Multi-Photon Quantum Interference

Multi-Photon Quantum Interference Zhe-Yu Jeff Ou Multi-Photon Quantum Interference Zhe-Yu Jeff Ou Department of Ph...

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Multi-Photon Quantum Interference

Zhe-Yu Jeff Ou

Multi-Photon Quantum Interference

Zhe-Yu Jeff Ou Department of Physics Indiana University-Purdue University Indianapolis Indianapolis, IN 46202

Library of Congress Control Number: 2007922125 ISBN 978-0-387-25532-3

e-ISBN 978-0-387-25554-5

c 2007 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 4 3 2 1 springer.com

For my parents, who did so much for me.

Preface

Quantum interference, as a fundamental phenomenon of quantum mechanics, is what makes quantum physics different from classical Newtonian physics. Optical interference has played some essential roles in the understanding of light. It has fascinated Dirac, the pioneer of the quantum theory of light, since the very beginning, as seen in his famous statement on photon interference: “Each photon ... interferes only with itself. Interference between different photons never occurs.” Feynman, who was one of the founders of quantum electrodynamics, wrote, in his well-known lecture series on physics, that the interference phenomenon “has in it the heart of quantum mechanics..., it contains the only mystery.” As we explore into the quantum regime in the 21st century, we will find even more presence of quantum interference in our life. Most commonly-occurring interference phenomena are optical interference, in the form of some beautiful interference fringe patterns. These phenomena have been well-studied by the classical theory of coherence, and documented in Born and Wolf’s classic book Principle of Optics. In terms of the language of photon, these phenomena can be categorized as the single-photon interference effect and described by the first part of Dirac’s statement given above. On the other hand, in the situation when more than one photon is involved, the second part of Dirac’s statement is not correct. In this book, we try to understand the phenomena of quantum interference through the multi-photon effects of photon correlation. Our major concern is the temporal correlation among photons and how it influences the interference effect. Because of this, we resort to the multi-frequency description of an optical field. The multi-photon interference effects discussed in this book will find their applications in many of the quantum information protocols, such as, quantum cryptography and quantum state teleportation. However, the emphasis of this book is on the fundamental physical principle in those protocols. Therefore, we will not cover the broad topics of quantum information processing. Nevertheless, readers may find the multi-frequency description of optical fields to be a good complement to the single-mode treatment found in most discussions on quantum information, and closer to a real experimental environment.

VIII

Preface

This book is organized into two parts. The first part deals mainly with the two-photon interference effect. The second part studies the effects of more than two photons. In addition to the interference effects, Chapter 2 is devoted to the generation and the spectral properties of a two-photon state in the process of parametric down-conversion, which is the main photon source for the effects studied in this book. We also investigate the coherence of the multi-photon source in Chapter 7, which is the preparation for Chapters 810. The complementary principle of quantum mechanics is demonstrated in a quantitative fashion in Chapter 9, when we discuss the relation between photon distinguishability and multi-photon interference effects. This book is based on a tutorial lecture series held during the Yellow Mountain Workshop on Quantum Information in 2001. I would like to thank Professor Guang-can Guo of the University of Science and Technology of China for inviting me to the workshop and for his generous support.

Indianapolis, September, 2006

Zhe-Yu Jeff Ou

Contents

Part I Two-Photon Interference 1

Historical Background and General Remarks . . . . . . . . . . . . . . 3 1.1 Interference between Independent Lasers: Magyar-Mandel and Pfleegor-Mandel Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Two-Photon Interpretation of Pfleegor-Mandel Experiment . . . 6 1.3 Two-Photon Interference with Quantum Sources . . . . . . . . . . . . 8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2

Quantum State from Parametric Down-Conversion . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Spontaneous Parametric Down-Conversion Process . . . . . . . . . . 2.3 Phase Matching Condition and Spectral Bandwidth . . . . . . . . . . 2.3.1 Type-I Phase Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Type-II Phase Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Quantum State with a Narrow Band Pump Field . . . . . . . . . . . . 2.5 Quantum State with a Wide Band Pump Field . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 21 27 27 30 34 39 41

3

Hong-Ou-Mandel Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Single-Mode Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Multi-Mode Treatment and Hong-Ou-Mandel Dip . . . . . . . . . . . 3.2.1 Narrow Band Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Wide Band Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Dispersion Cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 A Nonlocal Two-Photon Interference Effect . . . . . . . . . . . . . . . . . 3.4 Photon Bunching in Hong-Ou-Mandel Interferometer . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 43 45 49 52 55 56 59 61

X

4

5

6

Contents

Phase-Independent Two-Photon Interference . . . . . . . . . . . . . . 4.1 Two-Photon Polarization Entanglement . . . . . . . . . . . . . . . . . . . . 4.1.1 Polarization Hong-Ou-Mandel Interferometer and Violation of Bell’s Inequalities . . . . . . . . . . . . . . . . . . . 4.1.2 Photon Anti-Bunching Effect and Bell State Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Two-Photon Frequency Entanglement and Spatial Beating . . . . 4.3 Two-Photon Interference Fringes and Ghost Fringes . . . . . . . . . . 4.3.1 Two-Photon Interference Fringes . . . . . . . . . . . . . . . . . . . . 4.3.2 Spatial Correlation and Ghost Fringes . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase-Dependent Two-Photon Interference: Two-Photon Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Two-Photon Interferometer with Two Down-Converters . . . . . . 5.1.1 Phase Memory by Entanglement with Vacuum . . . . . . . . 5.1.2 Multi-Mode Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Franson Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Entanglement in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Two-Photon Coherence versus One-Photon Coherence: Multi-Mode Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Two-Photon De Broglie Interferometer . . . . . . . . . . . . . . . . . . . . . 5.3.1 Maximally Entangled Photon State – the NOON State . 5.3.2 Detailed Analysis of Two-Photon De Broglie Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63 63 67 70 73 74 78 82 83 83 83 85 89 89 92 95 95 96 99

Interference between a Two-Photon State and a Coherent State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.1 Anti-Bunching by Two-Photon Interference . . . . . . . . . . . . . . . . . 101 6.2 Multi-Mode Analysis I: CW Case . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.3 Multi-Mode Analysis II: Pulsed Case . . . . . . . . . . . . . . . . . . . . . . . 106 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Part II Quantum Interference of More Than Two Photons 7

Coherence and Multiple Pair Production in Parametric Down-Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.1 Coherence Properties of Spontaneous Parametric Down-Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.1.1 Field Correlation Functions and Coherence of Parametric Down-Converted Fields . . . . . . . . . . . . . . . . . . 114 7.1.2 Generation of Transform-Limited Single-Photon Wave Packet by Gated Photon Detection . . . . . . . . . . . . . . . . . . 118

Contents

XI

7.2 Multi-Pair Production and Stimulated Pair Emission . . . . . . . . . 121 7.2.1 Pair Statistics and Photon Bunching . . . . . . . . . . . . . . . . . 122 7.2.2 Stimulated Pair Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.2.3 Induced Coherence without Induced Emission . . . . . . . . . 130 7.3 Distinguishable or Indistinguishable Pairs of Photons . . . . . . . . 132 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 8

Quantum Interference with Two Pairs of Down-Converted Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 8.1 Hong-Ou-Mandel Interferometer for Independent Photons . . . . 137 8.1.1 Two-Photon Interference without Gating . . . . . . . . . . . . . 138 8.1.2 Two-Photon Interference with Gating: HongOu-Mandel Interferometer for Two Independent Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 8.2 Quantum State Teleportation and Swapping . . . . . . . . . . . . . . . . 142 8.2.1 Quantum State Teleportation: Single-Mode Case . . . . . . 143 8.2.2 Quantum State Teleportation: Multi-Mode Case . . . . . . . 145 8.2.3 Entanglement Swapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 8.3 Distinguishing a Genuine Polarization Entangled Four-Photon State from Two Independent EPR Pairs . . . . . . . . . . . . . . . . . . . . 149 8.3.1 Single-Mode Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 8.3.2 Multi-Mode Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 8.4 Hong-Ou-Mandel Interferometer for Two Pairs of Photons . . . . 155 8.4.1 Symmetric Beam Splitter . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 8.4.2 Asymmetric Beam Splitter . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.5 Generation of a NOON State by Superposition . . . . . . . . . . . . . . 163 8.5.1 Interference between a Coherent State and Parametric Down-Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 8.5.2 A Special N-Photon Interference Scheme . . . . . . . . . . . . . 166 8.6 Multi-Photon De Broglie Wavelength by Projection Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 8.6.1 Projection by Asymmetric Beam Splitters . . . . . . . . . . . . 169 8.6.2 NOON State Projection Measurement . . . . . . . . . . . . . . . 172 8.7 Stimulated Emission and Multi-Photon Interference . . . . . . . . . 177 8.8 Remarks on E and A and General Discussion . . . . . . . . . . . . . . . 181 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

9

Temporal Distinguishability of a Multi-Photon State . . . . . . 185 9.1 Hong-Ou-Mandel Interferometer for Characterizing Two-Photon Temporal Distinguishability . . . . . . . . . . . . . . . . . . . 186 9.2 Characterizing an N-Photon State in Time . . . . . . . . . . . . . . . . . 187 9.2.1 General Description of a Multi-Mode N-Photon State . . 188 9.2.2 Direct Photon Detection from Glauber’s Coherence Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

XII

Contents

9.2.3 A NOON-State Projection Measurement and Generalized Hong-Ou-Mandel Interferometer . . . . . . . . . . 195 9.3 The First Example of |1H , 2V . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9.4 The General Case of |1H , NV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 9.5 The General Case of |kH , NV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 9.5.1 General Formula for the Visibility . . . . . . . . . . . . . . . . . . . 206 9.5.2 The Special Cases of |2H , 2V , |2H , 3V , and |2H , 4V . . 206 9.5.3 The Special Case of |3H , 3V . . . . . . . . . . . . . . . . . . . . . . . . 208 9.6 The Scheme for Characterizing the Temporal Distinguishability by an Asymmetric Beam Splitter . . . . . . . . . . 208 9.6.1 The Temporal Distinguishability of |1H , NV . . . . . . . . . . 210 9.6.2 The Case of |2H , NV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 9.7 Experimental Realization of the Cases of |2H , 1V , |2H , 2V with Two Pairs of Down-Converted Photons . . . . . . . . . . . . . . . . 214 9.7.1 Generation of the State of |2H , 1V with Tunable Temporal Distinguishability . . . . . . . . . . . . . . . . . . . . . . . . 214 9.7.2 Distinguishing 4 × 1 Case from 2 × 2 Case for the State of |2H , 2V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 10 Homodyne of a Single-Photon State: A Special Multi-Photon Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 10.1 Interference with a Single-Photon State and an N-Photon State at a Symmetric Beam Splitter . . . . . . . . . . . . . . . . . . . . . . . 225 10.2 Interference of a Single-Photon State and an Arbitrary State . . 228 10.3 Multi-Mode Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 A

Lossless Beam Splitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

B

Derivation of the Visibility for |kH , NV . . . . . . . . . . . . . . . . . . . 245 B.1 The Case of |2H , NV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 B.1.1 The Scenario of 2HmV + (N − m)V . . . . . . . . . . . . . . . . . 245 B.1.2 The Scenario of 1HmV + 1HnV + (N − n − m)V . . . . . 249 B.2 The Case of |3H , NV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 B.2.1 The Scenario of 3HmV + (N − m)V . . . . . . . . . . . . . . . . . 251 B.2.2 The Scenario of 2HmV + 1HnV + (N − m − n)V . . . . . 255 B.2.3 The Scenario of 1HmV + 1HnV + 1HpV + (N −m−n−p)V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 B.3 The General Case of |kH , NV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 B.3.1 The Scenario of k1 V mH + k2 V nH . . . . . . . . . . . . . . . . . . 262 B.3.2 The Most General Scenario . . . . . . . . . . . . . . . . . . . . . . . . . 265

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Part I

Two-Photon Interference

1 Historical Background and General Remarks

For many years, the phenomena of the violations of local realism by photons [1.1, 1.2] and those of photon interference [1.3, 1.4, 1.5] were not intertwined. In the first few experimental demonstrations [1.2] of quantum nonlocality in the sense of the violations of Bell’s inequalities, the EPR-Bohm type [1.6] of polarization singlet state of two photons in the form √ (1.1) |Ψ 12 = (|x1 |y2 − |y1 |x2 )/ 2 is naturally produced from atomic cascade due to angular momentum conservation, where x, y represent the two orthogonal polarizations. In demonstrating the violation of Bell’s inequalities, a two-photon polarization correlation measurement is made. However, even though two-photon polarization interferometry is involved in the process, no one has attempted to interpret the polarization correlation of Eq.(1.1) in terms of two-photon interference until the experiments of Shih-Alley [1.7, 1.8] and Ou-Mandel [1.9], who created a two-photon polarization entangled state in Eq.(1.1) by superposing two fields from parametric down-conversion. Investigation of multi-photon entanglement truly began with the introduction of the two-photon correlation technique in parametric down-conversion by Burnham and Weinberg in 1970 [1.10], where entanglement in various degrees of freedom is demonstrated. These include polarization [1.7, 1.8, 1.9], frequency [1.11], space [1.4, 1.12], phase and momentum [1.13], etc. Many quantum information protocols are based on multi-photon interference effects. To better understand photon entanglement, we start with the simplest case: two-photon interference. Historically, the investigation of two-photon interference began as early as 1967 with the classic Pfleegor-Mandel experiment [1.14, 1.15]. Early studies by Mandel [1.3] emphasized the nonclassical effects in two-photon interference. This emphasis follows naturally the demonstrations of a series of nonclassical effects of light such as photon anti-bunching [1.16], sub-Poissonian photon statistics [1.17], and violation of Cauchy-Schwartz inequality [1.18].

4


In this chapter, we will recall some of the experiments, and their interpretation in the early development of two-photon interference. Then, we will make some general remarks before proceeding to the subject of parametric down conversion.

1.1 Interference between Independent Lasers: Magyar-Mandel and Pfleegor-Mandel Experiments In the early development of coherence theory in quantum optics, one important milestone was the Magyar-Mandel [1.19] experiment that demonstrated the interference effect between two independent lasers. This experiment showed that besides the high brightness, a laser also has a long coherence time. This property is quite different from a thermal source. This interference effect can be well understood with the classical optical coherence theory [1.20]: the second-order amplitude correlation function between the two lasers, Γ12 (τ ) = V1∗ (t)V2 (t + τ ),

(1.2)

is non-zero when the observation time τ is within the long coherence time of the two lasers. However, in terms of the language of photon, this immediately poses a serious challenge to the second part of Dirac’s famous statement on photon interference: each photon interferes only with itself; different photons never interfere with each other [1.21]. The early experiment by Taylor [1.22] where he introduced an extremely weak source of light in Young’s double slit interference experiment, supported the first part of Dirac’s statement. But the light level in the Magyar and Mandel experiment was too high to rule out the possibility of photon interaction in interference. Under this circumstance, Pfleegor and Mandel [1.14, 1.15, 1.23] performed an experiment similar to the one by Taylor but with two independent lasers.

Laser 2

θ ^ κ 1

Laser 1

∆k = k2−k1

^ κ 2

..

x2 x1

Fig. 1.1. The scheme for interference between two independent lasers in Magyar-Mandel and Pfleegor-Mandel experiments.

In the Pfleegor-Mandel experiment, however, due to low light levels, long exposure time well over the coherence time of the lasers is needed and as a result, the fringe pattern constantly moves, due to the random phase diffusion of each laser within the coherence time. It would, therefore, be difficult to

1.1 Interference between Independent Lasers

5

directly observe the fringe pattern at low light levels in the traditional way with just one detector, i.e., the intensity of light varies as a function of the detector position. In order to reveal the interference fringe pattern, Pfleegor and Mandel invented an ingenious method based on intensity correlation, or, more precisely, anti-correlation, as they first described in Ref.[1.14]. Pfleegor and Mandel noticed that with a moving fringe pattern, intensities at locations separated by half the fringe spacing tend to go in opposite directions. At locations with one full fringe spacing, on the other hand, the intensities tend to go in the same direction. These two correlated effects are illustrated in Fig.1.2. So when two detectors are placed half a fringe spacing apart, they should exhibit negative intensity correlation, while at one fringe spacing separation, the two detectors will have positive correlation.

L/2 A

L/2 B

L/2 C

L/2 A

B

D

C L/2

Direction of fringe motion

L/2 D

Fig. 1.2. Intensity correlation at full fringe separation (between points of same letters A,or B, or C, or D) and intensity anti-correlation at half fringe spacing (between A and B or C and D) for a moving fringe pattern.

More quantitatively, it is straightforward to explain the above using the classical wave theory. Let us describe the fields from the two lasers by two plane waves denoted as V1 (r, t) and V2 (r, t) with Vn (r, t) = Aei(kn ·r−ωt+ϕn (t)) (n = 1, 2).

(1.3)

Here, ϕn (t) (n = 1, 2) is the phase of the lasers. It diffuses with a time scale in the order of the coherence time of the lasers. The amplitudes of the lasers are represented by A and, for simplicity, we assume they are the same. When we superpose the two fields, we have the intensity at a certain location r as: 2 I(r, t) = Aei(k1 ·r−ωt+ϕ1 (t)) + Aei(k2 ·r−ωt+ϕ2 (t)) = 2A2 [1 + cos((k1 − k2 ) · r + Δϕ(t))] = 2A2 [1 + cos(2πx/L + Δϕ(t))],

(1.4)

where L = λ/θ is the fringe spacing with θ as the small angle between k1 and k2 . x is the distance along the direction of k1 − k2 . Δϕ(t) = ϕ1 (t) − ϕ2 (t) is the phase difference between the two independent lasers. It is constant in a short time scale, as in the experiment of Magyar and Mandel [1.19]. But in a long time scale ( >> coherence time), it is randomly fluctuating due to the independent phase diffusions of the two lasers, as in the experiment of Pfleegor and Mandel [1.14, 1.15, 1.23].

6


Therefore, for long time observation, there is no fringe: I(r, t)Δϕ = 2A2 . But for the intensity correlation at two locations, we have: (2)

G12 = I(x1 )I(x2 )Δϕ = 4A4 [1 + 0.5 cos 2π(x1 − x2 )/L],

(1.5)

which shows a modulation despite the fluctuations of Δϕ. The normalized intensity correlation function is given by: (2)

g12 =

I(x1 )I(x2 ) = 1 + 0.5 cos 2π(x1 − x2 )/L. I(x1 )I(x2 )

(1.6)

(2)

More specifically, for x1 − x2 = L/2, we have g12 = 0.5 < 1, which gives rise to the anti-correlation effect. Notice that in Eq.(1.6), the relative depth of modulation of the fringe pattern, namely the visibility, is only 50%. This is actually a general conclusion, which we will prove for classical fields later in Sect.1.4. Pfleegor and Mandel observed the above predicted intensity correlation effects with a light level so low that on average, a photon is detected well before the next one is emitted from the lasers. This seems to support the second part of the Dirac statement, i.e., different photons never interfere. However, Dirac’s statement on photon interference is too crude to account for details such as the 50% visibility in Eq.(1.5). In the following section, we will see how Dirac’s statement can be modified to suit the Pfleegor-Mandel experiment and provide a quantitative explanation in the language of twophoton correlation.

1.2 Two-Photon Interpretation of Pfleegor-Mandel Experiment The major difference between the Magyar-Mandel and Pfleegor-Mandel experiments is that the former records interference fringe in intensity, whereas the observed interference effect in the latter case is in intensity correlation with two detectors. To measure intensity of a field, we need only one detector which responds to single photons. On the other hand, in intensity correlation, the measurement device will produce a signal only when there are two photons, one at each detector. So intensity measurement registers single-photon events, whereas intensity correlation measures two-photon events (Fig. 1.3). With this picture in mind, we find that Dirac’s statement on single-photon interference still applies to the Magyar-Mandel experiment. But for twophoton detection in the Pfleegor-Mandel experiment, it is not appropriate. We need to modify Dirac’s statement to suit two-photon detection as follows: In interference involving two−photon detection, a pair of photons only interferes with the pair itself .

1.2 Two-Photon Interpretation of Pfleegor-Mandel Experiment

(a)

Coincidence Unit

(b)

7

Fig. 1.3. The difference between intensity measurement and intensity correlation measurement: (a) single-photon will produce a signal out (b) Only two photons registered at each of the two detectors will produce a coincidence signal.

Here, the pair and the pair itself correspond to two parts of a wave that is associated only with two photons. The two parts are the two indistinguishable paths for the two photons together. Thus, the interference is between the two possibilities. The special wave that is related only to two photons is different from the traditional waves that we usually encounter in a Michelson interferometer, which is associated only with one photon, since only one detector is involved. As we will see in Chapt.5, the two kinds of waves have different coherence times. Thus, in order to separate these two different waves, we refer to them as “two-photon wave” and “single-photon wave”, respectively. The above picture for two-photon interference was discussed briefly by Glauber [1.24], with an equivalent view of interference of two-photon amplitude. The generalization to an N -photon case is straightforward using the N -photon wave packet concept. Next, we will attempt to interpret the Pfleegor-Mandel experiment with the modified Dirac statement, or the two-photon wave concept. Consider the situation depicted in Fig.1.4, where two photons are detected with one at each detector. There are four possibilities for the two photons that produce the two-photon coincidence signal: the two photons are both from one of the two lasers in cases (A) and (B) or one from each laser in cases (C) and (D). The first two cases do not produce interference because of the phase diffusion (fluctuation of Δϕ in a time interval much longer than the coherence time of the lasers). Later in Chapter 5, we will see a situation when cases (A) and (B) do produce interference. But here, cases (A) and (B) will simply add a constant and raise the baseline. Cases (C) and (D) are indistinguishable and correspond to the cases of a pair of photons and the pair itself in the modified Dirac statement. So they will produce interference and give the fringe pattern in two-photon coincidence. Because of the randomness of photon statistics of a laser, all four cases have equal chances if the two lasers have the same intensity. So, if we denote A4 as the two-photon probability from one laser, we can write the intensity correlation function from the above discussion as (2)

G12 = A4 + A4 + 2A2 A2 [1 + cos 2π(x1 − x2 )/L],

(1.7)

8


(A)

(B)

(C)

(D)

Fig. 1.4. Four possible origins for the two photons detected by two detectors: no interference between (A) and (B) but 100% visibility interference between (C) and (D).

where the first two terms are from cases (A) and (B) and the last term is from the two-photon interference of cases (C) and (D). Eq.(1.7) is exactly same as Eq.(1.5), derived from classical wave theory. From the above argument leading to Eq.(1.7), we find that the reason for 50% visibility in Eq.(1.7) is due to the existence of cases (A) and (B), i.e., some nonzero two-photon probability from one source only. As a matter of fact, for any classical source, the two-photon probability P2 is always greater than or equal to the square of one-photon probability or the so-called accidental twophoton probability P12 . For coherent state from a laser, we have P2 = P12 as described above in the Pfleegor-Mandel experiment. On the other hand, for a thermal source, we have P2 = 2P12 , which leads to the so-called photon bunching effect. With this source, cases (A) and (B) are twice as probable as cases (C) and (D). This will give rise to a visibility of 1/3, from the argument above [1.3, 1.25]. To obtain a visibility over 50%, we must consider quantum sources with sub-Poissonian photon statistics or the anti-bunching effect. For example, if both fields are in single-photon state, the probabilities for cases (A) and (B) are simply zero. This leads to (2)

G12 = 2A4 [1 + cos 2π(x1 − x2 )/L],

(1.8)

which shows a visibility of 100% in two-photon interference. Next we will consider two-photon interference with general sources and treat optical fields and optical detection in a quantum mechanical fashion. We will show rigorously that the upper bound of the visibility for classical states is 50% and certain quantum sources with photon anti-bunching effect can produce a visibility exceeding this classical limit.

1.3 Two-Photon Interference with Quantum Sources Two-photon interference with quantum light was first studied by Fano [1.26], who used a complicated QED treatment for the problem of detection by two atom-detectors of the light emitted from two other simultaneously excited

1.3 Two-Photon Interference with Quantum Sources

9

atoms. Fano showed for the first time that it is possible to achieve 100% visibility for the interference effect in intensity correlation. This was also mentioned briefly by Richter [1.27] with regard to a similar problem. But it was Mandel [1.3] who first pointed out the difference between the predictions of quantum and classical theories on the visibility of two-photon interference. Although the early theoretical quantum treatments of this problem were applied to light emitted from two simultaneously excited atoms, the parametric down-conversion process proved to be more suitable for an experimental demonstration of the quantum nature of two-photon interference. Two-photon interference in the parametric down-conversion process was first analyzed by Ghosh et al. [1.28], and the subsequent experiments by Ghosh and Mandel [1.4] and by Hong, Ou, and Mandel [1.5] demonstrated two-photon interference with a visibility of more than 50%. In this section, we will show that the general classical limit is 50% for the visibility of two-photon interference and derive the necessary condition for the fields to have a nonclassical two-photon interference effect with visibility larger than 50%. Then we will examine a couple of special cases. We start with a quantum description of a free optical field [1.29] by the positive frequency part of an electric field operator: 1 ˆ (+) (r, t) = E ˆk ei(k·r−ωt) . (1.9) d3 k a (2π)3/2 Here, we treat the field as quasi-monochromatic and assume that all the modes in the field have the same polarization so that we can treat them as scalar fields. In the interference of two optical fields, we further assume that most of the modes in the integral are in the vacuum state and only modes in two directions with unit vectors κ ˆ 1 and κ ˆ 2 are excited (see Fig. 1.1). For simplicity, we omit all the unoccupied modes and re-write the field operator in Eq.(1.9) in one-dimensional form: ˆ (+) (r, t) + Eˆ (+) (r, t), ˆ (+) (r, t) = E E 1 2 with ˆ (+) (r, t) = √1 E n 2π

dωˆ an (ω)ei(kn ·r−ωt) (n = 1, 2),

(1.10)

(1.11)

where kn = κ ˆ n ω/c. In order to produce macroscopic fringe pattern, we assume the angle θ between the two directions κ ˆ 1 and κ ˆ 2 is small, i.e., cos θ ≡ κ ˆ1 · κ ˆ2 ≈ 1. Let the bandwidths of the two interfering fields (characterized by directions κ ˆ 1 and κ ˆ 2 ) be Δω1 and Δω2 , respectively. We assume that the two fields are quasi-monochromatic and have the same center frequency, that is, ω10 = ω20 >> Δω1 and Δω2 . We now evaluate the joint probability of detecting one photon at position r and at time t and the other at r at t + τ . This probability P2 (r, t; r , t + τ ) is given by

10


P2 (r, t; r , t + τ ) ∝ Eˆ (−) (r, t)Eˆ (−) (r , t + τ )Eˆ (+) (r , t + τ )Eˆ (+) (r, t) ˆ t)I(r ˆ , t + τ ) :, = T : I(r, (1.12) ˆ (+) ]† is the where T is time ordering and :: is normal ordering. Eˆ (−) = [E negative frequency part of the field operator. Before we proceed, let us now introduce the assumption made earlier that there is no interference effect in the field intensity (single-photon detection) so that we concentrate only on a genuine two-photon effect. The easiest way to achieve this is to assume that the two fields have independent phase fluctuations. We then find, after substituting Eq.(1.10) into Eq.(1.12) and making (−) (−) (+) (+) expansion of the product, that all unpaired terms like Eˆ1 Eˆ1 Eˆ2 Eˆ2 , (−) (−) (+) (+) ˆ E ˆ E ˆ ,etc. vanish. Only six terms survive and Eq.(1.12) becomes: ˆ E E 1 2 2 2 P2 (r, t; r , t + τ ) ∝ T : Iˆ1 (r, t)Iˆ1 (r , t + τ ) : + T : Iˆ2 (r , t + τ )Iˆ2 (r, t) :+ +T : Iˆ1 (r, t)Iˆ2 (r , t + τ ) : + T : Iˆ1 (r , t + τ )Iˆ2 (r, t) :+ ˆ (−) (r, t)Eˆ (−) (r , t + τ )Eˆ (+) (r , t + τ )Eˆ (+) (r, t)+ +E 1 2 1 2 ˆ (−) (r, t)Eˆ (−) (r , t + τ )Eˆ (+) (r , t + τ )Eˆ (+) (r, t). +E 2

1

1

2

(1.13)

It is obvious from the above expression that the first two terms are the contribution from cases (A) and (B) in Fig.1.4, i.e., two photons are all from the field of one direction, and the rest correspond to one photon from each field, as in cases (C) and (D). For the nearly parallel plane waves described by Eq.(1.12) and the nearly perpendicular observation plane where the detectors are located (see Fig.1.1 for detailed geometry), the first four terms in Eq.(1.13) are independent of r and r and are always positive. Actually, they are the auto-correlation and cross correlation of the corresponding fields, that is, ⎧ ⎪ T : Iˆ1 (r, t)Iˆ1 (r , t + τ ) : = Iˆ1 (t)Iˆ1 (t + τ )[1 + λ1 (t, τ )], ⎪ ⎨ T : Iˆ2 (r , t + τ )Iˆ2 (r, t) : = Iˆ2 (t)Iˆ2 (t + τ )[1 + λ2 (t, τ )], (1.14) ⎪ T : Iˆ1 (r, t)Iˆ2 (r , t + τ ) : = Iˆ1 (t)Iˆ2 (t + τ )[1 + λ12 (t, τ )], ⎪ ⎩ T : Iˆ1 (r , t + τ )Iˆ2 (r, t) : = Iˆ2 (t)Iˆ1 (t + τ )[1 + λ21 (t, τ )], where λ-functions are independent of t for stationary fields. λn (0)(n = 1, 2) is non-negative for classical fields but can be −1 for some quantum fields [1.30]. The last two terms in Eq.(1.13) involve mixed amplitudes of the two fields at positions r and r and change rapidly with r and r . These are the interference terms which give rise to modulation. Hence, the relative magnitude of these two terms, compared with the first four terms in Eq.(1.13), will determine the relative depth of modulation or the visibility of the interference pattern. To evaluate their relative magnitude, we consider the operator ˆ (+) (r , t + τ )Eˆ (+) (r, t)eiϕ , ˆ≡E ˆ (+) (r , t + τ )Eˆ (+) (r, t) − E O 1 2 2 1

(1.15)


11

where ϕ is an arbitrary phase. We then construct the non-negative quantity ˆ † O ˆ ≥ 0, O

(1.16)

where the average is on an arbitrary quantum state of the fields. Substituting Eq.(1.15) into Eq.(1.16) and expanding the product, we obtain: T : Iˆ1 (r, t)Iˆ1 (r , t + τ ) : + T : Iˆ2 (r , t + τ )Iˆ2 (r, t) : ˆ (−) (r, t)Eˆ (−) (r , t + τ )Eˆ (+) (r , t + τ )Eˆ (+) (r, t)e−iϕ + c.c. ≥ E 1 2 2

(1.17)

Since ϕ is arbitrary, we can choose it to be 0 or π. Then Eq.(1.17) becomes: ˆ ˆ ˆ T : Iˆ1 (r, t)I2 (r , t + τ ) : + T : I1 (r , t + τ )I2 (r, t) : ˆ (−) (−) (+) (+) ≥ E1 (r, t)Eˆ2 (r , t + τ )Eˆ1 (r , t + τ )Eˆ2 (r, t) + c.c..

(1.18)

This inequality ensures that the joint probability P2 is non-negative for any state of light. Next we will prove, for classical fields only, that the first two terms in Eq.(1.13) are larger than or equal to the last two interference terms, that is, ˆ ˆ ˆ T : Iˆ1 (r, t)I1 (r , t + τ ) :c + T : I2 (r , t + τ )I2 (r, t) :c ˆ (−) (−) (+) (+) ≥ E1 (r, t)Eˆ2 (r , t + τ )Eˆ1 (r , t + τ )Eˆ2 (r, t) + c.c.,

(1.19)

where the subscript c indicates that the quantum average is only over classical states. To prove Eq.(1.19), we write an arbitrary quantum state described by a density operator in the Glauber-Sudarshan P-representation [1.31, 1.32]: ρˆ = d{α}P ({α})|{α}{α}|, (1.20) where {α} is a set of variables covering all the excited modes of the fields and P ({α}) satisfies the normalization relation: d{α}P ({α}) = 1. (1.21) In general, P ({α}) may be negative. But classical fields are those with wellbehaved and non-negative P ({α}) so that P ({α}) can be treated as a true probability distribution. We now use Eq.(1.20) to express the first two terms in Eq.(1.13) as ˆ ˆ T : I1 (r, t)I1 (r , t + τ ) : = d{α}P ({α})|E1 (r, t)|2 |E1 (r , t + τ )|2 (1.22) T : Iˆ2 (r , t + τ )Iˆ2 (r, t) : = d{α}P ({α})|E2 (r, t)|2 |E2 (r , t + τ )|2 (1.23) and the last interference terms as

12

1 Historical Background and General Remarks (−) ˆ (−) ˆ (+) ˆ (+) Eˆ1 (r, t)E2 (r , t + τ )E1 (r , t + τ )E2 (r, t) = d{α}P ({α})E1∗ (r, t)E2∗ (r , t + τ )E2 (r, t)E1 (r , t + τ ),

where

1 En (r, t) = √ 2π

(1.24)

dωαn (ω)ei(kn ·r−ωt) .

(1.25)

Consider the following quantity with an arbitrary phase ϕ: O ≡ E1 (r, t)E1∗ (r , t + τ ) − E2 (r, t)E2∗ (r , t + τ )eiϕ .

(1.26)

|O|2 = O∗ O ≥ 0.

(1.27)

Obviously,

So, we have |E1 (r, t)|2 |E1∗ (r , t + τ )|2 + |E2 (r, t)|2 |E2∗ (r , t + τ )|2 ≥ E1∗ (r, t)E2∗ (r , t + τ )E2 (r, t)E1 (r , t + τ )eiϕ + c.c.

(1.28)

Since P ({α}) ≥ 0 for classical fields, we can multiply it to the above inequality and integrate over {α}. We then obtain an inequality exactly the same as Eq.(1.19), if we choose ϕ = 0 or π. Therefore, we may conclude generally from Eqs.(1.18, 1.19) that classical fields can only give rise to two-photon interference with a maximum visibility of 50%, whereas quantum fields may achieve 100% visibility. We have already seen one example of this in Sect.1.2 [Eq.(1.8)]. From inequalities (1.18) and (1.19), we may obtain a necessary condition for the nonclassical effect to occur in two-photon interference as ˆ ˆ ˆ T : Iˆ1 (r, t)I1 (r , t + τ ) : + T : I2 (r , t + τ )I2 (r, t) : ˆ (−) (−) (+) (+) < E1 (r, t)Eˆ2 (r , t + τ )Eˆ1 (r , t + τ )Eˆ2 (r, t) + c.c. , (1.29) M

where the subscript M stands for the maximum value. The reason for using the maximum value in the inequality (1.29) is that the interference terms on the right side of Eq.(1.19) modulate as r and r change. With the inequality in Eq.(1.18), we can write the necessary condition in Eq.(1.29) in a different form as T : Iˆ1 (r, t)Iˆ1 (r , t + τ ) : + T : Iˆ2 (r , t + τ )Iˆ2 (r, t) : < T : Iˆ1 (r, t)Iˆ2 (r , t + τ ) : + T : Iˆ1 (r , t + τ )Iˆ2 (r.t) :.

(1.30)

By using Eqs.(1.14) for stationary fields in the expression above, we can further simplify the necessary condition to Iˆ1 2 [1 + λ1 (τ )] + Iˆ2 2 [1 + λ2 (τ )] < Iˆ1 Iˆ2 [2 + λ12 (τ ) + λ21 (τ )]. (1.31)


13

If λn (τ ) = −1(n = 1, 2), the left hand side of the above expression has a minimum value of (1.32) Iˆ1 Iˆ2 [1 + λ1 (τ )][1 + λ2 (τ )], when the intensities Iˆ1 , Iˆ2 satisfy Iˆ1 2 [1 + λ1 (τ )] = Iˆ2 2 [1 + λ2 (τ )]. Using this minimum value in Eq.(1.31), we arrive at the necessary condition in its simplest form:

[1 + λ1 (τ )][1 + λ2 (τ )] < 1 +

λ12 (τ ) + λ21 (τ ) . 2

(1.33)

For the special case of λn (τ ) = −1 for any of the two fields (n = 1 or 2), the condition in Eq.(1.31) is always satisfied if we allow the intensity to be large enough for the field with λn (τ ) = −1(n =1 or 2). In this case, the necessary condition (1.33) is certainly satisfied. So, the condition in Eq.(1.33) is a more general necessary condition for any two fields to exhibit a nonclassical twophoton interference effect of larger than 50% visibility. The above-mentioned special case with, say, λ1 (τ ) = −1 presents an interesting phenomenon in two-photon interference. If we set Iˆ1 >> Iˆ2 , i.e., one field is much stronger than the other field, the condition in Eq.(1.31) is well satisfied and we will have a visibility larger than 50% in two-photon interference. This is in sharp contrast to the same situation in one-photon interference (fringe pattern is exhibited in intensity), where if one field is dominant, the other field can hardly disturb the final intensity distribution, resulting in nearly zero visibility. As a matter of fact, the visibility of two-photon interference is nearly 100% in the above case. To see the physical picture of this, let us go back to Fig.1.4. When λ1 (τ ) = −1 or Iˆ1 Iˆ1 = 0, there is no contribution from case (A). Since the contribution from case (B) is proportional to Iˆ2 2 and those from case C and D are proportional to Iˆ1 Iˆ2 , the contributions to two-photon coincidence from (C) and (D) will be much larger than (A) and (B) if Iˆ1 >> Iˆ2 . Therefore, we obtain nearly 100% visibility, according to the discussion in Sect.1.2. Two-photon interference of nearly 100% visibility with Iˆ1 >> Iˆ2 was demonstrated by Ou and Mandel [1.12]. In a real experiment, we measure the joint two-photon detection probability with some finite time resolution. So, the coincidence count registered in the lab will be dt dτ P2 (r, t; r , t + τ ), (1.34) Nc ∝ T

TR

where T is the total time of data-taking for the stationary cw case or the duration of the field for the non-stationary pulsed case, and TR is the coincidence window (time) for the two photoelectric pulses after the detection of the two photons. TR is normally limited by the detectors’ resolving time.

14


In the case where the detectors are fast enough so that we may choose a small TR within which P2 (r, t; r , t + τ ) ≈ P2 (r, t; r , t), i.e., we can set τ = 0 in all the expressions from Eq.(1.13) to Eq.(1.33), in particular, Eq.(1.33) becomes [1 + λ1 (0)][1 + λ2 (0)] < 1 + λ12 (0). (1.35) Note that if we use Eq.(1.14), Eq.(1.35) is just the opposite of the following Schwartz inequality for classical fields: I12 I22 ≥ I1 I2 2 .

(1.36)

So, the necessary condition for having over 50% visibility in two-photon interference is that the two interfering fields must be nonclassical fields that violate the Schwartz inequality in Eq.(1.36). In order to show that some quantum sources can give rise to 100% visibility in two-photon interference, let us consider a two-photon quantum state of the form |Ψ = |1k1 |1k2 ,

(1.37)

that is, one photon from each direction. The contributions from cases (A) and (B) in Fig.1.4 are zero because there is simply no two-photon event from field 1 or 2 alone in the quantum state depicted in Eq.(1.37). From the simple picture of Sect.1.3, we should expect 100% visibility in two-photon interference. We will confirm this by calculation. Because only one mode is excited (occupied) in field 1 or 2, we can simplify Eq.(1.10) as ˆ (+) = a E ˆk1 ei(k1 ·r−ωt) + a ˆk2 ei(k2 ·r−ωt) .

(1.38)

The change from continuous mode integral in Eq.(1.10) to discrete sum here √ will take away the 1/ 2π coefficient in Eq.(1.10). We can easily find the intensity I(r) ≡ Eˆ (−) Eˆ (+) as I(r) = 2.

(1.39)

So there is no interference effect in intensity. This is because the photon number Fock state in Eq.(1.37) does not have definite phases for the two fields. For intensity correlation, however, we have: Eˆ (−) (r1 , t)Eˆ (−) (r2 , t + τ )Eˆ (+) (r2 , t + τ )Eˆ (+) (r1 , t) 2 = ei(k1 ·r1 −ωt) ei[k2 ·r2 −ω(t+τ )] + ei[k1 ·r2 −ω(t+τ )] ei(k2 ·r1 −ωt)

= 2 1 + cos(k1 − k2 ) · (r1 − r2 ) (1.40) = 2[1 + cos 2π(x1 − x2 )/L], where x1 , x2 are the coordinates along the direction of k1 − k2 and L ≡ 1/2π|k1 −k2 | = λ/θ (θ is the small angle between k1 and k2 ). Indeed, Eq.(1.40) shows two-photon interference with 100% visibility.

References

15

The interference phenomena that we have discussed above only involve parameters of the position of the detectors. They are the most commonly seen interference phenomena. The other kind of interference phenomena, known as beating, are associated with time. As is well-known, the observation of beating effects requires a fast detection system. However, recent experiments on spatial beating [1.11, 1.33] have shown that this is not necessarily so. The phenomenon of spatial beating, which exhibits beat note in space domain, only exists in two-photon interference and will be studied in detail in Sect.4.2. Notice that the interference fringe exhibited in Eq.(1.40) does not depend on the phases of each field. This can be understood again with the picture in Fig.1.4 where the two interfering paths of cases (C) and (D) are overlapping, so that any phase change in one field will influence both paths. When the two interfering paths can be separated, two-photon interference phenomena will depend on the phases of the interfering fields [1.34, 1.35, 1.36, 1.37, 1.38]. The first experimental demonstration of phase-dependent two-photon interference was carried out by Ou et al. [1.39]. This will be the subject of Chapt.5 and Chapt.6. The quantum state in Eq.(1.37) can be produced in many ways. Early proposals are based on two atoms simultaneously excited [1.3, 1.26, 1.27]. Atomic cascade can also produce two photons of different frequencies [1.18]. Others are based on single photon on-demand [1.40]. However, none of these is as easy as the parametric down-conversion process. It can produce a twophoton state with entanglement in many degrees of freedom. This book is devoted to the understanding and applications of multi-photon interference with the parametric down-conversion process.

References 1.1 1.2 1.3 1.4 1.5 1.6 1.7

1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15

J. S. Bell, Physics (N. Y.) 1, 195 (1965). J. F. Clauser and A. Shimony, Rep. Prog. Phys. 41, 1881 (1978). L. Mandel, Phys. Rev. A28, 929 (1983). R. Ghosh and L. Mandel, Phys. Rev. Lett. 59, 1903 (1987). C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987). D. Bohm, Quantum Theory (Prentice Hall, Englewood Cliffs, N. J., 1951). C. O. Alley and Y. H. Shih, Proceedings of the Second International Symposium on Foundations of Quantum Mechanics in the Light of New Technology, edited by M. Namiki et al. (Physical Society of Japan, Tokyo, 1987). Y. H. Shih and C. O. Alley, Phys. Rev. Lett. 61, 2921 (1988). Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 61, 50 (1988). D. C. Burnham and D. L. Weinberg, Phys. Rev. Lett. 25, 84 (1970). Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 61, 54 (1988). Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 62, 2941 (1989). J. G. Rarity and P. R. Tapster, Phys. Rev. Lett. 64, 2495 (1990). R. L. Pfleegor and L. Mandel, Phys. Lett. 24A, 766 (1967). R. L. Pfleegor and L. Mandel, Phys. Rev. 159, 1084 (1967).

16 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24

1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.40

1 Historical Background and General Remarks H. J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev. Lett. 39, 691 (1977). R. Short and L. Mandel, Phys. Rev. Lett. 51, 384 (1983). J. F. Clauser, Phys. Rev. D 9, 853 (1974). G. Magyar and L. Mandel, Nature (London) 198, 255 (1963). M. Born and E. Wolf, Principle of Optics, (Pergamon, Oxford, 1st ed., 1959; 7th ed., 1999). P. A. M. Dirac, The Principles of Quantum Mechanics (Clarendon, Oxford, 1st ed., 1930; 5th ed., 1958). G. I. Taylor, Proc. Camb. Phil. Soc. 15, 114 (1909). R. L. Pfleegor and L. Mandel, J. Opt. Soc. Am. 58, 946 (1968). R. J. Glauber, Quantum Optics and Electronics (Les Houches Lectures), p.63, edited by C. deWitt, A. Blandin, and C. Cohen-Tannoudji (Gordon and Breach, New York, 1965). S. J. Kuo, D. T. Smithey, and M. G. Raymer, Phys. Rev. A43, 4083 (1991). U. Fano, Am. J. Phys. 29, 539 (1961). G. Richter, Abh. Acad. Wiss. DDR, 7N, 245 (1977). R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. A34, 3962 (1986). L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, New York, 1995). R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1st ed., 1973; 3rd ed., 2000). R. J. Glauber, Phys. Rev. 130, 2529 (1963); Phys. Rev. 131, 2766 (1963). E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963). Z. Y. Ou, E. C. Gage, B. E. Magill, and L. Mandel, Opt. Comm. 69, 1 (1988). P. Grangier, M. J. Potasek, and B. Yurke, Phys. Rev. A 38, 3132 (1988). B. J. Oliver and C. R. Stroud, Jr., Phys. Lett. 135A, 407 (1989). Z. Y. Ou, L. J. Wang, and L. Mandel, Phys. Rev. A 40, 1428 (1989). J. D. Franson, Phys. Rev. Lett. 62, 2205 (1989). M. A. Horne, A. Shimony, and A. Zeilinger, Phys. Rev. Lett. 62, 2209 (1989). Z. Y. Ou, L. J. Wang, X. Y. Zou, and L. Mandel, Phys. Rev. A 41, 566 (1990). C. Santori, D. Fattal, J. Vukovi, G. S. Solomon, and Y. Yamamoto, Nature 419, 594 (2002).

2 Quantum State from Parametric Down-Conversion

As we discussed in Sect.1.3, the situation when λn (0) = −1(n = 1, 2) gives the largest quantum effect in two-photon interference. We showed briefly in Sect.1.3 that the quantum state |1k1 , 1k2 gives a visibility of 100% in twophoton interference. This is not a surprise because λn (0) = −1(n = 1, 2) for this state. In this chapter, we will see how to generate a quantum state of the form |1a , 1b with a, b representing various kinds of modes of light fields such as polarization and frequency. This will also give rise to two-photon entanglement of various degrees of freedom.

2.1 Introduction The parametric process was initially developed in radio wave and microwave as low noise amplifiers, well before the invention of the laser [2.1]. With the emergence of nonlinear optics, most of the concepts and even the terminologies have been adapted in the optical range of electromagnetic wave. Parametric amplification was first demonstrated in the optical region by Giordmaine and Miller [2.2]. It soon became an important technique for generating intense, coherent, and tunable radiation. Usually, the amplified input field (labelled as “signal”) is accompanied by another field (called “idler”), which is required by the principle of quantum mechanics to preserve the commutation relation. The energy is from another field or fields (labelled as “pump”). In the optical range, there are two basic processes that can give rise to the parametric gain. They are three-wave mixing and four-wave mixing. Both of them are well studied in nonlinear optics with classical wave theory [2.3]. Quantum mechanically, the parametric process is described simply by the interaction Hamiltonian with the signal and idler fields in single-mode: ˆ I = i¯hχˆ H a†s a ˆ†i + H.c.,

(2.1)

18

2 Quantum State from Parametric Down-Conversion

where a ˆs(i) is the annihilation operator for the signal (idler) field and χ is a parameter related to the pump fields (which are usually treated as classical fields described by numbers). In the Heisenberg picture, the evolution of the annihilation operator can be solved, and follows the Bogoliubov transformation: a ˆs (t) = (μˆ as + νˆ a†i )e−iωt , (2.2) a î (t) = (μˆ ai + νˆ a†s )e−iωt , with μ = cosh |χ|t,

ν = (χ/|χ|) sinh |χ|t.

(2.3)

The above expressions describe the parametric amplification process with an a†s in the second line of Eq.(2.2)] gives amplitude gain of μ. The νˆ a†i term [or νˆ rise to the accompanying “idler” field. In the interaction picture, the quantum state evolves as ˆ (t)|Ψ (0) |Ψ (t) = U

(2.4)

ˆ = exp(−iH ˆ I t/¯h) = exp(ηˆ U a†s a ˆ†i − H.c.),

(2.5)

with

where η ≡ χt. This is the two-mode squeezed state [2.4] and for the generation of large squeezing, it is normally operated in the regime of |η| >> 1. This is the high gain regime of parametric amplifier. However, to generate the twophoton state, we work in the regime of low gain, with μ ≈ 1 or |η| 1/L|Δk |, the sinc-function dominates in the integral in Eq.(3.47) and can be approximated by a δ-function. Eq.(3.47) then becomes: 4 −1 V(Δz) = 2Γ (1/4) dye−y e−iy(σΔz/c) , (3.48)

1.0 0.8 0.6 0.4 0.2 0.0 −10

−5

0

σ ∆ z/c (a)

5

10

Normalized Coincidence

Normalized Coincidence

with σ = 2 |Δk σp /ko |. Γ (1/4) is the Euler Gamma function. Fig.3.7a shows the Hong-Ou-Mandel dip for this case. On the other hand, if σp > 1/L|Δk |; (b) Type-II, (i) σp /2Ω+ > 1, M . The extra normalization factor of 2 in the approximated expressions in Eqs.(10.11, 10.12) is because P (M ) = 0 for every other value of M . For a coherent state, Pnin = n ¯ n e−¯n /n!, with n ¯ being the average photon number, and we have, from Eqs.(10.9, 10.10), that n) P0 (M ) = e−¯n IM (¯ 2 M −¯n e IM (¯ P1 (M ) = n), n ¯

(10.13) (10.14)

230

10 Homodyne of a Single-Photon State: A Special Multi-Photon Interference

where IM (¯ n) is the Bessel function with a purely imaginary argument. Similar results as Eqs.(10.13, 10.14) were obtained in Refs.[10.2, 10.5]. For large n ¯, 2 1 e−M /2¯n , P0 (M ) ≈ √ 2¯ nπ 1 M 2 −M 2 /2¯n e P1 (M ) ≈ √ , ¯ 2¯ nπ n

(10.15) (10.16)

which has the same form as Eqs.(10.11, 10.12) for large N besides the factor of 2. This is not surprising if we consider the fact that when the photon number is large, the interference scheme discussed above becomes the homodyne detection scheme. Since both a vacuum state and a single-photon state have random phase distribution, homodyne detection with an N -photon state (N >> 1) and coherent state as local oscillators are equivalent. As a matter of fact, the output photon distributions will always have the form of Eqs.(10.15, 10.16) for any state as local oscillator, provided that its average photon number is large and the photon number fluctuation is much less than the average photonnumber ( Δn2 1,

(10.17)

and, similarly, M 2 /¯ n −M 2 /2¯n P1 (M ) ≈ √ e 2¯ nπ

when n ¯ >> 1.

(10.18)

We can also understand this result from the fact that any fluctuation in local oscillator is canceled in a balanced homodyne detection scheme [10.3]. Furthermore, if we set n ¯ → ∞, we √ can replace the discrete variable M with a continuous one defined by x = M/ n ¯ and the probability distributions in Eqs.(10.17, 10.18) lead to probability densities of continuous variable x as 2 1 P0 (x) = √ e−x /2 , 2π x2 −x2 /2 e P1 (x) = √ 2π

(10.19) (10.20)

which correspond to the square of the absolute value of the wave function for the vacuum state and single-photon state, respectively. Thus, by measuring

10.2 Interference of a Single-Photon State and an Arbitrary State

231

P (M ) in homodyne detection, we can deduce the wave function of the input state at port 2 besides a phase factor which can be fixed by the technique of optical tomography [10.6, 10.7]. However, there is an exception to the above. It is well-known that for thermal light, ¯ (¯ n + 1), (10.21) Δn2 = n ¯ and we cannot use the approximation in Eqs.(10.17, so that Δn2 ≈ n 10.18). For thermal light, Pnin = n ¯ n /(¯ n + 1)n+1 , so from Eq.(10.9, 10.10), we have: √ P0 (M ) = q M / 2¯ n+1 (M ≥ 0), (10.22) M n (M ≥ 0), (10.23) P1 (M ) = M q /¯ √ with q = 1 + 1/¯ n − 2¯ n + 1/¯ n. Therefore, the output photon distribution for thermal light input is different from that of coherent state input even when n ¯ >> 1. But, the general trend in the change of the shape from P0 (M ) to P1 (M ) is similar in both states (Fig.10.4) and the quantum interference effect due to a single-photon is still there.

0.025 0.020

0.04

P0

(a)

0.015

P(M)

P1

0.010

P0

0.02

P1

0.01

0.005 0.000

(b)

0.03

- 60

- 40

- 20

0

M

20

40

60

0.00

- 60

- 40

- 20

0

20

40

60

M

Fig. 10.4. Probability distribution P0,1 (M ) for balanced homodyne detection of vacuum state and single-photon state with (a) coherent state or (b) thermal state as local oscillator. n ¯ = 300. Reprinted figure with permission from Z. Y. Ou, Quan. c Semicl. Opt. 8, 315 (1996). 1996 by the Institute of Physics.

It is interesting to note that the quantum interference effect studied here has similarities with another type of intensity-independent interference effect where two interfering fields are substantially different in intensity (see Sect.1.3). In both cases, the presence of the weak field can dramatically change the outcome of the result. However, the underlying principles are quite different in the two cases. Here, all the N photons participate in the interference (N + 1-particle interference), whereas in Sect.1.3, only two photons are involved and the rest of the photons are not counted. Even though the nonclassical field is weak here, the result is very nonclassical in the sense that the probability of detecting equal intensities at the two outputs is zero

232


[P1 (M = 0) = 0]. It can be proved that in the similar situation (one field is weak and the other is strong), classical wave theory predicts that the probability is largest for equal intensity output at the two ports.

10.3 Multi-Mode Consideration The analysis in the previous section is based on a single-mode model in which we assumed that there is a perfect spatial and temporal mode match between the input field and the local oscillator (LO) field. Normally, spatial mode can be matched easily if the input field is generated in a well-defined cavity with Gaussian mode. The temporal mode, on the other hand, is determined by many factors and is harder to match. An important fact is that we cannot use filters for mode match because they will introduce vacuum noise in homodyne detection and are equivalent to losses. With vacuum noise, the probability distribution will be a combination of P0 and P1 and if the loss is large enough, we will lose the double peak feature in P1 . In the following, we will consider the same multi-photon interference effect for the case of a pulsed single-photon state as the input and a pulsed coherent state as the LO, and explore the possibility of observing this effect with a down-converted field as the singlephoton state. First of all, let us consider a non-stationary single-photon state described by ˆ† (ω)|vac, (10.24) |Ψ = dωΨ (ω)ejωt0 a with normalization condition this state has the form of

dω|Ψ (ω)|2 = 1. The intensity of the field in

ˆ † (t)E(t) ˆ I(t) = E = |A(t)|2 , with 1 A(t) = √ 2π

dωΨ (ω)e−jω(t−t0 ) .

(10.25)

(10.26)

It can be easily checked that the quantity γ(t1 , t2 ) = 1 for the state in Eq.(10.24). Thus, it is a transform-limited single-photon pulse centered at t = t0 . It is straightforward to show from Eq.(10.25) that the total photon number is ∞ ∞ dτ I(τ ) = dτ |A(τ )|2 = 1. (10.27) −∞

−∞

From the discussion in Sect.7.1.2, this state can be obtained from a pulsepumped parametric down-conversion process by conditional detection, as long as narrow filtering is performed on the gating field.

10.3 Multi-Mode Consideration

ε (t) E(t)

233

Coherent Local Oscillator

Signal D2

D1

∆i

Fig. 10.5. Homodyne detection with a strong coherent state as the local oscillator (LO).

Next, we mix this single-photon field with a strong coherent pulse by a 50:50 beam splitter, as shown in Fig.10.5. Let us denote the field operator for ˆ the coherent field by E(t), which can be expressed in terms of the annihilation ˆ operator b(ω) as ˆ = √1 (10.28) E(t) dω ˆb(ω)e−jωt . 2π For coherent pulses, the quantum states |B can be described by ˆb(ω)|B = β(ω)ejωT0 |B,

(10.29)

where β(ω) is a slowly varying function of ω with a definite phase relation for a different ω, in order to form a coherent pulse whose width is determined by the reciprocal of the bandwidth Δω of β(ω). The coherent pulse is centered at t = T0 . The intensity of the coherent pulse is given by ˆ = |B(t)|2 , IE (t) = Eˆ† (t)E(t) with 1 B(t) = √ 2π

dωβ(ω)e−jω(t−T0 ) .

(10.30)

(10.31)

So the pulse is also transform-limited and is centered at t = T0 . The output fields of the beam splitter are described by the field operators: √ ˆ + E(t)]/ ˆ Eˆ1 (t) = [E(t) √2, (10.32) ˆ − E(t)]/ ˆ Eˆ2 (t) = [E(t) 2. We will be interested in the probability distribution for photon number difference between the two output ports. It can be calculated from the joint probability PN1 N2 of finding N1 photons in port 1 and N2 photons in port 2, which is given by [10.4] ( ) ˆ N1 (T ) ˆ ˆ N2 W ˆ 2 (T ) 1 −W1 (T ) W2 (T ) −W : , (10.33) PN1 N2 = T : e e N1 ! N2 !

234


ˆ 1,2 (T ) = T dτ Iˆ1,2 (τ ). T is the time ordering and : : is the normal with W 0 ordering. However, this quantity is not easy to calculate for the multi-mode states in Eqs.(10.24, 10.29). On the other hand, photo-detectors make a quantum measurement of the photon number of optical fields, that is, the photoelectrons have the same statistics as the photons that fall on the detector [10.6, 10.12]. So, we can equivalently check the photocurrent fluctuations for the two detectors located at the outputs of the beam splitter and calculate the probability distribution of the photocurrents. The general formula for the characteristic function of the photocurrent fluctuations has been calculated in Ref.[10.12]. Under the large intensity condition, it has the following form for two detectors {Eq.(44) of Ref.[10.12]}: Ci1 ,i2 (r * 1 , r2 )

+ exp{jr1 i1 + jr2 i2 } & ∞ ∞ ≈ T : exp jαr1 dτ Iˆ1 (t − τ )Q(τ ) + jαr2 dτ Iˆ2 (t − τ )Q(τ )− 0 0 % ' αr22 ∞ ˆ αr12 ∞ ˆ 2 2 dτ I1 (t − τ )[Q(τ )] − dτ I2 (t − τ )[Q(τ )] − : , 2 0 2 0 (10.34)

=

where α and Q(τ ) are the quantum efficiency and the response function for the two detectors, respectively. For simplicity, we assume the two detectors are identical. In the detection of a non-stationary field, especially for ultra-fast laser pulses, the response of the detectors is usually slow, so that their action is simply an average of the photo-current over a long period of time T (longer than optical pulse width). Hence, we can choose the response function as for 0 < τ < T, i0 (10.35) Q(τ ) = 0 for other τ, where i0 is the average photocurrent in the detectors during the period T . For a balanced homodyne detection scheme, as shown in Fig.10.5, the photocurrents from the two detectors are subtracted to produce a difference current Δi = i1 − i2 , which represents the photon number difference between the two outputs of the beam splitter. Next, we find the probability distribution for the photocurrent difference Δi. Its characteristic function can be obtained from Eq.(10.34) by setting r1 = −r2 ≡ r. With Eq.(10.35) for Q(τ ), we have: CΔi (r) = exp{jrΔi} & T

≈ T : exp jrαi0 dτ Iˆ1 (t − τ ) − Iˆ2 (t − τ ) − % ' 0

r2 αi20 T ˆ dτ I1 (t − τ ) + Iˆ2 (t − τ ) : . (10.36) − 2 0 From Eq.(10.32), we find:

10.3 Multi-Mode Consideration

ˆ ˆ † Eˆ − Eˆ† E, Iˆ1 − Iˆ2 = E †ˆ †ˆ ˆ ˆ ˆ ˆ I1 + I2 = E E + E E.

235

(10.37)

So, Eq.(10.36) becomes: & T

CΔi (r) = T : exp − rαi0 dτ B(t − τ )Eˆ † − B ∗ (t − τ )Eˆ − 0 % ' 2 2 T r2 αi20 T ˆ − τ ) : exp − r αi0 dτ I(t dτ |B(t − τ )|2 , − 2 2 0 0 (10.38) where we replace the operator Eˆ with the c-function B(t) for the coherent pulse because of the normal ordering : :. It is now straightforward to calculate CΔi (r). By expanding the exponential function inside the angle brackets in Eq.(10.38) and using Eq.(10.24) for the single-photon state, we find: & T 2 '%

1 2 2 2 † ∗ ˆ ˆ CΔi (r) = 1 + r α i0 : B(t − τ )E − B (t − τ )E dτ : 2 0 T 1 dτ |B(t − τ )|2 , (10.39) × exp − r2 αi20 2 0 where higher-order terms in the expansion are zero due to the single-photon ˆ because we assume property of the state |Ψ , and we neglect the term with I(t) the coherent field is much stronger than the single-photon field. With the state |Ψ in Eq.(10.24), the quantity inside the angle brackets can be easily calculated and we have: CΔi (r) = (1 − ar2 ) e−br

2

/2

,

(10.40)

with a≡

α2 i20

0

T

2 2 dτ B (t − τ )A(t − τ ) , b ≡ αi0

T

∗

0

dτ |B(t − τ )|2 . (10.41)

The probability distribution for the photocurrent difference Δi can be easily obtained from the characteristic function in Eq.(10.40) by a Fourier transformation. Hence, 1 PΔi (x) = drCΔi (r)e−jrx 2π

2 1 a ax2 = √ (10.42) 1 − + 2 e−x /2b . b b 2πb The distribution PΔI (x) has a twin peak feature with a minimum at x = 0. This is the multi-photon quantum interference effect when a single photon state and a strong coherent state are superposed. When a = b, complete cancellation of probability is achieved at x = 0. However, by using Schwartz inequality, we have:

236


a ≤ αb

0

T

dτ |A(t − τ )|2 .

(10.43)

The equal sign occurs when B(τ ) = CA(τ ) with C a constant number, that is, the shape of the coherent pulse matches exactly with that of the single-photon state. Because α ≤ 1 and T ∞ dτ |A(t − τ )|2 ≤ dτ |A(t − τ )|2 = 1, (10.44) 0

−∞

we always have a ≤ b, which ensures the positiveness of the probability distribution PΔi (x) for arbitrary x. Therefore, in order to observe complete probability cancellation, we need a 100% quantum efficiency for the detectors and a long integration time and, most importantly, the overlap of the temporal modes of the two interfering fields. Of course, these are the ideal conditions that can never be met in practice. Any imperfection in the experimental set-up will result in a/b < 1, which can be equivalent to a less-than 100% quantum efficiency for the detectors (αe ≡ a/b). Although no perfect cancellation can be achieved, there is still a minimum at x = 0 unless αe ≤ 1/3 ≈ 33.3%, for which the twin peak in PΔI (x) will merge into a single peak and the signature for multi-photon quantum interference is completely lost. 33.3% quantum efficiency for a detector is easily achievable (quantum efficiency higher than 90% has been reported). However, what makes αe small is the temporal mode mismatch as a result of Schwartz inequality in Eq.(10.43). This will be the most important factor in the experiment investigation of this effect. Experimentally, the multi-photon interference effect discussed in this chapter was observed by Lvovsky et al. [10.13], with a single-photon state produced from parametric down-conversion by gated detection.

References 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13

B. Yurke and D. Stoler, Phys. Rev. A 36, 1955 (1987). W. Vogel and J. Grabow, Phys. Rev. A 47, 4227 (1993). H. P. Yuen and V. W. S. Chan, Opt. Lett. 8, 177 (1983). L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York 1995). S. L. Braunstein, Phys. Rev. A 42, 474 (1990). K. Vogel and H. Risken, Phys. Rev. A40, 2847 (1989). D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. 70, 1244 (1990). R. J. Glauber, Phys. Rev. 130, 2529 (1963); 131, 2766 (1963). E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963). C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987). R. A. Campos, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A 40, 1371 (1990). Z. Y. Ou and H. J. Kimble, Phys. Rev. A 52, 3126 (1995). A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, Phys. Rev. Lett. 87, 050402 (2001).

A Lossless Beam Splitter

An optical beam splitter plays an important role in optical interference. It usually acts as a device that splits the amplitude of an incoming wave or combines two waves for interference. Therefore, it is a 4-port device: 2 input ports and 2 output ports, where a beam splitter then connects the four coupled modes. Its behavior with waves is well-documented in classical electromagnetic wave theory [A.1]. Quantum mechanically, the relationship between the four input and output operators has been studied by a number of researchers [A.2, A.3, A.4, A.5, A.6] and it has a simple relation for the annihilation operators of the four modes in Heisenberg picture as ˆb1 = tˆ a1 + rˆ a2 , (A.1) ˆb2 = t a ˆ2 + r a ˆ1 . t, r, t , r are the complex amplitude transmissivity and reflectivity of the beam splitter, respectively. From the commutation relations [ˆbk , ˆb†l ] = δkl , (k, l = 1, 2)

(A.2)

we obtain |t|2 +|r|2 = 1, |t |2 +|r |2 = 1, |t| = |t |, |r| = |r |, and t∗ r +r∗ t = 0, which leads to ϕt + ϕr − ϕt − ϕr = π.

(A.3)

The above phase relation is universal and is independent of the specifics of the beam splitter. This relation can also be derived via input-output energy conservation from classical wave theory [A.7]. In general, t and r are complex numbers. However, by carefully choosing the reference point, we may arbitrarily change ϕt , ϕr , ϕt , ϕr within the restriction in Eq.(A.3). For simplicity, we let ϕt , ϕr , ϕt be zero and ϕr = −π, according to Eq.(A.3). Therefore, we have the beam splitter relation: ˆb1 = tˆ a1 + rˆ a2 , (t, r > 0) (A.4) ˆb2 = tˆ a2 − rˆ a1 .

238


The argument above holds for one polarization. When the incident fields have different polarizations, we need to decompose them into polarizations parallel (labeled by x) and perpendicular (labeled by y) to the incident plane. The phase relations between these two polarizations can be derived for a dielectric beam splitter from the Fresnel formulae (see, for example, Ref.[A.1]) and they have the form: ϕrx − ϕry = π.

(A.5)

for an incident angle smaller than the Brewster angle. Therefore, we have at near normal incidence: ˆb1x = tx a ˆ1x + rx a ˆ2x , (tx , rx > 0), (A.6) ˆb2x = tx a ˆ2x − rx a ˆ1x ,

ˆb1y = ty a ˆ1y − ry a ˆ2y , ˆb2y = ty a ˆ2y + ry a ˆ1y .

(ty , ry > 0).

(A.7)

The operator relationship in Eq.(A.4), together with the input state, is usually enough to determine the properties at the output ports. However, the above approach with operators lacks the visual connections to such interesting phenomena in quantum information as quantum entanglement and other nonclassical effects in the output ports. So, to see what emerges from the beam splitter, it is better to work in the Schr¨ odinger picture and find the output state of the beam splitter. For this purpose, there are at least a couple of papers [A.5, A.8] in the literature dealing with the output state for a general input state. In these papers, a general formalism is given that connects the input state with output state. For example, Ou et al. [A.5] used the GlauberSudashan P-representation [A.9, A.10] for the connection. This presentation, however, is best used to describe states with coherent state connection. For the number state input, the P-representation becomes very complicated. To circumvent this, Campos et al. [A.8] made use of the theory of angular momentum to derive a general formula for the output density operator in the photon number base. But this approach is complicated and requires some effort to review the theory of angular momentum. Here, we will present a more direct method to obtain the output state for the input of number states or their corresponding superposition states. We start by working in the Heisenberg picture, in which the output operators are connected to the input operators by a unitary transformation: ˆb1 = U ˆ †a ˆ = tˆ ˆ1 U a2 , a1 + rˆ (A.8) ˆb2 = U ˆ †a ˆ = tˆ ˆ2 U a1 . a2 − rˆ ˆ is a function of a ˆ Here, U ˆ1 and a ˆ2 . (We will give a simple derivation of U at the end of this Appendix.) The state is unchanged and is the same as the input state in this picture:


|Ψ = |φin .

239

(A.9)

All the properties at the output can be calculated by averaging the operators ˆb1 , ˆb2 in Eq.(A.8) over the state in Eq.(A.9) [A.5]. On the other hand, all these properties can also be equivalently calculated in the Schr¨ odinger picture in which the output operators ˆb1 , ˆb2 are the same as the input operators a ˆ1 , a ˆ2 , but the output state is connected to the input by ˆ |φin . |Ψ out = U

(A.10)

Generally, in order to find the output state, we need the explicit form of ˆ . However, for special states such as a number state or a coherent state, it is U not necessary, as illustrated below. For simplicity, let us first consider a single photon state input at port 1: |φin = |11 ⊗ |02 = (ˆ a†1 |01 ) ⊗ |02 .

(A.11)

The output state then becomes: â â ˆ † U|0 ˆ 1 ⊗ |02 , |Ψ out = U ˆ†1 |01 ⊗ |02 = U ˆ†1 U

(A.12)

ˆ = 1 between a ˆ †U ˆ1 and |01 . where we insert the unitary relation U ˆ It is easy to see that U |01 ⊗ |02 = |01 ⊗ |02 , that is, vacuum input gives vacuum output for a beam splitter. (This can be easily confirmed directly ˆ at the end of this Appendix.) To find when we have the explicit form of U † ˆ ˆ ˆ1 , a ˆ2 in terms of ˆb1 , ˆb2 : Ua ˆ1 U , we invert Eq.(A.4) to relate a ˆ ˆb1 U ˆ † = tˆb1 − rˆb2 , a ˆ1 = U ˆ ˆb2 U ˆ † = tˆb2 + rˆb1 . a ˆ2 = U So, the principle of reversibility gives: â ˆ † = tˆ U ˆ1 U a1 − rˆ a2 , ˆ ˆ † = tˆ a2 + rˆ a1 . Ua ˆ2 U

(A.13)

(A.14)

ˆ given The above relation can be derived directly from the explicit form of U later. Therefore, we have, for the output state: |Ψ out = (tˆb†1 − rˆb†2 )|01 ⊗ |02 = t|1, 0 − r|0, 1.

(A.15)

ˆ2 by the same output operators Note that we replaced the input operators a ˆ1 , a ˆb1 , ˆb2 in the Schr¨ odinger picture. Likewise, we can find the output state of an input state of |1, 1 in the Hong-Ou-Mandel interferometer (see Chapt.3): ˆ |11 |12 = U â â |Ψ out = U ˆ†1 |01 a ˆ†2 |02 = U ˆ†1 a ˆ†2 |0

240


ˆ †U â ˆ † U|0 ˆ â ˆ † )(U ˆ † )|0 â â ˆ†2 U = (U ˆ†1 U =U ˆ†1 U ˆ†2 U † † † † a2 )(tˆ a2 + rˆ a1 )|0 = (tˆ a1 − rˆ 2 2 † † a1 a ˆ2 |0 √ + tr(ˆ a†2 ˆ†2 = (t − r )ˆ 1 −a 2 )|0 = (t2 − r2 )|1, 1 + 2tr(|2, 0 − |0, 2),

(A.16)

and for a 50:50 beam splitter we obtain the two-photon entangled state: √ (A.17) |Ψ out = (|2, 0 − |0, 2)/ 2. For a general input state of |M, N , we find the output state as ˆ 1 |N 2 = √ 1 â ˆ†N |Ψ out = U|M U ˆ†M 1 |01 a 2 |02 M !N ! 1 ˆ † ˆ ˆ†N U ˆ † U|0 ˆ â = √ U ˆ†M 1 U Ua 2 M !N ! 1 ˆ † )M (U ˆ † )N |0 â â (U ˆ†1 U = √ ˆ†2 U M !N ! 1 (tˆ a†1 − rˆ a†2 )M (tˆ a†2 + rˆ a†1 )N |0 = √ M !N ! √ N M (−1)m M !N !tM+N −m−n rm+n †M−m+n †N −n+m a ˆ1 a ˆ2 |vac = (M − m)!m!(N − n)!n! m=0 n=0 M N (−1)m M !N !(M − m + n)!(N − n + m)! = (M − m)!m!(N − n)!n! m=0 n=0 ×tM+N −m−n rm+n |M − m + n, N − n + m,

(A.18)

which can be regrouped as |Ψ out =

M+N

ck |k, M + N − k,

(A.19)

k=0

where ck collects the coefficients of the common terms of |k, M + N − k in Eq.(A.18) and is in a very complicated form for the general case. But for some special cases, we can derive its explicit form. For example, for a 50:50 beam splitter with M = N , the above is simplified as 1 N (ˆ a†2 − a ˆ†2 2 ) |0 N !2N 1 N †2(N −k) 1 a ˆ†2k a ˆ = N |0 (−1)N −k 1 2 2 k!(N − k)! k=0 N (2k)!(2N − 2k)! 1 N −k (−1) = N |2k, 2N − 2k. 2 k!(N − k)!

|Ψ out =

(A.20)

k=0

√ Another special case of M = 1 with t = r = 1/ 2 is given in Eq.(10.5) of Chapt.10.


241

The above formalism can also be used in a coherent state representation, such as the Glauber-Sudarshan P-representation [A.9, A.10], to derive a general relation between the input and the output states for an arbitrary case. In the Glauber-Sudarshan P-representation, the input and output states are described by the density matrices as (A.21) ρîn = d2 α1 d2 α2 Pin (α1 , α2 )|α1 , α2 α1 , α2 |, ρôut = d2 α1 d2 α2 Pout (α1 , α2 )|α1 , α2 α1 , α2 |, (A.22) where Pin/out (α1 , α2 ) is a quasi-probability distribution and can completely describe the incoming/outgoing fields at the beam splitter. |α1 , α2 is the coherent state base. Our goal is to find the connection between Pin and Pout . From Eq.(A.10), the output density matrix is then given by ˆ ˆ† ρôut = U ρîn U ˆ |α1 , α2 α1 , α2 |U ˆ †. = d2 α1 d2 α2 Pin (α1 , α2 )U

(A.23)

ˆ |α1 , α2 is the output state corresponding to a coherent state Obviously, U input state of |α1 , α2 , and, from classical optics and Eq.(A.4), we know the output is also a coherent state of the form: ˆ |α1 , α2 = |β1 , β2 , U with

β1 = tα1 + rα2 β2 = tα2 − rα1 .

(A.24)

(A.25)

The above relation can also be derived with the method discussed before. For a coherent state, we have [A.9]: ˆ |α = D(α)|0,

(A.26)

ˆ D(α) = exp(αˆ a − α∗ a ˆ† ).

(A.27)

with

Then, we have: ˆ |α1 , α2 = U ˆD ˆ 1 (α1 )D ˆ 2 (α2 )|0, 0 U ˆ ˆ ˆ 2 (α2 )U ˆ † |0, 0. = U D1 (α1 )D But, with ˆD ˆ 1 (α1 ) D ˆ 2 (α2 )U ˆ† U

(A.28)

242


ˆ exp(α1 a ˆ† U ˆ1 − α∗1 a ˆ†1 + α2 a ˆ2 − α∗2 a ˆ†2 )U † † ∗ † â ˆ − α1 U â ˆ + α2 U â ˆ † − α∗2 U â ˆ †) exp(α1 U ˆ1 U ˆ1 U ˆ2 U ˆ†2 U † † a1 − rˆ a2 ) − α∗1 (tˆ a1 − rˆ a2 ) exp[α1 (tˆ a2 + rˆ a1 ) − α∗2 (tˆ a†2 + rˆ a†1 )] +α2 (tˆ ∗ ∗ † a1 − (tα1 + rα2 )ˆ a1 = exp[(tα1 + rα2 )ˆ +(tα2 − rα1 )ˆ a2 − (tα∗2 − rα∗1 )ˆ a†2 ] ˆ 1 (β1 )D ˆ 2 (β2 ), = D (A.29)

= = =

we have Eq.(A.24) and Eq.(A.25). Substituting Eq.(A.24) into Eq.(A.23) and making a change of variables from α to β by Eq.(A.25), we find the output state as ρôut = d2 β1 d2 β2 |β1 , β2 β1 , β2 |Pin (tβ1 − rβ2 , tβ2 + rβ1 ). (A.30) Therefore, we have: Pout (β1 , β2 ) = Pin (tβ1 − rβ2 , tβ2 + rβ1 ).

(A.31)

This is exactly the same function as in Ref.[A.5]. ˆ Derivation of an Explicit Expression for U ˆ bears some resemblance to angular momentum operThis derivation of U ators in rotation. If we note that t2 + r2 = 1, then we can assign t as cos θ and r as sin θ. And Eq.(A.8) becomes: ˆb1 = U ˆ †a ˆ = cos θˆ ˆ1 U a2 , a1 + sin θˆ (A.32) ˆb2 = U ˆ †a ˆ = cos θˆ ˆ2 U a1 , a2 − sin θˆ which is similar to the transformation of the two-dimensional rotation of angle θ. As in any transformation, we consider an infinitesimal transformation of δθ k

The part of the operators acting on the input state is given by ˆV (t1 )...EˆH (tk )...EˆH (tj )...EˆV (tN +2 )|2V mH E = G2 (Pk1 Pj2 {t1 , t2 ; t3 , ..., tN +2 })|vac, where Pk1 exchanges tk with t1 and Pj2 exchanges tj with t2 , and G(t1 , t2 ; P {t3 , ..., tN +2 }), G2 (t1 , t2 ; t3 , ..., tN +2 ) = 2

(B.7)

(B.8)

P

where we used Eq.(B.4) for the exchange between t1 , t2 . Note that the Gfunction is symmetric among the variables {t3 , ..., tN +2 }. The joint (N + 2)photon detection probability is then proportional to 2 PN +2 = dt1 ...dtN +2 ei(δk +δj ) G2 (Pk1 Pj2 {t1 , t2 ; t3 , ..., tN +2 }) . =

j>k j >k

j>k

e

i(δk +δj −δk −δj )

dt1 ...dtN +2 G2 (tk , tj ; t3 , ..., t1 , ..., t2 , ..., tN +2 ) ×G2∗ (tk , tj ; t3 , ..., t1 , ..., t2 , ..., tN +2 ).

The sum can be broken up into three parts as = A1 + A2 + A3 , j >k j>k

with the diagonal term equal equal to

(B.9)

(B.10)

B.1 The Case of |2H , NV

A1 =

,

247

(B.11)

j =j>k =k

and the cross terms to A2 =

+

j>k j >k j =j k =k

+

j>k j >k k =k j =j

+

j>k j >k j =k

,

(B.12)

j>k j >k k =j

and to

A3 =

.

(B.13)

j>k j >k j =j=k =k

The first diagonal sum is straightforward: 2 A1 = dt1 ...dtN +2 G2 (tk , tj ; t3 , ..., t1 , ..., t2 , ..., tN +2 ) j>k

= 4NN +2 N !(N + 2)(N + 1)/2.

(B.14)

Here, NN +2 is similar to NN +1 in Eq.(9.100), but with Φ(ω1 ; P {ω2 , ..., ωN +1 }) replaced by Φ(ω1 , ω2 ; P {ω3 , ..., ωN +2 }). The second part A2 consists of four sums that all have only two equal indices among (k, j, k , j ), with one from {k, j} and the other from {k , j }. Therefore, the time integral part in Eq.(B.12) is the same as that in Eq.(9.105) and gives 4NN +2 m(N − 1)!. The sum over the phases can be evaluated as a2 ≡ ei(δk −δk ) + ei(δj −δj ) + j>k j >k j =j k =k

+

=

j>k j >k k =k j =j

ei(δj −δk ) +

j>k>2 j >k >2 j =k

ei(δk2 −δk1 ) = N

k3 =k2 =k1

ei(δk −δj )

j>k>2 j >k >2 k =j

ei(δk2 −δk1 ) .

(B.15)

k2 =k1

But similar to Eq.(9.106), we have: ei(δk2 −δk1 ) = −(N + 2),

(B.16)

k2 =k1

where the indices in the sum now run up to N + 2. So, we obtain: a2 = −N (N + 2), and

(B.17)

248

B Derivation of the Visibility for |kH , NV

A2 = 4NN +2 m(N − 1)!a2 = −4NN +2 m(N − 1)!N (N + 2). (B.18) The last term A3 involves a time integral with all four indices (k, j, k , j ) unequal. The time integral part can be evaluated as dt1 ...dtN +2 G2 (tk , tj ; t3 , ..., t1 , ..., t2 , tN +2 ) =4

×G2∗ (tk , tj ; t3 , ..., t1 , ..., t2 , ..., tN +2 ) dt1 ...dtN +2 G(tk , tj ; P {t3 , ..., t1 , ..., t2 , tN +2 }) P G∗ (tk , tj ; P {t3 , ..., t1 , ..., t2 , ..., tN +2 }). (B.19) × P

Because of the permutation properties in Eq.(B.4) and tk , tj = tk , tj , nonzero results for the time integration require tk , tj in the first m location in P {t3 , ..., t1 , ..., t2 , ..., tN +2 }. The rest is arbitrary. This leads to a total of m(m − 1)(N − 2)! terms that are nonzero and equal in the sum over P . So, the time integral in Eq.(B.19) is 4NN +2 m(m − 1)(N − 2)!. On the other hand, the part of the sum over the phases is, for A3 : ei(δk +δj −δj −δk ) a3 ≡ j>k j >k j =j=k =k

=

ei(δk +δj −δj −δk ) −

j >k j>k

ei(δk2 −δk1 ) −

k3 =k2 =k1

1

j>k

= 0 − [−N (N + 2)] − (N + 2)(N + 1)/2 = (N + 2)(N − 1)/2, where we used j>k ei(δk +δj ) = 0. Hence, we obtain A3 as A3 = 4NN +2 m(m − 1)(N − 2)!a3 = 4NN +2 m(m − 1)(N − 2)!(N + 2)(N − 1)/2.

(B.20)

(B.21)

Finally, we have, after combining Eqs.(B.14, B.18, B.21): PN +2 (2HmV ) = 4NN +2 N !(N + 2)(N + 1)/2 − m(N − 1)!N (N + 2)+ +m(m − 1)(N − 2)!(N + 2)(N − 1)/2 (B.22) = 2NN +2 (N + 2)! 1 − VN +2 (2HmV ) , with VN +2 (2HmV ) =

m(m − 1) 2m − . N + 1 N (N + 1)

(B.23)

Again, the baseline for calculating the visibility is obtained when we set m = 0.


249

B.1.2 The Scenario of 1HmV + 1HnV + (N − n − m)V In this scenario, there are three groups: the first one consists of one H-photon and m V-photons that are indistinguishable; the second one is formed by the other H-photon and n V-photons and they are temporally indistinguishable; the rest of the V-photons make up the third group. Between different groups, photons are well-separated in time and are completely distinguishable. The quantum state in this case is given in Eq.(B.1), but with the permutation symmetry: Φ(ω1 , ω2 ; ω3 , ..., ωN +2 ) = PI PII Φ(ω1 , ω2 ; ω3 , ..., ωN +2 ),

(B.24)

where PI , PII are permutation operations that act on the groups of I = {ω1 ; ω3 , ..., ωm+2 } and II = {ω2 ; ωm+3 , ..., ωm+n+2 }, respectively. This symmetry relation indicates that one H-photon and m V-photons are completely indistinguishable in time and another H-photon and other n V-photons are also indistinguishable in time. The third group of frequency variables is denoted by III = {ωm+n+3, ..., ωN +2 }. Then, we may impose the orthogonal relation dωk dωj Φ∗ (ω1 , ω2 ; ω3 , ..., ωN +2 )Φ(Pkj {ω1 , ω2 ; ω3 , ..., ωN +2 }) = 0, (B.25) where Pkj interchanges ωk with ωj , and k, j belong to different groups of I, II, III, respectively. This orthogonal relation ensures that photons of different groups are completely distinguishable in time. We can also write similar relations in time domain:

and

G(t1 , t2 ; t3 , ..., tN +2 ) = PI PII G(t1 , t2 ; t3 , ..., tN +2 ),

(B.26)

dtk dtj G∗ (t1 , t2 ; t3 , ..., tN +2 )G(Pkj {t1 , ..., tN +2 }) = 0,

(B.27)

where G(t1 , t2 ; t3 , ..., tN +2 ) has the form similar to Eq.(9.34). The permutation operators now act on time variables. Similar to Eq.(B.9), we may write the joint (N + 2)-photon detection probability, which is proportional to 2 ei(δk +δj ) G2 (Pk1 Pj2 {t1 , t2 ; t3 , ..., tN +2 }) . PN +2 = dt1 ...dtN +2 j>k i(δk +δj −δk −δj ) e = dt1 ...dtN +2 G2 (tk , tj ; t3 , ..., t1 , ..., t2 , ..., tN +2 }) j>k j >k

×G2∗ (tk , tj ; t3 , ..., t1 , ..., t2 , ..., tN +2 }).

But, different from Eq.(B.8), we have, instead:

(B.28)

250


G2 (t1 , t2 ; t3 , ..., tN +2 ) G(t1 , t2 ; P {t3 , ..., tN +2 }) + G(t2 , t1 ; P {t3 , ..., tN +2 }) . ≡

(B.29)

P

Similar to Eq.(B.10), the sum in Eq.(B.28) can be broken up into three parts as = A1 + A2 + A3 . (B.30) j>k j >k

The first sum is straightforward: 2 dt1 ...dtN +2 G2 (tk , tj ; t3 , ..., t1 , ..., t2 , ..., tN +2 ) A1 = j>k

= 2NN +2 N !(N + 2)(N + 1)/2.

(B.31)

Note that because of the orthogonal relation in Eq.(B.27), the cross terms in the expansion of the absolute value in Eq.(B.31) give zero result. Similar to A2 , A2 consists of four sums that all have only two equal indices, with one from {k, j} and the other from {k , j }. From Eq.(B.29) for G2 , we then calculate the time integral part as dt1 ...dtN +2 G2 (ts , tr ; t3 , ..., t1 , ..., t2 , ..., tN +2 ) =

×G2∗ (ts , tr ; t3 , ..., t1 , ..., t2 , ..., tN +2 ) G(ts , tr ; P {...}) + G(tr , ts ; P {...}) dt1 ...dtN +2 P × G∗ (ts , tr ; P {...}) + G∗ (tr , ts ; P {...}) , (B.32) P

with r, r , s = k, j, k , j and r = r = s. Because of the orthogonal relation in Eq.(B.27), the time integrals of G(ts , tr ; P {...})G∗ (tr , ts ; P {...}) and G(tr , ts ; P {...}) G∗ (ts , tr ; P {...}) are zero. So, we have: dt1 ...dtN +2 G2 (ts , tr ; t3 , ..., t1 , ..., t2 , ..., tN +2 ) =

×G ∗ (t , t ; t , ..., t1 , ..., t2 , ..., tN +2 ) 2 s r 3 G(ts , tr ; P {...}) G∗ (ts , tr ; P {...}) dt1 ...dtN +2 P

+

P

G(tr , ts ; P {...})

P

= (m + n)(N − 1)!NN +2 .

Hence, similar to Eq.(B.18), we obtain: A2 = NN +2 (m + n)(N − 1)!a2

G∗ (tr , ts ; P {...})

P

(B.33)


= −NN +2 (m + n)(N − 1)!N (N + 2).

251

(B.34)

Noting that all four indices (k, j, k , j ) are unequal in the sum in A3 , we evaluate the time integral part as dt1 ...dtN +2 G2 (tk , tj ; t3 , ..., t1 , ..., t2 , ..., tN +2 ) =

×G2∗ (tk , tj ; t3 , ..., t1 , ..., t2 , ..., tN +2 ) dt1 ...dtN +2 G(tk , tj ; P {...}) + G(tj , tk ; P {...}) P × G∗ (tk , tj ; P {...}) + G∗ (tj , tk ; P {...}) P

= 4mn(N − 2)!NN +2 .

(B.35)

Using a3 in Eq.(B.20) for the sum, we obtain A3 as A3 = 2NN +2 mn(N − 2)!(N + 2)(N − 1).

(B.36)

Finally, we have, after combining Eqs.(B.31, B.34, B.36): PN +2 (1HmV + 1HnV ) = NN +2 (N + 2)! − (m + n)(N − 1)!N (N + 2)+

+2mn(N − 2)!(N + 2)(N − 1)

= NN +2 (N + 2)!(1 − VN +2 ),

(B.37)

with VN +2 (1HmV + 1HnV ) =

2mn m+n − . N + 1 N (N + 1)

(B.38)

B.2 The Case of |3H , NV We now consider the even more complicated situation of three H-photons and N V-photons for the NOON state measurement. We start with the scenario when all three H-photons are indistinguishable. B.2.1 The Scenario of 3HmV + (N − m)V The quantum state for this case is given by |3HmV = dω1 dω2 ...dωN +3 Φ(ω1 , ω2 , ω3 ; ω4 , ..., ωN +3 ) a†V (ω4 )...ˆ a†V (ωN +3 )|vac, a ˆ†H (ω1 )a†H (ω2 )a†H (ω3 )ˆ with

(B.39)


252

Φ(ω1 , ω2 , ω3 ; ω4 ..., ωN +3 ) = Φ(P {ω1 , ..., ωm+3 }, ωm+4 , ..., ωN +3 ),

(B.40)

where P is an arbitrary permutation operation. This expression is from the indistinguishability among the three H-photons and m V-photons. But, because other N − m V-photons are well separated from the 3HmV photons, we have the orthogonal relation (B.41) dωk dωj Φ∗ (ω1 , ..., ωN +3 )Φ(Pkj {ω1 , ..., ωN +3 }) = 0, where Pkj interchanges ωk with ωj , and 1 ≤ k ≤ m + 3, m + 4 ≤ j ≤ N + 3. Or, we can write similar relations in time domain: G(t1 , t2 , t3 ; t4 , ..., tN +3 ) = G(P {t1 , ..., tm+3 }, tm+4 , ..., tN +3 ),

dtk dtj G∗ (t1 , t2 , t3 ; t4 , ..., tN +2 )G(Pkj {t1 , ..., tN +3 }) = 0,

(B.42)

(B.43)

where G(t1 , t2 , t3 ; t4 , ..., tN +3 ) is similar to Eq.(9.34). The NOON state projection measurement has now (N+3) detectors and the phase shifts are δj = 2π(j−1)/(N +3) with j = 1, ..., N +3. The expression for PN +3 is similar to Eq.(9.91). After the expansion of the operator product, the terms with non-zero contributions are: ˆH (tl )...EˆH (tk )...EˆH (tj )...EˆV (tN +3 )|3HmV ˆV (t1 )...E ei(δk +δj +δl ) E j>k>l

(B.44) with ˆV (t1 )...EˆH (tl )...EˆH (tk )...E ˆH (tj )...E ˆV (tN +3 )|3HmV E = G3 (Pl1 Pk2 Pj3 {t1 , t2 , t3 ; t4 , ..., tN +3 })|vac, (B.45) where Pl1 exchanges tl with t1 , etc., and we have, due to Eq.(B.42): G(t1 , t2 , t3 ; P {t4 , ..., tN +2 }). G3 (t1 , t2 , t3 ; t4 , ..., tN +3 ) = 6

(B.46)

P

The joint (N + 3)-photon detection probability is then proportional to 2 PN +3 = dt1 ...dtN +3 ei(δl +δk +δj ) G3 (Pl1 Pk2 Pj3 {t1 , ..., tN +3 }) j>k>l dt1 ...dtN +2 G3 (Pl1 Pk2 Pj3 {t1 , ..., tN +3 }) = j >k >l j>k>l

×G3∗ (Pl 1 Pk 2 Pj 3 {t1 , ..., tN +3 })ei(δl +δk +δj −δl −δk −δj ) . (B.47)


The sum can be broken up into four parts now as = C1 + C2 + C3 + C4 ,

253

(B.48)

j >k >l j>k>l

with

C1 =

,

(B.49)

j =j>k =k>l =l

C2 =

,

(B.50)

,

(B.51)

.

(B.52)

j>k>l j >k >l two pairs equal

C3 =

j>k>l j >k >l one pair equal

C4 =

j>k>l j >k >l no equal indices

The first sum is straightforward: 2 dt1 ...dtN +3 G3 (tl , tk , tj ; t4 , ..., t1 , ..., t2 , ..., t3 , ..., tN +3 ) C1 = j>k>l

= 36NN +3 N !(N + 2)(N + 1)(N + 3)/6.

(B.53)

Here NN +3 is in a similar form as that of NN +1 in Eq.(9.100) and NN +2 in Eq.(B.14), but with Φ replaced by that in Eq.(B.39). In the sum of the second part C2 , the indices have two equal pairs among l, k, j, l , k , j with j > k > l, j > k > l , leaving two unequal indices, with one from {l, k, j} and the other from {l , k , j }. So, the time integral part is similar to that in Eq.(9.105) and gives 36NN +3 m(N − 1)!. On the other hand, for the sum of the phase terms, we have: ei(δl +δk +δj −δl −δk −δj ) = ei(δk2 −δk1 ) c2 ≡ j>k>l j >k >l two pairs equal

= [N (N + 1)/2]

k4 >k3 =k2 =k1

ei(δk2 −δk1 )

k2 =k1

= [N (N + 1)/2] × [0 − (N + 3)] = −N (N + 1)(N + 3)/2.

(B.54)

254


So, we arrive at C2 = 36NN +3 m(N − 1)!c2 = −18NN +3 m(N − 1)!N (N + 1)(N + 3). (B.55) The third term C3 has one pair of equal indices and the time integral part has the form: dt1 ...dtN +3 G3 (ts , tr , tq ; t4 , ..., tN +2 )G3∗ (ts , tr , tq ; t4 , ..., tN +3 ) G(ts , tr , tq ; P {t4 , ..., tN +3 }) = 36 dt1 ...dtN +2 P × G∗ (ts , tr , tq ; P {t4 , ..., tN +3 }), (B.56) P

where q = q = r = r . Because of the permutation properties in Eq.(B.42) and tq = tq = tr = tr , non-zero results for the time integration require tq , tr in the first m location in P {t4 , ..., tN +3 }. The rest is arbitrary. There are m(m−1)(N −2)! terms that are nonzero and equal in the sum over P . So, similar to Eq.(B.19), the time integral in Eq.(B.56) is 36NN +3 m(m − 1)(N − 2)!. On the other hand, we have, for the sum of the phase terms in C3 : ei(δl +δk +δj −δl −δk −δj ) = ei(δk4 +δk3 −δk2 −δk1 ) c3 ≡ j>k>l j >k >l one pair equal

k4 >k3 ,k2 >k1 k5 =k4 =k3 =k2 =k1

= (N − 1)

ei(δk4 +δk3 −δk2 −δk1 )

k4 >k3 ,k2 >k1

k3 =k2 =k1 k4 = = (N − 1) ei(δk4 +δk3 −δk2 −δk1 ) −

1

k4 =k2 >k3 =k1

k4 >k3 k2 >k1

−

ei(δk4 +δk3 −δk2 −δk1 )

%

k4 >k3 ,k2 >k1 one pair equal

= (N − 1){0 − (N + 3)(N + 2)/2 − [−(N + 3)(N + 1)]} = (N + 3)N (N − 1)/2.

(B.57)

Hence, we obtain C3 as C3 = 36NN +3 m(m − 1)(N − 2)!c3 = 18NN +3 m(m − 1)N !(N + 3).(B.58) Since all time variables are different, the time integral in the last sum C4 simply gives 36NN +3 m(m − 1)(m − 2)(N − 3)!. The sum can be calculated, similar to Eq.(B.20), as % i(δl +δk +δj −δl −δk −δj ) e = c4 ≡ − c3 − c2 − c1 j>k>l j >k >l no equal indices

j >k >l j>k>l


= 0 − (N − 1)N (N + 3)/2 − [−(N + 3)(N + 1)N/2] −(N + 3)(N + 2)(N + 1)/6 = −(N + 3)(N − 1)(N − 2)/6,

255

(B.59)

so that we have: C4 = −6NN +3 m(m − 1)(m − 2)(N − 3)!(N + 3)(N − 1)(N − 2). (B.60) Finally, we have, after combining Eqs.(B.53, B.55, B.58, B.60): PN +3 (3HmV ) = [(N + 3)! − 3m(N + 1)!(N + 3) + 3m(m − 1)N !(N + 3) −m(m − 1)(m − 2)(N − 1)!(N + 3)]6NN +3 = 6N (N + 3)!(1 − VN +3 ), (B.61) with VN +3 (3HmV ) =

3m 3m(m − 1) m(m − 1)(m − 2) − + . N + 2 (N + 1)(N + 2) N (N + 1)(N + 2)

(B.62)

B.2.2 The Scenario of 2HmV + 1HnV + (N − m − n)V The next scenario has one H-photon separated from the other two H-photons. Similar to the case in Sect.B.1.2, we can divide the photons into three groups: The first (I) has the two H-photons and m V-photons and the second (II) has the one H-photon and n V-photons. Photons in each of the two groups are indistinguishable in time. The third group consists of the rest of the V-photons which are well-separated from the photons in groups I and II. The quantum state for this case is given by |2HmV 1HnV = dω1 dω2 ...dωN +3 Φ(ω1 , ω2 ; ω3 ; ω4 , ..., ωN +3 ) a ˆ†H (ω1 )a†H (ω2 )a†H (ω3 )ˆ a†V (ω4 )...ˆ a†V (ωN +3 )|vac,

(B.63)

with Φ(ω1 , ω2 ; ω3 ; ω4 ..., ωN +3 ) = Φ(PI PII {ω1 , ..., ωN +3 }),

(B.64)

where PI is a permutation operation acting on I ≡ {ω1 , ω2 ; ω4 , ..., ωm+3 } and PII on II ≡ {ω3 ; ωm+4 , ..., ωm+n+3 }, and the orthogonal relation (B.65) dωk dωj Φ∗ (ω1 , ..., ωN +3 )Φ(Pkj {ω1 , ..., ωN +3 }) = 0, where Pkj interchanges ωk with ωj and ωk,j are not to be simultaneously in any one of subsets I, II, III with III ≡ {ωm+n+4 , ..., ωN +3 }. Or, we can write similar relations in time domain: G(t1 , ..., tN +3 ) = G(PI PII {t1 , ..., tN +3 }),

(B.66)

256


dtk dtj G∗ (t1 , ..., tN +2 )G(Pkj {t1 , ..., tN +3 }) = 0,

(B.67)

where G(t1 , ..., tN +3 ) is similar to Eq.(9.34). Similar to Eq.(B.44), to calculate joint (N+3)-photon detection probability, we need to evaluate ei(δk +δj +δl ) EˆV (t1 )...EˆH (tl )...EˆH (tk )... j>k>l

ˆH (tj )...EˆV (tN +3 )|2HmV 1HnV , ×E

(B.68)

where δj = 2π(j − 1)/(N + 3) and the operator part can be deduced as ˆV (t1 )...EˆH (tl )...E ˆH (tk )...EˆH (tj )...EˆV (tN +3 )|2HmV 1HnV E = G3 (Pl1 Pk2 Pj3 {t1 , t2 , t3 ; t4 , ..., tN +3 })|vac, (B.69) where Pl1 exchanges tl with t1 , etc., and we have, due to Eq.(B.66): G3 (t1 , t2 , t3 ; t 4 , ..., tN +3 ) =2 G(t1 , t2 ; t3 ; P {t4 , ..., tN +2 }) + G(t1 , t3 ; t2 ; P {t4 , ..., tN +2 }) P

+G(t3 , t2 ; t1 ; P {t4 , ..., tN +2 }) . (B.70) The joint (N + 3)-photon detection probability is then proportional to 2 ei(δl +δk +δj ) G3 (Pl1 Pk2 Pj3 {t1 , ..., tN +3 }) PN +3 = dt1 ...dtN +3 j>k>l = dt1 ...dtN +2 G3 (Pl1 Pk2 Pj3 {t1 , ..., tN +3 }) j>k>l j >k >l

×G3∗ (Pl 1 Pk 2 Pj 3 {t1 , ..., tN +3 })ei(δl +δk +δj −δl −δk −δj ) . (B.71)

Similar to Eq.(B.48), the sum can be broken up into four parts as = C1 + C2 + C3 + C4 .

(B.72)

j >k >l j>k>l

We calculate each terms as follows: C1 = [(N + 3)(N + 2)(N + 1)/6] dt1 ...dtN +3 |G (t1 , ..., tN +3 )|2 = [(N + 3)(N + 2)(N + 1)/6]4 × 3(N !)NN +3 = 2NN +3 (N + 3)!. (B.73) C2 = c2

dt1 ...dtN +3 G3 (ts , tr , tq ; ...)G3∗ (ts , tr , tq ; ...),

(B.74)


257

with q = q . With Eq.(B.70) for G3 (ts , tr , tq ; ...), we have the time integral as dt1 ...dtN +3 G3 (ts , tr , tq ; ...)G3∗ (ts , tr , tq ; ...) G(ts , tr ; tq ; P {...}) + G(ts , tq ; tr ; P {...})+ = 4 dt1 ...dtN +3 P G∗ (ts , tr ; tq ; P {...})+ +G(tq , tr ; ts ; P {...}) P +G∗ (ts , tq ; tr ; P {...}) + G∗ (tq , tr ; ts ; P {...}) G(ts , tr ; tq ; P {...}) G∗ (ts , tr ; tq ; P {...})+ = 4 dt1 ...dtN +3 P P + G(ts , tq ; tr ; P {...}) G∗ (ts , tq ; tr ; P {...})+ P P ∗ G(tq , tr ; ts ; P {...}) G (tq , tr ; ts ; P {...}) . + P

P

(B.75) The first term requires an exchange between tq , tq within the subset II while the second and third terms are the same and require exchange within the subset I. So, the time integral becomes: dt1 ...dtN +3 G3 (ts , tr , tq ; ...)G3∗ (ts , tr , tq ; ...) = 4(2m + n)(N − 1)!NN +3 .

(B.76)

With c2 in Eq.(B.54), we have: C2 = [−N (N + 1)(N + 3)/2] × 4(2m + n)(N − 1)!NN +3 = −2NN +3 (2m + n)(N + 1)!(N + 3). For C3 , we have: C3 = c3


(B.77)

(B.78)

where ts = tr = tq = tr = tq . Expanding G (ts , tr , tq ; ...), we have the time integral as dt1 ...dtN +3 G3 (ts , tr , tq ; ...)G3∗ (ts , tr , tq ; ...) G(ts , tr ; tq ; P {...}) + G(ts , tq ; tr ; P {...})+ = 4 dt1 ...dtN +3 P G∗ (ts , tr ; tq ; P {...})+ +G(tq , tr ; ts ; P {...}) P ∗ +G (ts , tq ; tr ; P {...}) + G∗ (tq , tr ; ts ; P {...})

258


=4

dt1 ...dtN +3 4 G(ts , tr ; tq ; P {...}) G∗ (ts , tr ; tq ; P {...})+ P P G(tr , tq ; ts ; P {...}) G∗ (tq , tr ; ts ; P {...}) + P

= [4mn + m(m − 1)](N − 2)!4NN +3 .

P

(B.79)

Hence, from Eq.(B.57) for c3 , we obtain: C3 = [(N + 3)N (N − 1)/2] × 4[4mn + m(m − 1)](N − 2)!NN +3 = 2NN +3 [4mn + m(m − 1)]N !(N + 3). (B.80) For C4 , we have: C4 = c4


(B.81)

where none of the time variables are equal. Substituting G3 from Eq.(B.70) and making an expansion of the product, we find that all the nine terms are the same and equal to m(m − 1)n(N − 3)!NN +3 so that the integral is 4 × 9m(m − 1)n(N − 3)!NN +3 . With Eq.(B.59) for c4 , we obtain: C4 = [−(N + 3)(N − 1)(N − 2)/6]4 × 9m(m − 1)n(N − 3)!NN +3 = −6m(m − 1)n(N − 1)!(N + 3). (B.82) Combining Eqs.(B.73, B.77, B.80, B.82), we have: PN +3 (2HmV 1HnV ) = 2NN +3 [(N + 3)! − (2m + n)(N + 1)!(N + 3) +[4mn + m(m − 1)]N !(N + 3) −3m(m − 1)n(N − 1)!(N + 3)] = 2NN +3 (N + 3)![1 − VN +3 (2HmV 1HnV )],

(B.83)

with VN +3 (2HmV 1HnV ) 2m + n 4mn + m(m − 1) 3nm(m − 1) = − + . N +2 (N + 1)(N + 2) N (N + 1)(N + 2)

(B.84)

B.2.3 The Scenario of 1HmV + 1HnV + 1HpV + (N −m−n−p)V The three H-photons in this scenario are all well-separated. Depending on the temporal overlap, we divide all the photons into four groups with the first one (I) containing m V-photons with one H-photon in one temporal mode, the second one (II) having n V-photons with one H-photon, and the third one (III) p V-photons with one H-photon. The rest of the V-photons belong to the fourth group (IV). The V-photons in the fourth group (IV) may or may not be indistinguishable in time.


259

As before, we write the quantum state for this case as |1HmV 1HnV 1HpV = dω1 dω2 ...dωN +3 Φ(ω1 ; ω2 ; ω3 ; ω4 , ..., ωN +3 ) a ˆ†H (ω1 )a†H (ω2 )a†H (ω3 )ˆ a†V (ω4 )...ˆ a†V (ωN +3 )|vac,

(B.85)

with Φ(ω1 ; ω2 ; ω3 ; ω4 ..., ωN +3 ) = Φ(PI PII PIII {ω1 , ..., ωN +3 }),

(B.86)

where the permutation PI acts on the first group I ≡ {ω1 ; ω4 , ..., ωm+3 }, PII on the second group II ≡ {ω2 ; ωm+4 , ..., ωm+n+3 }, and PIII on the third group III ≡ {ω3 ; ωm+n+4 , ..., ωm+n+p+3 }. Distinguishability among different groups is guaranteed by the orthogonal relation (B.87) dωk dωj Φ∗ (ω1 , ..., ωN +3 )Φ(Pkj {ω1 , ..., ωN +3 }) = 0, where Pkj interchanges ωk with ωj and ωk,j are not to be simultaneously in any one of groups I, II, III, IV with IV ≡ {ωm+n+p+4 , ..., ωN +3 }. Or, we can write similar relations in time domain: G(t1 , ..., tN +3 ) = G(PI PII PIII {t1 , ..., tN +3 });

(B.88)

dtk dtj G∗ (t1 , ..., tN +2 )G(Pkj {t1 , ..., tN +3 }) = 0,

(B.89)

where G(t1 , ..., tN +3 ) is similar to Eq.(9.34). The joint (N+3)-photon detection probability from the NOON state projection can be evaluated in the same way as in Eqs.(B.47, B.71), but with the following G3 function: G3 (t1 , t2 , t3 ; t4 ,..., tN +3 ) G(t1 ; t2 ; t3 ; P {t4 , ..., tN +2 }) + G(t1 ; t3 ; t2 ; P {t4 , ..., tN +2 }) = P

+G(t2 ; t3 ; t1 ; P {t4 , ..., tN +2 }) + G(t2 ; t1 ; t3 ; P {t4 , ..., tN +2 }) +G(t3 ; t2 ; t1 ; P {t4 , ..., tN +2 }) + G(t3 ; t1 ; t2 ; P {t4 , ..., tN +2 }) . (B.90) Again, we break up the sum into four parts as = C1 + C2 + C3 + C4 .

(B.91)

j >k >l j>k>l

We calculate each terms as follows: C1 = [(N + 3)(N + 2)(N + 1)/6]

dt1 ...dtN +3 |G (t1 , ..., tN +3 )|2

260


= [(N + 3)(N + 2)(N + 1)/6]6N !NN +3 = NN +3 (N + 3)!.

(B.92)

Note that the time integral is different due to the different G3 and, similarly: C2 = c2 dt1 ...dtN +3 G3 (ts , tr , tq ; ...)G3∗ (ts , tr , tq ; ...), (B.93) with q = q . With Eq.(B.90) for G (ts , tr , tq ; ...), it is straightforward to derive the time integral using the method described previously. Then, it becomes dt1 ...dtN +3 G3 (ts , tr , tq ; ...)G3∗ (ts , tr , tq ; ...) = (2m + 2n + 2p)(N − 1)!NN +3 ,

(B.94)

so that we have, with Eq.(B.54) for c2 : C2 = [−N (N + 1)(N + 3)/2] × (2m + 2n + 2p)(N − 1)!NN +3 = −NN +3 (m + n + p)(N + 1)!(N + 3). (B.95) For C3 , we have: C3

= c3


(B.96)

where ts = tr = tq = tr = tq . The time integral can be evaluated in a way similar to Eq.(B.79), but with six terms for G3 in Eq.(B.90). The result is dt1 ...dtN +3 G3 (ts , tr , tq ; ...)G3∗ (ts , tr , tq ; ...) = 4(mn + mp + np)(N − 2)!NN +3 .

(B.97)

Hence, from Eq.(B.57) for c3 , we obtain: C3 = [(N + 3)N (N − 1)/2] × 4(mn + mp + np)(N − 2)!NN +3 = 2NN +3 (mn + mp + np)N !(N + 3). (B.98) For C4 we have C4 = c4


(B.99)

where none of the time variables are equal. The 36 terms in the expansion of the product of the G3 function in Eq.(B.90) are all same. So, we obtain: C4 = [−(N + 3)(N − 1)(N − 2)/6] × 36mnp(N − 3)!NN +3 = −6mnp(N − 1)!(N + 3)NN +3 , (B.100) where Eq.(B.59) is used for c4 . Combining Eqs.(B.92, B.95, B.98, B.100), we have:

B.3 The General Case of |kH , NV

261

PN +3 (1HmV 1HnV 1HpV ) = NN +3 [(N + 3)! − (m + n + p)(N + 1)!(N + 3) +2(mn + mp + np)N !(N + 3) −6mnp(N − 1)!(N + 3)] = NN +3 (N + 3)![1 − VN +3 (1HmV 1HnV 1HpV )], (B.101) with VN +3 (1HmV 1HnV 1HpV ) 6mnp m + n + p 2(mn + mp + np) − + . (B.102) = N +2 (N + 1)(N + 2) N (N + 1)(N + 2)

B.3 The General Case of |kH , NV We start with the case where all k H-photons are indistinguishable in the state of |kHmV (N − m)V . Before we deal with arbitrary k, let us go one step further from the current result of k = 3 to a k = 4 case. This step is straightforward from Sect.B.2.1. We present only the result as follows. The (N+4)-photon joint detection probability can be derived as (B.103) PN +4 (4HmV ) = 4!(N + 4)!NN +4 1 − VN +4 (4HmV ) , with the visibility as VN +4 (4HmV ) =

6m(m − 1) 4m(m − 1)(m − 2) 4m − + − N + 3 (N + 3)(N + 2) (N + 3)(N + 2)(N + 1) m(m − 1)(m − 2)(m − 3) − . (B.104) (N + 3)(N + 2)(N + 1)N

If we check the coefficients in each term in Eqs.(B.23, B.62, B.104) for k = 2, 3, 4, we find that they are simply the binomial coefficients of Ckj = k!/(k − j)!j!. So, we can generalize the expression to an arbitrary k and the (N+k)photon joint detection probability of the NOON state measurement thus has the form of (B.105) PN +k (kHmV ) = (N + k)!k!NN +k 1 − VN +k (kHmV ) , with the visibility as VN +k (kHmV ) =

k l=1

(−1)l−1 Ckl

m(m − 1)...(m − l + 1) . (N + k − 1)...(N + k − l)

(B.106)

Actually, the derivation of the above result is relatively straightforward if we follow the case of k = 3 in the previous section. Like the k = 3 case, the


262

(N + k)-photon joint detection probability can be broken up into k + 1 parts and is written as PN +k (kHmV ) =

k

dk−l Tk−l ,

(B.107)

l=0

where dl is the sum of the phase part with l pairs of equal indices, while Tl is its corresponding time integral part. It is easy to see that for the state of |kHmV , we obtain Tk−l as Tk = N !(k!)2 NN +k

(l = 0, or k pairs of equal indices),

(B.108)

and Tk−l = m(m − 1)...(m − l + 1)(N − l)!(k!)2 NN +k

(l ≥ 1). (B.109)

For dk−l with l = 0, we have, obviously: k dk = CN +k = (N + k)(N + k − 1)...(N + 1)/k!.

(B.110)

However, the calculation of dk−l with l ≥ 1 is not so trivial. We can obtain it following the same line of argument that leads to the values of the a and c coefficients in previous sections, but we present only its value here: dk−l = (−1)l (N + k)

(N + k − l − 1)! . (N − l)!(k − l)!l!

(B.111)

Substituting the above expressions into Eq.(B.107), we obtain the expression in Eq.(B.104). Notice that Tk−l is state-dependent while dk−l is not. Therefore, it all comes down to finding Tk−l for the states in different scenarios. The most general scenario is when the k H-photons are separated into r groups by the relation k = k1 + k2 + .. + kr with each group having mi (i = 1, ..., r) V-photons overlapping in time (m1 + m2 + ... + mr ≤ N ). It is very complicated to derive Tk−l for this general case. We will derive it only for the r = 2 case as an example and present the general result at the end. B.3.1 The Scenario of k1 V mH + k2 V nH In this scenario, we can group the variables in the Φ-function into three subsets, as in Sect.B.2.2: the first includes the k1 H-photons and the m V-photons, the second includes the k2 H-photons and the n V-photons, and the third includes the remaining V-photons. Because of the permutation symmetry within the first and the second groups, we may write the Gk -function as Gk (t1 , ..., tk 1 ; tk 1 +1 , ..., tk1 +k2 ; tk+1 , ..., tk+N ) ≡ G(Pk {t1 , ..., tk1 ; tk1 +1 , ..., tk1 +k2 }; PN {tk+1 , ..., tk+N }) Pk PN


= k1 !k2 !

G(Ck1 k2 {t1 , ..., tk1 +k2 }; PN {...}),

263

(B.112)

Ck1 k2 PN

where Pk is the permutation on the k variables for the k H-photons and PN for the N V-photons. Ck1 k2 is the combination operation of taking k1 variables out of the first k = k1 + k2 variables in the G-function and putting them the first k1 positions of the G-function and remaining variables in the remaining positions. There are a total of (k1 + k2 )!/k1 !k2 ! different terms in the sum of . It is straightforward to find for l = 0: Ck k 1 2

2 dt1 ...dtk+N Gk (t1 , ..., tk+N ) 2 2 (k1 + k2 )! dt1 ...dtk+N = (k1 !k2 !) G(...; PN {...}) k1 !k2 !

Tk =

= NN +k N !k1 !k2 !(k1 + k2 )!

PN

(l = 0),

(B.113)

where NN +k is similar to NN +1 in Eq.(9.100), but with N + k variables in the Φ-function. Tk−1 is the time integral part of the term in the sum in Eq.(B.107) with only one unequal pair {s, s } of indices in its corresponding phase sum, similar to that in Eq.(B.50). It has the form of Tk−1 = dt1 ...dtk+N Gk∗ (..., ts , ...; tk+1 , ..., tk+N )Gk (..., ts , ...; tk+1 , ..., tk+N ) 2 dt1 ...dtk+N = (k1 !k2 !) ∗ ∗ × G (..., ts ; ...; PN {...}) + G (...; ts , ...; PN {...}) PN

×

PN

C(k1 −1)k2

C(k1 −1)k2

Ck1 (k2 −1)

G(..., t

s

; ...; PN {...})

+

G(...; t

s

,

, ...; PN {...})

Ck1 (k2 −1)

(B.114) where we write Gk∗ , Gk in two parts: ts( ) among the first k1 variables of the G function and ts( ) among the next k2 variables of the G function. After expanding the product, all the cross terms are zero because the indices in these terms belong to groups I and II, respectively, and they are orthogonal. The non-zero direct product terms are evaluated in the same way as those in Eqs.(9.105,B.33), although results of the integration are different for s( ) in group I and in group II. The final result is then, for l = 1: (k1 + k2 − 1)! Tk−1 = (k1 !k2 !)2 m(N − 1)!NN +k (k1 − 1)!k2 ! (k1 + k2 − 1)! n(N − 1)!N + N +k (k2 − 1)!k1 ! = (k1 + k2 − 1)!k1 !k2 !(k1 m + k2 n)(N − 1)!NN +k . (B.115)

264


For l = 2, we have: Tk−2 = dt1 ...dtk+N Gk∗ (..., ts , ..., tr , ...; tk+1 , ..., tk+N ) ×Gk (..., ts , ..., tr , ...; tk+1 , ..., tk+N ),

(B.116)

with s = s = r = r . Because of Eq.(B.112), we may break Gk in Eq.(B.116) into three parts, similar to Eq.(B.114): Gk∗ (..., ts , ...,tr , ...; tk+1 , ..., tk+N ) G(..., ts , tr ; ...; PN {...}) = PN

C(k1 −2),k2

+

G(..., ts ; tr , ...; PN {...}) + G(..., tr ; ts , ...; PN {...})

C(k1 −1),(k2 −1)

+

% G(...; ts , tr , ...; PN {...}) .

(B.117)

Ck1 ,(k2 −2)

Substituting Eq.(B.117) into Eq.(B.116), we find again that only the direct product terms give non-zero integration, which is different for terms in different parts. The evaluations of the three different time integrals are similar to those in Eqs.(B.19, B.35). We then obtain: (k1 + k2 − 2)! Tk−2 = (k1 !k2 !)2 m(m − 1)(N − 2)!NN +k (k1 − 2)!k2 ! (k1 + k2 − 2)! 4mn(N − 2)!NN +k + (k1 − 1)!(k2 − 1)! (k1 + k2 − 2)! n(n − 1)(N − 2)!NN +k + k1 !(k2 − 2)! = [k1 (k1 − 1)m(m − 1) + 4k1 k2 mn + k2 (k2 − 1)n(n − 1)] ×(k1 + k2 − 2)!k1 !k2 !(N − 2)!NN +k . (B.118) Similarly, we have, for l = 3: (k1 + k2 − 3)! (k1 + k2 − 3)! (3) m + (C 1 )2 m(2) n(1) Tk−3 = (k1 !k2 !)2 (k1 − 3)!k2 ! (k1 − 2)!(k2 − 1)! 3 (k1 + k2 − 3)! (k1 + k2 − 3)! (3) (C32 )2 m(1) n(2) + n + (k1 − 1)!(k2 − 2)! k1 !(k2 − 3)! (B.119) ×(N − 3)!NN +k , where m(i) ≡ m(m − 1)...(m − i + 1) and m(0) = 1, and furthermore, for arbitrary l ≤ k = k1 + k2 : Tk−l = (N − l)!NN +k (k1 !k2 !)2 2 l l! (k1 + k2 − l)! × m(i) n(l−i) , (B.120) (k1 − i)!(k2 − l + i)! i!(l − i)! i=0

where, as before, the sum comes directly from the break-up of the Gk -function.


265

B.3.2 The Most General Scenario It is straightforward to generalize Eq.(B.120) to the most general scenario k1 Hm1 V...kr Hmr V , as described right before Sect.B.3.1. We present only the result as (k1 + ... + kr )! NN +k N ! k1 !...kr ! = NN +k N !k1 !k2 !...kr !(k1 + ... + kr )! (l = 0),

Tk = (k1 !k2 !...kr !)2

(B.121)

and for l ≥ 1, l

Tk−l = (k1 !k2 !...kr !)2

i1 ...ir i1 +...+ir =l

2 (k1 + ... + kr − l)! l! (k1 − i1 )!...(kr − ir )! i1 !...ir ! (i )

r) ×m1 1 ...m(i r (N − l)!NN +k .

(B.122)

After substituting the above results into Eq.(B.107), we obtain: PN +k (k1 Hm1 V...kr Hmr V ) = (N + k)!k1 !...kr !NN +k [1 − VN +k ], (B.123) with the visibility as VN +k =

k l=1

(−1)l−1

l i1 ...ir i1 +...+ir =l

(i ) (i ) Cki11 ...Ckirr m1 1 ...mr r l! . (B.124) i1 !...ir ! (N + k − 1)...(N + k − l)

Index

asymmetric beam splitter, 159, 169, 208–213 beam-like PDC, 33 Bell measurement, 69 Bell states, 68 de Broglie wavelength four-photon, 171 N-photon, 163, 174 three-photon, 169 two-photon, 95 dispersion cancellation, 55–56 distinguishability N-photon, 189 three-photon, 200 two pairs, 132, 149–155, 182, 187 two-photon, 186 entanglement swapping, 147 evolution operator, 18, 24–26 beam splitter, 167, 238, 242 parametric amplifier, 102, 165 exchange symmetry, 181, 188, 189, 200, 201, 205 four-photon interference, 156 four-wave mixing, 19, 20 Franson interferometer, 89–95, 99 ghost fringes, 78–81 Glauber-Sudarshan P-representation, 11, 241

indistinguishability N-photon, 189 three-photon, 200 two pairs, 132, 182, 187 two-photon, 186 NOON state four-photon, 166 N-photon, 163, 166–168 three-photon, 163–165 two-photon, 95 NOON state projection measurement, 172, 173, 195 normalization factor, 188–190, 192, 202 orthogonality of exchange, 182, 189, 200, 201 phase matching, 20, 21, 27–32, 41 extended, 31, 54 type-I, 19, 24, 27, 28, 30, 41 type-II, 19, 24, 27, 30, 31, 33, 41 photon anti-bunching effect, 67, 101–104, 106 photon bunching effect multi-photon, 177–181, 195 PDC, 122–125 two pairs, 159, 194, 195 two-photon, 60, 128, 142, 154 quantum state of PDC, 18, 26 broad band, 39 entangled, 33 four-photon, 26

268

Index

frequency entangled, 70 narrow band, 34 polarization entangled, 3, 19, 63 time entangled, 89 stimulated emission, 177–181 pair, 121, 125 teleportation, 142–147 three-wave mixing, 19 transform-limitedness, 115–118 single-photon, 118–121, 232, 235 two-photon wave, 7, 86, 88–92, 94–96, 99

visibility of interference classical maximum, 12 four-photon, 142, 147, 148, 154, 162, 171, 177, 248 N-photon, 204–208, 212, 213, 251, 255, 258, 261, 265 three-photon, 170, 199 two-photon, 50, 55, 57, 60, 66, 77, 85, 87, 99, 127, 129, 130, 140 coherent state, 8 Franson interferometer, 94 thermal source, 8, 127, 130, 140 type-I, 50 type-II, 51 wide band pumping, 52–55

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