cavities

for large modulation frequencies, we obtain 𝜙(−Ωpm ) = 𝜙(+Ωpm ) = 0 The phase diﬀerence at the output is Δ𝜙out (Δ𝜈) = (𝜙+ − 𝜙c ) − (𝜙c − 𝜙− ) = 𝜙(Δ𝜈 + Ωpm ) − 𝜙(Δ𝜈) − (𝜙(Δ𝜈) − 𝜙(Δ𝜈 − Ωpm ))

(8.22)

This output phase diﬀerence is shown in Figure 8.15c(ii). On resonance both terms are equal, 𝜙(Δ𝜈) = 0, and the two beat signals keep their original phase, Δ𝜙out (0) = Δ𝜙in (0). The modulation remains pure phase modulation. The error signal is zero. For any other detuning we have to evaluate the phase response explicitly. It crosses zero on resonance, Δ𝜈 = 0, and rises steeply on either side. Close to resonance it has a linear slope, which can be measured in rad/MHZ. The steeper the slope, the higher the gain in the feedback loop. At a detuning less than one cavity linewidth, the phase diﬀerence peaks and then rolls oﬀ gently. At such large detunings the feedback system still has the correct sign. However, it has a small gain and the system is only slowly pushed backed to the resonance. The regime of constant feedback sign is called the capture range. As long as the error signal stays within the capture range, the detuning will be reduced and will eventually arrive at resonance; outside it the system moves away from resonance. The capture range of this technique is large, as large as the modulation frequency. A diﬀerent interpretation of the result given in Eq. (8.22) is to say that the vector describing the reﬂected beam is rotated in the phasor diagram in regard to that of the input beam. However, the demodulated signal after the mixer corresponds to a measurement along the quadrature amplitude axis. Phase modulation of the input beam is orthogonal to this axis. The reﬂected light, with a rotated phasor diagram, now has some component of modulation in this direction. Phase modulation is partially converted to amplitude modulation; an error signal appears. The rotation angle 𝜃(Δ𝜈) is the half phase diﬀerence Δ𝜙out (Δ𝜈), and the maximum rotation is ±𝜋∕2. This rotation obviously applies to all types of modulations and also to ﬂuctuations. Error signals can be generated using AM or mixtures of AM and PM. The rotation of the quadrature is also used as a self-homodyning detector for quadrature squeezed light (Section 9.3). 8.4.2

Tilt Locking

An alternative technique for the generation of the error signal is tilt locking. This is based on the interference of spatial modes reﬂected by the cavity. The input ﬁeld is slightly misaligned with respect to the cavity, resulting in a TEM0,0 and other higher-order modes TEMi,j inside the cavity, which are all resonant at diﬀerent detunings Δi,j as a result of the Gouy phase shift (Sections 5.3.3 and 2.1.2). To ﬁrst order, the misaligned beam can be approximated by the sum of the TEM0,0 and TEM0,1 modes that have a phase shift Φ0,0 0,1 between them. Assuming the cavity is not confocal and that TEM0,0 is near resonance, the mode TEM0,1 will be far oﬀ resonance and will be fully reﬂected, like the sidebands in the PDH scheme. The mode TEM0,1 has a phase shift of 𝜋 between the two spatial components. The

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8 Quantum Noise: Basic Measurements and Techniques

mode TEM0,0 will be partially reﬂected, and we now have to combine the amplitudes of the two modes and detect the resulting beam with a split detector that is aligned to resolve the two parts of the TEM0,1 ﬁeld. The interference is critically dependent on Φ0,0 0,1 , which in turn depends on the detuning Δ0,0 . On resonance, Δ0,0 = 0, the two modes are exactly Φ0,0 0,1 = 𝜋∕2 out of phase, and the resulting amplitude on both sides of the split detector has equal magnitude. For a detuning to one side, Δ0,0 < 0, and the phase diﬀerence Φ0,0 0,1 is larger than 𝜋∕2, and this results in a reduction of the combined amplitude on one side of the split detector and an increase on the other side, as shown in Figure 8.15b. The amplitude 𝛼0,0 is rotated in the phasor diagram with respect to the amplitude 𝛼0,1 . The result is, for argument’s sake, a positive signal. For the opposite detuning, Δ0,0 > 0, the roles are reversed, and we will obtain a negative signal. Altogether we generate the error signal as displayed in Figure 8.15c(iii). We see that both techniques are equivalent and produce similar error signals. Tilt locking requires less equipment but is more sensitive to mechanical instabilities. PDH locking is the more commonly employed technique. A third alternative, after Hänsch and Couillaud, is based on the response of the cavity to two input polarizations. Provided the cavity has a higher ﬁnesse for one polarization than the other, the reﬂected light, when analysed in the appropriate polarization direction, will again produce an error signal with the required dispersion shape. Which particular technique should be chosen depends on the speciﬁc technical requirements. 8.4.3

The PID Controller

The other component is the feedback ampliﬁer. It processes the error signal. The gain should be as large as possible, and the bandwidth should span as far as possible to control slow and fast frequency excursions. However, there are severe limitations set by the requirements for the phase of the feedback gain. As already discussed in Section 8.3, a reliable rule for stable operation is that the gain rolls oﬀ with high frequencies and the phase shift at the upper unity gain point should be less than 𝜋. A more precise deﬁnition can be obtained using a Nyquist diagram (Figure 8.10). As a consequence the delay times have to be kept short, and any resonances, mechanical or electronic, have to be avoided. The gain is usually controlled by a PID ampliﬁer, which means a combination of proportional ampliﬁer (middle frequencies), integrator (low frequencies), and diﬀerentiator (high frequencies). All these components contribute to the electronic round-trip phase and have to be carefully optimized. Unfortunately, most electromechanical actuators have resonances in the acoustic range kHz (Figure 8.16), severely limiting the bandwidth of the ampliﬁer that has to reach unity gain at frequencies well below the resonance. One option is to split the control into several independent branches, for example, to use a piezo with large range (length changes of several optical wavelengths) at low frequencies to compensate for a drift of the frequencies and to use electro-optic components, such as a phase modulator inside the cavity, which has a smaller range but no resonance, for the control of the high frequency excursions. Extensive research

8.4 Frequency Stabilization and Locking of Cavities

kPZT, γPZT

Universe

(a)

Counter weight

PZT

kc,γc Universe

Counter weight

mm

mc xc(t)

x=0 Amplitude response

(b)

xm(t)

Phase response

20

0 –0.5 –1 –1.5

10 0 (i)

Amplitude (dB)

(iii)

ϕ control–π

10 0

Stability

Stability region

–10 region 20

0

–π ϕ control –0.5

10 0 Stability –10 region 20

–1 –1.5

Stability region

(iv)

–π 0

ol

ϕ contr

10 0

Stability

–10 region 10

0 –0.5 –1 –1.5

Stability region

20

30

40

50

10

Phase (rad/π)

(ii)

–10 20

20

30

40

–0.5 –1 –1.5 50

Frequency (kHz)

Figure 8.16 (a) Mirror mounted on a PZT modelled as a mass-and-spring system. The counterweight is the mass on which the PZT is pushing. This model includes the option of preloading. (b) Amplitude and phase response of a PZT (i) with small counterweight, (ii) with large counterweight, (iii) with large counterweight and low tension preloading, and (iv) with large counterweight and high tension preloading. Source: From W. Bowen [21].

has been carried out into these control systems, for example, for the applications in frequency standards and precision interferometry. 8.4.4

How to Mount a Mirror

Where the control is done using a lead zirconate titanate (PZT)-mounted mirror, the bandwidth of the feedback control system will be severely limited. The mirror, PZT, and mirror mount eﬀectively form a system with a series of mechanical resonances Ωres,j , from the lowest j = 1 to higher frequencies. Since this system is inside the feedback loop, we can expect stable operation only at frequencies well below the lowest resonance Ωres,1 . Otherwise the phase shift induced by the resonance will interfere with feedback control. The resonance frequency depends on the mass of the mirror, the mass of the mirror mount or counterweight, and the way the PZT and the mirror are connected. The ﬁrst improvement is to use the smallest possible mirror substrate and a PZT glued with a hard

295

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compound onto a massive backplate as the counterweight. This type of solution is found in many commercial systems, for example, many laser systems with an extended optical resonator. The next step is to modify the mount by preloading it, which means we contain the mirror and PZT in a rigid cage and exert a pressure force onto the mirror. In practical terms the cage is pushing via a rubber O-ring onto the mirror. The result is that the eﬀective spring constants are increased and that the resonance frequencies are increased. This trick has been used by many laboratories that specialize in laser frequency control, and many diﬀerent mechanical systems have been developed. One example was built and analysed by W. Bowen et al. [21] where the counterweight and the preloading tension can be systematically varied. They mounted the mirror as part of a cavity and recorded the cavity locking signal with an electronic network analyser that can record the amplitude and phase response. The results are shown in Figure 8.16b(ii). We can see that the resonances in the spectrum move towards higher frequencies. The shaded area indicates the useful locking range, and this is extended by more and more preloading. The preloading force is limited to values that cause no damage to the mirror. 8.4.5

The Extremes of Mirror Suspension: GW Detectors

This is only one of many examples of optimizing a mirror mount. Many special engineering solutions are being employed, ranging from very small and passive systems to large, complex active systems. The most extreme case is the mounting of the beamsplitter and end mirror in a gravitational-wave (GW) detector as outlined in Section 10.3. Here the mirrors are mounted on a most elaborate and optimized seismic suspension that includes both active and passive components located inside a vacuum system to suppress the vibrations in the environment by many orders of magnitude. The overall concept is to suspend the mirrors on a sequence of pendula, each with diﬀerent frequencies. The combined transfer function will show a large attenuation at all frequencies. The ground vibrations are the largest frequencies due to many seismic and man-made eﬀects from mHz to hundreds of Hz. The multi-pendula suspension in a GW detector has a very sharp cut-oﬀ and is typically used above10 Hz. Whilst very complex both mechanically and in regard to the feedback systems, such suspension systems can be extremely reliable and have been used in continuous long-term operation of GW detectors.

8.5 Injection Locking For a number of reasons, high-power lasers tend to have a much wider linewidth and not have a well-deﬁned output mode. The mechanical design is simple, allowing more mechanical vibrations; the control of the pump source is not very sophisticated, and the electric current driving the pump source could be noisy or more than one pump source may be used. On the other hand, very high-quality lasers of low power but with excellent frequency and intensity noise

8.5 Injection Locking

quality and perfect spatial mode can be built. The trick is to combine the high output power of one laser with the high beam quality of the other. This can be achieved through the technique of injection locking. The basic idea is to seed the high-power laser, referred to as the slave laser, with some of the output from the well-controlled, high-quality laser, the master laser, and force the high-power laser to take on most of the mode qualities of the master laser. The slave laser will change its output frequency and phase to that of the master, and it will remain locked to this frequency. As a consequence the laser linewidth of the slave will be reduced to that of the master. If the slave laser was already operating in a single spatial mode, no changes would occur, and the spatial mode is determined by the geometry of the resonator of the slave laser. However, if it was operating in several modes, one mode will get an advantage, through the injection of light, and the laser can change over to this speciﬁc spatial mode. The output power of the slave laser is basically unaﬀected by the injection locking. Injection locking is the optical equivalent of the locking of two oscillators, an eﬀect that was known for a long time for mechanical systems. It was reported that in 1865 Christiaan Huygens, whilst conﬁned to his bed by illness, observed that the pendula of two clocks in his room moved in synchrony if the clocks were hung close to each other, but became free running when hung apart. He eventually traced the coupling to mechanical vibrations transmitted through the wall from one clock to the other. This observation was later conﬁrmed using electrical oscillators and has developed into a widespread technique in electronic signal processing and microwave technology. Thus it was well known and established before the laser was invented. Optical injection locking will occur simply when the output beam of the master laser is matched into the resonator of the slave laser, both in the spatial waveform and by having the same polarization state. In addition, the optical frequency 𝜈m of the master laser has to be within the locking range 𝜈s ± Δl of the slave laser. Inside this range locking will happen completely. Outside this range no locking will occur at all, and the slave laser will operate as a free-running laser. The change from one state to another is abrupt. In practice this means that the slave laser will track the master laser as long as their independent free-running frequencies are within the locking range, |𝜈s − 𝜈m | = Δms < Δl . It is actually not easy to detect where the slave laser is within the locking range since the free-running frequency cannot be observed. The only indication would be a phase diﬀerence between the injection locked output beam and the master beam. To maintain locking, it might be necessary to add a control system that suppresses frequency drifts of the slave to keep the detuning Δms within the locking range. This can be achieved by a much simpler control system than would be necessary for a full frequency stabilization of the slave laser (Figure 8.17). An expression for the locking range can be derived by comparing the the power Ps of the free-running slave laser with the power of the master laser after ampliﬁcation inside the slave resonator. This later value is given by the product of the amplitude gain |G(𝜈)|2 and the power Pm injected from the master into the slave laser Pm . The gain |G(𝜈)|2 can be estimated as (2𝜅)2 ∕(2𝜋(𝜈 − 𝜈s ))2 where 𝜅 is the amplitude decay rate of the slave resonator that, in many small coupling cases, can be estimated as 𝜅 = (1 − R)∕t, R the reﬂectivity of the output coupler, and t

297

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8 Quantum Noise: Basic Measurements and Techniques

V1pm

Slave pump

Master pump V1master

V1slave

Master laser Slave laser

Figure 8.17 Schematic layout for injection locking.

the round-trip time. Setting the two values equal leads to the locking condition 2𝜅 √ |G(𝜈)|2 Pm ≥ Ps or Δl ≤ Pm ∕Ps (8.23) 2𝜋 As a practical example, if the slave laser has a power of 10 W, the laser resonator has a round-trip length L of 1 m, the cavity loss rate is 10% per round trip, and the √ master laser has a power of 0.1 W, the locking range will be Δl = 2(1 − R)c∕2𝜋L Pm ∕Ps = 0.5 MHz. The ﬁrst observation of optical injection locking was carried out in 1966 by Stover and Steier who used two He–Ne lasers for this purpose [22]. They controlled the frequencies of the two lasers and were able to conﬁrm the predicted locking range. The technique is now widespread, and it is used to control the properties of the light emitted by high-power lasers, to improve the mode quality, and to build powerful tunable lasers by locking a low cost powerful diode laser to well-controlled low-power diode lasers. On the other hand, frequency locking can also be an undesirable process. It prevents the measurement of very small frequency diﬀerences. For example, in laser gyroscopes, which are used in navigational applications, a beat signal between the two counter-propagating modes in a ring laser is measured. The modes experience diﬀerent cavity lengths, if the ring laser rotates in an inertial frame. They will beat and the frequency of the beat is a measure of the rate of rotation. These frequencies fbeat are very small, and they are given by 2𝜋fbeat = Ωrot A∕p where A is the area and p the perimeter of the gyroscope. Ωrot is the angular frequency of the rotation. For the Earth’s rotation the value, at the pole, is 2𝜋 in 24 hours or 75 × 10−6 rad/s. For a 1 m2 ring gyroscope located at the pole, the beat frequency is about 20 Hz. Locking of the two modes is very likely, but it obviously has to be avoided; otherwise the beat signal is zero, and it would appear that the Earth stood still. Extraordinarily small scatter losses are suﬃcient to provide enough light to cause locking. As a consequence the demand for laser gyroscopes has been the motivation for most of the recent improvements in low-loss optics and the advances in the technology of mirror manufacture. Quantum optics experiments have gained from these improvements. It is interesting to ask what the noise properties of the injection-locked laser system could be. Does the master laser dominate the system completely? Is it possible to build in this way a high-power quantum noise limited laser? This question was addressed by several researchers in the regime of classical ﬂuctuations [23]. More recently, complete laser models have been developed, and using the

References

65 (iii) Power (dB above QNL)

55 45 35

(ii)

25 15 (i)

5 QNL –5 0.01

0.1

1

10

Frequency (MHz)

Figure 8.18 The spectral intensity noise of an injection-locked laser. (i) A polynomial ﬁt to the experimental data. (ii) Experimental data. (iii) Excess noise due to ampliﬁcation of the master laser’s phase noise by the injection-locked slave laser. Source: From C.C. Harb et al. [25].

linearized quantum theory, it is possible to derive complete noise spectra for the injection locked systems. In these models the laser is literally described as an oscillator with two distinct internal modes, at the frequency of the master and the slave laser, and the combined system is solved within one laser model, similar to that described in Chapter 6 [24]. The outcome of these models and of experiments [25] is that the characteristic of the laser intensity noise varies with the detection frequency. There are clearly distinguishable regions of frequencies that show diﬀerent properties. A practical example is the locking of two Nd:YAG lasers. One region is centred roughly around the relaxation oscillation frequency of the slave laser (Figure 8.18). Here the slave laser operates eﬀectively like an ampliﬁer for the noise of the master laser. Note that if the master laser is quantum noise limited, this means output is considerably noisier than a QNL laser with the same power as the slave laser. At very high frequencies the output can reach the quantum limit, and it maintains the properties of the slave laser. At very low frequencies the dominant source of noise is the pump source of the slave laser. Diﬀerent strategies have to be followed to optimize the system at the diﬀerent frequencies.

References 1 Levenson, M.D., Shelby, R.M., and Perlmutter, S.H. (1985). Squeezing of clas-

sical noise by nondegenerate four-wave mixing in an optical ﬁber. Opt. Lett. 10: 514. 2 Yuen, H.P. and Shapiro, J.H. (1980). Optical communication with two-photon coherent states. I. Quantum-state propagation and quantum-noise reduction. IEEE Trans. Inf. Theory IT-26: 78.

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3 Walls, D.F. and Zoller, P. (1981). Reduced quantum ﬂuctuations in resonance

ﬂuorescence. Phys. Rev. Lett. 47: 709. 4 Robson, B.A. (1974). The Theory of Polarisation Phenomena. Oxford: Claren-

don. 5 Schnabel, R., Bowen, W.P., Treps, N. et al. (2003). Stokes operator-squeezed

continuous-variable polarization states. Phys. Rev. A 67: 012316. 6 Kane, T.J. and Byer, R.L. (1985). Monolithic, unidirectional single-mode

Nd:YAG ring laser. Opt. Lett. 10: 65. 7 Köchner, W. (1996). Solid-State Laser Engineering, 4e. Springer-Verlag. 8 Dorf, R.C. (1992). Modern Control Systems, 6e. Addison Wesly. 9 Wiseman, H.M., Taubman, M.S., and Bachor, H.-A. (1995).

10 11

12 13 14 15

16

17

18

19

20

21 22

Feedback-enhanced squeezing in second-harmonic generation. Phys. Rev. A 51: 3227. Wiseman, H.M. (1999). Squashed states of light: theory and applications to quantum spectroscopy. J. Opt. B: Quantum Semiclassical Opt. 1: 459. Masalov, A.V., Putilin, A.A., and Vasilyev, M.V. (1994). Sub-Poissonian light and photocurrent shot-noise suppression in a closed optoelectronic loop. J. Mod. Opt. 41: 1941. Troshin, A.S. (1991). Photon statistics in a negative-feedback optical system. Opt. Spektrosk. 70: 389. Taubman, M.S., Wiseman, H.M., McClelland, D.E., and Bachor, H.-A. (1995). Eﬀects of intensity feedback on quantum noise. J. Opt. Soc. Am. B 12: 1792. Mertz, J. and Heidmann, A. (1993). Photon noise reduction by controlled deletion techniques. J. Opt. Soc. Am. B 10: 745. Robertson, N.A., Hoggan, S., Mangan, J.B., and Hough, J. (1986). Intensity stabilization of an Argon laser using electro-optic modulator: performance and limitation. Appl. Phys. B 39: 149. Harb, C.C., Gray, M.B., Bachor, H.-A. et al. (1994). Suppression of the intensity noise in a diode pumped neodymium YAG nonplanar ring laser. IEEE J. Quantum Electron. 30: 2907. Kane, T.J. (1990). Intensity noise in a diode pumped single frequency Nd:YAG laser and its control by electronic feedback. IEEE Photon. Technol. Lett. 2: 244. Allan, D.W. (1987). Time and frequency (time-domain) characterization, estimation, and prediction of precision clocks and oscillators. ICEE Trans. Ultrason. Ferroelectr. Freq. Control 6: 647. Day, T., Gustafson, E.K., and Byer, R.L. (1992). Sub-Hertz relative frequency stabilisation of two diode laser pumped Nd:YAG lasers locked to a Fabry-Perot interferometer. Opt. Lett. 17: 1204. Shaddock, D.A., Gray, M.B., and McClelland, D.E. (1999). Frequency locking a laser to an optical cavity by use of spatial mode interference. Opt. Lett. 24: 1499. Bowen, W. (2003). Experiments towards a quantum information network with squeezed light and entanglement. PhD thesis. Australian National University. Stover, H.L. and Staier, W.H. (1966). Locking of laser oscillators by light injection. Appl. Phys. Lett. 8: 91.

References

23 Farinas, A.D., Gustafson, E.K., and Byer, R.L. (1995). Frequency and intensity

noise in an injection locked, solid state laser. J. Opt. Soc. Am. B 12: 328. 24 Ralph, T.C., Harb, C.C., and Bachor, H.-A. (1996). Intensity noise of

injection-locked lasers: quantum theory using linearized input-output method. Phys. Rev. A 54: 4359. 25 Harb, C.C., Ralph, T.C., Huntington, E.H. et al. (1996). Intensity-noise properties of injection locked lasers. Phys. Rev. A 54: 4370.

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9 Squeezed Light 9.1 The Concept of Squeezing So far we have worked with coherent states of light and the limits imposed by quantum noise that aﬀect all quadratures equally. However, it is possible to generate light that has less noise in one selected quadrature than the quantum noise limit (QNL) dictates. Light with this property is called squeezed light. This form of light is clearly the ﬁrst milestone in quantum optics with many photons. Squeezing technology made many applications of quantum optics possible and lets us create many improvements unique to quantum optics. In optical measurements and metrology, we can achieve a better signal-to-noise ratio than is possible with coherent light, and this is now state of the art in interferometry. Squeezed light is the tool for entanglement of laser beams and for techniques such as teleportation that play a role in communication and signal processing. Even quantum logic might be feasible. Combined with optimized detection techniques, squeezed light can be the source of speciﬁc eﬀects, such as Schrödinger’s cat states of light. In this chapter, we start by describing the eﬀect of non-linear optical processes on the photon statistics and the ﬂuctuations of the light. This is ﬁrst done in a classical approach – since it is possible to see some of the key results immediately. We use two examples that are intuitive and instructive. Next we introduce the concept of squeezed states of light rigorously. This will provide the mathematical tools to analyse the experiments. Finally, we discuss the ﬁrst experiments that have been performed since 1985 to generate squeezed light and describe the state of the art in 2018. 9.1.1

Tools for Squeezing: Two Simple Examples

The properties of quantum noise have already been introduced in detail in Chapter 4, and now we wish to reduce the ﬂuctuations of the light. We cannot suppress the ﬂuctuations altogether that would violate the uncertainty principle. But we can rearrange them between the quadratures. We can build experiments, and protocols, where we measure only one quadrature at a time and only the noise in this one quadrature contributes to the ﬁnal result. Consequently, it will be suﬃcient to reduce the ﬂuctuations in this one quadrature alone. The challenge is: What are the tools that would reduce the noise in one quadrature? How can we generate states of light that have asymmetric quadrature noise A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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9 Squeezed Light

Figure 9.1 Tools for squeezing. X2

X1

distributions that are squeezed in one quadrature? We will see in this section that the trick is to start with a normal coherent state and to introduce correlations ̂ and X𝟐 ̂ through non-linear between the ﬂuctuations of the two quadratures X𝟏 interactions (Figure 9.1). 9.1.1.1

The Kerr Eﬀect

To illustrate how correlations can reduce the quantum noise, the case of the non-linear Kerr eﬀect is instructive [1]. Consider a beam of light travelling through a medium, which has an intensity-dependent refractive index. The beam can be described as an optical ﬁeld Ein with both amplitude and phase ﬂuctuations: 𝛼(t) = 𝛼in + 𝛿X1in (t) + 𝛿X2in (t)

(9.1)

This could be a QNL laser beam, in which case 𝛿X1in (t) and 𝛿X2in (t) represent the quadratures of the quantum noise. The non-linear refractive index n(𝛼) can be described by n(𝛼) = n0 (1 + n2 𝛼 2 )

(9.2)

As a consequence ﬂuctuations in the intensity will modulate the refractive index. The refractive index, in turn, will modulate the phase of the transmitted light. For a medium of length L and light at the wavelength 𝜆, the output phase is given by 2πL n 2πL = 𝜙in (t) + (n0 + n2 (𝛼in + 𝛿X1in (t))2 ) 𝜆 𝜆 For small ﬂuctuations this can be approximated by 𝜙out (t) = 𝜙in (t) +

𝜙out (t) = 𝜙Kerr +

4πL n (𝛼 𝛿X1in (t) 𝜆 2 in

(9.3)

(9.4)

9.1 The Concept of Squeezing

ϕin – ϕout αout

(a)

αin

(b)

αin

Figure 9.2 Eﬀect of a Kerr medium on the electric ﬁeld vector: (a) input ﬁeld and (b) output ﬁeld. Note that the ﬁeld illustrated here has a large amplitude, compared to the ﬂuctuations. Thus the origin of the diagram is far away and not shown. Consequently any changes in the phase are orthogonal to changes in the amplitude. The main result is that the change in the phase ﬂuctuations depends on the intensity proportional to 𝛼 2 .

where 𝜙Kerr is the time averaged phase added by the medium. This is illustrated in Figure 9.2. There is no change to the intensity of the light: 2 2 = 𝛼in = 𝛼2 𝛼out

and 𝛿X1out (t) = 𝛿X1in (t)

(9.5)

At the output of the medium, the phase ﬂuctuations are those of the input light plus a component that is linked to the input intensity ﬂuctuations. Using the relationship between the quadratures 𝛼𝛿𝜙 = 𝛿X2, which is valid for small angles, this can be written in terms of the quadrature amplitudes: 4πLn0 n2 2 𝛼 𝛿X1in (t) 𝜆 = 𝛿X2in (t) + 2rKerr 𝛿X1in (t) 2πLn0 n2 𝛼 2 (9.6) with rKerr = 𝜆 where rKerr is a parameter that includes the non-linearity and the average amplitude 𝛼 of the input ﬁeld. Higher-order terms in 𝛿X1 have been neglected since they are exceedingly small, and only the terms linear in 𝛿X1 have been kept. We see from Eq. (9.6) that the non-linear medium couples the ﬂuctuations in amplitude and phase quadratures. The amplitude 𝛼 drives the non-linear refractive index. This in turn controls the quadrature phase X2. At the output of the medium, the ﬂuctuations are no longer independent. The two quadratures are now correlated. What is the eﬀect of this process on the statistics of the ﬂuctuations? On the one hand, we have made the light noisier by adding ﬂuctuations. On the other hand, we now have correlated ﬂuctuations. We can estimate the consequences by plotting the eﬀect not just for one point but for a whole distribution of classical amplitudes. This is done in Figure 9.3. The input and the output ﬁelds are represented by density plots, already discussed in Chapter 4 and shown in Figure 2.13. The axes of the plot for Kerr eﬀect (Figure 9.3A(a)) are the input quadratures X1in and X2in . The mean value is given by the point (𝛼in , 0). The ﬂuctuating amplitude is represented by a group of dots, wherein each dot gives one instantaneous value of (𝛿X1in (t), 𝛿X2in (t)). The density of the dots indicates the probability of ﬁnding a speciﬁc value of the quadrature amplitude ﬂuctuation, as introduced in 𝛿X2out (t) = 𝛿X2in (t) +

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9 Squeezed Light

Output E2

Input δX2

δY1

δY2 (a)

(a) δX1in δX1

(b)

δX1 δX2

(b)

Histogram

δY1 Histogram

306

δY2

(A)

(B)

Figure 9.3 Simulation of the eﬀect of a Kerr medium on the ﬂuctuations of an electromagnetic ﬁeld. Left-hand side (A) is the input ﬁeld and right-hand side (B) the output ﬁeld. The top part (a) shows the probability distribution of the ﬁeld, and the bottom part (b) shows the histograms of the samples above.

Section 4.1. This simulation uses a random distribution, with two-dimensional (2D) Gaussian weighting, as described in Appendix A. The probability of measuring a certain value X1(t) is given by a histogram of the projection of all dots onto one of the axes. Two such histograms, for both the X1 and X2 coordinates, are shown in Figure 9.3A(b). Both of them are Gaussian curves with the same width. Figure 9.3B shows the distribution at the output of a non-linear Kerr medium for a parameter rKerr = 0.375. Note that the total phase shift 𝜙out has been ignored. The axes are again the quadrature amplitudes X1out and X2out . The distribution is no longer circularly symmetric. The area covered by the distribution is actually larger since we have added ﬂuctuations ((Figure 9.3B(a)). But note that the distribution is narrower in one special direction. The histogram of this distribution, taken for a projection axis Y 1 rotated by the angle 𝜃s to the X1 axis (Figure 9.3B(b)) is narrower than the histogram for the input state (Figure 9.3A(b)). In this particular case the full width at half maximum (FWHM), or standard deviation, is reduced from 1 to 0.79. The histogram for the projection Y 2 (Figure 9.3B(b)), orthogonal to Y 1, is wider, by a factor 1.23, than at the input. The width of the distribution in the squeezing direction is given by √ 2 2 VY 1 = 1 − 2 rKerr 1 − rKerr + 2 rKerr (9.7)

9.1 The Concept of Squeezing

well below the QNL. This result is derived on page 343. This concept can also be used for a QNL laser beam or coherent state. The dot diagram represents the ̂ and X𝟐. ̂ The variance of the rotated measured probabilities of the observables X𝟏 ̂ ̂ quadrature Y𝟏, rotated by 𝜃s with respect to X𝟏, is smaller than that for a coherent state. Just by introducing this correlation between the quadratures, we have created light with noise reduced below the QNL in one speciﬁc quadrature. Note that in this example the uncertainty area has been increased. The output is no longer a minimum uncertainty state, but this is not of great importance for squeezing. ̂ The details of the We are only interested in measurements of the quadrature Y𝟏. size of the noise suppression and of the squeezing angle in a real experiment are described in Section 9.4.4. The Kerr eﬀect acts on all ﬂuctuations, independent of their frequency. The model can be applied to each individual Fourier component of the ﬂuctuations. For each detection frequency the process correlates the quadratures of the modes. The correlations are introduced separately for each Fourier component. In this way, the Kerr eﬀect correlates the amplitude and the phase quadrature of the light, and this applies to modulations as well as the quantum noise. 9.1.1.2

Four-Wave Mixing

A second example of a squeezing tool is four-wave mixing (4WM). This is a well-known non-linear process that occurs in many resonant media, such as atoms. Imagine three laser beams that are interacting with the medium and are tuned close to resonance with one absorption line, as illustrated in Figure 9.4a. This process is best described in the frequency domain. Two of the laser beams, the pump beams, have the optical frequency 𝜈L . They enter from diﬀerent angles. The third beam is at the frequency 𝜈L + Ω. The medium responds with the generation of a new beam at 𝜈L − Ω. This process is energetically allowed; the energy comes from the abundance of photons at 𝜈L . The strength or probability, of this process depends on three factors: (i) a non-linear coeﬃcient 𝜒 (3) that varies from material to material, (ii) the density of the sample, and (iii) a condition that describes how well the momentum of the photons is preserved. The latter is also known as phase matching and determines the geometry of the input and output beams [1]. Conventionally, this process is used as a technique to produce a new beam that emerges at a diﬀerent frequency or direction. This new beam can be used to investigate the properties of the non-linear medium or as a source of light at the new wavelength. This process also works if we have only one single input beam that has a single modulation sideband at 𝜈L + Ω. In this case all frequency components are co-linear, and 4WM produces a new sideband at 𝜈L − Ω, which is phase-locked to the original sideband. The eﬃciency of the process is given by the relative size of the new sideband. It can be quantiﬁed by the ratio C between the amplitude of the new and the original sideband. This applies to any sideband. Now consider the eﬀect of 4WM on an amplitude-modulated laser beam. That means we have pairs of sidebands, each pair in phase with each other, i.e. their respective beat signals are in phase and constructively interfere. Each sideband is partially transferred to the other sideband respectively by 4WM (Section 2.1.7). This will increase the sidebands and thus the modulation by a factor (1 + C).

307

308

9 Squeezed Light

4WM vL

Non-linear medium

vL – Ω

vL + Ω

C

vL

vL – Ω (b)

(a)

δX1in

vL

vL + Ω

δX1in vL

vL More AM Less FM

δX1out (c)

δX1out vL

(d)

vL

Figure 9.4 The eﬀect of a 4WM medium. (a) The geometrical layout: a new beam is generated. (b) The eﬀect of 4WM in the frequency domain is a transfer from one sideband to another. (c) As a consequence amplitude modulation (AM) is increased. (d) Phase and frequency modulation (FM) is reduced.

The medium provides gain for amplitude modulation, as shown in Figure 9.4c. In contrast, phase modulation consists of two sidebands 180∘ out of phase. Again, 4WM creates additional components at the opposite sideband frequency. The transfer from one sideband to the other will decrease the phase modulation by a factor (1 − C), which means 4WM reduces phase modulation (Figure 9.4d). In conclusion, the non-linear process of 4WM provides a selective gain to one quadrature and reduces the other. 4WM is an example of a phase-sensitive ampliﬁer (Section 6.2), which has diﬀerent gain for each quadrature. This process applies both to modulation and to noise sidebands. Any amplitude modulation noise is ampliﬁed; any phase modulation noise is reduced. Consequently, a beam that initially has equal amounts of both types of noise will turn into light with enhanced amplitude modulation and reduced phase modulation noise. This classical argument shows us that an input beam that is in a coherent state will turn into a state with suppressed phase noise. This result is conﬁrmed by a full quantum treatment [2]. It is useful to consider the 4WM process in terms of a phasor diagram, much like we have done for the Kerr eﬀect. Again, we will consider one Fourier component of the ﬂuctuations at a time. Any periodic excursion from the long-term average value corresponds, in the phasor diagram, to an oscillation at the modulation frequency Ω. It can be a mixture of amplitude and phase modulation. This can be drawn, as shown in Figure 9.4c, as an additional vector, the sum of the vectors 𝛿X1(Ω) and 𝛿X2(Ω). The 4WM process will increase 𝛿X1(Ω) by a factor (1 + C) and reduce 𝛿X2(Ω) by a factor (1 − C) (Figure 9.4d). This is called phase-sensitive gain. We can test the eﬀect of 4WM on a truly noisy state, which

9.1 The Concept of Squeezing

Input

Output

δX2

δY2

δY1 (a)

(a)

δX1 δX2

(b)

Histogram

(b)

Histogram

δX1

δY2

δY1 (A)

(B)

Figure 9.5 Simulation of the eﬀect of a 4WM medium on the ﬂuctuations of an electromagnetic ﬁeld. Part (A) shows the input state. The distribution is circularly symmetric. Part (B) shows the distribution after the 4WM medium. It is stretched. The top part (a) shows the probability distribution of the ﬁeld, and the bottom part (b) shows the histograms of the samples above.

is represented by a 2D random Gaussian distribution in time. The transformation we apply point by point is 𝛿X1out (t) = (1 + C) 𝛿X1in (t) and 𝛿X2out (t) = (1 − C) 𝛿X2in (t)

(9.8)

Figure 9.5 shows an example of both input and output distributions, Figure 9.5A andB, respectively, based on the random sample from Figure 9.3 and a value of C = 0.3. Clearly the output state has increased amplitude ﬂuctuations and reduced phase ﬂuctuations. The size of the uncertainty area is preserved. This phasor diagram represents a QNL input; clearly the output has phase ﬂuctuations below the QNL. It is worth noting that the arguments used here are exactly the same as those applied in phase conjugation. The only diﬀerence is that there the spatial phase contributions are reduced by the generation of a reﬂected wave with opposite phase whilst in 4WM we consider temporal ﬂuctuations. These two examples, Kerr and 4WM, are only illustrations of a concept. They show that correlations introduced by non-linear processes can be used to generate a new type of light with smaller ﬂuctuations, in one speciﬁc quadrature, as compared with the input light. The above classical examples can be used as a guide for a complete quantum model of this new type of light. In Chapter 4, it was shown that the quadrature probability distribution for a quantum state can be

309

310

9 Squeezed Light

described by the Wigner function. The coherent light has a Gaussian distribution of the ﬂuctuations around the time averaged value, and this will not change for the strength of non-linearities that are available. In particular, for a coherent state, the Wigner function is a 2D Gaussian function with circular symmetry. The new state has a Wigner function that is compressed in one direction and expanded in the other. Certainly squeezed state is a ﬁtting description. The ﬁrst step for a complete description of this light is a formal and rigorous deﬁnition. 9.1.2

Properties of Squeezed States

With squeezed states we mean states that have the following property: the distribution function of the quadratures is not symmetric, that is, the uncertainty area is no longer a circle, and the width of the distribution function in one particular direction Y 1 is narrower than the QNL. The direction Y 1 can be at any angle with respect to the quadratures X1 and X2. Measurements that are designed to be sensitive only to Y 1 and not the orthogonal quadrature Y 2 contain noise less than the QNL. For this reason we call it non-classical light, meaning that it cannot be described by the normal theory of coherent light, given in Chapter 4, which prescribes a minimum uncertainty area that is independent of the quadratures. Quantum mechanics and non-linear optics have to be combined to avoid the symmetry and to go below the standard uncertainty limit (Figure 9.6). X2

X2 δX2

δX1

δX2

X1

Coherent state

(a)

δX1

(b)

X2

Minimum uncertainty state X2

δX2

δX1

(c)

X1

δX2

X1

Squeezed and excess noise

δX1

(d)

X1

Noisy and not squeezed

Figure 9.6 Phasor diagram comparing squeezed and other states. (a) Coherent state. (b) Minimum uncertainty squeezed state that is narrower than the coherent state in one direction. (c) Squeezed state with excess noise. (d) An asymmetric, noisy, but not squeezed state – it is described by an ellipse, but no projection is narrower than the coherent state.

9.1 The Concept of Squeezing

δX2

δX2

(a)

δX1

(b)

δX1

Figure 9.7 Wigner function of (a) coherent state and (b) squeezed state.

The uncertainty area for a squeezed state is generally drawn as an ellipse. Why does it have this particular shape? The Wigner function for any coherent state is a Gaussian distribution in every direction. The contour line for this state is a circle. We found that this corresponds to ﬂuctuations in the measurements of the variable such that V (Ω, 0) = V (Ω, 90) = 1 for all detection frequencies. It was already shown by W. Pauli [3] that any minimum uncertainty state, even squeezed, will have 2D Gaussian distribution functions. The only requirement is that the widths of these functions satisfy the uncertainty principle V (Ω, 𝜃) V (Ω, 𝜃 + 90) = 1 for any direction 𝜃. One of the two variances can be less than 1. The contour line for such a 2D Gaussian function is always an ellipse. An ellipse can be described by exactly two parameters: the direction of the narrowest axis Y 1 and the ellipticity. Note that the area inside the contour can never be smaller than the area inside the uncertainty circle of a coherent state (Figure 9.7). There are states with excess noise, that is, states with V (Ω, 𝜃) V (Ω, 𝜃 + 90) > 1 for some or even all frequencies. Such states need not be described by 2D Gaussian with elliptical contours. However, we ﬁnd that most realistic optical systems can be described by theoretical models that are linearized in the ﬂuctuations (Section 4.5). These systems can have non-linear properties that are essential for the generation of squeezed light, but its description requires only ﬁrst-order terms in regard to the ﬂuctuations. All such models result in Gaussian distribution functions – and the contours are ellipses. It would be of great interest to generate light with more exotic contours, with parts of the Wigner functions that have negative values and strong non-classical properties – but this has so far only been achieved in the single photon counting domain. Further details are given in Section 9.7. For the remainder we shall restrict ourselves to squeezed states that are described by elliptical contours. Squeezed states require a 2D description. The quadrature amplitudes Y 1 and Y 2 provide the most useful coordinate system – the theoretical model is

311

312

9 Squeezed Light

X2

X2

δY2 θs = 0

X1

θs =

δX1

(a)

X1

π 2

δX1

(b)

X2

X2

δY2

δY1

δY2 δY1

θs

X1 (c)

X1 (d)

Figure 9.8 Diﬀerent types of squeezed states. (a) Amplitude. (b) Phase squeezed state. (c) Quadrature squeezed state. (d) Vacuum quadrature squeezed state.

best derived using these parameters. Direct intensity measurements will only determine the intensity ﬂuctuations. Whilst this determines the width of the variance V 1(Ω) of the amplitude, measured along the X1 axis, it does not measure the quadrature ﬂuctuations in any other direction. We require experiments that can measure rotated quadratures, in particular V (Ω, 𝜃s ) and V (Ω, 𝜃s + 90). This is usually achieved by using a homodyne detector (Section 8.1.4). There are diﬀerent types of squeezed states, as shown in Figure 9.8. Each squeezed state ellipse has a narrow axis (Y 1) and, at right angles, the axis with the widest distribution (Y 2). If the narrow axis is aligned with the amplitude axis, i.e. (Y 1) parallel to (X1), we call the light amplitude squeezed. If it is aligned with the quadrature phase axis, i.e. (Y 1) parallel to (X2), it is called phase squeezed light. In general, (Y 1) could be rotated at some arbitrary angle 𝜃s relative to the axis X1. This is called quadrature squeezed light. 9.1.2.1

What Are the Uses of These Various Types of Squeezed Light?

Amplitude squeezed light is useful in absorption or modulation measurements. It can be detected simply by direct detection. Due to the reduced amplitude noise, the signal-to-noise ratio for amplitude modulation is improved. The ﬂuctuations in the other quadrature, the phase quadrature, are above the QNL. There could even be excess noise in the phase quadrature. None of this would aﬀect the direct intensity measurement. Amplitude squeezed light is obviously related to light with sub-Poissonian counting statistics. It has a Fano factor of less than unity. The detailed connection between squeezing and photon statistics has to be determined in each individual case. It depends on the actual photon probability distribution that could be very diﬀerent to a

9.1 The Concept of Squeezing

Poissonian distribution. However, for the generally observed moderate degree of squeezing, the link is simple: F𝛼 = V 1𝛼 . This light is frequently called bright squeezed light. The generation of bright squeezed light using second harmonic generators (SHGs) [4–7] or diode lasers is now a reliable experimental technique and is described in Section 9.4.3. Quadrature phase squeezed light can be used to improve the signal-to-noise ratio of optical measurements. Care has to be taken that only the phase quadrature is measured. This can be achieved in interferometric arrangements where amplitude ﬂuctuations are cancelled out and phase modulations are turned into modulation of the intensity at the output. In such an arrangement only the noise of the light in the phase quadrature will contribute to the ﬁnal signal-to-noise ratio. Finally, there is a peculiar state, centred at the origin of the phasor diagram, that has a squeezing ellipse. This state has an axis of reduced ﬂuctuations. But for each ﬂuctuation there is an equally likely ﬂuctuation in the opposite direction. This state is given the confusing name squeezed vacuum state. We will see that it is closely related to the coherent vacuum state |0⟩. An attempt to detect squeezed vacuum state directly would not result in any measurable intensity. To be precise, this state contains a rather small number of photon pairs (Eq. (9.13)), and the number of such pairs increases the more the state is squeezed. However, this eﬀect is far too small to be detected directly with photodiode technology. Only with photon counting could such small intensities be seen. In any other regard this state has normal properties: it is a beam with a well-deﬁned mode, direction, size, frequency, and polarization. However, once this beam is aligned with a homodyne detector and beats with a strong local oscillator beam, we can see its properties: the quantum ﬂuctuations vary with the detection phase. The ﬂuctuations corresponding to the Y 1 quadrature are smaller than those of the Y 2 quadrature. This squeezed vacuum state can have signiﬁcant practical applications, as shown in Section 5.2. All interferometers, and other instruments based on beamsplitters, contain input ports that are not used. That means there are locations in the experiment that could be used as the input for a real laser beam that would propagate right through the instrument, probably with some attenuation, and ﬁnally reach the detector. Such an input port has the properties of a real mode of light. In practice this port is not illuminated, and it can be ignored in classical calculations. However, in a quantum model of such an instrument, we cannot ignore this port, as shown in Chapter 5, and we have to represent it with a vacuum mode, the coherent state |0⟩. These vacuum input modes are the origin of the quantum noise in the output, as was shown in Section 5.1. Consequently the performance of the instrument can be improved by illuminating the unused input port by a squeezed vacuum mode. The squeezing quadrature Y 1 has to be aligned such that it is now the driver of the quantum ﬂuctuations that we measure in our apparatus. The result will be reduced quantum noise and a better signal-to-noise ratio. This was proposed by C.M. Caves [8] for the case of interferometers and has been demonstrated in an impressive landmark experiment by M. Xiao et al. [9] who achieved an improvement of the signal-to-noise ratio of 3 dB. The details and limitations of this technique will be discussed in the Chapter 10.

313

314

9 Squeezed Light

9.2 Quantum Model of Squeezed States In Chapter 4 we described pure states of light with Gaussian quadrature statistics as coherent states. We now want to expand this to include a much larger class of states, the squeezed states. They have to be based on a wider deﬁnition and can be derived from the lowest energy state, the vacuum state. 9.2.1

The Formal Deﬁnition of a Squeezed State

Squeezed states are represented by |𝛼, 𝜉⟩

and 𝜉 = rs exp(i2𝜃s )

(9.9)

where |𝛼|2 is the intensity of the state, 𝜃 the orientation of the squeezing axis, and rs the degree of squeezing. The squeezed states can be generated from the vacuum state: ̂ ̂ S(𝜉)|0⟩ |𝛼, 𝜉⟩ = D(𝛼) ̂ where the squeezing operator S(𝜉) is deﬁned as ) ( 1 1 ̂ S(𝜉) = exp 𝜉 ∗ â 2 − 𝜉 â †2 2 2 The properties of the squeeze operator are

(9.10)

(9.11)

̂ Ŝ † (𝜉) = Ŝ −1 (𝜉) = S(−𝜉) ̂ = â cosh(rs ) − â † exp(−2i𝜃s ) sinh(rs ) Ŝ † (𝜉)̂aS(𝜉) ̂ = â † cosh(rs ) − â exp(2i𝜃s ) sinh(rs ) Ŝ † (𝜉)̂a† S(𝜉)

(9.12)

̂ in Eq. (9.10) is possible. This results ̂ Note that the reverse order of D(𝛼) and S(𝜉) in the so-called two-photon correlated state, which was introduced by H.P. Yuen even before the name squeezed state was coined [10]. However, this distinction has little relevance for experiments. It makes no detectable diﬀerence for any realistic state. The formal description allows us to derive the noise properties of a squeezed state exactly. The squeezed states have the following expectation values for the annihilation and creation operators: ⟨𝛼, 𝜉|̂a|𝛼, 𝜉⟩ = 𝛼 ⟨𝛼, 𝜉|̂a2 |𝛼, 𝜉⟩ = 𝛼 2 − cosh(rs ) sinh(rs ) exp(2i𝜃s ) ⟨𝛼, 𝜉|̂a† â |𝛼, 𝜉⟩ = |𝛼|2 + sinh2 (rs )

(9.13)

The last equation means that the intensity of a squeezed state is slightly larger than that of the coherent state with the same value |𝛼|. However, this is a minute eﬀect for any laser beam with detectable intensity. Using these properties we can ̂ derive the variances of the generalized quadrature X(𝜃) = exp(−i𝜃)̂a + exp(i𝜃)̂a† , the quadrature that would be measured at the rotation angle 𝜃. This variance is given by ̂ Var(X(𝜃)) = cosh(2 rs ) − sinh(2 rs ) cos(2 (𝜃 − 𝜃s ))

(9.14)

9.2 Quantum Model of Squeezed States

10

V(θ) (dB)

5

0

–5

–10

(a)

π/2

0

π

3π/2

2π

1

0.75 >7 [5, 7] [3, 5] [1, 3] [−1, 1] [−3, −1] [−5, −3] [−7, −3]

θs X1

n δX1(Ω)

δX2(Ω)

V1(Ω) 1

Ω

0

δX2

δX1

Ω S(Ω)

V(θ)

θ

1

1

0 θs

–Ω

0

Ω

Figure 9.15 Summary of the various graphical representations of a squeezed state.

of losses is easy to predict since all passive losses behave in the same way and can be simulated by beamsplitters. Only non-linear processes are excluded from this model. In terms of the sideband picture, the eﬀect of the beamsplitter is easy to describe. The squeezed light corresponds to light with partially correlated sidebands 𝛿E+Ω and 𝛿E−Ω . For a beamsplitter with reﬂection 𝜖, transmission √ 𝜂 = (1 − 𝜖), the correlated part of the noise sidebands is reduced by 𝜂, in the same way as modulation sidebands are reduced (Section 2.1), whilst simultaneously the uncorrelated part is increased to keep the magnitude of the quantum noise sidebands constant. This is equivalent to including the quantum noise sidebands of the vacuum state from the empty port of the beamsplitter. The combined sidebands 𝛿E+Ω,out , 𝛿E−Ω,out at the output are less correlated than those at the input, and the degree of squeezing is reduced (Figure 9.16a). In terms of the phasor diagrams, the beamsplitter replaces the squeezed Wigner function with one that is less squeezed (Figure 9.16b), which means the contour ellipse

9.3 Detecting Squeezed Light

X2

Partially αin correlated noise

δY2 Ω

δY1 θs

V(θ) θ

1

X1 θs X2

αout Less correlated noise

V(θ) δY2

Ω (a)

δY1 θs

θ

1

X1 (b)

(c)

θs

Figure 9.16 The eﬀect of a beamsplitter on squeezed light, input at the top and output at the bottom. Three representations are used: (a) correlated sidebands, (b) phasor diagrams, and (c) quadrature variances. The top row is representative of the light at the input, and the bottom row at the output of the beamsplitter.

is not as narrow. The corresponding variance V (𝜃) at the output is to the QNL (Figure 9.16c). Loss transforms the contour from an elongated ellipse towards a circle, as shown in Figure 9.17. This might not be obvious if one considers that the ﬂuctuations of the input of the beamsplitter simply add. Just adding the two Wigner functions would suggest a rather more complicated result, as indicated in Figure 9.17b, for the case of adding two squeezed states with diﬀerent squeezing angle 𝜃s . However, all features of the Wigner function have to be taken into account, including the phase terms – and the answer is actually simple: the result is another ellipse with the new orientation 𝜃s and smaller ellipticity. It has to be an ellipse since we know that any solution of the linearized model will result in a 2D Gaussian quasi-probability distribution (Section 9.2). Thus the only parameter that can vary is the variance VY 1 of the squeezed state after the beamsplitter. If the input state is a squeezed state with the general parameters (𝛼, rs , 𝜃s ), the output state, after the beamsplitter, is (𝛼 ′ , rs′ , 𝜃s′ ). We know that the intensity is reduced such that |𝛼 ′ |2 = 𝜂|𝛼|2 . The beamsplitter has a second input port for vacuum ﬂuctuations and is described in the standard way: √ √ ̃ in + 1 − 𝜂 𝜹X ̃ vac ̃ out = 𝜂 𝜹X(𝜃) (9.21) 𝜹X(𝜃) The calculation can be carried out independently for each angle 𝜃. As a result, the minimum will remain at 𝜃s since |0⟩ has no angular dependence. Thus the squeezing angle is unchanged, 𝜃s′ = 𝜃s . We have to evaluate the variances and this is a linear operation √ that leads to the sum of two contributions. The second term simply results in 1 − 𝜂 since any contribution from the vacuum state results in 2 ̃ ⟩ = 1. As a result we obtain Vvac = ⟨|𝜹X(𝜃)| VY 1out = 𝜂VY 1in + (1 − 𝜂) or (1 − VY 1out ) = 𝜂(1 − VY 1in )

(9.22)

327

328

9 Squeezed Light

(a)

No

Yes (b)

Figure 9.17 Wigner functions cannot be added directly; the phase terms must be taken into account. (a) Attenuation of a squeezed state by a beamsplitter results in a state with reduced squeezing. (b) The combination of two squeezed states with diﬀerent 𝜃s values results in a normal squeezed state, not a Wigner function with four contributions.

This means the noise suppression (1 − VY 1out ) is reduced linearly with loss 𝜂. The noise suppression is aﬀected in the same way as the intensity. Note that the noise suppression is commonly quoted on a logarithmic scale in dB for the normalized variance and this value is not reduced linearly. The eﬀect of 𝜂 is shown in Figure 9.18 where the output variance VY 1out is plotted in dB versus the input variance VY 1in . The eﬀect of losses is more severe on strongly squeezed light. A loss of 50%, (𝜂 = 0.5) transforms VY 1in = −3 dB into −1.3 dB, whilst for the much better squeezing of VY 1in = −10 dB, the output is very similar, namely, −2 dB. Strong squeezing is easily lost in a system with limited eﬃciencies. The minimum variance of a squeezed state is VY 1𝛼,𝜉 = exp(−2rs ). One could formally try to link the squeezing parameter rs′ to rs . The link is exp(−2rs′ ) = exp(−2rs )𝜂 + (𝜂 − 1). This is not a simple relationship, and the squeezing parameter is not practical for this type of calculation. Since the conversion of the squeezing due to losses is constantly required in the evaluation

9.3 Detecting Squeezed Light

Output log (VY1(α′,ξ′)) (dB)

0

2

0.5 0.6

4

0.7 0.8

6

0.9 η = 1.0

8

10

0

2

4

6

8

10

Input log (VY1(α, ξ)) (dB)

Figure 9.18 The quantitative eﬀect of diﬀerent eﬃciencies 𝜂 on the minimum variance of the squeezed state.

of squeezing data, the squeezing parameter is rarely used by experimentalists. The most common parameter cited is (1 − VY 1) either on a linear or logarithmic scale, and this scales easily, as stated in Eq. (9.22). Note that a beamsplitter, or any other phase-independent loss, does not change the spectrum of the squeezing. Equation (9.22) holds equally well for all detection frequencies. Both the shape of the spectrum and the squeezing angle are not aﬀected by losses, and only the degree of squeezing is reduced. The squeezed light produced by some devices, for example, a below threshold OPO, can be minimum uncertainty state, a pure state with VY 1 VY 2 = 1. Does it remain a minimum uncertainty state after losses? The beamsplitter will mix in a fraction of the vacuum state, itself a minimum uncertainty state, and the combined output will be a mixed state, above the minimum uncertainty value. How much above? We can directly calculate VY 1out VY 2out = (𝜂VY 1in + (1 − 𝜂)) (𝜂VY 2in + (1 − 𝜂)) but do not obtain a simple answer. The result is easiest interpreted from a diagram. Figure 9.19 shows the values for the product of the variances as a function of 𝜂 for three degrees of squeezing: VY 1in = 0.5, 0.2, and 0.1. We see that the product rises quickly and for 𝜂 ≈ 0.5 reaches a maximum that depends on the initial degree of squeezing. For large attenuation all these states approach again a minimum uncertainty, the coherent state. If the light is too intense for the detector, we should not simply attenuate the light but reﬂect it oﬀ a cavity. If a cavity with smaller linewidth than the detection frequency is chosen, the cavity will fully reﬂect the noise sidebands whilst the carrier component is partially transmitted. In this way the optical power can be attenuated without losing the squeezing. An interesting extension would be to incorporate an active gain medium into the cavity [6]. This now becomes

329

9 Squeezed Light

3 VY1in = 0.1 VY1out * VY2out

330

2.5

2

VY1in = 0.2

1.5 VY1in = 0.5 1 0

0.2

0.4

0.6

0.8

1

Loss η

Figure 9.19 The uncertainty product for a squeezed state after loss. Three diﬀerent degrees of squeezing are shown.

a noiseless optical power ampliﬁer – it ampliﬁes the energy but leaves the noise sidebands outside the cavity linewidth unaﬀected. Alternatively we could detune the cavity and therefore rotate the squeezing ellipse, as discussed in the Section 9.4.4. Amplitude quadrature squeezing can be turned into phase quadrature squeezing and vice versa. We have all the components available to modify the squeezing parameters, but the only thing we cannot do is to increase the degree of squeezing.

9.4 Early Demonstrations of Squeezed Light 9.4.1

Four Wave Mixing

We are now discussing the pioneering squeezing experiments in detail. By the middle of 1984 several research groups were competing to measure the ﬁrst non-classical noise suppression – to detect the ﬁrst squeezed state. Each group favoured a diﬀerent non-linear process, and eventually all these attempts were successful. The ﬁrst group to report experimental observations of squeezing was led by R.E. Slusher at the AT&T Bell Laboratories in Murray Hill. Their experiment used 4WM in a Na atomic beam [15]. This is a resonant medium. The non-linearity was enhanced by an optical cavity. The ﬁrst proposal of squeezing with a 4WM mixing medium had been published as early as 1974 by H.P. Yuen and J.H. Shapiro [16]. The diﬃculty was to achieve suﬃcient 4WM gain whilst avoiding all other sources of noise. The analysis of squeezing by 4WM conﬁrmed that we require a non-degenerate situation [2]. That means the pump beam, which provided the energy for the process, and the signal and idler beams should have separate frequencies. This led to the classical interpretation of 4WM as correlating two sidebands, each detuned by ±Ω from the pump frequency, as discussed in Section 9.2.3

9.4 Early Demonstrations of Squeezed Light

One important noise source that could swamp any squeezing was spontaneous emission. Light from the pump beam, which was absorbed by the atoms, was re-emitted either at the same frequency or at the atomic resonance frequency. Such light produced ﬂuctuations in the photocurrent that raise the noise level above the QNL. It was necessary to detune the pump wave from the atomic resonance by many atomic linewidths. However, this reduced the non-linearity and a compromise between non-linearity and spontaneous emission noise had to be achieved. One technique to reduce the spontaneous emission, at a given detuning, was to reduce the Doppler broadening. Consequently most experiments were performed with atomic beams that are essentially Doppler-free as long as all optical beams are aligned at right angles to the propagation axis of the atoms. The atomic beams had to be optimized for high densities. In Slusher’s experiment an atomic absorption of exp(−0.5) on line centre was measured. The experiment consisted of a beam of Na atoms that was illuminated, pumped by an optical pump beam, and detuned by about 2 GHz away from the hyperﬁne components of the Na D1 transition at 590 nm. Figure 9.20 shows the experimental layout and Figure 9.21 the frequencies of all the optical beams in the experiment. A second beam, the probe, was aligned through the same atoms at

ϕLO

Dye laser

BS1 EO2 FB2

AO3,4

LO

AO1,2

vL+3vSC

vL+2vSC

PM1

BS2

SM2

DB

EO1 S1

SM1 Na

PM2

DA –

FB1

SA

Figure 9.20 Layout of the atom 4WM experiment. Source: After R.E. Slusher et al. [15].

vatom

vsignal

vprobe

vidler

vlock

Figure 9.21 The diﬀerent frequencies involved in the 4WM experiment. The cavity modes are shown as thin solid lines. They are shifted by the dispersion of the atoms.

331

332

9 Squeezed Light

a small angle to the pump beam. This probe beam was aligned into a heterodyne detection system together with a local oscillator beam that was split oﬀ from the pump beam. Two photodiodes detected the beams and generated a beat signal at the diﬀerence frequency between probe and pump, in this case at 595 MHz. In the absence of the atoms, and with the probe beam blocked, the noise level recorded was the QNL (1 Hoover). It can be interpreted as the noise due to the vacuum state entering the beamsplitter of the heterodyne detector. With the atoms present and the probe beam turned on, the 4WM medium generated a second sideband with exactly the opposite detuning to that of the probe. Now two beams were present and were detected together. In this way the probe beam was used to measure the classical 4WM gain and to align the detection system. The 4WM process can be interpreted as phase-sensitive ampliﬁcation: it increases amplitude modulation and reduces phase modulation sidebands. This gain correlates the two sidebands, as discussed in Section 9.2. Even with the probe beam switched oﬀ, this correlation exists. The ever-present quantum noise sidebands, which enter the medium on the axis that was deﬁned by the probe beam, will be correlated, resulting in a squeezed vacuum state. For a single transit of the Na beam, this eﬀect was far too small to be detectable. The 4WM process, and the resulting correlation, could be built up by a cavity that is tuned to both sidebands simultaneously. Each sideband was on resonance with a diﬀerent cavity mode. In this experiment they were not even adjacent modes, but they were separated by nine free spectral ranges (FSRs) (Figure 9.21). Again, the probe beam was used for the alignment of the cavity. A locking system kept the cavity on resonance. This used an optical beam, which probed the cavity at yet another optical frequency, avoiding the pump and the probe frequency. One practical problem was the fact that the atoms produced a strongly dispersive refractive index that shifts all cavity resonances. The cavity modes were asymmetric to the probe frequency, but by using acousto-optical modulators (AOMs), the locking beam could be shifted until both the probe and the newly generated beam were in resonance. The pump intensity was raised until it approached the level of the saturation intensity of the sodium vapour at a detuning of 2 GHz, equivalent to 400 atomic linewidths. and the pump beam was retro-reﬂected to provide an even higher pump intensity. Finally, the probe beam is blocked. The vacuum mode, which is collinear with the probe beam, experiences the correlation at the sidebands at +Ω and −Ω induced by the atoms inside the single-sided cavity. A squeezed beam is created that travels towards the detectors. The light is squeezed in a band of frequencies around the cavity resonance frequency. The strength and the quadrature of the squeezing are measured by scanning the phase of the local oscillator. This ﬁrst squeezing result is shown in Figure 9.22a. On the right-hand side, Figure 9.22b shows the result after continuous improvements 15 months after the ﬁrst observation [15]. The dotted line in Figure 9.22b gives the QNL, obtained with the cavity blocked. The measured noise increases 1.3 dB above and decreases −0.7 dB below the SQL. The electronic noise (ampliﬁer noise) is −10 dB below the SQL and contributes about 10% to the noise level. The detection frequency is 594.6 MHz, resolution bandwidth 300 kHz, and video bandwidth 100 Hz. The theoretical model predicts ±2 dB (solid curve) for an ideal measurement. The eﬃciency of the photodetectors is 𝜂det = 0.8, and the eﬃciency of the

9.4 Early Demonstrations of Squeezed Light

–58

Noise level (dBm)

–59

–60

–61

–62 0 (a)

(b)

π/2 ϕLO

π

3π/2

Figure 9.22 The ﬁrst squeezing results. (a) Photograph of the ﬁrst trace [15]. (b) Quantitative comparison between experimental results and theoretical predictions after months of further work. Source: After R.E. Slusher et al. [2].

interferometer is 𝜂vis = 0.81, resulting in a total eﬃciency of 0.65. Taking this and the electronic noise into account, the theoretically predicted noise is given as the dashed-dotted line in Figure 9.22b. It shows good agreement with the enhanced noise and moderate agreement with the squeezed noise quadrature, namely, −0.7 dB observed versus −1.05 dB predicted. One practical limitation is the mechanical stability of the interferometer required for the homodyne detection. Any mechanical vibration can wash out the noise suppression. This eﬀect, described as phase jitter, is cited as the reason for the disagreement with the prediction. It can explain this particular set of results. However, at higher intensities, the disagreement gets even larger, and other processes have to be considered. Non-linear coupling between the light used for the locking of the cavity to the laser frequency and the mode of the squeezed light is considered by the authors as a possible source of excess noise. This experiment clearly demonstrates the existence of a squeezed state of light, in this case a squeezed vacuum state. All predictions about the properties of the squeezed light and the detection systems were conﬁrmed qualitatively. The observed noise reduction is modest. Despite the predictions of squeezing of up to −10 dB, 4WM has never been really successful as a squeezing process. The technical diﬃculties with the atomic beam were frustrating, and this type of experiment has not been repeated by any other group. It remains a milestone in the early exploration of squeezed light. 9.4.2

Optical Parametric Processes

Some of the most successful results in the early period of squeezing came from the group led by H.J. Kimble at the University of Texas [17]. He used an

333

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9 Squeezed Light

M2

M1 αp

A1 vp

Detunings

ζ

αd κp1 κd1 κp = Total loss at vp

Aout

Δp Δd

vd = 1/2 vp

κp2 κd2 κd = Total loss at vd

Figure 9.23 A parametric medium inside a cavity. The various input, output, and intra-cavity modes are labelled. This system has a threshold and can operate below threshold as a phase-sensitive ampliﬁer of the vacuum modes and above threshold as an oscillator. The subscript p refers to the fundamental frequency, and the subscript d refers to the subharmonic frequency.

optical parametric ampliﬁer (OPA) to generate squeezing. We have already seen in Chapter 6 that the OPA is ideally suited since it is phase sensitive and ampliﬁes one quadrature whilst it attenuates the other as shown in Figure 9.24. The trick to enhance the process is to build a cavity around the OPA as shown in Figure 9.23. The experiments with the non-linear OPO are carried out with a cavity containing a 𝜒 2 material, usually a crystal such as LiNbO3 . A single-mode laser illuminates the cavity. Following the derivations in Section 6.3.3, we know that a parametric oscillator below threshold will produce quite diﬀerent gain for the two quadratures, similar to 4WM, and therefore suppresses the noise selectively. Figure 9.24 symbolically shows how the quadratures are aﬀected and the conversion of a vacuum input state (circle) into a squeezed vacuum (ellipse). From Eq. (6.43) we can derive a noise spectrum and ﬁnd that the eﬀect is greatest at low frequencies (Ω ≪ cavity linewidth). For such an oscillator the crucial c , experimental parameter is given by the normalized pump amplitude d = 𝛼in ∕𝛼in c where 𝛼in is the input amplitude required to reach threshold. We wish to operate with d < 1 and can optimize the performance by approaching d → 1 by changing the input intensity. In terms of the cavity parameters, losses and non-linear gain, Figure 9.24 Symbolic representation of parametric gain in the phase diagram.

δX2

δX1

9.4 Early Demonstrations of Squeezed Light

1

S(Ω)

0.8 0.6 0.4 d = 0.5

0.2 0

4

0

2

2

4

Ω

Figure 9.25 The squeezing spectrum for an OPO below threshold for a ﬁxed value of d = 0.5. The modulation frequencies are scaled in units of the cavity bandwidth.

we ﬁnd d = 𝜒∕𝜅a , and consequently we can derive the variance from Eq. (6.35): V 2(Ω) = 1 −

4d (1 + d)2 + (2πΩ)2

(9.23)

which is identical to the squeezing spectrum, S(Ω) = V 2(Ω), since the noise suppression always occurs in the phase quadrature. An example is shown in Figure 9.25. At very low frequencies, Ω → 0, we get the optimum value V 2(Ω) =

(1 − d)2 (1 + d)2

(9.24)

and at the same time the noise enhancement in the orthogonal quadrature V 1(Ω) =

(1 + d)2 (1 − d)2

(9.25)

We see that the ﬂuctuations in the X2 quadrature tend to zero. On the other hand, the ﬂuctuations in the X1 quadrature diverge. The eﬀect for the two quadratures is shown in Figure 9.26. The system has a critical point at the 40

100

VY1

30 20 V (dB)

Gain

10 Amplitude

0

1 Phase 0.1 (a)

10

20 0

0.2

0.6

0.4 d2

0.8

VY2

10 1 (b)

0

0.2

0.6

0.4 d

0.8

1

2

Figure 9.26 Phase-dependent gain and resulting variances V1 and V2 for the two quadratures of a OPO below threshold as a function of the normalized input power d2 . This calculation is for very low detection frequencies less than the linewidth of the cavity Ω = 0 and for a ﬁnite quantum eﬃciency 𝜂 ≤ 1.

335

336

9 Squeezed Light

Ba2NaNb5O15 SHG

MgO:LiNbO3 0.53 μm OPO

1.06 μm Laser Nd:YAG

θlo +_

Figure 9.27 The schematic layout of the experiment by L.-A. Wu et al. [17] generating a squeezed vacuum state with a sub-threshold OPO.

threshold. Perfect squeezing is possible at threshold, at least in theory, but in practice the operation gets unstable and in most situations values no larger than d = 0.8 are chosen. Interestingly the noise suppression improves only slowly for intensities larger than half the threshold value (d2 = 0.5). The eﬃciency 𝜂 of the apparatus has to be taken into account, and the complete equation for the noise suppression is V 2(Ω) = 1 −

𝜂 4d (1 + d)2 + (2πΩ)2

(9.26)

The ﬁrst realization of this experiment was carried out by Wu et al. in 1986. The layout of their experiment is shown schematically in Figure 9.27. The light source is a continuous-wave (CW) Nd:YAG ring laser, in this case lamp pumped. The light is frequency doubled using an intra-cavity Ba2 NaNb5 O15 crystal. The laser cavity has two output beams, at 1064 and 532 nm, with orthogonal polarizations. The green light is sent to the parametric oscillator, which is a MgO-doped LiNbO3 crystal inside a linear cavity. When operated above threshold as a degenerate OPO, the cavity sends out a beam at exactly the laser frequency. This beam is recombined in a Mach–Zehnder interferometer with the infrared output from the laser cavity. This arrangement forms the homodyne detector for the output of the parametric oscillator. The laser provides the local oscillator. When operated below threshold, the output ceases, but the apparatus is still aligned. Now the OPO emits a squeezed vacuum state that is detected by the homodyne detector. All measurements were carried out at detection frequencies less than the cavity linewidth. The phase angle 𝜃 of the local oscillator is scanned. In this way the quadratures can be probed. The vacuum noise level is determined by blocking the beam from the OPO. The results are shown in Figure 9.28a. The minimum of the trace gives the optimum noise suppression. It was diﬃcult to stabilize the phase of the local oscillator; thus mainly scans of the projection angle 𝜃 were published by L.-A. Wu et al. [17]. We can investigate the properties of the OPA in diﬀerent ways. The noise suppression will improve when the threshold is approached. This has been tested and results are given in Figure 9.28a. The agreement with the theoretical predictions (Eq. (9.26)) is excellent. The best noise suppression observed is

9.4 Early Demonstrations of Squeezed Light

(ii) 2.0 (iv) V (θ) 1.0 (i) (iii) 0 (a)

θ0

θ0 + π

θ0

θ

θ0 + π

θ0 + 2π

θ0 + 3π

d2 = P2 /P0 0

0

0.5

1.0

S_(r) –0.5

–1.0

(b)

Figure 9.28 Experimental results. (a) Dependence of noise power on the local oscillator phase 𝜃. With the output of the OPO blocked, the vacuum ﬁeld entering the detector produces trace (i). With the OPO input present, trace (ii) exhibits phase-sensitive deviations both below and above the SQL. Trace (iii) is the ampliﬁer noise. (b) Noise suppression for diﬀerent normalized intensities d2 . Comparison of the inferred noise suppression V2(d2 ) = 1 + S− (r) after correction for the quantum eﬃciencies derived from measurements such as shown in (a) with optimized phase angle with those given by theory (solid curve). Detection frequency Ω is ﬁxed. Perfect squeezing corresponds to V2 = 0 and is indicated by the dashed line. Source: After L.-A. Wu et al. [17].

V1 = 0.33. Once the detector eﬃciencies and the losses are taken into account, the best inferred squeezing corresponds to a factor of 10, or −10 dB, for the light emitted by the cavity. Finally, this experiment is also a most impressive demonstration of the concept of a minimum uncertainty state. In Figure 9.28b both the minimum and the maximum variances for each input amplitude are plotted.

337

9 Squeezed Light

10.0

Squeezed states

Minimum uncertainty (1 + S+ )(1 + S_ ) = 1

(1 + S+ )

338

5.0

Vacuum state 1.0 0

0

0.5

1.0

1.5

(1 + S_ )

Figure 9.29 Variances V1 and V2 determined from measurements. The solid curve is the hyperbola V1 V2 = 1, which deﬁnes the class of minimum uncertainty states. Squeezed states are those states for which either V1 or V2 < 1 and that lie in the region bounded by the hyperbola and the dashed lines. Source: From L.-A. Wu et al. [17].

According to the theory, the product of the two variances should be unity, since the parametric ampliﬁcation preserves the minimum uncertainty area. The data points should lie on a hyperbola, and this is indeed the case. Note that this result can only to be expected for data describing directly the OPO, with perfect eﬃciency, since any loss would have turned these minimum uncertainty states into mixed states. The data for Figure 9.29 are inferred after removing the eﬀect of the detector loss. This was a landmark experiment. Some of the most striking features of squeezed light have been demonstrated in a single experiment: noise suppression, noise enhancement in the opposite quadrature, and preservation of the minimum uncertainty. And the light generated, a squeezed vacuum beam, is certainly not classical in any regard. The work of Liang Wu et al. is a remarkable achievement and remained unsurpassed for many years. The main advance has been in the reduction of losses inside the OPO. The limitations are now the eﬃciencies of the detector and technical ﬂuctuations in the phase of the local oscillator 𝜃 that wash out the very sharp minima. Work at Caltech, Konstanz, ANU, Shanxi University, Tokyo, Paris, and Hannover have established the OPO/OPA as the most reliable source for squeezed light.

9.4 Early Demonstrations of Squeezed Light

9.4.3

Second Harmonic Generation

The process of frequency doubling or SHG has been used to generate amplitude squeezed light. It is presently the most reliable and eﬃcient process to generate light with high intensity and good noise suppression. A qualitative description why SHG leads to amplitude squeezed light is obtained from the photon model. A coherent input beam can be thought of as a stream of photons with Poissonian distribution. The time separation between photons is random. This beam illuminates a non-linear crystal, which converts a fraction of the light into a new beam with twice the optical frequency. Two beams emerge, the remaining fundamental (f ) beam and an SH beam. The crystal converts pairs of photons from the incoming fundamental beam into single photons of the SH beam. If the probability for the conversion were constant, the statistics of the two emerging beams would be identical to that of the input beam. However, the conversion probability is proportional to 𝛼 2 𝜒 (2) , that is, the instantaneous intensity of the fundamental beam multiplied by the second-order non-linearity. This means the conversion is more likely when the pump intensity is high, when the photons in the pump are closely spaced. Closely spaced pairs of photons are most likely to be converted into a single SH photon. As a consequence the statistics of the SH photons is closely linked to the statistics of pairs in the fundamental beam. The SH photons come in a more regular stream than the pump photons. If the pump beam has a Poissonian photon statistics, the SH beam has a sub-Poissonian distribution. Similarly, the remaining fundamental light from which the pairs have been deleted has a more regular stream of photons; it also has a sub-Poissonian photon statistics. Similar arguments apply to non-linear absorption, which could also produce squeezing. This simple interpretation is suitable only for a single pass through the non-linear SHG material. In CW experiments the non-linearity is not large enough, and the crystal is placed inside a cavity to build up a mode with high intensity at the fundamental frequency. The ﬁrst successful observations were carried out by S.F. Pereira et al. [18] who detected a noise suppression of −0.6 dB in the fundamental beam. These early experiments were plagued by additional technical noise sources inside the build-up cavity. A breakthrough occurred when it became possible to polish and coat the non-linear crystal directly. In such a monolithic design, the internal losses can be kept to a minimum, the fundamental light is built up to increase the non-linearity, and the SH ﬁeld is reﬂected at one end to have all of the SH ﬁeld escape on one side. Care has to be taken that the fundamental and SH waves stay in phase, which is achieved by selecting the correct phase matching temperature. The coatings are designed such that phase shifts upon reﬂection preserve the optimum phase relation for a large number of round trips. Such a monolith was ﬁrst designed and operated by the group in München [4]. A. Sizmann et al. measured a noise suppression of −1.0 dB in the fundamental light in the presence of signiﬁcant electronic noise from which they inferred a noise suppression of at least −2.2 dB. This was one of the earliest applications of the balanced detector system (see Section 8.1). Unfortunately the system was not stable, but squeezing was observed only for milliseconds at a time. Systematic

339

340

9 Squeezed Light

1064 nm

532 nm

High Refl 1064 High Refl 532

High Refl 1064 Low Refl 532

Figure 9.30 The monolithic design for second harmonic generation (SHG) using CW lasers. The fundamental ﬁeld (FF) is enhanced by the cavity, and the second harmonic ﬁeld is reﬂected at one end and forms one single output beam. Source: From R. Paschotta et al. [5].

PD1 + –

SA

BS PD2 532 nm Oven 110 °C Laser

F1

1064 nm

M1

1064 nm Lock-in 12 MHz

Figure 9.31 Typical layout of an SHG squeezing experiment. The monolithic resonator is pumped through an isolator (FI) by a single-mode ND:YAG laser at 1064 nm. The reﬂected second harmonic light is detected by a balanced detector, and the sum and diﬀerence currents from the two detectors (D1 and D2) are measured with a spectrum analyser (SA). The parameters used by R. Paschotta et al. are MgO:LiNbO3 , cavity length 7.5 mm, radius of curvature 10 mm, input reﬂectivities of RFH = 99.9%, RSH < 1%, and output reﬂectivities close to perfect for both ﬁelds. Source: After R. Paschotta et al. [5].

improvements were carried out by the group in Konstanz who achieved very reliable squeezing of considerable magnitude. They used a monolithic cavity (Figure 9.30). R. Paschotta et al. [5] measured −0.6 dB squeezing (−2.0 dB inferred) on the SH beam. This system showed great reliability (Figure 9.31). A quantitative analysis of the ﬂuctuations can be obtained from a description ̂ in at frequency of the cavity using quantum operators. The input mode A 𝜈L enters the cavity through a mirror with decay constant 𝜅f 1 . It builds up an intra-cavity mode â f . A mode â sh at frequency 2 𝜈L is generated via the second-order non-linearity of strength 𝜒 (2) and leaves the cavity via a mirror of decay constant 𝜅sh1 . The total decay of the internal fundamental mode is described by 𝜅f = (𝜅f 1 + 𝜅ﬂ ), which includes losses due to scattering, whilst that of the SH cavity is 𝜅sh = (𝜅sh1 + 𝜅shl ). A corresponding amount of vacuum noise is coupled in to each cavity. The quantum Langevin equations describing the

9.4 Early Demonstrations of Squeezed Light

–22

Noise power (dBm)

–24

Signal Shot noise

–26

–28

–30 0

10

20

30

Noise frequency (MHz)

Figure 9.32 Squeezing spectrum obtained by R. Paschotta et al. Source: From R. Paschotta et al. [5].

generation of the SH ﬁeld are given by √ dâ f √ ̂ﬂ ̂ in + 2𝜅ﬂ 𝜹A = −𝜅f â f − 2𝜒 (2) â †f â sh + 2𝜅f 1 A dt √ √ dâ sh ̂ sh + 2𝜅shl 𝜹A ̂ shl (9.27) = −𝜅sh â sh + 𝜒 (2) â 2f + 2𝜅sh1 𝜹A dt ̂ i are all vacuum ﬁelds. Equations like these were used by M. Collett where the 𝜹A and later R. Paschotta and co-workers [5, 19]. They concluded that the squeezing was best in the limit of low detection frequencies (Ω → 0) and that the variance of the SH beam could ideally reach the value of V 1SH = 0.11 or −9.5 dB. The squeezing extended up to frequencies of the order of the linewidth of the cavity. This model assumed that the laser was driven by a coherent state. However, in the experiment, the laser noise swamped the noise suppression at very low frequencies. The best noise suppression observed (Figure 9.32) was only V 1SH = 0.8, or −1.0 dB, at a frequency of 16 MHz. The full spectrum can be calculated using the formalism of linearization. It was shown that the spectrum of the SH light (assuming low scattering losses) is given by a noise transfer function [6]: V 1SH (Ω) = 1 −

8𝜅nl2 − 8𝜅nl 𝜅f (V 1las − 1) (3𝜅nl + 𝜅f )2 + (2πΩ)2

2

where 𝜅nl =

2𝜒 (2) |𝛼f |2 𝜅sh (9.28)

̂ in , the driving ﬁeld where 𝛼f = ⟨̂af ⟩ and V 1las is the amplitude noise spectrum of A of the laser at the fundamental. At low frequencies the large excess noise of the laser V 1las , derived in Chapter 6, dominates the measurement. An optimum detection frequency exists. The noise that masks the squeezing is not a technical imperfection, but it is intrinsic to the laser. It is possible to reduce the laser noise by placing a mode cleaner cavity between the laser and the squeezing cavity. This cavity will reﬂect the high

341

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9 Squeezed Light

Piezo A

PID

Mode cleaner (optional)

±

Lens

Mixer

Dichroic

Photodiode

Signal generator

Power monitor

λ 2

λ 4

Adder/subtracter

Monolithic frequency doubler

λ Faraday 2 rotator

Transmitted 1064 nm detector

Laser Slow control

Fast control

PBS

PBS PBS Reflected pump beam from doubler

28 MHz

45 MHz

BS

BS Reflected 1064 nm self-homodyne detector Highpass filter

Combiner

Laser controller

A

±

532 nm self-homodyne detector

±

PID

Figure 9.33 Schematic layout of the improved SHG experiment with mode cleaner. Source: After A.G. White et al. [20].

frequency noise for frequencies larger than its linewidth. The system can now be modelled by two sequential transfer functions, one for each cavity. This theoretical model was tested by A.G. White et al. [20]. The experimental layout shown in Figure 9.33 is similar to that used by R. Paschotta (see Figure 9.31). The light source is a diode laser-pumped monolithic Nd:YAG laser. The non-linear material is MgO-doped LiNbO3 . The crystal is phase-matched by tuning the temperature. The operating temperature is at about 110 ∘ C and has to be optimized to within ±2 mK. The SH light is separated by a dichroic mirror and detected by a balanced detector. The detectors receive optical powers of up to 15 mW each. The eﬃciency of the photodetectors (EG&G FND100) is low (𝜂det = 0.55). In a repeat of the experiment, a mode cleaner cavity, locked to the laser frequency, is placed between the laser and the crystal. The experimental noise spectra are shown in Figure 9.34 for the frequency range 2–34 MHz. The QNL was determined by measuring the diﬀerence of the currents from the two detectors. The electronic noise, typically −6 dB, has been subtracted. The experimental trace (a) shows the noise suppression for the monolithic cavity. The optimum of −0.6 dB is measured at 13 MHz. A theoretical prediction, trace (b), based on Eq. (9.28) and using measured values for the non-linearity, losses, intensities, and cavity reﬂectivities agrees very well. In a second experiment the mode cleaner cavity is inserted. As shown in trace (c), the high frequency noise from the laser (2–15 MHz) is substantially rejected. The best observed squeezing is increased to −1.6 dB at 10 MHz (3 dB inferred). Trace (d) shows the theoretical prediction obtained by including the transfer function for the mode cleaner cavity, which means replacing V 1las in Eq. (9.28) with the output variance from the cavity. The good agreement shows that all

10

8.36

7.5

6.03

5

3.81

(a)

1.78

2.5 (b) Quantum noise limit

0 (d)

–2.5

–1.45

(c)

(e)

0

Normalized noise power (dB) (measured)

Normalized noise power (dB) (inferred)

9.4 Early Demonstrations of Squeezed Light

–2.55

–5 2

7

12

17

22

27

32

Frequency, f (MHz)

Figure 9.34 Comparison between theory and experiment for the squeezing generated by second harmonic generation as shown in Figure 9.33. The linearized model describes the complete system in a modular approach through the use of transfer functions. (a, b) No cavity. (c, d) With mode cleaner cavity. (e) Model with no laser noise. Source: After A.G. White et al. [20].

eﬀects are included in the combined transfer function. How can these results be improved? Can we get closer to the limit of V 1SH = 1∕9? A systematic study has shown that the main obstacles are the limited ratio of non-linearity to absorption in available non-linear materials and competing non-linearities [21]. One appealing, and technically simple, alternative to the external cavity frequency doubler would be to incorporate the crystal into the laser cavity. Such internal, or active, systems show very high conversion eﬃciencies for SHG. Early theoretical studies predicted large squeezing in these systems, but a careful analysis by White et al. [20], including the intensity and dephasing noise of the laser, shows that in practice no squeezing can be expected. This prediction applies to a wide a range of lasers, and thus active doubling is not worth considering for experiments. 9.4.4

The Kerr Eﬀect

A third process for squeezing that had been proposed very early is the Kerr eﬀect. It was favoured by the group of M.D. Levenson and R.M. Shelby at IBM Almaden Research Laboratories. It led to some of the earliest published squeezing results and was later extended by H.A. Haus and K. Bergman at MIT to the pulsed regime where it achieved impressive −3.5 dB noise reduction. The responses of the Kerr medium was brieﬂy discussed in Section 9.1. Here we analyse it in detail. 9.4.4.1

The Response of the Kerr Medium

The classical description of light propagating through a non-linear Kerr medium has already been given in Eqs. (9.1) and (9.6). The ﬂuctuations of the quadrature

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9 Squeezed Light

phase of the output ﬁeld can be described by quadrature operators, either in the time or as here the frequency domain: ̃ in ̃ out = X𝟏 X𝟏 2πn0 n2 L𝛼 2 (9.29) 𝜆 This output ﬁeld is detected using a homodyne detector by beating the output signal against a local oscillator. The detected quadrature is a linear superposition of 𝛿X1 and 𝛿X2 given by ̃ in + 2rKerr X𝟏 ̃ out = X𝟐 ̃ in X𝟐

with rKerr =

̃ ̃ cos(𝜃) + X𝟐 ̃ sin(𝜃) X(𝜃) = X𝟏

(9.30)

The variance is a measure of the noise in the quadrature and is calculated as shown in Section 4.1.1. It depends on the Kerr parameter rKerr and is normalized to the variance of a coherent state. Note that the amplitude remains unchanged and that for 𝜃 = 0, the amplitude quadrature variance is always equal to the QNL. This particular projection is independent of the non-linearity: 2 sin2 (𝜃) V (𝜃, rKerr ) = 1 + 2rKerr sin(2𝜃) + 4rKerr

(9.31)

where V (𝜃 = 0, rKerr ) = 1 is the QNL, and for V (𝜃, rKerr ) < 1 we have achieved squeezing. The angle 𝜃s for the best squeezing, the minimum of V (𝜃, rKerr ), can be found by evaluating d∕d𝜃(V (𝜃, rKerr )) = 0, which leads to ) ( 1 −1 (9.32) cot(2𝜃s ) = −rKerr or 𝜃s = arctan 2 rKerr In the limit of a small non-linear parameter (rKerr → 0), the best squeezing is observed for 𝜃s = −0.25π. For large squeezing parameters the angle approaches asymptotically the value −0.5π. Figure 9.35 shows an example of the normalized variance V (𝜃, rKerr ) for three values of the non-linearity, namely, rKerr = 0.25, 0.5, 0.75, and 1.0. It can be seen that the noise suppression is improving with larger non-linearity and that the angle of best squeezing is rotated. The noise suppression increases steadily with the non-linearity, and the optimum squeezing is given by √ 2 2 V (𝜃s ) = 1 − 2 rKerr 1 + rKerr + 2 rKerr (9.33) 4

1

3

rKerr = 0.5

2

rKerr = 0.25

rKerr = 0.75

0.8 V(θs)

V(θ)

344

1 0 (a)

0.6 0.4 0.2

0

π/4

π/2 θ (rad)

3π/4

0

π

(b)

0

0.5

1

1.5

2

rKerr

Figure 9.35 (a) V(𝜃) for three values of rKerr . (b) Dependence of optimum value V(𝜃) on rKerr .

9.4 Early Demonstrations of Squeezed Light

The calculation did not assume any speciﬁc detection frequency Ω. This is correct for a situation where the non-linearity does not depend on Ω. In practice, this applies only to the case of a travelling wave through a non-resonant Kerr medium. The two diﬀerent cases of (i) a broadband Kerr medium inside a cavity and (ii) of a resonant Kerr medium, such as atoms close to an absorption line, will require a description of n(𝛼 2 ) as a function of Ω and have to be evaluated for each detection frequency individually. It is useful to consider the Kerr eﬀect in view of quantum noise sidebands. At the core of the above calculation is the correlation between the quadratures stated in Eq. (9.29). This correlation enforces a link between the noise sidebands. Without such a correlation 𝛿EΩ and 𝛿E−Ω would be completely random. There would be independent contributions from 𝛿EΩ and 𝛿E−Ω to both 𝛿X1 and 𝛿X2. The correlation requires that for each component of 𝛿EΩ , there is also an additional component −2 rKerr 𝛿EΩ at the other sideband, and vice versa. The sidebands at the output are larger than at the input, as can be seen in Figure 9.4. In detection the correlation leads to a reduction of the noise in one quadrature. 9.4.4.2

Optimizing the Kerr Eﬀect

The Kerr eﬀect has been used in both CW and pulsed systems. It was actually one of the ﬁrst successful demonstrations of squeezing. The main practical problem is to achieve a large Kerr parameter rKerr , and this requires the largest possible values for the non-linear coeﬃcient 𝜒 (3) , interaction length L, and pump intensity I. There are four strategies for obtaining a large value for rKerr : (i) Optical ﬁbre. The coeﬃcient for the non-linearity is small, but the length of the medium and the intensity can be increased by using several hundred metres of single-mode ﬁbre. Values for rs in the range 0.1–0.5 have been obtained for the case of a CW squeezing and in the range 0.5–2.0 have been achieved using pulses (see Section 9.5). (ii) Non-resonant medium inside a cavity. The same material (quartz) as used in the optical ﬁbre can be used in a high ﬁnesse resonator. Whilst the physical length is short (typically 10 mm), the eﬀective length and the high intra-cavity intensity in a high ﬁnesse cavity could compensate this shortcoming. No results have yet been published. (iii) Resonant media. The non-linear coeﬃcient 𝜒 (3) can be enhanced several orders of magnitudes by operating close to an atomic resonance. Every absorption line shows a strong dispersive eﬀect, which means a strong linear refractive index. The spectral lineshape of the refractive index is aﬀected by the driving ﬁeld. The transition can be saturated and power broadened, which in turn provides a non-linear refractive index. In the experiments this eﬀect is again enhanced by a cavity around the medium. (iv) The use of cascaded 𝜒 (2) processes. It is possible to create a process that has the properties of the Kerr eﬀect by combining two 𝜒 (2) processes. It was shown [22] that the simultaneous occurrence of OPO and SHG is equivalent to one single process. Provided the system is degenerate, the excitation of the pump and down-converted mode will be coupled in a way that resembles a Kerr eﬀect for the pump mode. This was demonstrated in single-pass experiments [23] and for the CW case with a non-linear crystal in a cavity [24].

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These strategies suggest that large Kerr parameters should be achievable. However, when we analyse the experiments in detail, we will ﬁnd that the reality of squeezing experiments is not that simple. All four approaches have severe limitations since, in all experiments, there are extra sources of noise and competing non-linear eﬀects that will degrade the squeezing. In the actual experiments special tricks are used to suppress the extraneous noise, and a maximum of the non-linearity is traded oﬀ against a minimum of degradation due to competing eﬀects. 9.4.4.3

Fibre Kerr Squeezing

Fibre-optic quantum noise reduction was eventually developed to reliable and strong photon number noise reduction through three following generations of experiments [25]. After the pioneering experiments with CW laser light described here, the next generations used ultrashort pulses to achieve a larger non-linear phase shift with less Brillouin scattering noise in shorter ﬁbres. With short pulses the distortion of pulse shape and phase becomes an issue in coherent detection schemes. The elegant solution is to employ solitons that propagate free of chirp. For solitons, the local oscillator phase can easily be optimized for the entire pulse in the phase-sensitive detection of quadrature amplitude squeezing. No phase-matched local oscillator was needed in the two latest generations of experiments in ﬁbre-optic Kerr squeezing. Using the methods of spectral ﬁltering and asymmetric non-linear interferometers with picosecond or sub-picosecond solitons, directly detectable photon number squeezing was produced. These experiments are described in Section 9.5.4 on soliton squeezing. A comprehensive summary of ﬁbre-optic Kerr squeezing experiments is given in the tables of Ref. [25]. One of the pioneering experiments was carried out by M.D. Levenson, R.M. Shelby, and their collaborators at IBM in 1985 [13]. The idea was very straightforward: use a quartz ﬁbre with a Kerr coeﬃcient to turn a coherent state, generated by a krypton laser, into quadrature squeezed light. The experimental arrangement is shown in Figure 9.36. The detection system used the scheme of a detuned cavity that provides attenuation for the light without destroying the squeezing and provides a variable projection of the uncertainty area as a function of detuning. The details of this scheme are described in Section 9.3.3. Phase-dependent noise was readily observed. However, the initial squeezing results were rather disappointing. The apparatus appeared to be dominated by classical noise, frequently signiﬁcantly larger than the QNL. The reason is that optical ﬁbres show Brillouin scattering, which is driven by the thermal excitations of the solid-state material. This drives small ﬂuctuations in the refractive index. Inside the ﬁbre, which has a well-deﬁned refractive index guiding geometry, acoustic waves can be set up that have a complex spectrum of resonances. A consequence of these eﬀects is a broadband spectrum of phase noise, called guided acoustic wave Brillouin scattering (GAWBS) (Section 5.4.7). This adds to the noise in the squeezing quadrature and can mask all squeezing. The GAWBS noise can be partially suppressed by cooling the ﬁbre; thus the experiments by Levenson and co-workers

9.4 Early Demonstrations of Squeezed Light

DC

DVM

AC

100 Ω Laser

Spectrum analyser

D

Lens Faraday rotation isolation Phase modulator ωM

Amp

Cavity ωM

LHe Fibre

Figure 9.36 The schematic layout of the pioneering Kerr squeezing experiment in an optical ﬁbre at IBM. Source: After [26].

were performed at 4 K using liquid helium-cooled ﬁbres [26]. The GAWBS eﬀect also varies with the materials and the geometry used in the ﬁbre. Improved materials have more recently reduced the eﬀects of GAWBS [27]. It is possible to suppress many of these phase ﬂuctuations by cancelling the classical phase noise due to GAWBS in an interferometric arrangement. Here, the local oscillator and the squeezed beam both propagate through the ﬁbre in opposite directions, and both beams experience the same classical linear phase ﬂuctuations that then are cancelled in the detection scheme. Finally, there is stimulated Brillouin scattering in a ﬁbre. This is a non-linear phenomenon with a threshold above which a signiﬁcant fraction of light in the ﬁbre is backscattered. The transmitted light is extremely noisy, thereby limiting the intensity that can be used in a ﬁbre experiment. This eﬀect can be partially overcome using not a single-mode but a multimode input, where each mode is squeezed individually and has its own stimulated Brillouin threshold. The experiment has to be arranged such that all these squeezing features add up in phase. The results of the experiment are shown in Figure 9.37. The data clearly show squeezing at a squeezing angle of 28∘ . The best noise suppression observed is V 1(𝜃s ) = 0.85 (−0.7 dB), which corresponds to an inferred squeezing of V 1(𝜃s ) = 0.7 for the light emitted by the ﬁbre. What are the alternatives? For the non-resonant medium in a cavity, the main obstacle is the problem associated with achieving a suﬃciently high ﬁnesse and the eﬀects of losses in the cavity. The cavity will have to be locked actively to the laser frequency using the techniques described in Section 8.4. Squeezing can be achieved only at sidebands around each cavity resonance. That means the squeezing spectrum will show maxima at multiples of the FSR of the cavity. Any dispersion eﬀects have to be carefully avoided. Otherwise the FSR will change with the optical frequency, and the sidebands at plus and minus one FSR cannot be brought into resonance simultaneously.

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Figure 9.37 Early squeezing results from the ﬁbre Kerr squeezing experiment at IBM. Source: After [26].

2.5 Normalized noise level V

348

2.0

1.5

Standard quantum limit = 1.0 –100

–50

0

50

100

Phase shift θ (°)

How high a ﬁnesse is required? In the ﬁbre the interaction length is typically 100 m, the input intensity is 100 mW, and the waist size is 3 μm. In a resonator with the same input power, with a waist size of 10 μm and a material length of 10 mm, a ﬁnesse of about 100 000 would be required to achieve the same parameter rKerr for the interaction. For a single-port cavity, this requires a total loss of about 10−5 in optical power, including all absorption and scatter losses. Such low values require extremely pure and homogeneous materials that are not yet available. 9.4.4.4

Atomic Kerr Squeezing

The use of a resonant medium relies on the ability to tune the input light, which is to be squeezed, close to resonance with an absorption line of the medium. The non-linear refractive index is enhanced. However, for these small detunings, the spontaneous emission becomes an extraneous source of noise, as with 4WM (Section 9.4.1). The light absorbed and re-emitted by the atoms produces intensity ﬂuctuations that are detected alongside the ﬂuctuations due to the quantum noise of the input beam. This produces uncorrelated noise at the detection frequency and swamps the carefully arranged correlations between the sidebands that we want to observe. Therefore the squeezing experiment requires detuning larger than the atomic linewidth absorption. For thermal atoms this would mean the Doppler linewidth. To avoid this the experiments by A. Lambrecht et al.[28] were carried out in laser-cooled atoms. A calculation for the Kerr eﬀect generated by a sample of cold atoms inside an atomic trap predicts signiﬁcant noise suppression. In practice several complications arise: The high intensities inside the cavity lead to instabilities. The cavity with a non-linear medium displays bistability and irregular switching. In addition, real atoms have many states due to hyperﬁne and Zeeman splitting – the prediction of a simple two-level model can frequently not be achieved, and optical pumping has to be taken into account. After overcoming all these diﬃculties, good noise suppression of more than −1.8 dB below the QNL

9.5 Pulsed Squeezing

has been observed [28]. These are diﬃcult experiments, and despite an intensive optimization, such higher atomic densities in magneto-optical traps or even the proposal of using a Bose–Einstein condensate as the non-linear sample and much smaller detunings, the atoms are not a promising Kerr medium. The best results were probably achieved by the group led by P. Grangier and co-workers [29] who optimized the medium for the demonstration of the quantum non-demolition (QND) eﬀect, as described in Chapter 11. 9.4.4.5

Atomic Polarization Self-Rotation

A related mechanism for squeezing is the rotation of the polarization of the light induced by an atomic vapour that occurs at high intensities. Interaction of the near-resonant atoms with elliptically polarized light can cause the initially isotropic 𝜒 (3) non-linear medium to display circular birefringence: the two orthogonal circularly polarized components of the light will propagate at diﬀerent velocities. The result is a rotation of the linear polarization of a light beam. This in turn corresponds to a shear in phase space and the distortion of the initial symmetric Wigner function into an ellipse [30]. The eﬀect was demonstrated in rubidium and the experiment can be very simple: an atomic vapour cell, heated to achieve the correct atomic density and shielded from stray magnetic ﬁelds, is illuminated by carefully controlled polarized light. The output beam is separated with a polarizing beamsplitter into a squeezed vacuum beam and a local oscillator beam that are combined, after rotation of the polarization of the local oscillator, in a standard homodyne detector. Whilst a simple theory predicts squeezing of up to −6 to −8 dB [31], the experiment achieved so far a squeezing of −0.85 dB [30]. The limitations are largely due to self-focussing of the vapour, resonance ﬂuorescence, and phase mismatch. It remains to be seen if this tantalizingly simple experiment can achieve its full potential.

9.5 Pulsed Squeezing Some of the best squeezing results have been achieved using the very high intensities and non-linearities that can be achieved with optical pulses. 9.5.1

Quantum Noise of Optical Pulses

The concepts of quantum noise and squeezing can be extended to pulsed light sources. Consider a series of optical pulses, each pulse with a length tpulse repeated at time intervals trep ≫ tpulse , as shown in Figure 9.38a. The spectrum of the photocurrent will contain frequency peaks at the repetition frequency Ωrep and its higher harmonics. The width of each peak will depend on the length of the pulse train measured or the integration time of the detection. The distribution of power amongst, or the envelope of, the frequency peaks is set by the Fourier transform of the pulse shape of the individual pulses. This is schematically shown in Figure 9.38b. For the short pulses that can be generated with mode-locked lasers, (Ωrep typically 100 MHz, tpulse 0.2–100 ps), the envelope is very wide, but only

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9 Squeezed Light

tpulse

Intensity

trep

Time

(a) Intensity

350

(b)

Envelope due to pulse shape Quantum noise δΩ Ω rep

2 Ω rep

Frequency Ω

Figure 9.38 Optical pulses in the time domain (a) and the frequency domain (b). Quantum noise is present at frequencies between the fundamental components associated with the pulses.

a few of the fundamental peaks ﬁt within the detection bandwidth. In between these peaks the spectrum contains a quantum noise ﬂoor. This is equivalent to the quantum noise spectrum of a CW laser. Thus we can call this quasi-CW . One can interpret this noise as being the sum of many pairs of quantum noise sidebands, each beating with one of the fundamental peaks in the optical spectrum. Any part of the recorded spectrum at frequency Ω < Ωrep represents the Fourier component of the variation in the energy of many pulses. For example, the pulsed source could be intensity modulated at a frequency Ωmod . This would create spectral sidebands to all fundamental peaks of the optical spectrum, which means signals at nΩrep ± Ωmod . The quantum noise is the noise ﬂoor in such a sideband. In these experiments noise represents the lack of reproducibility of the pulses. An ideal classical light source would produce a sequence of perfectly reproducible pulses, and thus no noise outside the fundamental components would be detected. Coherent light cannot have exactly identical pulses, but the energy will vary from pulse to pulse. We are comparing the variation of one group of pulses with that of the next group – and observing the noise ﬂoor. This quantum noise is independent of the detection frequency, as long as it is less than the repetition frequency, and a QNL pulsed laser will have a ﬂat, white quantum noise spectrum between the fundamental peaks. The detection scheme for pulsed light is almost identical to that for CW laser light. The same photodiodes and SA are used. A single detector can reveal intensity ﬂuctuations. A balanced detector will be able to distinguish between classical and quantum intensity noise, and a balanced homodyne detector can detect noise of speciﬁc quadratures. However, three additional requirements have to be satisﬁed in the detection of pulses: (i) The detector should not respond to fundamental frequency components and their harmonics; otherwise saturation of the detector and the electronic ampliﬁers will occur even at very low optical powers. This can be achieved by using sophisticated ﬁlter networks with notch ﬁlters suppressing any

9.5 Pulsed Squeezing

response at Ωrep . Such detectors require extensive development. The only exception is the detection at very low frequencies, Ω ≪ Ωrep , where the detector sensitivity can simply be rolled oﬀ with low-pass ﬁlters. (ii) The pulses have to be carefully aligned in the time domain. For example, the pulses used as the signal and local oscillator beams in the homodyne detector have to arrive simultaneously, with a time diﬀerence of much less than the pulse length. This requires an accurate adjustment of all distances in the experiment. Any cavity would have to have round-trip times that are an integer of trep . (iii) The pulses contain very high intensities. Whilst this is an advantage in that the non-linear response is signiﬁcantly enhanced, it can introduce at the same time other non-linear processes, such as self-focussing, which limit the beam quality and thus the detectability of the light. The fundamental advantages of pulsed experiments are the high intensities and consequently the very large non-linear eﬀects that can be observed. The intensity during a pulse makes the build-up cavities, which are necessary in CW experiments, obsolete. Single-pass SHG or OPO operation is possible. Similarly, the Kerr eﬀect in single pass is suﬃciently strong to introduce a correlation between noise sidebands and therefore induce squeezing. However, there are some eﬀects that limit noise suppression. The squeezing occurs only during the pulse if we assume that the response of the material is instantaneous. When calculating the degree and the quadrature of the measurable squeezing, it is necessary to take the time development of the squeezing into account and to average over the temporal development of the noise properties and the squeezing during the duration of a pulse. Figure 9.39 shows an example where the degree of squeezing and the orientation vary during the pulse. The measurement averages the noise contributions, which are weighted by the intensity at the detector at any one time. Although the squeezing at the peak intensity gets the highest weighting, the observed total squeezing will be considerably less than predicted for the peak intensity alone. In addition the squeezing angle is again a weighted average, and squeezing in diﬀerent quadratures can partially cancel each other. Figure 9.39 The development of the squeezing during a light pulse, indicated by the change of the squeezing ellipse. Source: From H.A. Haus et al. [32]. Time

Squeezed axis

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9 Squeezed Light

Input 85/15 coupler

3 dB coupler

Squeezed vacuum

Local oscillator

Balanced detector –

Fibre (non-linear)

Output

Stabilization circuit (a)

15 10 Relative noise level (dBm)

352

5 Shot noise level

0 –5 –10 –15 –20 0

(b)

1

2

3

4

5

6

Pulse non-linear phase (rad)

Figure 9.40 The pulsed Kerr experiment. Source: After H.A. Haus et al. [32]. (a) The schematic layout. (b) Noise suppression results from the pulsed Kerr experiment.

9.5.2

Pulsed Squeezing Experiments with Kerr Media

A very successful series of squeezing experiments with optical pulses was carried out by the MIT group [32, 33] Figure 9.40b. Here the optical medium is a single-mode ﬁbre, and the non-linear process is the Kerr eﬀect. The light source is a mode-locked Nd:YLF laser generating 100 ps pulse with a repetition frequency of 100 MHz, Ωrep = 10 ns. The wavelength is 1.3 μm, which is the zero-dispersion wavelength of the ﬁbre and assures that the pulses maintain their pulse shape during the experiment. The experiment is shown schematically in Figure 9.40a. The experimental arrangement is equivalent to a non-linear Sagnac interferometer. The pulses from the laser enter the interferometer via a 3 dB ﬁbre coupler, equivalent to a 50/50 beamsplitter. There they are separated into two pulses of

9.5 Pulsed Squeezing

Time (s) Noise with swept mirror

Figure 9.41 The temporal trace of the noise from the pulsed experiment. Detection frequency is ﬁxed at Ω = 55 kHz. Source: After K. Bergman and H.A. Haus [33].

identical intensity, which move in opposite directions through the ﬁbre. After a full traverse, the pulses arrive simultaneously back at the coupler. They interfere and form a pulse that is moving back towards the laser. A pulse of squeezed vacuum light leaves through the other port of the interferometer. This pulse is detected with a homodyne detector. A fraction of the reﬂected light is used as a local oscillator, generated by the 85/15 coupler. The arrival time of the local oscillator has to be synchronized with the pulse of squeezed light. The phase of the local oscillator is set and stabilized with a piezo controller. The eﬀect is seen in diﬀerent experiments in two frequency bands: at low frequencies (80–100 kHz) and at high frequencies (≥ 100 MHz). The Sagnac interferometer is essential for such measurements. All ﬂuctuations induced by the ﬁbre, such as phase shifts and intensity variations due to mechanical vibrations, are experienced by both counter-propagating pulses. They are cancelled when the pulses interfere after one round trip and do not appear on the output. This also applies to phase noise induced by the ﬁbre, such as GAWBS. Equally, noise in the laser light frequency or intensity is compensated by the interferometer. In this particular example the suppression was so good that all noise with the exception of large relaxation oscillations were cancelled. What remains is the correlation between the quadratures, induced by the Kerr eﬀect. The squeezing in this case is so strong that they did not even need a SA, but it was directly detectable with an oscilloscope in the time traces of the laser intensity. Figure 9.41 shows alternating noise traces with the squeezing process turned on and oﬀ. The diﬀerence between these traces gives directly the amount of squeezing. A noise suppression of −3.5 dB was achieved at frequencies as low as 55 kHz. 9.5.3

Pulsed SHG and OPO Experiments

The high peak intensities of short laser pulses allow the realization of direct single-pass travelling-wave second harmonic generation (TWSHG) and the

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associated squeezing. Consider that in a single pass through a crystal of length , a beam of input power P(0) generates a beam of power PSHG () = ΓP(0). We call Γ the single-pass conversion eﬃciency. At the same time the beam will experience an eﬀective non-linear phase shift 𝜙eﬀ between the SH light generated at position z = and SH light generated at position z = 0 and propagating to z = . Best SHG is achieved with complete phase matching, or 𝜙eﬀ = 0. It was shown by the group of P. Kumar [22] that in this case the predicted noise suppression at low frequencies is S(Ω → 0) = 1 − Γ

(9.34)

This indicates the possibility of almost perfect squeezing provided that most of the light is converted into the SH. Even with a moderate amount of phase mismatch, the noise suppression should be very strong, and detailed calculations predict squeezing as large as 10 dB [22]. However, the corresponding experiments were so far not successful, but a noise suppression of only 0.3 dB has been observed [34]. This was achieved with a Na-doped potassium titanyl phosphate (KTP) crystal. An SHG eﬃciency of Γ = 0.15 was obtained at a peak intensity of about 250 MW/cm2 . The main technical challenge has been in isolating the squeezed from the fundamental light. Using a type II phase-matched crystal, it was possible to separate the SHG beam from the fundamental beam using polarization optics. But the achievable extinction ratio was too small. Excessive leakage of the input ﬁeld into the detector led to saturation of the detection system and limited the observed noise suppression. This type of squeezing requires further technical developments. Far more successful has been the generation of squeezed vacuum state using a degenerate optical parametric ampliﬁer (DOPA). The group of P. Kumar used a KTP crystal in single-pass operation. The ﬂuctuations are measured with a homodyne detector. Special attention was given to the matching of the local oscillator beam to the squeezed beam in both spectral and temporal domains. Noise suppression of 5.8 dB was observed [35], matching the best results from the corresponding CW cavity experiment. 9.5.4

Soliton Squeezing

One of the requirements for quadrature squeezing in the pulsed experiments is to maintain the temporal shape of the pulse throughout the medium. This is commonly achieved by working at the zero-dispersion wavelength of an optical ﬁbre. However, the material shows self-phase modulation that results in a chirp of the frequencies. Consequently a non-soliton pulse does not allow the local oscillator phase to be optimized for the pulse as a whole. This limited early pulsed ﬁbre squeezing experiments where Gaussian zero-dispersion pulses were used. A clever alternative was proposed by P. Drummond and his group [36] and in parallel by the group of H.A. Haus [37]. A special pulse form can be used to form temporal solitons that preserve their shape by a balance between dispersion and non-linear refractive eﬀects and propagate free of chirp. Solitons have almost particle-like properties, are robust against a variety of perturbations, and preserve their shape, energy, and momentum even after numerous interactions with

9.5 Pulsed Squeezing

each other [38]. The pulse shape is a hyperbolic secant. The pulse width tFWHM is dependent on the peak power Ppeak because shorter pulses experience stronger dispersive broadening and consequently require a higher peak power to balance dispersion. A fundamental soliton is obtained for a wide range of pulse energies and wide varieties of initial pulse shapes such as Gaussian or square pulses. After some initial self-reshaping, the exact balance of pulse width and peak power is eventually attained with a smooth soliton shape. 2 , It is convenient to introduce the pulse parameter N 2 = LD ∕LNL ∝ Ppeak tFWHM which indicates whether dispersive pulse broadening (N < 1) or non-linear pulse compression (N > 1) dominates the initial pulse evolution. The N parameter is the ratio of the length scale for dispersive pulse broadening, LD ∝ (tFWHM )2 , and the length per one radian non-linear phase shift in the pulse peak, LNL ∝ (Ppeak )−1 . These length scales are given by the dispersive and non-linear coeﬃcients of the ﬁbre. A fundamental soliton is also called a N = 1 soliton because it exactly balances dispersion with non-linear eﬀects. The quantum mechanical description, using a non-linear Schrödinger equation, was able to describe the soliton’s noise properties and unique quantum soliton eﬀects were found [37, 39]. The ﬁrst such eﬀect was amplitude quadrature squeezing, predicted by Carter et al. [40]. It was observed at IBM by M. Rosenbluh and R.M. Shelby [41], who used a colour centre laser to produce soliton pulses with a pulse length of 200 fs at 1500 nm propagating through a polarization-preserving single-mode ﬁbre. The Kerr eﬀect in the ﬁbre generated the squeezing. Noise suppression of −1.67 dB was measured with a homodyne detector. A second quantum eﬀect is the ‘entanglement’ of pairs of solitons. Their quadratures get linked, which can be exploited for QND experiments (Chapter 11) and quantum information experiments (Chapter 13). 9.5.5

Spectral Filtering

A third quantum eﬀect is soliton photon number squeezing by spectral ﬁltering, which was unexpectedly found by S.R. Friberg et al. [42] in 1996. The experimental method was fairly simple: a pulse launched into a ﬁbre was spectrally bandpass ﬁltered after emerging from the ﬁbre end and was then directly detected – an unusual idea since ﬁltering deliberately introduces losses to achieve squeezing. The eﬀect can be heuristically explained by considering the propagation of solitons in the ﬁbre. As they propagate, self-phase modulation and group velocity dispersion interact to cause a periodic broadening and shrinking of the soliton spectrum. For a ﬁxed length of ﬁbre the soliton energy, or photon number, determines the periodicity. When the soliton, after leaving the ﬁbre, is attenuated by a ﬁxed bandwidth spectral ﬁlter, the losses depend on the soliton’s pulse energy. The combination of propagation and ﬁltering results in a non-linear input–output relationship, i.e. non-linear absorption. At speciﬁc lengths, or pulse energies, the intensity noise is suppressed below the QNL – photon number squeezing is possible. These simple ideas were conﬁrmed by full quantum models [43]. The underlying squeezing mechanism can be best understood in terms of the multimode

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quantum structure of solitons. Spectrally resolved quantum noise measurements showed that anti-correlations in photon number ﬂuctuations emerge within the spectrum of the soliton as it propagates down the ﬁbre [44, 45]. If these anti-correlated spectral domains are selectively transmitted through a spectral ﬁlter and then detected simultaneously and coherently, their ﬂuctuations cancel each other in the observed photon ﬂux. This anti-correlation produces a photocurrent noise reduction far below shot noise. The pioneering experiment was carried out with 2.7 ps pulses with a repetition rate of 100 MHz centred at 1455 nm with a bandwidth of 1.25 nm. After propagating through 1.5 km of ﬁbre, the pulses emerge with a pulse width of 1.3 ps and a spectral bandwidth of 1.65 nm. They were ﬁltered with a variable bandpass ﬁlter to remove the high and low cut-oﬀ frequencies, resulting in the observation of −2.3 dB squeezing with N = 1.2 solitons [42]. The method was further investigated in Erlangen by A. Sizmann and G. Leuchs et al. with sub-picosecond solitons and a variety of ﬁlter functions, ﬁbre lengths, and pulse energies below and above the fundamental soliton energy [45]. When optimized in the experiment, up to −3.8(±0.2) dB of photocurrent noise reduction below shot noise was achieved in direct detection. If corrected for linear losses, a reduction in the photon number uncertainty by −6.4 dB can be inferred [45]. This technique is very elegant and has signiﬁcant practical advantages: It requires no coherent noise measurements and thus is insensitive to phase (GAWBS) noise. It is broadband and all that is required is a pulsed soliton source and suitable ﬁbre. By using an in-line ﬁlter, this device can be made exceedingly lossless and robust. 9.5.6

Non-linear Interferometers

The most promising and most successful method of photon number noise reduction with solitons so far are highly asymmetric non-linear interferometers (Figure 9.42a). The CW [46] and soliton-pulse analysis [47–49] show that the beam splitting ratio in the interferometer must be highly asymmetric in order to produce strong photon number noise reduction, in contrast to earlier experiments with symmetric non-linear ﬁbre interferometers that produced a squeezed vacuum. More than 10 dB of noise reduction was predicted. After the ﬁrst experimental demonstration in Erlangen by S. Schmitt et al. using 130 fs solitons, the interferometer parameters were optimized [48], and the observation of less than −5 dB in the photocurrent noise was reported (Figure 9.42c). This and parallel work reported by K. Bergman and co-workers [50] are the strongest noise reductions observed for solitons to date. When corrected for linear losses, the inferred photon number squeezing was −7.3 dB [51]. A simple model shows a steplike input–output energy transfer function. Around each minimum the transmission is non-linear, the output energy is stabilized at a ﬁxed value, and the photon number uncertainty is reduced below the shot noise limit. However, the squeezing mechanism can be better understood in terms of number–phase correlations induced by the Kerr non-linearity [52], similar to the CW case studied by M. Kitagawa and Y. Yamamoto [46]. A Kerr squeezed state from the high-power path of the interferometer is slightly rotated by the

9.5 Pulsed Squeezing

Fibre loop

X2

BS1 Laser (2) (1) BS2 + – Relative noise power (dB) Transmitted pulse energy (pJ)

(a)

120

(b)

α/(1–α) = 90/10 Lfibre = 6.40 m τFWHM = 126 fs fexp = 20 MHz

100 80 60 40 20 0 6 4 2 0 –2 –4

(c)

X1

Spectrum analysers

0

50

100

150

Total launched pulse energy (pJ)

Figure 9.42 Intensity squeezing of solitons with the non-linear interferometer. (a) Experimental arrangement. (b) The concept of projecting the squeezing ellipse after the Kerr eﬀect. (c) Experimental results with solitons of diﬀerent orders N. The top trace shows the transmission and the bottom trace the measured noise suppression. Source: From A. Sizmann and G. Leuchs [25].

addition of the weak ﬁeld from the low-power path of the interferometer, such that the amplitude quadrature of the total output ﬁeld is squeezed (Figure 9.42b). This squeezing mechanism implies that noise reduction grows with the non-linear phase shift in one arm of the non-linear interferometer [52]. This is in contrast to spectral ﬁltering where noise reduction originates from multimode photon number correlations, which does not grow with propagation distance. Today the non-linear loop interferometer is seriously considered in technical applications for stabilizing communication systems operating with Terabaud

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information rates. It is also employed in basic research as an eﬀective squeezing source for generating entangled beams in quantum information experiments (Chapter 13). To summarize, optical solitons in ﬁbres have provided the perspectives of building ﬁbre-integrated sources of noise reduction and given new insight into multimode quantum correlations with a terahertz bandwidth through spectrally resolved quantum measurements.

9.6 Amplitude Squeezed Light from Diode Lasers The simplest intensity squeezing experiment is to produce amplitude squeezed light directly with a laser of high eﬃciency that is driven by a sub-Poissonian current. The laser can be driven inside a high impedance circuit with a very quiet current (V idr ≪ 1). The conversion eﬃciency of the diode laser can be as high as 𝜂las = 0.5 − 0.75. An important distinction is the fact that a diode laser has an internal cavity to build up the radiation. The laser also involves both spontaneous and stimulated emission. Consequently, it has laser dynamics and spontaneous emission noise. The earliest models for diode lasers were developed by Y. Yamamoto and co-worker [53]. More recently linearized laser models have been developed [54]. It is possible to apply them to the diode laser system (see Chapter 6) and to calculate directly the transfer function between the ﬂuctuations of the pump source (idr ) and the light. As Eq. (6.17) shows, the noise of the laser intensity is completely dictated by the pump noise. For an electrically pumped laser, the eﬃciency of pumping is close to unity, and, in diode lasers, the output coupling is large. Thus the eﬃciency 𝜂las is basically given by 𝜅m ∕𝜅. The variances V 1 of the amplitude quadrature and VN of the intensity are equivalent, and it is common practice to use the Fano factor for this situation. With these assumptions, Eq. (6.17) turns into (1 − V 1las ) = 𝜂las (1 − V idr )

(9.35)

This is the same result as for any other passive device. The maximum noise reduction for a perfectly quiet drive current is V 1las = 1 − 𝜂las . Two further important assumptions have been made in this model, namely, that the laser emits all light into a single output mode and that all the current entering the laser is contributing to the lasing process. Bias currents have been neglected. In practice, however, both these assumptions are not easy to justify. An experiment by W.H. Richardson and Y. Yamamoto [55] showed a noise suppression of 6.5 dB. This is a most impressive result that was actually better than predicted by Eq. (9.35) and the overall eﬃciency of the laser used. No deﬁnite explanation has been given to date for this result. However, it is possible that the diode laser had signiﬁcant internal parallel currents that did not participate in the lasing process. Thus the actual laser eﬃciency 𝜂las might have been better than measured directly. Alternatively, some unknown process might have correlated the drive current with the light. Unfortunately, later experiments were not able to reproduce these early measurements, but they showed less noise reduction and tended to conﬁrm the prediction of Eq. (9.35). In particular Inoue observed noise reduction up to −4.5 dB in diode lasers cooled to 20 K [56].

9.6 Amplitude Squeezed Light from Diode Lasers

It has been observed that diode lasers will excite more than one laser mode. These might not contribute much to the intensity of the output beam, but they can inﬂuence the noise behaviour dramatically. A particularly systematic study was carried out by the group in France [57]. They showed that a free-running diode laser was emitting light into over 50 individual modes. The total intensity was found to be sub-Poissonian. However, by measuring only some of the modes, rejecting the others with a spectrograph, the noise increased drastically. The noise of the single central mode, which carried the vast majority of the power, was up to 40 dB above the Poissonian level. This suggests that the various modes have anti-correlated noise and the noise terms cancel each other when all modes are detected simultaneously. This behaviour can be predicted by a simple multimode laser model that includes the competition between the laser modes [57]. Only some of the modes will be operating well above threshold, whilst others will remain at threshold and only contribute ﬂuctuations. The competition incorporates a limitation of the total gain based on the available number of excited atoms. The division between the modes is not stochastic but balances the ﬂuctuations, similar to a conservation of the total number of available photons. The result is anti-correlation between the modes. More detailed theoretical models have been published to take mode competition properly into account [58], but further study is still required. It is possible to enforce single-mode operation, and simultaneous noise suppression, by either injection locking or optical feedback from an external cavity. When these techniques are optimized, the laser system will operate on a truly single-mode output and generate amplitude squeezed light. M.J. Freeman et al. [59] used injection locking, and H. Wang et al. [60] used external cavities to achieve single-mode operation. They achieved a noise suppression of 1.8 dB with lasers at room temperature. More recently, D.C. Kilper et al. observed a noise reduction by 4.5 dB in an injection-locked laser at 15 K. On the whole, these results ﬁt well to Eq. (9.35). An overview of all these results is given in Figure 9.46. However, it appears that not all diode lasers show reproducibly the same degree of noise suppression. The group led by Y. Yamamoto has succeeded for the ﬁrst time in designing and manufacturing diode lasers that operate in single mode due to their geometry and that reliably operate at −2.8 dB below the Poissonian limit [61]. In contrast, most commercial systems do not perform as well as earlier prototype versions of the same laser design. Large variations in the noise performance have been found even within one commercial series of lasers with otherwise identical speciﬁcations. The detailed reasons for this variation can be many. Finally, one experimental problem is that the maximum eﬃciency of the lasers occurs at the highest drive currents, well above threshold, where the output powers are large. For a measurement of the noise suppression, all the light has to be detected, unless complex structures are used to selectively attenuate the power but not the squeezed sidebands. Thus the beam cannot be attenuated or otherwise the noise suppression is lost. The limit of detectable power is set by the saturation properties of the detector; presently only a maximum of 50 mW can typically be detected with fast detectors capable of measuring ﬂuctuations at radio frequencies (RF) (≥ 10 MHz). This means that some of the best noise suppression cannot actually be detected.

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In summary, the expectation of simple and reliable diode lasers as low cost sources of non-classical light has not yet come to fruition.

9.7 Quantum State Tomography So far we have concentrated only on the measurements of the variances of quantum noise. It has been shown that this is fully suﬃcient for a description of squeezed states, or to be more general, any states that can be described by linearizable models. Figure 9.43 shows a very clear example of the noise suppression that was available already in 1996 after optimization of the experimental parameters. This type of scan provides a good knowledge of the properties of the quantum state. We saw in Section 4.5 that these states have 2D Gaussian Wigner functions. For a quadrature squeezed state, with unknown squeezing angle, we have to measure the variance V (𝜃) for a whole range of angles, and from these data the minimum value V (𝜃s ) is determined. We actually quote only this one number, and the entire squeezed state is described in this way. At least as long as it is a minimum uncertainty state, otherwise a second number, V (𝜃s + 90), is required to describe the state completely. On the other hand, techniques have been developed to measure the Wigner function. This is of fundamental interest since some quasi probability distributions (QPDs) exhibit unusual features and it would be nice to show diagrams of completely reconstructed QPDs. The key idea to these techniques is to measure higher-order moments of the noise statistics. Rather than measuring only the variance for a certain projection angle 𝜃, the full noise trace is recorded (see Figure 9.45). This was described by U. Leonhardt and H. Paul [63] and further elaborated by U. Leonhardt [64] and others. The idea was ﬁrst applied to pulsed –75 (a)

–80 Noise power (dBm)

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–90 (b) –95

–100 –0.5

0

0.5

1

1.5

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ϕ (π rad)

Figure 9.43 Experimental data for CW squeezing using a monolithic OPO: the typical scan of the variances indicating squeezing and anti-squeezing. For comparison a theoretical curve with a noise suppression of −7.1 dB is shown. Source: After P.K. Lam et al. [62].

9.7 Quantum State Tomography

light. M.G. Raymer’s group at Oregon [65] recorded histograms of the energy of many pulses and in this way the full noise characteristics of the pulses. Many such histograms were recorded for diﬀerent angles 𝜃, and the underlying Wigner function can be reconstructed using tomographic techniques [65]. For CW experiments the process is similar. The full time history of the ﬂuctuations is recorded. This can be achieved by directly measuring the noise ﬂuctuations of the photocurrent and demodulating the signal with a ﬁxed single detection frequency. Alternatively an electronic SA is used, as shown in Figure 9.45a. By leaving the video bandwidth suﬃciently large, the full noise trace is recorded. For a squeezed state these traces vary dramatically with the projection angle, as shown Figure 9.45b. The many traces are then transformed into histograms of the ﬂuctuations for this speciﬁc projection angle and detection frequency Ω. The restriction to speciﬁc values of Ω is particularly important for those squeezing sources that have a distinct squeezing spectrum S(Ω). Here it is important to detect only those sidebands that are strongly squeezed. For each detection frequency a function W (Ω) can be reconstructed using the tomographic technique [63, 67]. Note that all the measured Wigner functions will be centred around zero and they show the ﬂuctuations that are independent of the mean value of the intensity. The quadrature of the squeezing is only determined as a rotation angle, which is measured either with a homodyne detector or in the case of a bright squeezed source with large intensity, in comparison with the phase of this beam. Apart from a rotation of the Wigner function, the phase and amplitude squeezed states look the same. In addition, the shape of the Wigner function is frequency dependent and can vary considerably with Ω. The same light source can simultaneously be squeezed at one frequency and noisy at another. The complete beam is described by a whole spectrum of Wigner functions (Figure 9.44).

X2 X1

Figure 9.44 The concept of quantum state tomography: complete noise traces are recorded for diﬀerent projection angles, and one such example is given here. These data can be recombined to reconstruct the Wigner function of the quantum state. Source: After U. Leonhardt [64].

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0.2 Local oscillator

Phase control θ Homodyne detector

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Figure 9.45 Quantum tomography experiment. Part (a) shows the schematic layout. Results for a squeezed vacuum state are given: (b) the phase-dependent noise, (c) the squeezed two-dimensional Wigner function, and (d) the reconstructed photon probability distribution. Source: From S. Schiller et al. [66].

How does this compare with the pulsed situation that seems to provide one unique answer for the Wigner function? In this case the selection of the detection frequencies is made automatically; they are given by the Fourier components of the pulses, the pulse repetition frequency, and its higher harmonics. Only these speciﬁc detection frequencies contribute. Only one combined answer, a weighted average of several single frequency Wigner functions, was recorded in the measurements by M.G. Raymer’s group [65]. Using their OPO, the Konstanz group demonstrated the capability of tomography [66] very early on. They investigated a squeezed vacuum, measured the noise projections, and converted them into Wigner functions and photon distributions. One example is given in Figure 9.45. Their results, the complete squeezed Wigner function (c) and the photon statistics (d) with enhanced probabilities for even photon numbers, are a beautiful demonstration of quantum optics. This is now a routine diagnostic technique, both for the quality of the light that is generated and for the quality of the apparatus that is processing the squeezed light and using it for a variety of applications. The comprehensive review by A.I. Lvovsky and M.G. Raymer [68] provides many details of the experiments and the data analysis. It is interesting to note that we can also apply tomography usefully in situations where the light is not squeezed but at the QNL or weakly squeezed and contains

9.8 State of the Art of CW Squeezing

some well-known modulation. After testing the light at several locations in the apparatus, a comparison of the tomographic results will show quite clearly the performance and losses imposed by the apparatus. In this way the quality and capability of the apparatus can be tested and veriﬁed. This would, for example, mean that in the case of commercial production of components that should be able to handle squeezed light or entangled beams, tomography is a useful tool for quality testing and assurance of components and without the need to use non-classical light. As long as we stay within the limit of Gaussian states, tomography should in principle not reveal more than the measurement of the two orthogonal quadratures. However, can we use tomography to show even more exotic results? This depends not so much on the measurement techniques but on the types of states that we can manufacture, and this has not been easy. As pointed out in Section 4.5, we would require a system that cannot be linearized and ﬂuctuations that do not follow Gaussian statistics. One successful approach is to consider a state with a very small photon number, which is heralded by photon counting techniques. Alternatively photon number selective post-processing can be used to select the data for such a speciﬁc state of light imbedded in the optical beam. In these cases strongly non-classical Wigner functions have been recorded, as discussed in Chapter 13. An example is shown on the front cover of this book, where the non-Gaussian Wigner functions are made visible using a modiﬁed detection scheme, where the homodyne detection is conditional on a photon counting measurement. Alternatively we would require a CW system that is extremely non-linear, for example, close to an instability point, and where the coeﬃcients for the non-linearities (e.g. 𝜒 (2) or 𝜒 (3) ) change within the uncertainty limit of the state. These conditions are extremely diﬃcult to achieve since we do not have such non-linearities, and this remains a challenge for the future.

9.8 State of the Art of CW Squeezing Squeezing moved from the stage of a cute theoretical idea to the ﬁrst experimental demonstration in a remarkably short time. From 1980 to 1985 the details of possible squeezing were explored theoretically; then within one year several squeezing experiments were performed more or less simultaneously. Since the ﬁrst demonstration of squeezed states, we have seen a steady improvement in the size and the reliability of the noise suppression. During the ﬁrst few years, several unexpected new noise sources were discovered that hampered the progress. These had to be avoided, in most cases through improvements of the materials used. Examples are as follows: the new ﬁbres used by K. Bergman et al. had significantly less GAWBS noise than the earlier ﬁbres used by M.D. Levenson et al., the early noise problems encountered by H.J. Kimble et al. with external cavity SHG were avoided in the more recent monolithic devices, and the limitation in squeezing with the OPO was set by noise created inside the crystal, which can now be avoided by lower-loss materials. Very quiet monolithic diode-pumped Nd:YAG lasers have replaced the much noisier lamp-pumped lasers used in the earlier

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Squeezing history 3 0 Noise supression (dB)

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4WM/Kerr Pulsed SHG OPO/OPA Diode laser Twin beams TEM 01

–6 –9 –12 –15 1980 –18

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Figure 9.46 Best observed noise suppression in diﬀerent squeezing experiments in the time 1985–2016. The symbols describe diﬀerent processes: Boxes = CW vacuum squeezing using a sub-threshold OPO, diamonds = CW bright squeezed light (SHG, atoms, Kerr), triangles = diode laser noise suppression, crosses = pulsed squeezing experiments (OPO, SHG, solitons, etc.), and circles = twin-photon beam experiments.

experiments. And the detectors have been improved with a quantum eﬃciency of more than 0.95 now available for the visible (Si) as well as the near-infrared spectrum (InGaAs). All this technology had matured by about 2000. Some processes did not show suﬃcient non-linearities to produce signiﬁcant and reliable squeezing, including bistability [69] and atom–cavity interactions [70] and did not stand the test of time. New processes were found, such as the non-linear interferometer for pulsed squeezing. Now hundreds of diﬀerent experiments have been reported, too many to list separately. Instead, Figure 9.46 shows the historical development of the experiments. The observed noise suppression is plotted for several groups of experiments producing squeezed light. As can be seen, the OPO generating a squeezed vacuum state was the ﬁrst to show large squeezing and still remains the best process. At this point in time a suppression to −15 dB has reliably been achievable, albeit only by very few groups, since it requires an amazing attention to detail. If the long-term trend holds up, one could expect a suppression to −20 dB once all the components and materials are optimized. The CW experiments with atomic systems, both 4WM and strong cavity coupling, and with Kerr media were beautiful demonstrations of concepts and test of the theory. They were essential for our understanding but are less likely to play a role in applications. The singly resonant SHG was rapidly improving and is a simple and reliable source of bright amplitude squeezed light. Noise suppression down to −5 dB is a realistic expectation. These systems can be made very robust, small, and reliable, and with the use of periodically poled materials, we might see even simpler and better devices. For the pulsed experiments the results look equally impressive. Here the Kerr eﬀect, soliton squeezing, and the non-linear interferometer all have the

9.9 Squeezing of Multiple Modes

potential to achieve a noise suppression to −10 dB. The potential of new ﬁbres with engineered band-gap structures has not been explored. The only exception to this positive trend has been the diode lasers. Some outstanding results have been achieved very early on, but they could not be conﬁrmed. The best of the present results is supporting the direct theoretical prediction, and further understanding and reﬁnements of the manufacturing process could make −3 dB of noise suppression available in commercial low cost lasers. Squeezed vacuum states can be transformed into various types of squeezed states, the squeezing axis can be rotated, and coherent amplitude can be added through reﬂection oﬀ a cavity. We can control the squeezing independently of the coherent amplitude. In addition we can produce polarization squeezed light and spatially squeezed light. Fibre-based squeezing and diode lasers have the potential to be readily integrated into practical systems. In parallel the other main advance is reliability. In 1984 squeezing light was regarded as a ‘Hero’ experiment, which would normally work no earlier than 2 a.m. ‘Daylight squeezing’ during normal business hours became a norm by about 1996. Now we have systems squeezers that operate continuously for many days, ‘24/7’, for example, within a gravitational-wave observatory with both very high duty cycle and reliability. In addition we have single-pass squeezers as part of integrated optics that are directly built into custom-made devices. Technically this is all possible and commercialization is a clear option. The future use is limited only by our creativity for applications and the demand for such devices.

9.9 Squeezing of Multiple Modes 9.9.1

Twin-Photon Beams

Closely related to amplitude squeezed light is the generation of twin-photon beams. In this case two optical beams are generated by one above threshold OPO (Sections 9.4.2 and 6.3.3). The single input beam is converted, in a non-linear crystal inside a resonant cavity pumped above threshold, into two output beams that have identical statistical properties. In a simple picture one can imagine that each input photon generates two identical photons, one for each output beam, but unlike the parametric down-converter we have here not correlated pairs of photons but two single-mode beams with correlated photon ﬂux. Using type II materials, which means the two modes have orthogonal polarizations (Section 9.4.2), the beams can easily be separated and detected individually. The OPO in most cases is non-degenerate, which means the optical frequencies of the beams are diﬀerent but this has no consequences since they are not recombined optically. Theoretical calculations [71, 72] show that each beam individually has intensity ﬂuctuations that depend on the dynamics of the OPO, as discussed in Section 6.3.3. In particular, close to threshold, each beam has very large ﬂuctuations, V 1(Ω) ≫ 1, at four times the threshold the beams are at the QNL, and far above the threshold each beam should be squeezed, V 1(Ω) = 0.5. This has not yet been observed.

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However, if we detect the diﬀerence between photocurrents, these ﬂuctuations cancel because of their strong correlations, and we can observe a noise level well below the QNL. The QNL in this case corresponds to the noise measured on the diﬀerence between the intensities of ‘classical twin beams’, obtained by splitting any light beam on a 50∕50 beamsplitter. It is equal to the shot noise level for the combined photocurrent. This experiment was ﬁrst carried out with CW lasers by the Paris group, led by E. Giacobino and co-workers [73] who recorded a noise suppression of 27 % (V 1 = 0.73). Next came an experiment with mode-locked Q-switched lasers by O. Aytür and P. Kumar [74]. C.D. Nabors and R.M. Shelby [75] used a doubly resonant OPO to generate two beams of diﬀerent colours and reported a noise suppression of −3.1 dB. Systematic improvements of their apparatus by the group in Paris led to observations of noise suppression by −8.5 dB, after correction for the dark currents, which is a record that stood for a long time. It was beaten by the Taiyuan group [76] with a noise reduction of −9.2 dB and ﬁnally improved again by the Paris group to −9.5 dB [77]. Since these experiments involve only intensity measurements, and no local oscillators, they avoid the complications of mode matching and beam overlap, and the data are limited purely by the quantum eﬃciency of the detectors and the escape eﬃciency. Note also that twin-photon beams are obtained at the two outputs of a 50∕50 beamsplitter, which has at its two input ports a squeezed vacuum state and a local oscillator of appropriate phase. This is actually what is done in any homodyne measurement of squeezing. It is possible to generate one beam that has sub-Poissonian statistics by measuring one beam and using this information to control, via electro-optic feedforward, the intensity of the other. This was demonstrated by J. Mertz et al. [78] who achieved a noise suppression of −4 dB of one beam compared to the QNL of this individual beam. This new output beam can be directly used for all the applications of sub-Poissonian light, which forms a direct alternative to the quiet diode lasers mentioned previously. Twin-photon beams can also be directly used to beat the QNL in the spectroscopy of very weak absorption lines by −1.9 dB [79] and the measurement of very weak absorptions by −7.0 dB [76]. Finally, the Paris group showed that twin beams oﬀer the opportunity to introduce post-selection processes, commonly used in the photon counting regime, to the regime of continuous variables [77]. If the ﬂuctuations of the two photocurrents at a ﬁxed detection frequency Ωdet are recorded directly and stored, as it was described in the section on state tomography (Section 9.7), we can now produce a sequence of recordings, each containing short intervals of the ﬂuctuations. Thanks to the quantum correlations, we can use one beam to select intervals where V 1(Ωdet ) is within a narrow, well-deﬁned band and select from the other beam the corresponding intervals. The selected light describes a beam with unusual properties. For example, we could select information that describes a beam with strong sub-Poissonian distribution, which was demonstrated experimentally and produced an intensity noise reduction of −4.4 dB below the QNL. Alternatively the selection could be done with an electro-optical modulator. The resulting beam is now no longer continuous but intermittent, depending on the selection process that rejects the majority of the time intervals but has the advantage of representing a strongly non-classical quantum state.

9.9 Squeezing of Multiple Modes

9.9.2

Polarization Squeezing

The polarization of a beam of light is another property that is limited by quantum noise. In order to detect the polarization, we require a combination of two detectors. Each is producing quantum noise, and consequently one expects a QNL for the polarization. The polarization is well described by the Stokes parameters S0 , S1 , S2 , S3 (Section 3.5.2), which describe the light in terms of three degrees of freedom for linear and circular components and one normalization factor. The state of polarization can be described by a three-dimensional diagram, the 1 Poincaré sphere, deﬁned by the radius S = (S12 + S22 + S32 ) 2 . Each point on the sphere is one polarization, and the radius of the sphere is given by the intensity of the beam. An analysis of the quantum properties of polarization shows that they obey the cyclical commutation relation [Ŝ 1 , Ŝ 2 ] = 2iŜ 3 ,

[Ŝ 2 , Ŝ 3 ] = 2iŜ 1 ,

[Ŝ 3 , Ŝ 1 ] = 2iŜ 2

(9.36)

This means that the polarization has an uncertainty volume and the polarization is only deﬁned within a small sphere around each point on the Poincaré sphere, as shown in Figure 9.47A. The possibility of achieving polarization ﬂuctuations below the QNL was investigated extensively by the group around A.S. Chirkin in Moscow. In particular N.V. Korolkova and A.P. Alodjants investigated the possibility of using amplitude squeezed light to inﬂuence the polarization [81, 82]. It was shown that two squeezed beams can be combined on a beamsplitter and by selecting the correct polarization and phase between the two beams, the output beam will have one or more of the Stokes variables measurable below the QNL [83]. The uncertainty volume is compressed and can take the shape of a cigar, with the two Stokes parameters Ŝ 𝟏 Ŝ 𝟑 suppressed at the same time, or alternatively like a pancake, with suppression of only one parameter, Ŝ 𝟐 . The ﬁrst experimental demonstrations were made by the group at Bell labs [84] and the Aarhus group [85]. The ANU group, including R. Schnabel, W. Bowen, and P.K. Lam, achieved the goal to squeeze several Stokes parameters simultaneously and below the −3 dB limit. They generated two independent squeezed beams [80] with two seeded OPAs driven by one Nd:YAG laser and frequency doubler. The squeezed beams were combined on one beamsplitter. The apparatus is shown in Figure 9.48 as an example of the actual complexity of quantum optics experiments. The multitude of components, all individually aligned and optimized, is required to produce the pump, seed, and local oscillator beams, to produce the two squeezed beams, to combine them on a beamsplitter, and to detect the properties of the output state. The main challenges are to reduce the loss of all components, to optimize the mode match between all the beams, and to maintain the phase locking between all the cavities, such as lasers, mode cleaners, and OPOs. For this purpose this experiment contains 4 temperature control systems and 7 optical locking systems. By generating two bright squeezed beams with about 1 mW power and about −4 dB of squeezing each, they were able to achieve both the cigar and pancake states, with suppression of the Stokes parameters simultaneously by better than −3 dB at detection frequencies around 8 MHz. Figure 9.47B shows the experimental results in the form of reconstructed

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Figure 9.47 (A) Diagram of (a) the classical and (b) quantum Stokes vectors mapped on a Poincaré sphere. The ball at the end of the vector visualizes the quantum noise of the polarization states. (B) Measured quantum polarization noise at 8.5 MHz from diﬀerent combinations of input beams: (a) single coherent beam, (b) coherent beam and squeezed beam, (c) bright squeezed beams, (d) two phase squeezed beams, and (e) two amplitude squeezed beams. Source: After R. Schnabel et al. [80].

uncertainty areas on the Poincaré sphere. This type of light can now be used to couple the quantum properties of the light to the quantum properties of atoms, as discussed in Chapter 14. 9.9.3

Degenerate Multimode Squeezers

Following the lead of the polarization squeezer, it is possible to use the output of two squeezers within one experiment. We will see that, for example, teleportation requires two squeezed inputs, and for quantum gates and error correction, we might need up to eight squeezed inputs. This can be achieved by building several devices, each producing a separate beam, and these can be combined using a whole array of beamsplitters, resulting in several output beams that are detected individually. This technique has been developed into a ﬁne art by the group of A. Furusawa at Tokyo University, and many more details can be found in his book [86].

9.9 Squeezing of Multiple Modes

2011 ANU double squeezer, entangler, tweezer

1989 IBM Kerr squeezer QND

Figure 9.48 Squeezing experiments are complex. They combine low loss optical systems and nonlinear devices, high eﬃciency detectors and high quality opto-mechanical control. They are operated in well controlled environments, built on optical tables with minimum mechanical noise, excellent thermal and dust control. These photos shows two examples: on the right had side an experiment at IBM in 1989 for the demonstration of squeezing and QND in optical ﬁbres. On the left hand side an apparatus at ANU in 2011 with two OPO squeezers for entanglement, spatial mode squeezing and applications to tweezers with biological samples. The design principles remained, the complexity and performance of the devices had greatly improved. The future will see integrated opto-mechanical designs with low loss wave guides, custom manufactured materials and opto-mechanical systems providing reduced size, ruggedness and higher reliability. Source: After R.Schnabel et al. [80].

Alternatively, one could search for one device that can generate squeezing in several modes simultaneously. This is in principle possible and has been used for OPOs that operate simultaneously in both output directions and in this way produce two squeezed beams. This apparatus has considerably fewer components than the complex machine that has two separate squeezers, as shown in Figure 9.48. The goal of producing a device with many modes has for a long time been elusive. Early attempts of using degenerate OPOs in special cavity conﬁgurations to generate several spatial modes simultaneously were found to be too delicate. However, since 2010, there has been considerable progress, in particular with temporal modes, where more than 10 squeezed modes have been generated and detected by the Paris group in a femtosecond frequency comb [87] using one OPO and creating orthogonal modes where each is a diﬀerent combination of a large set of individual frequencies in the comb. This is now being applied to several ideas from quantum information, communication, and sensing. At the same time the team in Virginia led by O. Pﬁster has produced and veriﬁed 60 modes of carefully engineered combinations of frequencies within an OPO with two non-linear crystals [88]. Each combination represents a whole entity, or mode, which combines the simple quantum states at individual frequencies of the

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9 Squeezed Light

optical ﬁeld and their magnitudes and phases in such a way that the combined mode is part of an elaborate basis set. The apparatus is carefully designed and crafted to create and then let us test quantum correlations not only between two of these modes but a whole set of multimode correlations, following carefully set out rules for graph states. These states are the realization of complex rules designed to represent mathematical algorithms and the realization of quantum logic protocols.

9.10 Summary: Quantum Limits and Enhancement In summary, the tools for optical quantum engineering have now emerged. Squeezed light is now being used in a number of applications to improve the detection of light beyond the QNL associated with the photon statistics of the best classical light, with the details in Section 10.2. There are several other situations in which the apparatus is limited by the QNL, for example, attenuators and ampliﬁers. In all cases the QNL, which we experience with a coherent state of light produced by a perfect laser, represents the limit of the performance of the instrument. Fortunately there are ways in which these limits can be overcome through the use of quantum correlations of the optical ﬂuctuations created by non-linear optical systems. Table 9.1 provides an overview of these situations. Table 9.1 Other eﬀects that are related to squeezing. Limit of a coherent state

Quantum eﬀect

QNL

Squeezing

V1 = V2 = 1

Vsq < 1, Vantisq > 1

Attenuation

Squeezing

Vout = 𝜂Vin + (1 − 𝜂)

V1out = 𝜂V1in

Ampliﬁcation

Anti-squeezing

Vout = G Vin + 1

V2out = GV2in

Independent beams

Entangled beams

V1(A|B) V2(A|B) = 1

V1(A|B) V2(A|B) < 1

Measurement

Quantum non-demolition

Vout > Vin

V1out = V1in

Classical communication

Teleportation

V1out ≥ V1in + 2

V1in ≤ V1out < V1in + 2

V2out ≥ V2in + 2

V2in ≤ V2out < V2in + 2

Beam displacement

Quantum laser pointer

VΔx = Vdx 1

VΔx < Vdx 1

VΔ𝜃 = Vdx 2

VΔ𝜃 < Vdx 2

References

The ﬁrst column states the quantum limits as set by the coherent state in terms of the variances of the ﬂuctuations. One aim of quantum optics is to go beyond these limits. The second column of the table names the devices that use quantum correlations and states the limits that can be achieved. If only one squeezer is used, we see an improvement in one selected quadrature Vsq . By using two squeezed sources, we can arrange an improvement in both quadratures of the same mode, for example, in the case of entanglement between two beams and teleportation. The improvement applies to all aspects of the information that is encoded in these orthogonal quadratures. In the case of spatial measurements, the detection of a displacement or tilt in a transverse direction x is limited by the quantum noise in the quadratures of a separate orthogonal mode with quadratures Xdx 1 and Xdx 1. This mode contains all the information of the displacement and tilt, and squeezing this mode will yield improved measurements of this particular degree of freedom. The following chapter shows examples of the use of squeezed light for measurement applications. Presently the quality and reliability of the devices is being improved. After more creative engineering, the complexity and usefulness of the state generators will improve to a point where they are simply sources of a desired quantum correlation pattern. More and more complex tests are being devised and will be used for in-principle demonstrations and quality control. The results of today will appear to be just the beginning – and it is now time to consider real applications of muitlmode entanglement, based on squeezing of light, or an equivalent set of modes based on particles, on photons, or ions, or atoms, or other particle-like quantum eﬀects. We shall now progress to see in Chapter 10 the eﬀects such quantum correlations have on actual devices, real sensors, communication, and imaging and in Chapter 13 on quantum logic protocols and devices.

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asymmetric ﬁber interferometer. Opt. Lett. 23: 1390. 51 (a) Schmitt, S., Ficker, J., Sizmann, A., and Leuchs, G. (1998). Investiga-

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66 Schiller, S., Breitenbach, G., Pereira, S.F. et al. (1996). Quantum statistics

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of the squeezed vacuum state by measurement of the density matrix in the number state representation. Phys. Rev. Lett. 77: 2933. Leonhardt, U., Muroe, M., Kiss, T. et al. (1996). Sampling of photon statistics and density matrix using homodyne detection. Opt. Commun. 127: 144. Lvovsky, A.I. and Raymer, M.G. (2009). Continuous-variable optical quantum-state review. Rev. Mod. Phys. 8: 299. Orozco, L.A., Raizen, M.G., Xiao, M. et al. (1987). Squeezed-state generation in optical bistability. J. Opt. Soc. Am. B 4: 1490. Hope, D.M., Bachor, H.-A., McClelland, D.E., and Stevenson, A. (1992). The atom-cavity system as a generator of quadrature squeezed states. Appl. Phys. B 55: 210. Fabre, C., Giacobino, E., Heidmann, A., and Reynaud, S. (1989). Noise characteristics of a non-degenerate optical parametric oscillator – application to quantum noise reduction. J. Phys. 50: 1209. Fabre, C., Giacobino, E., Heidman, A. et al. (1990). Squeezing in detuned degenerate optical parametric oscillators. Quantum Opt. 2: 159–187. Heidmann, A., Horowicz, R.J., Reynaud, S. et al. (1987). Observation of quantum noise reduction on twin laser beams. Phys. Rev. Lett. 59: 2555. Aytür, O. and Kumar, P. (1990). Pulsed twin beams of light. Phys. Rev. Lett. 65: 1551. Nabors, C.D. and Shelby, R.M. (1990). Two-color squeezing and sub-shot noise signal recovery in doubly resonant optical parametric oscillators. Phys. Rev. A 42: 556. Gao, J., Cui, F., Xue, C. et al. (1998). Generation and application of twin beams from an optical parametric oscillator including an 𝛼-cut KTP crystal. Opt. Lett. 23: 870. Laurat, J., Coudreau, T., Treps, N. et al. (2003). Conditional preparation of quantum states in the continuous variable regime: generation of sub-Poissonian states from twin beams. Phys. Rev. Lett. 91: 213601. Mertz, J., Heidmann, A., and Fabre, C. (1991). Generation of sub-Poissonian light using active control with twin beams. Phys. Rev. A 44: 3229. Schwob, C., Ribeiro, P.H.S., Maître, A., and Fabre, C. (1997). Sub shot noise high sensitivity spectroscopy with OPO twin beams. Opt. Lett. 22: 1893. Schnabel, R., Bowen, W.P., Treps, N. et al. (2003). Stokes operator-squeezed continuous-variable polarization states. Phys. Rev. A 67: 012316. Korolkova, N.V. and Chirkin, A.S. (1996). Formation and conversion of the polarization-squeezed light. J. Mod. Opt. 43 (5): 869. Alodjants, A.P., Arakelian, S.M., and Chirkin, A.S. (1998). Polarization quantum states of light in nonlinear distributed feedback systems. Appl. Phys. B 66: 53. Korolkova, N., Leuchs, G., Loudon, R. et al. (2002). Polarization squeezing and continuous-variable polarization entanglement. Phys. Rev. A 65: 052306. Grangier, P., Slusher, R.E., Yurke, B., and LaPorta, A. (1987). Squeezed-light-enhanced polarization interferometer. Phys. Rev. Lett. 59: 2153.

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85 Soerernsen, J.L., Hald, J., and Polzik, E.S. (1998). Quantum noise of an atomic

spin polarization measurement. Phys. Rev. Lett. 80: 3487. 86 Furusawa, A. and vanLook, P. (2011). Quantum Teleportation and Entangle-

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quantum networks with ultrafast frequency combs. Nat. Photonics. https://doi .org/10.1038/NPHOTON.2013.340. 88 Chen, M., Menicucci, N.C., and Pﬁster, O. (2014). Experimental realisation of multipartite entanglement of 60 modes of a quantum optical frequency comb. Phys. Rev. Lett. 112: 120505.

Further Reading Drummond, P. and Ficek, Z. (ed.) (2003). Quantum Squeezing. Springer. Furusawa, A. (2015). Quantum States of Light. Springer. ISBN: 978-4-431-55958-0. Schleich, W.P., Mayr, E., and Krähmer, D. (1997). Quantum Optics in Phase Space. Wiley-VCH. Walls, D.F. and Milburn, G. (1993 and 1997). Quantum Optics. Springer-Verlag.

377

10 Applications of Quantum Light Squeezed states and entangled pairs of photons allow us, in principle, to circumvent some of the quantum limits that apply to coherent states of light. Quantum measurements provide opportunities that classical systems do not have. This has practical applications and four areas stand out as the most likely applications: optical sensors beneﬁt from squeezed light in gaining sensitivity. Optical communication beneﬁts when information can be transmitted with non-classical light. Spatial eﬀects and imaging as well as timing are worth discussing as multimode examples for quantum enhanced measurements. Finally, interferometric detectors for gravitational waves (GWs) are the most impressive example for the use of squeezed light to date. For all these cases we will discuss the standard quantum limits (SQLs) that apply to coherent states and ways to overcome them. A diﬀerent class of applications based on quantum information concepts is discussed in Chapter 13.

10.1 Quantum Enhanced Sensors 10.1.1

Coherent Sensors and Sensitivity Scaling

An optical sensor is a device that monitors a property and translates a change of this property into an optical signal. The property under investigation could be a position, orientation, acceleration, temperature, pressure, concentration, chemical composition, light level, electric or magnetic ﬁeld, or even a GW, to name the most common. The sensor could be a local device, sometimes with very high microscopic spatial resolution, or it could be a remote sensing device allowing measurements of a property at a distance. Frequently, the sensor is used to produce the data for a 2D or 3D spatial or temporal map of the property. In all these cases we want to measure the property with the best possible sensitivity, low noise, and big dynamic range. We may wish to monitor the eﬀect continuously, and it is then normally an advantage to push the limit of sensitivity and signal-to-noise ratio (SNR). It is this goal where quantum optics makes a contribution (Figure 10.1). Technically speaking a coherent optical sensor can be regarded as a modulator (see Section 5.4.5), where the modulation of a beam of light is aﬀected by just one external parameter; let us call it ℘. The concept is that a change in ℘ leads to a A Guide to Experiments in Quantum Optics, Third Edition. Hans-A. Bachor and Timothy C. Ralph. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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Parameter ρ (x,y,z,t) changing in time and space

Parameter ρ (x,y,z,t) changing in time and space

DET Input beam optical power A in ^2(t)

(a)

One- or two-photon excitation

Output A out ^2(t)

DET

(b)

Figure 10.1 Coherent optical sensor (a) and incoherent sensor based on ﬂuorescence (b).

change of the optical output power, or intensity for a ﬁxed beam size, which can then can be detected. Given an input mode â (t), we can write the output mode as √ √ â o (t) = 𝜖(℘(t))̂a(t) + 1 − 𝜖(℘(t))̂v(t) (10.1) where the eﬀective optical transmission, 𝜖, is a function of ℘ and v̂ (t) is the vacuum mode that inevitably couples for non-unit transmission. In a high sensitivity situation, we may typically want to estimate a small change in our parameter, 𝛿℘, away from a known value ℘o . Given that a small change in ℘ leads to a small change in the transmission, we can write ) ( √ K ′ 𝛿℘ â o (t) = â (t) 𝜖(℘o ) − √ 2 𝜖(℘o ) ( ) √ K ′ 𝛿℘ + v̂ (t) (10.2) 1 − 𝜖(℘o ) + √ 2 1 − 𝜖(℘o ) 𝜕𝜖 where K ′ = 𝜕℘ |℘o . We will discuss the ultimate limits of this type of sensor in Section 10.1.3. A related situation is where we want to use a sensor to continuously monitor small time-dependent changes 𝛿℘(t) of the parameter. The changes can be due to changes of ℘(x, y, z, t) in time at a ﬁxed location or due to scanning the sensor in space or both. The changes in the photocurrent at the output iout (t) = â †o (t)̂ao (t) are seen as a proxy for the changes in ℘(t). The design of the sensor can vary widely – we can have very direct links, such as the measurement of concentration of a compound by absorption or a displacement measurement in an interferometer. However, we also have some quite complex sensors, for example, for the measurement of magnetic ﬁelds, temperature, or pressure. We also assume that any change 𝛿℘(t) can occur with any time dependence, which means there could be harmonic signals with ﬁxed or variable frequencies, step functions, or bursts. Consequently we want to sample the data stream from the sensor at regular intervals, Δtsamp , fast enough to capture the dynamics of interest. On the other hand we assume that the underlying noise, from the input beam and from other sources such as the electronics, is of Gaussian nature.

10.1 Quantum Enhanced Sensors

As a concrete example let us take a simple case where our external parameter is a harmonic signal and there is a linear relationship between 𝜖 and the parameter, i.e. 𝜖 = K℘ = K(℘o + 𝛿℘o cos(2𝜋Ωsig t)), where K and 𝛿℘o ≪ ℘o are constants. ̃ = 𝛿℘o(𝛿(Ω + Ωsig ) + 𝛿(Ω − The Fourier transform of the ﬂuctuating term is 𝛿℘ ′ Ωsig )) and K = K. We may also assume that the ﬂuctuations and the signal are small compared to the average power such that the input mode can be writ̃ ̃ ̂ ten as â (t) = 𝛼 + 𝜹a(t) with |𝛼| ≫ 1 and hence A(Ω) = 𝛿(Ω)𝛼 + 𝜹A(Ω). Taking the Fourier transform of Eq. (10.2), substituting and neglecting small terms (see Section 5.4.5) gives √ √ ̃ o (Ω) = ℘o A(Ω) ̃ ̃ A + 1 − ℘o V(Ω) K 𝛼 𝛿℘ (10.3) + √ o (𝛿(Ω − Ωsig ) + 𝛿(Ω + Ωsig )) 4 ℘o where we have taken 𝛼 real. We see that the harmonically oscillating parameter maps onto frequency sidebands. The spectral power as measured via direct detection is proportional to the amplitude quadrature (see Section 4.5.1), so we ﬁnd ̃ o (Ω) + A ̃ †o (Ω)|2 ⟩ VI (Ω) = ⟨|A = 1 + ℘o (V 1A (Ω) − 1) K 2 𝛼 2 𝛿℘2o + (𝛿(Ω − Ωsig ) + 𝛿(Ω + Ωsig )) 4℘o

(10.4)

The signal to noise of, for example, the upper sideband is SNR =

K 2 𝛼 2 𝛿℘2o 4℘o (1 + ℘o (V 1A (Ωsig ) − 1))

(10.5)

If our input beam is in a coherent state, then V 1A (Ωsig ) = 1. However, we see that in this simple example amplitude squeezing, i.e. V 1A (Ωsig ) < 1, can improve the signal to noise of the sensor and hence improve the precision of our estimate of the parameter. Notice that in the coherent state case, the SNR scales with the input power Pin ∝ |𝛼|2 = n. This scaling is sometimes referred to as the quantum noise limit (QNL) for sensors. However, if we use squeezed light at the input, we ﬁnd that the quantum noise term is reduced, and, under ideal conditions, for the same parameter ℘, the signal now scales proportional to the optical power squared, 2 . This limit is known as the Heisenberg limit in the sensitivity. This can be Pin seen in the above example by setting ℘o ≈ 1 and assuming V 1A ≪ 1. We also require that the number of photons arising from the squeezing, ns ≈ 1∕(4V 1A ), is equal to the number of photons from the coherent amplitude, na = |𝛼|2 , such that na = ns = 1∕2 n ∝ 1∕2 Pin . Substituting these conditions into Eq. (10.6) gives 2

K 2 n 𝛿℘2o (10.6) 4 thus exhibiting the expected Heisenberg scaling. One test for the operation below the quantum limit is to measure the SNR as a function of the optical power Pin SNR ≈

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at the input, which would be proportional to Pin for the quantum noise lim2 for the quanited case and approach, but not exceed the proportionality to Pin tum enhanced case. The fact that this is the best possible scaling can be derived rigorously using the concept of information theory, Fisher information, and the Cramer–Rao Bound as will be discussed in Section 10.1.3. 10.1.2

Practical Examples of Sensors

In all of these cases, the sensor is measuring and tracking small changes in a property with high sensitivity. To reach its full potential, the sensor should be ﬁrst optimized to operate at the QNL, commonly referred to as the shot noise limit, and then the sensitivity can be improved by increasing the laser power. Once this is no longer possible, most likely through the onset of non-linear eﬀects, squeezing can provide an additional improvement. At the moment typically by a factor of 2 (or 3 dB) in the SNR. Potentially, this could be expanded to a factor 10 (or 10 dB) but at a tremendous cost in reﬁnement since even the smallest optical losses have to be avoided. Here the GW detectors stand out as the shining future example [1], since already all other improvements have been made and the scientiﬁc gain of even a factor of 2 improvement allows the detection in 3D from an eight times larger volume, which is very important for cosmology. The most notable examples of quantum enhanced sensors are described in more detail in the following. It has been shown that absorption spectra can be recorded with improved SNR [2]. Interferometers can be improved, as ﬁrst predicted by C.M. Caves et al. [3] and ﬁrst implemented for table top instruments [4, 5]. It is now a routine component of some of the large-scale GW detectors [6], as will be discussed in more detail in Section 10.3. Squeezed light improves the measurement of the polarization of a laser beam [7] and any eﬀect that introduces changes to the polarization, the macroscopic spin of a sample of atoms [8], and the displacement or position of a laser beam [9]. Squeezed light enhances the tracking of particles within living cells [10]. It can be used for the improvement of the uncertainty in timing pulses and frequency combs [11, 12]. Other technical applications will follow. Very early in the game, the concept was applied to the detection of absorption, see Figure 10.2(a,b). Amplitude squeezed light was used, as generated with quiet diode lasers (see page 213 or, from a second harmonic generator (SHG) experiment, Section 9.4.3. The experiments were identical to conventional absorption sensors or absorption spectroscopy with the only change being that the input beam is squeezed. The absorption has to be modulated at a frequency Ωem within the bandwidth of the squeezing spectrum. Care has to be taken that the overall eﬃciency of the apparatus and the detectors is very high; otherwise the degree of squeezing and the improvement in SNR rapidly disappears. One of the ﬁrst demonstrations was made by E.S. Polzik et al. [2], who demonstrated an improvement of 2 dB for the SNR for saturated absorption spectroscopy of a Cs vapour. They used an optical parametric oscillation (OPO) as a source for squeezed light, and the major challenge was to make this light source tunable. Another ﬁne demonstration was carried out by C.D. Nabors and R.M. Shelby [13] who showed that a modulation can be sensed with improved SNR using

10.1 Quantum Enhanced Sensors

SHG

Intensity squeezed laser

Absorber

Laser

OPO

Detector

Absorber

(a)

(b)

SHG

Laser PBS

Modulator

OPO Absorber

(c)

Intensity squeezed laser

Moving scattering particles

(d)

Figure 10.2 Diﬀerent schemes for using squeezed light for improved SNR in optical sensors. (a) Absorption using amplitude squeezed light. (b) Absorption using tunable OPO. Source: After E.S. Polzik et al. [2]. (c) Measuring DC absorption using twin photon beams. Source: After Hai et al. [14]. (d) Detecting scattered laser light for motion detection.

the bright squeezed light from an above threshold doubly resonant monolithic OPO. One of the diﬃculties is to modulate the absorption at a suﬃciently high frequency. The experiments by the group in Shanxi University [14] and in Paris and now many others use an elegant trick: twin photon beams generated by an above threshold OPO. One beam is used as a reference. The other is modulated at Ωmod and split into another pair of beams, probe 1 and probe 2, and only probe 1 is sent through the sample (see Figure 10.2c). By ﬁrst forming the sum of the two probe photocurrents and then the diﬀerence between the total probe and the reference, a current is generated that contains an AC signal at Ωmod , which is proportional to the strength of the absorption. In this way a slowly varying absorption can be observed with high sensitivity. A similar approach can be used to detect the intensity of scattered light (see Figure10.2d). It was shown that the light scattered by fast-moving smoke particles can be detected using an optical homodyne detectors and tracking the frequency sidebands. The Doppler shift provided the necessary frequency shift, and the spectrum was used to detect the density and the velocity distribution of the sample. This technique is known as Doppler velocimetry. Using amplitude squeezed light from a diode laser, Y.Q. Li et al. were able to demonstrate an increase in the SNR of 1 dB compared to an experiment with classical light [15]. More recently this was used, for example, for tracking particles inside biological samples with quantum enhancement (see page 418).

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10.1.3

Ultimate Sensing Limits

In Section 10.1.1 we found measurement precision limits from SNRs given a particular detection strategy, i.e. direct detection. Here we discuss a more general approach that ﬁnds the ultimate precision limit for any detection strategy. We also discuss recent suggestions for non-linear quantum sensors. As discussed earlier, in general, optical sensors use some interferometric arrangement as an actuator such that changes in a parameter of interest lead to a modulation of the intensity of a probe ﬁeld. Hence the beamsplitter interaction described by Eq. (10.2) can be considered a generic model of a quantum optical sensor, with diﬀerent types of sensors being characterized by diﬀerent functional dependences of 𝜖(℘) on the parameter of interest, ℘. Previously we used a simple signal-to-noise approach to analyse the precision of the parameter estimation. More generally the lower bound on the variance of an unbiased estimator, maximized over all possible measurements on the output probe, can be found by calculating the quantum Fisher information, H(𝜖). The variance of the estimator, V (℘), is bounded by the Cramer–Rao inequality [16], generalized to the quantum case [17]: V (℘) ≥ (

1 )2 𝜕𝜖 H(𝜖)N 𝜕℘

(10.7)

where N is the number of measurements taken. The Fisher information is the second moment of the classical logarithmic derivative of the likelihood function. The quantum Fisher information is further optimized to coincide with the best measurement strategy allowed by quantum mechanics. The quantum Fisher information can be evaluated via several diﬀerent methods. One often convenient ̂ 𝝈), ̂ between the two density operators 𝝆̂ method is via the quantum ﬁdelity, (𝝆, ̂ where and 𝝈, | [√√ √ ]||2 | = |Tr 𝝆̂ 𝝈̂ 𝝆̂ | | | | |

(10.8)

The quantum ﬁdelity is a well-known quantiﬁcation for the similarity of quantum states. For pure states, the ﬁdelity reduces to the overlap between states: = |⟨𝜓|𝜓 ′ ⟩|2 . The ﬁdelity can be used to calculate the quantum Fisher information via √ 8(1 − (𝝆̂ 𝜖 , 𝝆̂ 𝜖+𝛿𝜖 )) H(𝜖) = lim (10.9) 𝛿𝜖→0 𝛿𝜖 2 where 𝝆̂ 𝜖 is the output state of the probe after the interaction with the beamsplitter, whilst 𝝆̂ 𝜖+𝛿𝜖 is the output state after the interaction given a small perturbation, 𝛿𝜖, to 𝜖. For example, a coherent state probe remains a pure state after transmission through the beamsplitter, so the ﬁdelity is easily calculated as ( 2 2 ) √ √ |𝛼| 𝛿𝜖 ) (𝜌𝜖 , 𝜌𝜖+𝛿𝜖 ) = |⟨ 𝜖𝛼| 𝜖 + 𝛿𝜖𝛼⟩|2 = exp − (10.10) 8𝜖

10.1 Quantum Enhanced Sensors

Substituting Eq. (10.10) into Eq. (10.9) and hence into Eq. (10.7) gives the precision bound for a coherent probe: V (℘) ≥ (

𝜕𝜖 𝜕℘

1 )2 4|𝛼|2 N

(10.11)

Given that the variance of the estimator is inversely proportional to the SNR, we can note that Eq. (10.11) has the same 1∕n scaling as obtained from the signal-to-noise analysis. As we observed earlier a squeezed probe can do better, 2 leading ideally to a 1∕n scaling [18]. By using the general expression for ﬁdelity between Gaussian states (see Section 13.6.2), we can arrive at the same conclusion from the Fisher information analysis. Under more practical conditions, the addition of squeezing to a coherent probe can give a constant improvement factor as can also be shown from the Fisher information analysis. Determining the measurement basis that will saturate the quantum Cramer–Rao bound is more involved, but for Gaussian states it has been shown in general to be a displaced, squeezed number basis [19]. Recently it has been observed that if there is a strong non-linear coupling to 2 the probe, then variance scalings better than the 1∕n can be achieved [20], and an experimental example has been demonstrated [21]. For example, consider the following non-linear beamsplitter relationship: â o =

† † † † 1 1 (1 + ei𝜒∕2(̂a −̂v )(̂a−̂v) ) â + (1 − ei𝜒∕2(̂a −̂v )(̂a−̂v) ) v̂ 2 2

(10.12)

where, as in Eq. (10.1), â represents the probe mode and v̂ a vacuum mode. Here we wish to estimate the parameter 𝜒; however the transmission depends both on 𝜒 and the intensity of the probe in a non-linear way. A signal-to-noise analysis can be carried out by assuming that the ﬂuctuations and the signal are small compared with the average power such that the input mode can be written as ̂ with 𝛼 ≫ 1 and 𝜒 = 𝜒o + 𝛿𝜒 with 𝛿𝜒 ≪ 1 and linearizing the ﬂuctuâ = 𝛼 + 𝜹a ations. This leads to an approximate power spectrum for the ﬂuctuations of the output ﬁeld of 𝜒o 𝜒o 2 2 1 1 Vo = V𝜒 |𝛼|6 + Va |1 + ei 2 |𝛼| (1 + i𝜒o |𝛼|2 )|2 + |1 − ei 2 |𝛼| (1 + i𝜒o |𝛼|2 )|2 4 4 (10.13)

Now suppose we are looking for ﬂuctuations around the point 𝜒o = 0. Under these conditions, with a coherent probe, we ﬁnd SNR = V𝜒 |𝛼|6 ∕16, and so the 3 estimation precision scales as 1∕n , exceeding the Heisenberg scaling. These claims have generated some controversy. In particular it should be noted that: 1. The enhanced precision typically only applies locally, i.e. around the operating point (𝜒o = 0 in our example), and the range in which an enhancement is obtained may be too narrow to be useful [22]. 2. In order to have practical applications, an eﬃcient way to map some physical parameter of interest onto a non-linear parameter, such as 𝜒2 or 𝜒3 , is required. This is a technical challenge.

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Nevertheless, such non-linear devices are an interesting ﬁeld of future research in quantum optical sensors. 10.1.4

Adaptive Phase Estimation

Finally, we note that so far we have only discussed sensors that are measuring small variations starting with a known value. That means we do not need to estimate the full value of the parameter ℘. All we need to record in each time step is the change 𝛿℘(t). In contrast it is possible to consider sensors that estimate parameter values for which we have no previous knowledge, and thus we have to start in each time interval from the same point, e.g. ℘ = 0. For example, in an interferometric sensor we normally consider situations where we track the small changes in the phase 𝛿𝜙(t) of the light and compare it to the value of 𝜙(t − 𝜏) at the previous time step. In contrast one could start with no knowledge of 𝜙 in the range 0 to 2𝜋 in each time step. This task requires an adaptive approach and more resources and photons per time interval than a sensor, which constantly tracks small changes of the phase. A considerable amount of theory work has been done in this ﬁeld, in particular by H. Wiseman, G. Milburn, and H. Mabuchi and their respective teams [23, 24]. In more recent times experiments by a combined Tokyo and Canberra team, led by A. Furusawa and E. Huntington, have explored adaptive optical phase estimation using time-symmetric quantum smoothing [25] and combined with squeezed light [26], and they have veriﬁed the theoretical predictions. It is important to establish the correct quantum limit and minimum number of quantum resources, which are required to achieve the phase estimation. This team later studied the quantum-limited estimation of a mirror position. A detailed study was made to optimize which data should be used in post-processing. The authors concluded that using only data recorded before the measurement was not as eﬃcient as using the same number of data points recorded before and after the timing of the signal [27]. These studies are the beginning of more complex analysis and the design of devices that can be used in more sophisticated optical control situations at and below the QNL. This work has been successful with squeezed light and also with the corresponding experiments using single photons and the corresponding phase estimation algorithms. See, for example, the work at Griﬃths University by G.J. Pryde and co-workers [28], which summarizes the approaches to measurements and estimation and identiﬁes which are the most appropriate quantum states and how best to use them.

10.2 Optical Communication Optical communication has been one of the motivations for the development of quantum optics technology. In the last three decades, there has been a tremendous advance in optical communication technology: low loss ﬁbres, high quantum eﬃciency photodiodes, diode lasers, and laser ampliﬁers have all improved the system performance. The speed has increased relentlessly over the decades by a factor of 10 about every 4 years, as seen in Figure 10.3.

10.2 Optical Communication

1017 y ver 0e 1 × :

1016 1015 Capacity (bit/s)

1014

Capacity limit for current technology

nd Tre

1013 1012

s ear 4y

Space division multiplexing

WDM

1011 High spectral efficiency coding

1010 109 108 107 106 105 1980

EDFA Improved transmission fibres 1990

2000 Year

2010

Figure 10.3 Communication speed and multiplexing. Source: From Richardson et al. [29].

In optical systems the information is encoded on one or more channels or modes of light that are propagated via free space or more commonly via an optical ﬁbre and decoded at the receiver. The attraction of optical systems is the wide modulation bandwidth and the ability to multiplex the wavelengths, which means carrier frequencies, the polarization, and, very soon, spatial modes in the ﬁbre. The communication speed can be measured by bits per second (bits/s), and an overview of the communication speed can be seen in Figure 10.3. We have now systems approaching a speed of 1014 bit/s, which was hard to imagine only a few years ago, and this requires the simultaneous use of many advanced techniques. One trick is multiplexing, which means in our language that information is encoded in a series of orthogonal modes, with minimum crosstalk between the modes. Multiplexing technology is currently one of the major topics of international research and development. Combined with all the tricks used in coherent detection, using a known detection phase reference via a local oscillator, and homodyne detection, the information density and thus also communication bandwidth can be increased further. The technology of the fastest communication systems and quantum optics experiments is remarkably similar [29, 30], and we ﬁnd increasingly an overlap and a demand from the communication industry and engineers. It can be foreseen that we will soon reach the situation where the quantum properties of light become an important limit for the performance of optical communication systems. At that stage squeezed light might play a role in overcoming these limits and in enhancing the performance. In this chapter we want to estimate the size of such possible improvements. In principle, a communication system contains the components shown in Figure 10.4. Diﬀerent encoding and detection schemes are in use, with the simplest and most commonly used one direct incoherent modulation. In this case the transmitter is a diode laser, the

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Data in

Laser

Transmission line Demodulator

Fibre laser amplifier

Modulator

Data out

Coupler

Figure 10.4 Components of an optical communication system without coherent detection.

information is encoded by an amplitude modulator, and the receiver is a simple photodetector. More advanced systems use coherent schemes such as phase shift keying (PSK). This requires a phase modulator in the transmitter and a homodyne, or heterodyne, detector with a local oscillator beam that is phase-locked to the input mode. These systems can operate within a few decibels of the QNL. The question of how non-classical light can improve the performance of the system was the motivation for the early work by H. Takahashi [31] and H.P. Yuen and J.H. Shapiro [32]. The techniques for the encoding, propagating, and decoding of information on a quantum system were reviewed by Y. Yamamoto and H.A. Haus [33] and by C.M. Caves and P.D. Drummond [34]. Non-classical light can be used in several ways to improve components of the system: (i) the laser can produce sub-Poissonian or quadrature squeezed light and thus possibly allow a better SNR. This is particularly attractive and practical when diode lasers are used. (ii) The phase-insensitive laser ampliﬁer can be replaced by a phase-sensitive parametric ampliﬁer. (iii) The beamsplitter, or coupler, can be equipped with a squeezed vacuum input that reduces the noise introduced at this point or alternatively a quantum non-demolition (QND) device can be used [35]. In all cases the same detection system can be used. The overall improvement can be illustrated by considering the amount of information that can be encoded on a mode of light with given maximum optical power. For an incoherent system, where only intensity information is processed, this information content can be estimated by evaluating the number of states with X2

α

δX1

X1

X2

δX1

X1

Figure 10.5 Illustration of the number of distinguishable states for incoherent detection. Top, coherent states; below, squeezed states.

10.2 Optical Communication

fGW

Two rotating neutron stars or similar

Time

Figure 10.6 A typical source for gravitational waves and its eﬀect on space and time. The eﬀects observed by the end of 2017 include colliding black holes and spinning neutron stars.

diﬀerent intensities that can be distinguished. This is illustrated in Figure 10.5, where the sequence of distinguishable coherent states are shown, all with one phase = zero. Each diﬀerent state is represented by an uncertainty area, symmetric circles in the case of coherent states. An incoherent system recognizes diﬀerent intensities; the phase information is ignored. With sub-Poissonian light the uncertainty in the amplitude is smaller, and for given maximum power a larger number of states can be distinguished with the same SNR. Hence more information can be encoded Figure 10.6. A quantitative ﬁgure of merit for a communication system is the channel capacity, a concept that describes the maximum amount of information that can be transmitted based on statistical arguments. The Shannon capacity [36] of a communication channel with Gaussian noise of power (variance) N and Gaussian distributed signal power S operating at the bandwidth limit is ] [ S 1 (10.14) C = log2 1 + 2 N where C is in units of bits per symbol. Equation (10.14) can be used to calculate the channel capacities of quantum states with Gaussian probability distributions such as coherent states and squeezed states. Consider ﬁrst a signal composed of a Gaussian distribution of coherent state amplitudes all with the same quadrature angle. The signal power Vs is given by the variance of the distribution. The noise is given by the intrinsic quantum noise of the coherent states, Vn = 1. Because the quadrature angle of the signal is known, homodyne detection can in principle detect the signal without further penalty, and thus the measured signal to noise is S∕N = Vs ∕Vn = Vs . In general the average photon number per bandwidth per second of a light beam is given by 1 + 1 (V + V − ) − (10.15) 4 2 where V + (V − ) are the variances of the maximum (minimum) quadrature projections of the noise ellipse of the state. These projections are orthogonal quadratures, such as amplitude and phase, and obey the uncertainty principle V + V − ≥ 1. In the above example one quadrature is made up of signal plus n=

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quantum noise such that V + = Vs + 1, whilst the orthogonal quadrature is just quantum noise, so V − = 1. Hence n = 1∕4Vs , and so the channel capacity of a coherent state with single quadrature encoding and homodyne detection is [√ ] Cc = log2 1 + 4n (10.16) Establishing in an experiment that a particular optical mode has, this capacity would involve (i) measuring the quadrature amplitude variances of the beam, V + and V − , (ii) calibrating Alice’s signal variance, and (iii) measuring Bob’s signal to noise. If these measurement agreed with the theoretical conditions above, then Shannon’s theorem tells us that an encoding scheme exists that could realize the channel capacity of Eq. (10.16). An example of such an encoding is given in [37]. For photon numbers n > 2 improved channel capacity can be obtained by encoding symmetrically on both quadratures and detecting both quadratures simultaneously using heterodyne detection or dual homodyne detection. Because of the non-commutation of orthogonal quadratures, there is a penalty for their simultaneous detection that reduces the signal to noise of each quadrature to S∕N = 1∕2Vs . Also because there is signal on both quadratures, the average photon number of the beam is now n = 1∕2Vs . On the other hand the total channel capacity will now be the sum of the two independent channels carried by the two quadratures. Thus the channel capacity for a coherent state with dual quadrature encoding and heterodyne detection is [ ] [ ] S− 1 S+ 1 + log2 1 + Cch = log2 1 + 2 N 2 N (10.17) = log2 [1 + n] which exceeds that of the homodyne technique (Eq. (10.16)) for n > 2. The above channel capacities are the best achievable if we restrict ourselves to a semi-classical treatment of light. However the channel capacity of the homodyne technique can be improved by the use of non-classical squeezed light. With squeezed light the noise variance of the encoded quadrature can be reduced such that Vne < 1, whilst the noise of the unencoded quadrature is increased such that Vnu ≥ 1∕Vne . As a result the signal to noise is improved to S∕N = Vs ∕Vne , whilst the photon number is now given by Eq. (10.15) but with V + = Vs + Vne and V − = 1∕Vne where a pure (i.e. minimum uncertainty) squeezed state has been 2 assumed. Maximizing the signal to noise for ﬁxed n leads to S∕N = 4(n + n ) for a squeezed quadrature variance of Vne,opt = 1∕(1 + 2n). Hence the channel capacity for a squeezed beam with homodyne detection is ] [ Csh = log2 1 + 2n (10.18) which exceeds both coherent homodyne and heterodyne for all values of n. Although squeezing gives an improvement in channel capacity, this improvement is rapidly washed out by nonidealities in the system such as propagation loss and other ineﬃciencies. It still seems some way oﬀ before squeezing might be considered a viable option for increasing channel capacity. In principle, a ﬁnal improvement in channel capacity can be obtained by allowing non-Gaussian states. The absolute maximum channel capacity for a single mode is given by

10.3 Gravitational Wave Detection

the Holevo bound and can be realized by encoding in a maximum entropy ensemble of Fock states and using photon number detection. This ultimate channel capacity is CFock = (1 + n)log2 [(1 + n)] − nlog2 [n]

(10.19)

which is the maximal channel capacity at all values of n. Quantum optics can describe the ultimate limits of data communication, and we can ﬁnd ways to approach this limit. In 2018 such quantum limits ca