Laser Spectroscopy: Proceedings of the XVI International Conference, Palm Cove, Queensland, Australia 13-18 July 2003

Laser Spectrocopy

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Palm Cove, Queensland, Australia 13 - 18 July 2003

Laser Spectroscopy Proceedings of the XVI InternationalConference

editors

Peter Hannaford Andrei Sidorov Swinburne University of Technology, Australia

Hans Bachor Ken Baldwin Australian National University

K- World- Scientific

N E W JERSEY * LONDON

SINGAPORE

SHANGHAI

HONG KONG * TAIPEI * CHENNAI

Published by

World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA ofice: Suite 202,1060 Main Street, River Edge, NJ 07661 UK ofice: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.

LASER SPECTROSCOPY Proceedings of the XVI International Conference

Copyright 0 2004 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereof; may not be reproduced in any form or by any means, electronic or mechanical, includingphotocopying, recording or any information storage and retrieval system now known or to be invented, without wrinen permissionfrom the Publisher.

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Printed in Singapore

Preface The 16* International Conference on Laser Spectroscopy (ICOLSO3) was held from 14 to 18 July, 2003 in Palm Cove, a beautiful coastal village near the Great Barrier Reef in Tropical North Queensland, Australia. Following the tradition of previous ICOLS conferences - held in Vail, MCgkve, Jackson Lake, Rottach-Egern, Jasper Park, Interlaken, Mau, h e , Bretton Woods, Font-Romeu, Hot Springs, Capri, Hangzhou, Innsbruck and Snowbird - Palm Cove provided an informal and remote scenic setting for researchers in the field to meet and discuss the latest developments and applications in Laser Spectroscopy. ICOLSO3 was attended by 230 scientists from 23 countries, including Australia, Austria, Canada, Denmark, England, Estonia, France, Germany, Iran, Israel, Italy, Japan, Korea, the Netherlands, New Zealand, Pakistan, Poland, Russia, Sweden, Switzerland, Scotland, Taiwan and the USA. The scientific program comprised 12 topical sessions, with 46 invited talks chosen by the Program Committee and 154 poster contributions. The papers reflected the remarkable new developments that had taken place in the field during the previous 12 months. These included ultra-precise spectroscopic measurements on atoms and molecules based on optical frequency combs including tests of the stability of fundamental constants; the successful production of cold anti-hydrogen; the realization of Bose-Einstein condensation in caesium and ytterbium; the behavior of ultra-cold bosons and fermions in optical lattices; the production of ultra-cold caesium, helium and fermionic lithium molecules; the coherent transport of ultra-cold atoms on the surface of chips; quantum information processing with nonclassical light and in cavity QED experiments; and the implementation of quantum algorithms and experiments towards a scalable quantum computer based on trapped ions. These Proceedings comprise a collection of invited and selected contributions presented at ICOLSO3. We would like to thank the participants, particularly those who contributed talks, posters and manuscripts, for making ICOLSO3 such an exciting and memorable conference. We thank the Program Committee and the International Steering Committee for their expert help and advice in putting together an excellent scientific program. We gratefully acknowledge the financial support of our sponsors: the Australian V

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Academy of Science, the Australian Institute of Physics, the ARC Centre of Excellence for Quantum-Atom Optics, The Australian National University, Coherent Inc, Coherent Scientific, Lastek, Sacher Lasertechnik, Spectra-Physics, Swinburne University of Technology, Toptica Photonics, Wiley-VCH and World Scientific. Finally, we thank our Conference Secretariat, Maria Lamari, for her tireless efforts in the organization of ICOLSO3; David Lau for his help with the editing of the Proceedings; and the staff of the Palm Cove Novotel Resort, whose friendly and efficient service contributed much to the success of the conference. Peter Hannaford and Andrei Sidorov Swinbume University of Technology, Melbourne Hans Bachor and Ken Baldwin The Australian National University, Canberra August 2003

Steering Committee F.T. Arecchi, E. Arimondo, H.-A. Bachor, R. Blatt, N. Bloembergen, C.J. BordC, R.G. Brewer, S. Chu, W. Demtroder, M. Ducloy, M.S. Feld, J.L. Hall, P. Hannaford, T.W. Htinsch, S. Haroche, S.E. Harris, L. Hollberg, M. Inguscio, V.S. Letokov, A. Mooradian, Y.R. Shen, F. Shimizu, K. Shimoda, B.P. Stoicheff, S. Svanberg, H. Walther, Y .Z. Wang, Z.M. Zhang

Program Committee A. Aspect, H. Bachor, S. Bagayev, R.J. Ballagh, R. Blatt, S. Chu, W. Ertmer, A.I. Ferguson, J.L. Hall, P. Hannaford, T.W. Hiinsch, M. Inguscio, G. Milburn, D.E. Pritchard, C. Salomon, F. Shimizu, Y.Z. Wang

List of Sponsors The Australian Academy of Science The Australian Institute of Physics The ARC Centre of Excellence for Quantum-Atom Optics The Australian National University Coherent Inc Coherent Scientific Lastek Sacher Lasertechnik Spectra-Physics Swinburne University of Technology Toptica Photonics Wiley-VCH World Scientific

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Contents Precision Spectroscopy

1

Laser Frequency Combs and Ultraprecise Spectroscopy T.W. Hansch

3

Optical Clocks with Cold Atoms and Stable Lasers L. Hollberg, C.W. Oates, G. Wilpers, E.A. Curtis, C. W. Hoyt, S.A. Diddams, A. Bartels and T.M. Raymond

14

Optical Lattice Clock: Precision Spectroscopy of Neutral Atoms in Tight Confinement H. Katori and M. Takamoto

22

A Clock Transition for a Future Optical Frequenky Standard with Trapped Atoms I. Courtillot, A. Quessada, R.P. Kovacich, A. Brusch, D. Kolker, J.-J. Zondy, G.D. Rovera and P. Lemonde

30

Ultracold Atomic Strontium: From Unconventional Laser Cooling and Future Optical Standards to Photon-Free Anisotropic Many Body Physics T. Loftus, X.-Y. Xu, T. Ido, M. Boyd, J. L. Hall, A. Gallagher and J. Ye

34

Ultracold Calcium Atoms for Optical Clocks and Collisional Studies 37 U.Sterr, C. Degenhardt, H. Stoehr, G. Wilpers, T. Binnewies, F. Riehle, J. Helmcke, Ch. Lisdat and E. Tiemann Comparison of Two Single-Ion Optical Frequency Standards at the Sub-Hertz Level Chr. Tamm, T. Schneider and E. Peik

40

Limits on Temporal Variation of Fine Structure Constant, Quark Masses and Strong Interaction V.V. Flambaum

49

ix

X

Testing the Stability of Fundamental Constants using Atomic Fountains S. Bize, M. Abgrall, H. Marion, F. Pereira Dos Santos, I. Maksimovic, S. Zhang, Y. Sortais, C. Vian, J. Griinert, L. Cacciapuoti, C. Mandache, Ph. Laurent, P. Lemonde, P. Rosenbusch, G. Santarelli, A. Clairon and C. Salomon

58

Extending the Optical Comb Synthesizer to the Infrared: From He at 1.083 pm to COZat 4.2 pm P. De Natale, S. Borri, P. Cancio, G. Guisfredi, D. Mauotti, M. Prevedelli, C. de Mauro and M. Inguscio

63

Cold Atom Gyroscope for Precision Measurements F. Leduc, D. Holleville, J. Fils, A. Clairon, N. Dimarcq, A. Landragin, P. Bouyer and Ch. J. Bordk

68

New Optical Tests of Special Relativity H. Miiller, S. Herrmann, C. Braxmaier, A. Peters and S. Schiller

71

Ultrafast Spectroscopy

75

Ultra-Precise Phase Control of Short Pulses: Applications to Nonlinear Spectroscopy J. Ye, L. Chen, R.J. Jones, K. Holman and D.J. Jones

77

Optimal Control of Molecular Femtochemistry T. Brixner, G. Krampert, P. Niklaus and G. Gerber

85

Spectrally Resolved Femtosecond 2-Colour 3-Pulse Photon Echoes for Studies of Molecular Dynamics L.V. Dao, C.N. Lincoln, R.M. Lowe and P. Hannaford

96

Quantum Degenerate Gases

101

Experiments with a Bose-Einstein Condensate of Cesium Atoms T. Weber, J. Herbig, M. Mark, T. Kraemer, C. Chin, H.-C. Nagerl and R. Grimm

103

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Bose-Einstein Condensation of Ytterbium Atoms Y. Takahashi, Y. Takasu, K. Maki, K. Komori, T. Takano, K. Hondu, A. Yamaguchi, K. Kato, M. Mizoguchi, M. Kumakura and T. Yabuzaki

111

Momentum Spectroscopy of Phase Fluctuations of an Elongated Bose-Einstein Condensate A. Aspect, S. Richard, F. Gerbier, M. Hugbart, J. Retter, J. H. Thywissen and P. Bouyer

116

Experimental Study of a Bose Gas in One Dimension W.D. Phillips, M. Anderlini, J.H. Huckans, B. Laburthe Tolra, K.M. O’Hara, J.V. Port0 and S.L. Rolston

124

Quantum Degenerate Bosons and Fennions in a 1D Optical Lattice C. Fort, G. Modugno, F.S. Cataliotti, J. Catani, E. de Mirandes, L. Fallani, F. Ferlaino, M. Modugno, H. Ott, G. Roati and M. Inguscio

129

Dynamics of a Highly-Degenerate, Strongly-Interacting Fermi Gas 137 J.E. Thomas, S.L. Hemmer, J.M. Kinast, A.V. Turlapov, M.E. Gehm and K.M. O’Hara Spectroscopy of Strongly Correlated Cold Atoms A.J. Daley, P.O. Fedichev, P. Rabi, P. Zoller, A. Recati, J.I. Cirac, J. Von Deljl and W. Zwerger

145

Strong Correlation Effects in Cold Atomic Gases B. Paredes, G. Metalidis, V. Murg, J.I. Cirac and C. Tejedor

153

Stochastic Gauge: A New Technique for Quantum Simulations P. Drummond, P. Deuar, J.F. Comey and K. Kheruntsyan

161

Growth and Stabilization of Vortex Lattices in a Bose-Einstein Condensate C.W. Gardiner, A S . Bradley, A.A. Penckwitt and R.J. Ballagh

171

A Storage Ring for Bose-Einstein Condensates C.S. Garvie. E. Riis and A.S. Arnold

178

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Bragg Spectroscopy of an Accelerating Bose-Einstein Condensate K.J. Challis, R. W. Geursen, R.J. Ballagh and A.C. Wilson

181

Dispersion Management and Bright Gap Solitons for Atomic Matter Waves B. Eiermann, Th. Anker, M. Albiez, M. Taglieber, P. Treutlein and M.K. Oberthaler

184

All-Optical Realization of an Atom Laser Based on Field-Insensitive Bose-Einstein Condensates G. Cennini, G. Ritt, C. Geckeler and M. Weitz

187

Dynamical Effects of Back-Coupling on an Atom Laser N.P. Robins, J.E. Lye, C.S. Fletcher, S.A. Haine, J. Dugui, C. Breme, J.J. Hope and J.D. Close

191

Cold Molecules and Cold Collisions

195

Photoassociation Spectroscopy of Ultracold Metastable Helium Atoms: Numerical Analysis M. Leduc, M. Portier, J. Leonard, M. Walhout, F. Masnou-Seeuws, K. Wilner and A. Mosk

197

Production of Long-Lived Ultra-Cold Liz Molecules from a Fermi Gas J. Cubizolles, T. Bourdel, S.J.J.M. F. Kokkelmuns, C. Salomon and G. Shlyapnikov

205

Feshbach Resonances in Dilute Quantum Gases M.J. Holland

212

Atom Optics and Interferometry

221

Cold Atoms Near Metallic and Dielectric Surfaces M.P.A. Jones, C.J. Vale, D. Sahagun, B.V. Hall, B.E. Sauer, E.A. Hinds, C.C. Eberlein, F. Furusawa and D. Richardson

223

...

XI11

Coherent Atomic States in Microtraps Ph. Treutlein, P. Hommelhofi T.W. Hansch and J. Reichel

23 1

Atom Optics with Microtraps and Atom Chips: Assembling Tools for Quantum Information Processing L. Feenstra, K. Brugger, R. Folmn, S. Groth, A. Kasper, P. Kriiger, X . Luo, S. Schneider, S. Wildermuth and J. Schmiedmayer

237

On-Chip Laboratory for Bose-Einstein Condensation

242

J. Fortrigh, H. Ott, S. Kraf, A. Giinther, C. Truck and C. Zimmermann Atom Optics and Quantum Information Processing with Atoms in Optical Micro-Structures M. Volk, T. Miither, F. Scharnberg, A. Lengwenus, R. Dumke, W. Ertmer and G. Birkl

248

A Controllable Diffraction Grating for Matter Waves H. Oberst, S. Kasashima, F. Shimizu and V. I. Balykin

25 3

Cavity QED

257

Cavity QED by the Numbers H.J. Kimble, A. Boca, A.D. Boozer, W.P. Bowen, J.R. Buck, C.W. Chou, L.-M. Duan, A. Kuzmich and J. McKeever

259

Manipulating Mesoscopic Fields with a Single Atom in a Cavity S. Haroche, A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune and J. M. Raimond

265

Vacuum-Field Mechanical Action on a Single Ion

273

J. Eschner, P. Bushev, A. Wilson, F. Schmidt-Kaler, C. Becher, C. Raab and R. Blatt Coupling of Atoms, Surfaces and Fields in Dielectric Nanocavities 277 G. Dutier, I. Hamdi, P.C.S. Segundo, A. Yarovitski, S. Saltiel, M.-P. Gorza, M. Fichet, D. Bloch, M. Ducloy, D. Sarkisyan, A. Papoyan and T. Varzhapetyan

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Quantum Optics and Quantum Information

285

Ion Crystals for Quantum Information Processing F. Schmidt-Kaler, H. Hafier, W. Hansel, S. Gulde, M. Riebe, T. Deuschle, J. Benhelm, G.P.T. Lancaster, J. Eschner, C. Becher, C.F. Roos and R. Blatt

287

Building Blocks for a Scalable Quantum Information Processor Based on Trapped Ions D. Leibji-ied,M.D. Barrett, A. Ben Kish, J. Britton, J. Chiaverini, B. DeMarco, W.M. Itano, B. JelenkoviC,J.D. Jost, C. Langer, D. Lucas, V. Meyer, T. Rosenband, M.A. Rowe, T. Schaetz and D.J. Wineland

295

Controlled Transport of Single Neutral Atom Qubits D. Schrader, S. Kuhr, W. Alt, Y. Miroshnychenko, I. Dotsenko, W. Rosenfeld, M. Khuderverdyan, V. Gomer, A. Rauschenbeutel and D. Meschede

304

Ferreting out the Fluffy Bunnies: Entanglement Constrained by Generalised Super-selection Rules H.M. Wiseman, S.D. Bartlett and J.A. Vaccaro

307

Photon Number Diagram for Characterizing Continuous Variable Entanglement W.P. Bowen, M.T.L. Hsu, T. Symul, A.M. Lance, B.C. Buchler, R.S. Schnabel, N. Treps, H.-A. Bachor, P.K. Lam and T.C. Ralph

315

Continuous Variable Teleportation within Stochastic Electrodynamics H.J. Carmichael and H. Nha

324

Surpassing the Standard Quantum Limit for High Sensitivity Measurements in Optical Images Using Multimode Non Classical Light C. Fabre, S. Gigan, A. Maitre, M. Martinelli, N. Treps, U.Andersen, P.K. Lam, W. Bowen, B. Buchler, N. Grosse and H.-A. Bachor

334

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Novel Applications and New Laser Sources

343

Self-Organization of Atomic Samples in Resonators and Collective Light Forces A.T. Black, H. W. Chan and V. Vuletii

345

Photoionization of Cold and Ultracold Rubidium Atoms M. Anderlini, D. Ciampini, E. Courtade, F. Fuso, 0.Morsch, J.H. Miiller and E. Arimondo

353

Superluminal and Ultra-Slow Light Propagation in Room-TemperatureSolids R. W. Boyd, M.S. Bigelow and N.N. Ltpeshkin

362

Abraham’s Force on a Highly Dispersive Medium P.D. Lett and L.J. Wang

365

Optically Pumped VECSELs for High Resolution Spectroscopy: The New Ti:Sapphire? R.A. Abram, M. Schmid, E. Riis and A.I. Ferguson

369

Medical Applications

373

Seeing Small Biological Structures with Light G. Popescu, C.M. Fang-Yen, L.P. Dejlores, M. Chu, M. Hunter, M. Kalashnikov, K. Badizadegan, C. Boone, R.R. Dasari, M.S. Feld, H. Iwai, V. Backman and G. Stoner

375

Laser Cardiomagnetometry: Our Hearts Beat for Cesium R. Wynands, G. Bisen and A. Weis

383

Author Index

391

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Precision Spectroscopy

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LASER FREQUENCY COMBS AND ULTRAPRECISE SPECTROSCOPY* T.W. HANSCH Max-Planck-lnstitutfr Quantenoptik * Hans-Kopfermann-Str. I , 0-85748 Garching, and Sektion Physik, Ludwig-Maxirnilians- Universitiit Schellingstr. 4, 0-80799 Munich, Germany

For three decades, precision spectroscopy of atomic hydrogen has motivated advances in laser spectroscopy and optical frequency metrology. Recently, femtosecond laser optical frequency comb synthesizers have arrived as revolutionary tools for ultraprecise optical spectroscopy. Preliminary results of a new absolute frequency measurement of the hydrogen 1 S-2s two-photon resonance are reported.

1. Introduction The first International Conference on Laser Spectroscopy (ICOLS) was held in Vail, Colorado in 1973. As a newly appointed Professor at Stanford, I was then proud to present a Doppler-free saturation spectrum of the red hydrogen Balmera line, recorded with a pulsed tunable dye laser. The 2s Lamb shift appeared clearly resolved in the optical spectrum. This was the beginning of a long adventure in precision spectroscopy of the simple hydrogen atom, which permits unique confrontations between experiment and theory. Figure 1 illustrates how the accuracy of optical spectroscopy of atomic hydrogen has improved over time. Classical spectroscopists remained limited to about six or seven digits of precision by the large Doppler broadening of hydrogen spectral lines. In 1971, our group at Stanford overcame this barrier by nonlinear laser spectroscopy. Other groups, notably in New Haven, Oxford, and Paris, soon joined in to improve the accuracy by three orders of magnitude over the next two decades. Around 1990, a new barrier appeared: the limits of optical wavelength metrology due to unavoidable geometric wavefront errors. Progress beyond a few parts in 10'' has been achieved only because we have learned increasingly well how to measure the frequency of light rather than its wavelength. In 2000, the accuracy had reached 1.9 parts in 1014 (see Ref. 1). Extrapolating, we should expect an accuracy of a few parts in 10" at this ICOLS 2003 meeting. However, further progress is becoming difficult, because we are * www.mpq.mpg.de

3

4

again approaching a barrier: the limits of how well we know our unit of t h e , the second. Since 1967 the second has been defined in terms of the 9 GHz ground state hyperfine splitting of atomic cesium. Cesium atomic clocks have been continually refined, as shown by the dashed line in Fig. 1. With the latest generation of laser cooled cesium atomic fountain clocks, one can now reach an accuracy of a few parts in but the potential for further improvements seems almost exhausted. However, our optical frequency counting techniques make it now feasible to develop optical atomic clocks, based on sharp optical resonances in laser-cooled trapped ions, neutral atoms or molecules. With such clocks future spectroscopic measurements may reach accuracies of parts in 10" and beyond. 10-5 10 6 10-7 10-8 10-9 %

2 3 8 Q .-9 3 3 L

10-10 10-1' 10-12

1013 1014 1045 1016 1017 1940 1960 1980 2000 2020

year Figure 1 . Advances in the relative accuracy of optical spectroscopy of atomic hydrogen,

In atomic hydrogen, the highest resolution can be achieved on the ultraviolet 1s-2s two-photon resonance with a natural linewidth of only 1 Hz. At Garching, we observe this resonance by collinear excitation of a cold hydrogen atomic beam. The hydrogen atoms are produced by microwave dissociation of molecules and cooled to a temperature of about 6 K by collisions with the walls of a nozzle mounted to a helium cryostat. A collinear standing wave field at 243 nm for Doppler-free two-photon excitation is produced by coupling the

5

frequency-doubled output of a dye laser into a buildup cavity inside the vacuum chamber. Atoms excited to the 2s metastable state after a travel path of about 10 cm are detected by applying a quenching electric field and counting the emitted vacuum ultraviolet Lyman-a photons. The laser light is periodically blocked by a chopper, and the photon counts are sorted into bins corresponding to different delay times. With slow atoms selected by a delay time of 1.3 ms, the linewidth is now reduced to about 530 Hz at 243 nm corresponding to a To measure the line position to 1% of this width, we resolution of 4.3 would have to reach an accuracy of 5 parts in 10".

2.

Optical Frequency Measurements

As recently as 1996, an optical frequency measurement would have required a large and highly complex harmonic laser frequency chain.2 In 1997, we demonstrated a frequency interval divider chain, which can stay in a convenient region of the spectrum by working with frequency differences rather than with the frequencies themselves. We never built a complete optical frequency counter, but we used a 4-stage divider chain to bridge a 1 THz frequency interval. This approach made it possible to measure the hydrogen 1S2s frequency with a conventional harmonic frequency chain. A transportable CH4-stabilizedHe-Ne-laser served as an intermediate ref er en~ e. ~ Since 1999, optical frequency measurements have been enormously simplified with the advent of femtosecond laser optical frequency comb syr~thesizers.~~~ In a now common implementation, the pulse train from a Kerrlens mode-locked Ti:sapphire laser is sent through a microstructured silica fiber, which broadens the spectrum by nonlinear processes so that white light emerges which can be dispersed by a diffraction grating into a rainbow of colors. Remarkably, successive light pulses are so highly phase correlated that the spectrum presents an evenly spaced comb of several hundred thousand sharp spectral lines. The spacing is precisely given by the pulse repetition frequency. The entire comb is displaced by some offset frequency that arises from a slip of the carrier wave relative to the pulse envelope from pulse to pulse. With a frequency comb spanning more than an optical octave, it is straightforward to measure this carrier-envelope offset (CEO) frequency. We only need to produce the second harmonic of the red part of the comb spectrum and observe a beat note with the blue lines of the original comb. Once we can measure the offset frequency, we can control it or even make it go away, so that the frequencies of the comb lines become simply integer harmonics of the pulse repetition rate. Such a frequency comb provides a direct link between optical frequencies and microwave frequencies. This link can be used in either direction. We can measure or control the pulse repetition rate with a cesium atomic clock and synthesize a dense comb of optical reference frequencies which are directly linked to the primary standard of time. Or we can start with a sharp optical

6

reference line in some cold trapped ion, cold atoms, or slow molecules, and lock a nearby comb line to this optical reference. All the other comb line frequencies are thus rational multiples of the optical reference frequency, and the repetition frequency becomes a precisely known fraction. The comb synthesizer can thus act as a clockwork for future generations of optical atomic clocks. Such clocks will slice time into a hundred thousand times finer intervals than microwave cesium clocks. Standards laboratories in many industrialized nations have begun research programs aimed at the development and perfection of such optical atomic clocks, and several contributions at th~sconference are reporting on advances and new ideas in this intriguing field. Extensive review articles have been written on optical frequency comb synthesizer^.^ Here, I may perhaps add some personal perspective on this development. The idea of using the frequency comb of a mode-locked laser for high resolution spectroscopy is not new. Already in the late seventies, our group at Stanford had demonstrated that a mode-locked picosecond dye laser could produce a frequency comb which we used to measure fine structure intervals in atomic sodium.6 The origin of the comb spectrum is well explained in a classic textbook.’ Consider an arbitrary optical waveform circulating inside an optical cavity. During each roundtrip, an attenuated copy escapes through a partly transmitting mirror. A single copy will have a broad and more or less complicated spectrum. However, two identical copies end-to-end will produce interference fringes in the spectrum, somewhat reminiscent of Young’s double slit experiment. Three copies produce a spectrum that resembles the interference pattern of a triple-slit, and an infinite series of copies produces sharp lines which can be identified with the modes of the cavity. In a real laser, successive pulses will not be identical replicas. Because of dispersion, the carrier wave inside the resonator travels with a phase velocity that differs from the group velocity of the pulse envelope. The resulting carrierenvelope (CE) phase slip may amount to thousands of cycles during a roundtrip, but only the remainder modulo 2n is relevant. As a result, all the comb lines are displaced by a CEO frequency that equals this phase slip per pulse interval. T h s relationship has already been discussed in detail in the 1978 Stanford Ph.D. thesis of Jim Eckstein. A first interferometric observation of pulse-to-pulse phase shifts was reported in 1996 by the group of F. Krausz in Vienna.8 In the late 1970s, we did not seriously consider absolute frequency measurements with a laser frequency comb, because the spectrum of the available dye laser was much too narrow. In the early 1990s, the technology of ultrafast lasers advanced dramatically with the discovery of Kerr-lens mode locking. I remember a trade show in 1994, when I was captivated by an exhibit of a (Coherent Mira) mode-locked Tisapphire femtosecond laser with regenerative amplifier. The laser beam was focused into a glass slide to produce a white light continuum which a prism dispersed into a rainbow of colors. A striking feature was the laser-like speckle pattern, which indicated a high degree

7

of spatial coherence. However, the speckle did not imply anything about the allimportant phase correlations between successive pulses, and the pulse repetition frequency of a few hundred kHz remained inconveniently low for frequency comb experiments. Nonetheless, I felt sufficiently intrigued to acquire such a system for our frequency metrology laboratory in 1994. We did not pursue the femtosecond laser approach seriously right away, because we had come quite far in perfecting our alternative scheme of optical interval division. An accurate measurement of the 1s-2s frequency seemed almost within reach. We also felt that we would need an independent tool to verify any measurement with a femtosecond laser frequency comb, since the frequency metrology community would otherwise distrust any results. The measurements involving optical interval dividers took longer than anticipated. In 1997 we finally published a result for the 1s-2s frequency with an accuracy of 3.4 parts in 1013,a record for optical frequency measurements at the time.3 From this result, together with other spectroscopic measurements, we could determine a new value of the Rydberg constant and of the 1 s ground state Lamb shift. If one believes in QED, the same measurement also yields an accurate value for the mean quadratic charge radius of the proton, and the hydrogen deuterium isotope shft provides the structure radius of the deuteron. We were proud that our table-top experiment exceeded the accuracy of earlier measurements by electron scattering with large accelerators by an order of magnitude. Exploring alternatives to our frequency interval divider chain, we also began experiments with electro-optical frequency comb generators, kindly provided by M. Kourogi. Such cavity-enhanced phase modulators readily generate combs of modulation sidebands extending over several THz.' It seemed now compelling to try to produce even broader frequency combs with a mode-locked femtosecond laser, and Thomas Udem was getting ready to take a closer look at the frequency spectrum of our Mira laser, to be later joined by J. Reichert and R. Holzwarth. At that time, in March of 1997, I visited the European laboratory for nonlinear spectroscopy (LENS) in Florence, Italy, and watched M. Bellini working with an amplified femtosecond laser system. As is common in many ultrafast laboratories, he produced a white light continuum by focusing part of the laser pulse train into a plate of calcium fluoride. I asked what would happen if the laser beam were split into two parts and focused in two separate spots; would the two white light continua interfere? Most people in the laboratory did not expect to see interference fringes, but when we tried the experiment, using a Michelson interferometer to carefully adjust the relative timing of the two pulses, we soon observed stable interference fringes of high contrast for all the colors visible to the eye." The two white light sources had to be mutually phase-coherent. No matter how complicated the process of white light continuum generation might be, the process was reproducible. If the two pulses were separated in time rather than in space .... In the next few days I sat

8 down to write a detailed six page proposal for a universal frequency comb synthesizer which essentially described the now common self-referencing scheme. The idea seemed so close to reality now that I asked M. Weitz and Th. Udem to witness every page of this proposal on April 4, 1997. This vision provided a new motivation for our team to seriously explore the potential of Kerr-lens mode-locked femtosecond lasers for optical frequency comb synthesis. By that time, hundreds of such lasers were in use in laboratories around the world, but they were mostly used to study ultrafast phenomena. Nobody had ever looked for any comb lines, as far as we could tell. With a repetition frequency of 90 MHz, the comb spectrum of our Mira laser was so densely spaced that no spectrometer in our laboratory could resolve the comb lines. Therefore, we resorted to heterodyne detection, employing a cw diode laser as a local oscillator. The diode laser beam and the pulse train were superimposed with a beam splitter, and a beat signal was detected with an avalanche photodiode after some spectral filtering. After paying attention to the mechanical stability of the femtosecond laser, we did observe stable comb lines. Next, we investigated the spacing of the comb lines. We phase-locked two diode lasers to two arbitrarily chosen comb lines and used an optical interval divider stage to produce a new frequency precisely at the center. A beat note with the nearest comb line confirmed, much to our delight, that the comb lines were pefectly evenly spaced, way out into the wings of the emission spectrum, within a few parts in 10" (see Ref. 11). It was now certain that the frequency comb of such a mode-locked femtosecond laser could serve as a ruler in frequency space to measure large optical frequency intervals. In a first demonstration of a precision measurement with a femtosecond laser, we determined the frequency interval between the cesium D1 resonance line and the fourth harmonic of a transportable CH4stabilized 3.39 pm He-Ne-laser, which had been calibrated with a harmonic laser frequency chain at the PTB Braunschweig.12 The optical cesium frequency allows a determination of the fine structure constant from the atomic recoil energy as measured by atom interferometry in the group of Steve Chu at Stanford. Afterwards, we focused ow efforts on the more ambitious goal to measure an absolute optical frequency relative to a cesium atomic clock in our own laboratory. With a frequency comb spanning an entire octave, we could have simply measured the interval between a laser frequency and its second harmonic. However, in early 1999, we did not yet have such a broad comb. We could produce combs of some 60 THz width by broadening the spectrum of our Mira laser by self-phase modulation in a short piece of ordinary optical fiber. Therefore, we relied on some interval divider stages to produce 112 and 417 of the frequency of the 486 nm dye laser in the hydrogen spectrometer. We could then bridge the remaining gap with our frequency comb to determine the laser frequency itself. As a reference, we first used a commercial cesium atomic clock,

9

and then a highly accurate transportable cesium fountain clock (PHARAO), built at the LPTF in Paris. In June of 1999, this first absolute optical frequency measurement with a femtosecond laser frequency comb yelded a new value of the hydrogen 1s-2s frequency accurate to 1.4 loL4, surpassing all earlier optical frequency measurements by more than an order of magnitude.’ Members of the frequency metrology community, such as J.L. Hall in Boulder, who had remained extremely skeptical, soon became ardent evangelists for the new femto-comb approach. Just before the completion of these measurements, a new tool for the generation of octave spanning frequency combs appeared on the horizon. Researchers at Bell Laboratories demonstrated a microstructured “rainbow fiber”, which could broaden the spectrum of the nano-joule pulses of a modelocked femtosecond laser oscillator to a white light continuum. After the whte light interference experiments in Florence, I felt rather confident that this magic fiber would preserve the phase coherence of successive pulses and produce useable comb lines with a desirable large frequency spacing. However, our efforts to obtain a sample of this fiber were foiled by the lawyers at Lucent Technologies. Fortunately, we learned that the group of P. St. J. Russel at the University of Bath has long been producing similar photonic crystal (PC) fibers, and in November of 1999, we could finally try such a fiber in OUT laboratory. At that time, we had acquired a small Ti:sapphire ring laser (GigaOptics GmbH, GigaJet) producing pulses of about 25 fs at a repetition frequency of 625 MHz. Launchmg about 170 mW into a 30 cm long PC fiber, we immediately produced a spectrum spanning more than an octave. In the fiber, a small fiber core is surrounded by air-filled holes, which give a large change in the effective refractive index. Part of the light travels as an evanescent wave in air, so that the region of zero group velocity dispersion is shifted down to match the wavelength of the Ti:sapphire laser. The injected laser pulses can thus maintain a high intensity, without quickly spreading by dispersion. The detailed mechanism of spectral broadening is still under investigation, with self-phase modulation and soliton splitting identified as important contributors. However, the process is sufficiently reproducible that useable comb lines can be produced throughout the spectrum. Today we know that similar broad spectra can also be produced with tapered communication fibers, and a growing number of laboratories are demonstrating laser oscillators which produce almost an octave or even a full useable octave without any need for external spectral br~adening.‘~,’~ With such an octave-spanning comb, it is now a simple task to realize a self-referencing frequency comb synthesizer. We only need to select a few thousand comb lines from the red end of the specwhich form pulses that are intense enough to be frequency doubled in a nonlinear crystal. The comb lines in the doubled spectrum are displaced from the precise integer harmonics of the pulse repetition frequency by twice the CEO frequency. A collective beat note with corresponding lines in the original comb therefore directly reveals the

10

CEO frequency, The absolute frequency of each comb line is then determined by two radio-frequencies, which can be measured precisely, and an integer mode number, which can be identified uniquely by a wavelength measurement with a common wavemeter. This type of self-referencing frequency synthesizer was first realized by D. Jones et al. in Boulder,” who obtained a fiber sample from Bell Laboratories a few weeks before we received our fiber from the University of Bath. In a first stringent test, Ronald Holzwarth has compared such an octave spanning frequency comb synthesizer with the more complex frequency synthesizer used in the 1999 hydrogen frequency measurement.16 By starting with a common 10 MHz radiofrequency reference and comparing comb lines near 350 THz, he could verify agreement within a few parts in probably limited by Doppler shifts due to air pressure changes or thermal expansion of the optical tables. In 2002, a group at the PTB Braunschweig demonstrated how a femtosecond laser frequency comb generator can be used as a transfer oscillator to precisely measure optical frequency ratios.” As a test case, they measured the frequency ratio between the second harmonic of a Nd:YAG laser and the fundamental frequency, verifying the expected value of 2 with an uncertainty of 7 parts in More recently, M. Zimmermann in our laboratory has pushed a related experiment to an uncertainty of 6 parts in 10” (see Ref. IS). So far, we have not identified any systematic error that would limit the precision of spectroscopic measurements or the performance of future optical atomic clocks. Commercial frequency comb synthesizers are now being produced by Menlo Systems GmbH, founded by some of my former students.” 3.

New Frequency Measurement of Hydrogen 1s-2s

In February 2003, we used such an octave spanning comb synthesizer in a new measurement of the hydrogen 1S-2s transition frequency. M. Fischer and N. Kolachevsky have implemented many improvements in the hydrogen spectrometer. The interaction region is now differentially pumped to reduce line shifts and the loss of slow atoms due to collisions with background gas. The passive reference cavity for stabilization of the dye laser has been much improved with mirrors optically contacted to a monolithic spacer of ULE ceramic. The cavity housing with its temperature stabilization and acoustic shielding is mounted on an active vibration isolation stage. Light from the dye laser is sent through a fiber into the frequency metrology laboratory, where the optical frequency has been compared by M. Zimmermann and Th. Udem to the radio frequency of the Paris PHARAO atomic fountain clock, which has again been brought to Garching.

11 16000-

nozzle temperature: 7 K

2000-

-20

-10

10

Figure 2. Hydrogen IS-2s line profiles for different delay times between turning off the laser and counting Lyman-a photons.

With an immediate absolute frequency calibration, the hydrogen spectroscopy could be performed with unprecedented comfort. Figure 2 shows line profiles corresponding to different delay times between turning off the laser and observing the signal Lyman-a photons. A viable line shape model must predict all these line profiles correctly, starting from reasonable assumptions about the atomic trajectories and velocity distribution. Line shifts due to the ac Stark effect are readily observable, so that one can extrapolate to zero intensity. An initially somewhat worrysome aspect has been a seeming correlation between the observed line center and the nozzle temperature. Since the temperature of the nozzle is affected by the gas flow, such a correlation might give evidence for some small pressure shift. In the meantime, the experimental data have been analyzed more thoroughly by M. Haas and U. Jentschura at the University of Freiburg, using several different line shape models. Their analysis concludes that the dependence on nozzle temperature is not statistically significant, but they ignore data when the nozzle temperature deviates by an excessive amount. In this way, one obtains a preliminary value for the hyperfine centroid of the hydrogen 1s-2s frequency of 2 466 061 413 187 127 (18) Hz, with a relative statistical uncertainty of 7.10-15. The uncertainty due to systematic shfts is still under investigation. M. Fischer is currently conducting more tests to make sure that pressure shifts can really be ruled out. The preliminary new result agrees well with the 1999 measurement of 2 466 061 413 187 103 (46) Hz. A difference of 24 (50) Hz in 43 months corresponds to a relative drift of the 1s-2s transition frequency of 2.8 (5.7) lo-'' per year, i.e. it is compatible with zero drift. This experiment has recently attracted some attention because it can be considered as a test for a possible slow variation of the fine structure constant.

12

As discussed by V. Flambaum in this volume, astronomical observations of spectral lines in the light of distant quasars suggest that the fine structure constant a had a smaller value during the early age of the Universe. Assuming a per year. H. linear rate of change, a might be changing by about 1 part in Fritsch has recently argued that grand unification would imply that a cannot change simply by itself.” If all known forces are to remain unified at very high energies, other coupling constants must change as well. As a result, the masses and magnetic moments of hadrons should change relative to those of the electron. Fritsch is pointing out an intersecting magnifying effect: “we would expect that radiation emitted in a hyperfine transition should vary in time about 17 times more strongly than light emitted in a normal atomic transition, but in the opposite direction, i.e. the atomic wavelength becomes smaller with time, but the hyperfine wavelength increases.” So far, we have not found any evidence for such a change. However, it certainly remains an important goal to further improve the accuracy of spectroscopic measurements to establish more stringent limits on possible slow variations of fundamental constants. 4.

Outlook

So far, we have discussed applications of femtosecond laser frequency comb synthesizers in the frequency domain. However, frequency comb techniques are also providing intriguing new tools for ultrafast physics. By controlling the CE phase of intense light pulses lasting for only a few cycles, they make it possible to study novel phenomena in ultrafast light matter interactions, such as the production of single sub-femtosecond pulses of soft Xrays in high harmonic generation.” Spectroscopy of the simple hydrogen atom has sparked off the cross fertilization of two seemingly unrelated frontiers, precise optical spectroscopy and the study of ultrafast phenomena. Future laser spectroscopy conferences will likely deal with reports on new atomic clocks, fundamental tests and searches for time variations of fundamental constants, as they will highlight results from the new frontier of attosecond physics.

References 1. 2. 3. 4. 5. 6. 7.

M. Niering et al., Phys. Rev.Lett. 84, 5496 (2000). H. Schnatz et al., Phys. Rev.Lett. 76, 18 (1 996). Th. Udem et al., Phys. Rev.Lett. 79, 2646 (1997). J. Reichert et al., Opt. Commun. 172, 59 (1999). Th. Udem et al., Nature. 416,233 (2002). J.N. Eckstein et al., Phys. Rev.Lett. 40, 847 ( 1 978). A.E. Siegman, “Lasers”, University Science Books, Mill Valley, 1986.

13 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

L. Xu et al., Opt. Lett. 21, 2008 (1996). T. Udem et al., Opt. Lett. 23, 1387 (1998). M. Bellini and T.W. Hhsch, Opt. Lett. 25, 1049 (2000). Th. Udem et al., Opt. Lett. 24, 881 (1999). Th. Udem et al., Phys. Rev. Lett. 82, 3568 (1999). T.M. Fortier, D.J. Jones, and S. Cundiff, Opt. Lett., submitted. www.gigaoptics.de D. Jones et al., Science. 288, 635 (2000). R. Holzwarth et al., Phys. Rev. Lett. 85, 2264 (2000). J. Stenger et al., Phys. Rev. Lett. 88, 073601 (2002). M. Zimmermann et al., Opt. Lett., submitted. www.menlosystems.com X. Calmet and H. Fritzsch, Phys. Lett. 540, 173 (2002). A. Baltuska et al., Nature. 421,611 (2003).

OPTICAL CLOCKS WITH COLD ATOMS AND STABLE LASERS’

’,

L. HOLLBERG, C.W. OATES, G. WILPERS, E.A. CURTIS C.W. HOYT S.A. DIDDAMS, A. BARTELS, AND T.M. RAMOND

’,

National Institute of Standards and Technology 32.5 Broadway, Boulder, CO,8030.5, USA E-mail: hollberg @boulder.nist.gov The performance and prospects for neutral-atom optical frequency standards are discussed based on our recent progress with a calcium optical frequency standard. Second stage narrow-line cooling to microkelvin (and even 300 nK) temperatures, combined with launched atoms, should reduce Doppler frequency errors to about Ix Advanced femtosecond optical frequency combs allow direct comparisons between the Ca optical standard, the Hg’ optical standard and the Cs primary standard. These comparisons provide independent “reality checks” on both the stability and accuracy. Relative frequency measurements also constrain the possible time variation of atomic energy levels and fundamental constants.

1. Introduction NIST is developing optical frequency standards based on laser-cooled and trapped calcium atoms, and also single Hg’ ions.” Both systems are promising candidates for the next generation of frequency standards and atomic clocks. In this paper we focus primarily on recent progress in laser-cooling and high precision spectroscopy of the calcium optical clock transition. The results are relevant to other neutral-atom frequency standards, particularly with respct to factors that affect stability and accuracy, which motivates much of our work. Precision spectroscopy of narrow transitions in cold (but moving) atoms deals with the questions of determining the exact lineshape and finding the center of the natural atomic resonance. The push toward atomic frequency standards based on optical transitions is inspired by the predicted improvement in stability and accuracy. Our work builds on the pioneering ideas and work of visionary scientists over more than 40 years.3v Just now, the ideas are coming to fruition because of combined advances in three technical areas of laser physics: laser cooling and trapping of atoms, highly stabilized cw-lasers, and a convenient method for optical synthesis using mode-locked lasers.

* A contribution of NIST, an agency of the US government, and not subject to copyright.

Also, University of Colorado, Boulder, CO 80305, USA. f4

15

I . 1. Optical Frequency Standards: a Stability Advantage It is clear that the use of higher-frequency oscillations for a clock will divide time into smaller units, and thus provide more precise timing and higher frequency stability. The fractional frequency fluctuations of an atomic frequency standard in the quantum projection noise limit, with N atoms and averaging time z varies as ( A V / V ~ ) ( I / N A ~ ~simplistic ~~. interpretation of the atomic frequency stability suggests that an optical standard with lo6 atoms and a 1 Hz linewidth 19 -NZ will give’ q ( z ) =2x10- z . However, it might be foolish to extrapolate so far beyond our present experience. To reach the more fhdamental atomic limits will require efficient use of the atoms, and extremely stable lasers as localoscillators. Outstanding progress in that direction has been demonstrated with lasers locked to cavities at the mHz level,6 and also with the stability of the combined laser-plus-cavity system at the ~ x 1 O -level l ~ for 0.5 to 200 s.’ In our present Ca frequency standard there are several factors that limit the short-term stability, including a natural linewidth of about 470 Hz, and a very poor ratio of “clock time” (about 1 ms) to cycle time (= 30 ms). Nonetheless, an instability of about 2x 10‘l6 I”’should be possible.’** Fortunately, the stability of our laser when locked to a short, high-finesse optical cavity (= 2 Hz at 1 s) is reasonably well matched to the present Ca atomic stability.*

2.

Calcium Optical Frequency Standard

The calcium inter-combination transition at 657 nm (‘SOf) 3P1) was identified long ago for its potential use as an optical frequency standardclock. Our work benefits notably from the calcium project at PTB (U. Sterr et al., see t h s Proceedings) and in many ways is a collaborative effort. The 657 nm clock transition is spectrally sharp (470 Hz natural width) and can be detected with excellent signal-to-noise using shelving detection, factors which have led to demonstrated frequency i n s t a b i l i ~ of 4 x 10-’52-1’2. Furthermore, the calcium transition is remarkably immune from perturbing effects due to external electric ( = 3 ~ 1 0 Hz/(V/m)2) -~ and magnetic fields (0.5 Hz/G2), and any cold-atom collision shift is so small that it proves difficult to detect. The Ca resonance line at 423 nm (34 MHz linewidth) allows rapid cooling, but is limited to simple Doppler cooling temperatures of = 2 mK, corresponding to velocities of =70 c d s . T h ~ ssignificant thermal velocity and residual drift velocity when the atoms are released from the MOT, combined with experimental constraints on the geometry of the probe laser fields, put real limits on the accuracy that could be achieved. To date, the uncertainties in the clock frequency at 456 THz are 21 Hz at NIST, and 8 Hz at PTB, due primarily to 1st-order Doppler-related ~hifts.~,’’ While the Ca frequency standard has exceptional stability, it is not yet competitive in accuracy with the best

16

microwave standards and single-ion optical standards that now show (= 1 Hz at optical frequencies). reproducibility at about 1x 2.1. Doppler Problems The seriousness and complexity of the first-order Doppler-related frequency shfts have been addressed in a number of papers, including seminal work by BordC and Hall,'' and recent papers focusing on cold atom clocks and interferometers.12, l 3 The basic problem is that time-dependent phases of optical fields seen by the atom appear as frequency shifts to the observed line center. For an atom in free-flight with gravity, and interacting with a laser field, the optical phase that the atom sees is a h c t i o n of position r and interaction time ti, and is given approximately as @(?(ti))

=

1 k' -(f0+Go .ti + -g

2

. t ; > +k rperp (ti 1' 2R

where vo is the initial atomic velocity, g is the gravitational acceleration, rperpis the radial distance of the atom from the center of the laser beam and R is the radius of curvature of the fields.

Figure 1. Geometry for saturated absorption spectroscopy of a ball of cold atoms. In the present context - saturated absorption spectroscopy of a sample of cold atoms released from a MOT - the motional phase shifts can be separated into a number of different terms that have simple physical descriptions and specific fimctional dependencies on R, v, t* and g. For ow situation the dominant terms are the atom velocity coupled to the imperfect cancellation of the forward and backward k-vectors, and the wavefront curvature of the probe fields. With single-stage cooling and mK atoms we typically have: V,herma/ = 70 c d s , vdr# = 10 c d s , k-vector misalignment uncertainty = 40 p a d , angle to gravity (a-d2) < 1 mrad, and R 2 40 m. Using these parameters and interaction times of = 1 ms, the dominant terms each result in frequency shifts of 1 to 10 Hz, fractionally a few xlO-I4! Thus, our recent efforts (and similarly those of PTB) have focused on advanced laser cooling schemes to reduce the velocity, combined with atom interferometry to improve the probe beams.

17

2.2. Quenched Narrow-Line Cooling

Since Ca lacks hyperfine structure that can provide ‘‘free” sub-Doppler cooling, we instead take advantage of the narrow clock transition that offers excellent velocity selectivity (0.3 m m / s for a 470 Hz linewidth). The 370 ps lifetime is too long to be useful for simple Doppler cooling of mK atoms in gravity. Nevertheless, an additional laser can be used to quench the 3P1excited state more rapidly and allow a reasonable cooling rate with the narrow clock transition. By simultaneously applying 657 nm and 552 nm light we observe temperatures of =lo pK in 3-dimensions while retaining = 30 % of the atoms, as shown in Fig. 2.

Figure 2. Quenched narrow line cooling of calcium atoms using 657 nm plus 552 nm. The right panel shows the velocity distribution with just 423 nm cooling, and with

additional second stage cooling. A pulsed variant of the quenched-cooling method reduces the velocity by another factor of five and produces sub-recoil 1-D temperatures = 300 nK, v = 1 cm/s.’ This pulsed narrow-line cooling could be done in 3-D but it would be somewhat complex. With the Ca thermal velocity reduced by a factor of fifteen, the velocitydependent systematic errors are correspondingly reduced and the signal contrast is increased (Fig. 3). Unfortunately, even velocities of 5 cm/s will not be good enough to reach the accuracies that we strive for, and we are left with several Doppler frequency shifts of about 200 mHz, fractionally 5 ~ 1 0 - l ~ Obviously, . it would be advantageous to have the atoms confined to the Lamb-Dicke limit, as is possible with ions. Several people have proposed using optical lattices for this p u r p ~ s e . ’ ~Optical clocks based on these concepts look promising, but additional complications such as magnetic structure need to be worked out. In the mean time, substantial further progress can be made with our existing cold neutral atoms.

18

40 -

50 h

30

20

-

10 .

0

0 2 4 6 -600 -400 -200 0 200 400 60( (MW (kW Figure 3. Saturated absorption optical Ramsey fringes on the 657 nm calcium clock transition created by the four-pulse Bordt method. Both vertical axes are in percentage of atoms excited, and the horizontal axes show laser frequency minus calcium frequency (but note the different scales). The left-hand panel shows a low-resolution scan of the fringes taken with 2.5 mK atoms, while the graphs on the right use 10 pK atoms. The dashed trace without fringes shows the Doppler width of the 10 pK atoms. The underlying width of the Ramsey pedestal results from the Doppler width in the case of the mK atoms, while in the case of the 10 pK atoms it is the Fourier-transform limit of the probe pulses. -6

-4

-2

To W e r reduce the Doppler and beam geometry problems we are now using cold atoms to diagnose the quality of the probe laser fields. The idea is simple but powerful: by launching balls of cold atoms with r d s velocities, rather than c d s , we amplify by = 200 times the problems associated with non-ideal optical probe beams. To do tlus, we use a 3-pulse collinear non-frequency selective atom interferometer described by Trebst. l3 Launching the atoms through the probe laser fields, we enhance the phase-shifts resulting from wavefront errors and can use the results to adjust the wavefronts of the laser fields. Similarly, the 4-pulse Ramsey-BordC frequency-sensitive interferometer is used with launched atoms to adjust the probe beam alignment. Controlling both the launch velocity and the interaction time allows us to separate the wavefront errors from those associated with gravity and spatial offset. Preliminary results indicate that these methods allow correction of the laser wavefront to an effective radius of > 300 m,and the beam alignment to about 5 microradians. This implies that we should be able to reduce the velocity-dependent frequency shifts to a level of UN 1 ~ 1 0 - I ~If. so, they will be well below other frequency uncertainties. We must now more seriously address effects such as the 300 K blackbody radiation induced Stark shift, the Stark shift due to the clock laser itself, any collision shift, the residual phase-shifts in the probe fields due to AOMs and switches, inhomogeneity in optical probe fields, and other mechanical-, thermal- and gravity-induced optical phase changes. At thls point,

19

we don’t really know what the ultimate limitation in accuracy will be for the calcium optical frequency standard.

3.

Optical Frequency Combs and Comparisons of Atomic Standards

Optical frequency combs based on femtosecond mode-locked lasers are used with our optical frequency standards to realize optical atomic clocks, to coherently connect optical and microwave frequencies, and to make intercomparisons between the Ca, Hg’ and Cs standards at NIST. Recent advances in comb technology greatly extend their useful spectral coverage and make the systems quite reliable, allowing long-term operation. In particular, a 213-octave frequency comb is created with a broadband 1 GHz Ti:Sapphire laser that is self-referenced using the 2f-3f method without any microstructure fiber.15 Coherently linking t h s broadband laser with a = 500 MHz mode-locked Cr:Forsterite laser (1.3 pm) provides a comb of modes from 570 to 1450 nm, which can be referenced to the optical standards or a microwave source.16 In both the optical and microwave regions, we have explored the time- and frequency-domain characteristics of optically-controlled femtosecond combs, and find that they have remarkably high fidelity in accuracy and stability. In fact, many of our results are still limited by our measurement systems rather than by the performance of the combs. At the present time, the optically-referenced frequency combs at NIST show: Frequency reproducibility optical-to-optical I 4x lo-’’ Frequency instability optical-to-optical 6 . 3 1~0-l6t l Repetition rate instability by optical cross-correlation 2 2 x Z’ Timing jitter detected by optical cross-correlation 5 0.4 fs (1-100 Hz BW) 0 Photodiode-generated microwave instability I 1 x 1O-I47’ Phase-noise on photodiode-generated 1 GHz < -125 dBc/Hz (100 Hz offset) Reproducibility of repetition-rate detected on photodiodes < 2x These results indicate that femtosecond optical frequency combs are not a limitation to the performance of the current generation of optical atomic clocks. If we use atomic transitions as our references for time, frequency, and length measurements, we must be confident that these do not vary with the operating conditions (other than the effects predicted by Einstein’s relativity). There is always the hdamental question that Dirac raised, of whether the forces of nature and structure of atoms evolve with time; and thus whether our reference of time depends on time. Renewed interest in this topic comes from three areas: an increasing body of astronomical data suggesting” that about 10 billion years ago the fine structure constant was different from the present value by 1 part in lo5, new test theories that are being formulated that parameterize searches for physics beyond relativity and the standard model,‘*, and that now there is more than one type of atomic frequency standard that can provide accuracy at the to 1 0 - l ~level.

20

>

JulOO

J a n 01

JulOl

J a n 02

Ju102

J a n 03

Figure 4. Hg’ optical frequency relative to the cesium primary frequency standard.” Using optical frequency combs, we have made several comparisons of the relative frequency of Ca, Hg’ and Cs standards. Published frequency measurements of Hg’ relative to the NIST-F1 primary cesium standard are shown in Fig. 4. The average of all the data gives the Hg’ optical frequency with uncertainties of f l . 1 Hz statistical and +lo Hz systematic for the electric quadrupole shift not yet evaluated. A linear fit using the total uncertainty gives a slope of -0.24 f 1.3 Hz/yr, indicated by the bold line. Following the PrestageD ~ u b a ”2o~ model for evaluating frequency comparisons in terms of a possible time variation of the fine structure constant a, and assuming that gCs(m,/q) is constant, gives (l/a)(da/dt) 5 0.5 f l . l x lO-”/year. Similarly, measurements of calcium in terms of cesium have been made at PTB through the years using harmonic frequency chains, and more recently at NIST and PTB using optical combs as shown in Fig. 5.

1996

1997

1998

1999

2000

2001

Figure 5 . Ca optical frequency in Hz from published results from NIST (symbol-0) and PTB (symbol-+)for the past 6 years. The line represents the weighted linear fit.

21

The data show steady improvement in accuracy of the Ca standard over time, but no obvious temporal dependence. A weighted linear fit to the data gives a slope of 0.8 +12 Hz/yr. At the present accuracy, we find no systematic temporal variation of either the Hg' or Ca optical frequencies relative to cesium. 4.

Summary

Optical atomic clocks are no longer just a promise of the future, but are here today. The reported reproducibility for some the ion-based optical standards are similar to the best microwave atomic frequency standards, and optical neutrals look as though they can also be competitive. Short-term instabilities of optical standards are already orders of magnitude better than other sources. With rapid improvements in all the atomic standards due to cold atoms, better lasers, and microwave sources, we anticipate more stringent tests of fundamental physics, such as time variation of fundamental constants, Einstein's relativity, symmetry postulates, and searches for other forces, as well as new technical capabilities and applications. Support for this work has been provided by NIST and in part by ONRMUM and NASA. We gratefully acknowledge the important contributions by J.C. Bergquist, S. Bize and the Hg' group, U. Sterr and the PTB Ca group, S. Jefferts and the Cs fountain group, and long-term interactions with J.L. Hall.

References 1. T. Udem, et al., Phys. Rev. Lett. 86,4996(2001). 2. C.W. Oates, et al., Optics Lett. 25, 1603 (2000). 3. J.L. Hall, IEEE J. Sel. Top. Quantum Electron. 6,1136 (2000). 4. T. Udem, et al., Nature, 416,233 (2002). 5. L.Hollberg, et al., IEEEJ. Quantum Electron. 37, 1502 (2001). 6. C. Salomon, et al., J. Opt. SOC.Am. B 5, 1576 (1988). 7. B.C. Young, et al., Phys. Rev. Lett. 82,3799(1999). 8. G. Wilpers, et al., Phys. Rev. Lett. 89,230801 (2002). 9. E.A. Curtis, Thesis, Dept. ofphysics, University of Colorado, Boulder (2003). 10. J. Helmcke, et al., IEEE Trans. Instrum. Meas. 52,250(2003). 11. C.J. BordC, et al., Phys. Rev. A 14,236 (1976). 12. K. Bongs, et al., arXiv:quant-ph/0204102 v2 (2002). 13. T. Trebst, et al., IEEE Trans. Instrum. Measure. 50,535 (2001). 14. H. Katori, ICOLS-03 Proceedings (2003). 15. T.M. Ramond, et al., Optics Lett. 27,1842 (2002). 16. A.Bartels, et al., submitted for publication (2003). 17.V.V. Flambaum, ICOLS-03 Proceedings (2003). 18.R.Bluhm, et al., Phys. Rev. Lett. 88,090801 (2002). 19.S.G. Karshenboim, Can. J. Phys. 78,639 (2000). 20.J.D. Prestage, et al., Phys. Rev. Lett. 74,351 1 (1995). 21. S.Bize, et al., Phys. Rev. Lett. 90,150802 (2003).

OPTICAL LATTICE CLOCK: PRECISION SPECTROSCOPY OF NEUTRAL ATOMS IN TIGHT CONFINEMENT

HIDETOSHI KATORI AND MASAO TAKAMOTO Engineering Research Institute, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan

E-mail: katoriQamo.t.u-tokyo.ac.jp We report on the spectroscopy of the 5s2 ?SO(F = 9/2) -+ 5s5p 3 P ~ (F = 9/2)clock transition of s7Sr atoms (natural linewidth of 1 mHz) trapped in a one-dimensional optical lattice. A recoilless transition with a linewidth of 0.7 kHz was observed for the trap laser wavelength X M 813nm, where the light shift in the clock transition canceled out. We discuss the feasibility of realizing an optical frequency standard based on this scheme.

1. Introduction

Careful elimination of perturbations of electronic states and of motional effects has been considered as a prerequisite for realizing an atom frequency standard.' A single ion trapped in an RF quadrupole field is one of the ideal systems that satisfy these requirements,2 as the trap prepares a quantum absorber completely at rest in free space for an extended time and its electric field vanishes at the center of the trap. Employing this scheme, quantum projection noise (QPN) limited spectroscopy3 has been performed with an expected accuracy of 10-18.1,4 Despite its anticipated high accuracy, the stability of the single-ion based optical clock is severely limited by QPN; an inordinately long averaging time T is required to meet its ultimate a c ~ u r a c y One . ~ may think of increasing the number of quantum absorbers N as employed in neutral atom based optical standard^.^?^^^ In this case, however, the atom-laser interaction time sets upper bounds6 for the effective transition line &-factor, which is more than two orders of magnitude smaller than that in ion clocks, since an atom cloud in free space expands with finite velocity and is strongly accelerated by gravity during the measurement. Furthermore, it has been pointed out that residual Doppler shifts arising from an imperfect wavefront of the probe beam and atom-atom collisions during the measurement 22

23

affect its ultimate accuracy, which is predicted to approach up to 8 x in the future.8 In this paper, we discuss the feasibility of an “optical lattice clock,’” which utilizes millions of neutral atoms prepared in an optical latticelo as illustrated in Fig. l(a). Sub-wavelength localization of atoms in each lattice site suppresses the first order Doppler shift12i13i14as well as the collisional frequency shifts;* therefore the scheme simulates millions of singleion clocks operated simultaneously. In striking contrast with conventional approaches toward frequency standards,l this lattice scheme interrogates atoms while they are strongly perturbed by an external field: We will show that this perturbation can be cancelled out to below precision level15 by designing the light shift trap so as to adjust dipole polarizabilities for the probed electronic states. By applying the scheme on the ‘SO -+ 3P0 transition of Sr atoms, we demonstrate Doppler free spectroscopy with a linewidth of 0.7 kHz and determine the cancellation wavelength.16 2. Engineered Light Shift Trap

The transition frequency v of atoms exposed to a trap laser electric field of € is described as 1 1 tiv = A d o ) - -Aa(e, w)E2 - -AT(e, w)E4 - . . . , 4 64 where Aa(e,w) and Ay(e,w) are the difference of ac polarizabilities and hyperpolarizabilities of the upper and lower states that depends both on the laser frequency w and on the polarization vector e. By adjusting the polarizabilities of the upper and lower states to set Aa(e,w) = 0, the observed atomic transition frequency v will be equal to the unperturbed transition frequency do)independent of the laser intensity I 0: I€I2, as long as higher order corrections 0 ( E 4 ) are negligible. For better control of dipole polarizabilities, parameters are preferably specified only by the frequency w; the light polarization e should have less influence on the system. Our strategy is to employ the J = 0 state which exhibits a scalar light shift. We took the 5s’ ‘So -+ 5s5p3P0 forbidden transition (A0 = 698nm, see Fig. l(b)) of 87Sr with nuclear spin I = 9/2 as the “clock” t r a n ~ i t i o n in , ~ which the hyperfine mixing of the 3P0( F = 9/2) state with the lp3P1stated7 provides a finite lifetime of ril = 150s. Figure 2 shows the light shift for the ‘So and 3P0 states as a function of the trapping laser wavelength with an intensity of I = 10 kW/cm2. The calculation was performed by summing up the light-shift contributions

24

Figure 1. (a) 3D optical lattice provides atoms with Lamb-Dicke confinement while it prevents atom-atom interactions. (b) Relevant energy levels for Sr. The 5s2 ’SO and 5s5p 3P0 states are coupled to the upper respective spin states by an off-resonant standing wave light field to produce equal amounts of light shifts (M,and Me)in the electronic-vibrational clock transition. The excited atoms on the I’So) @ In) + 13P~)@ln) transitions in the light shift potentials were quenched into the 3P2 metastable state via the rapidly decaying 3 S ~state. The clock transition was then sensitively monitored on the IS0 - lP1 transition.

with electronic states up to n = 11 orbits.l1?l5We found the intersection wavelength to be XL M 800nm, where the light shift v,, changed with the trapping laser frequency w as dv,,/dw = -3.6 x and -1.3 x for the lSo and 3P0 state, respectively. This precision enhancement of more than a factor of lo9 allows one to control the light shift well within 1 mHz by defining the coupling laser within 1 MHz. Owing to the hyperfine interaction (HFI), the clock transition no longer consists of simple scalar states, hence the tensor light shift arises. Its contribution was calculated for the 3P0 state by taking into account the electricdipole coupling to the 5s6s 3S1and 5s4d 3D1 hyperfine manifolds that are energy shifted due to HFI; both of these manifolds provide half ( M 65 kHz) of the total light shift but dominate its tensor component.15 The inset of Fig. 2 shows the result with the light polarization parameterized as e = cos 0 e- i sin 6 e+ (e* represent the unit vector for polarization). The m = f 1 / 2 states can be best used for the “clock” transition, as they exhibit the smallest polarization-dependence of less than 1 Hz, which allows one to control the light shift within 1 mHz by defining B within 1 mrad. Higher order corrections are included in the hyperpolarizability y(e, w ) and in the higher-order multipole corrections to the polarizability cr(e,w).15>18 The fourth-order ac Stark shifts at the intersection wavelength are calculated to be AE:i:/h M -5.3 Hz and AEi;A/h M

,*

+

+

25 -1 00

h

N

I

5

-150

E r (I)

zm

tj -200 0

m

-250 700

750

800

850

900

950

Laser Wavelength (nm) Figure 2. Light shift as a function of the trapping laser wavelength for a laser intensity of I = 10kW/cm2. The solid and dashed lines show the light shifts for the lS0 and 3Po states, respectively, which intersect at XL sz 800nm. The inset demonstrates the insignificance of the polarization-dependent light shifts of the 3 P (~F =. 9/2) state by taking into account the dipole coupling to the 3S1and 3D1 states at X L In the presence of a magnetic field Bo = 30 mG.

-2.3 Hz for the lS0 and 3P0 states,15 respectively, for the trapping laser intensity of 10kW/cm2. The second order Stark shifts due to the magnetic dipole and electric quadrupole terms are estimated to be 7 orders of magnitude smaller15 than that of the electric dipole term. Therefore the higher order corrections can be as small as 5 x Further elimination of this systematic error can be achieved by extrapolating the trapping laser intensity to zero.14

3. Experiment 3.1. Spectroscopy of Bound Atoms Ultracold 87Sr atoms were produced and loaded into a 1D optical lattice as described previously.19 In search for the 1 mHz narrow 'SO- 3P0 transition,20i16we strongly saturation-broadened the clock transition. By guiding the clock and trap lasers in the same optical fiber as shown in Fig. 3(a), the clock laser ( P = 1 mW) was exactly focused onto the trapped atom cloud to achieve a peak power density of I p = 120 W/cm2, which provided a saturation broadening of 110 kHz. We facilitated the first search by frequency modulating the clock laser with a spectrum spread of tens of MHz as well as by optically coupling the 3P0 state to the rapidly de-

26

-

I.o

h

v)

.c

0.8

S

2

-

v

0.6

v)

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5

1D FORL & Pro 0.2

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L

-150

-

. , I

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-100

. ,,

-50

r L

,

. , . , . , . 0

50

100

,

150

l

200

Clock laser detuning (kHz) Figure 3. (a) The clock laser was superimposed on the Ti-Sapphire laser (used for trapping atoms) and coupled into a polarization-maintaining single-mode fiber. They were then focused onto the ultracold atom cloud and the trapping laser alone was retroreflected to form an optical lattice. (b) The ground state population as a function of the clock laser detuning. The base line fluctuation of M 15% was due to the shot-to-shot fluctuation of atoms loaded into the optical lattice. (c) The carrier component with a linewidth of 0.7 kHz (FWHM) was observed at AL = 813 nm.

caying 3Sl state with X = 679 nm laser radiation to quench the 3P0 state lifetime and to simultaneously transfer the population into the long-lived 3P2 metastable state via the 3Sl state (see Fig. l(b)). These techniques together allowed us to easily find the clock transition with a linewidth of tens of MHz and with nearly 100 % excitation efficiency. The excitation of the clock transition was sensitively detected by monitoring the ground state population employing the cyclic 'SO- 'PI transition. We then alternately chopped the clock and quenching lasers to completely remove the light shift and broadening due to the quenching laser. Figure 3(b) shows the 'So state population as a function of the clock laser detuning. Reduction of the higher order sidebands was observed because of the tight confinement of atoms with a Lamb-Dicke parameter q M 0.26. The upper and the lower sidebands at f R / 2 n M f 6 4 kHz correspond to the SO) 8 In) .+ I3Po) @ In f 1) transitions, where 0 and In) denote the oscillation frequency and the vibrational state of atoms in the lattice potential, respectively. The asymmetry in the heating and cooling sidebands inferred a mean vibrational state occupation of (n)M 0.5 or an atom temperature of T = 2.8 pK. The narrowest linewidth of 0.7 kHz, as shown in Fig. 3(c), was observed16 at XL M 813 nm by reducing the clock laser intensity down to I p = 0.1 W/cm2, where the saturation broadening of 0.5 kHz

27

was comparable to the clock laser frequency jitter.

3 . 2 . Determination of Degenerate Wavelength

The mismatch of the confining potentials in the clock transition introduces an additional linewidth broadening for the carrier component. The energy shift of atoms in the n-th vibrational state of the i = e (excited) or g (ground) electronic state of the clock transition is written as

where u i ( X ~< ) 0 is the light shift at the anti-node of the standing wave and C l i ( X ~ ) / 2 7 ~M ~ - / X L is the vibrational frequency of atoms in the fast axis of the lattice potential. Taking u, as the transition frequency for the ('SO)@In)4 l3PO) @ In) vibrational transition as shown in Fig. l(b), the transition frequency difference between adjacent vibrational transitions, 6w = w,+1 - u,, is calculated to be equal to the vibrational frequency mismatch 6R = SZ, - SZ, of the lattice potentials in the excited and the ground states. At a finite temperature T , the occupation probability p , of atoms in the n-th vibrational state obeys the Boltzmann distribution law, p , + l / p , = exp(-hfl,/lcBT) = fs. Therefore, the carrier spectrum for atoms in a non-degenerate light shift trap (Re# 0,) consists of several Lorentzian excitation profiles with frequency offset given by 6 0 and their peak height weighted by the Boltzmann factor fB. Figure 4(a) demonstrates the linewidth broadening of the carrier spectrum at the trapping laser wavelength XL = 820 nm: The profile was fitted by 4 Lorentzians corresponding to n = 0 , 1 , 2 , 3vibrational states to extract the differential vibrational frequency 6 0f 27r = 0.8 kHz and the Boltzmann factor fB M 0.5. By applying this fitting procedure, the trapping-laserwavelength dependent vibrational frequency mismatch SR f 2n was determined as shown by filled circles in Fig. 4(b). These data points were then interpolated by a quadratic polynomial, which approximated the wavelength dependence of the frequency mismatch, to find the degenerate wavelength to be X L = 813.5 f 0.9 nm.I6 The result was also confirmed by observing the reduction of the clock transition linewidth as shown in Fig. 4(c). This degenerate wavelength agreed with the calculation to within 2 % (see Fig. 2), in which the discrepancy may be attributed to the truncation in summing up the light shift contributions and to the limited accuracy of the available transition strengths.

28

Lattice laser wavelength (rim)

Wavelength (nm)

Figure 4. (a) The carrier spectrum (open circles) for the I?S'o) @ In) -+ I3Po) @ In) transition measured at the lattice laser wavelength XL = 820 nm. This lineshape was fitted by 4 Lorentzian profiles corresponding to 7t = 0,1,2,3 vibrational transitions to deduce a differential vibrational frequency SR/27r x 0.8 kHz. (b) The frequency mismatch was plotted as a function of the lattice laser wavelength to determine the degenerate wavelength ( 6 0 = 0). (c) The degenerate wavelength was also confirmed by measuring the reduction of the carrier linewidth.

4. Conclusion

We have discussed the feasibility of precision spectroscopy of neutral atom ensembles confined in an optical lattice, by applying a light-shift cancellation technique on the 'So ( F = 9/2) -+ 3Po ( F = 9/2) clock transition of 87Sr, which has negligibly small tensor as well as higher-order light shifts. These features will allow one to measure the bare atomic transition frequency at 1 mHz precision, or with an accuracy of better than in the presence of strong perturbation due to the trapping light fields. Since this scheme is equivalent to millions of single ion-clocks operated in parallel, a thousand times improvement in stability over state-of-the-art ion clocks can be expected, leading to an exceptionally low instability of C T ~ ( TM) with an interrogation time of only r = 1s. This may open up new applications in ultra precise metrology, such as the search for a time variation of fundamental constants" and real time monitoring of the gravitational frequency shift.

Acknowledgments The authors would like to thank M. Yasuda and K. Okamura for their experimental assistance and V. G. Pal'chikov for the calculation of higher

29 order corrections in t he light shift. H. K. acknowledges financial support from the Japan Society for the Promotion of Science under Grant-in-Aid for Young Scientists (A) KAKENHI 14702013, and from PRESTO, Japan Science an d Technology Corporation.

References 1. See articles in ''Requency Measurement and Control," edited by Andre N. Luiten, Springer Topics in Applied Physics (Springer-Verlag, Berlin, 2001). 2. H. Dehmelt, IEEE Trans. Instrum. Meas. 31,83 (1982). 3. D. J. Wineland, J. J. Bollinger, W . M. Itano, F. L. Moore, and D. J. Heinzen, Phys. Rev. A 46,R6797 (1992). 4. R. J. Rafac, B. C. Young, J. A. Beall, W. M. Itano, D. J. Wineland, and J. C. Bergquist, Phys. Rev. Lett. 85,2462 (2000). 5. D. J. Wineland, W. M. Itano, J. C. Bergquist, and R.G. Hulet, Phys. Rev. A 36,2220 (1987). 6. F. Ruschewitz, J. L. Peng, H. Hinderthur, N. Schaffrath, K. Sengstock, and W. Ertmer, Phys. Rev. Lett. 80, 3173 (1998). 7. Th. Udem, S. A. Diddams, K. R.Vogel, C. W. Oates, E. A. Curtis, W. D. Lee, W. M. Itano, R. E. Drullinger, J. C. Bergquist, and L. Hollberg, Phys. Rev. Lett. 86,4996 (2001) 8. G. Wilpers, T. Binnewies, C. Degenhardt, U. Sterr, J. Helmcke, and F. Riehle,

Phys. Rev. Lett. 89, 230801 (2002). 9. H. Katori, Proceedings of the 6th Symposium Frequency Standards and Metrology, edited by P. Gill (World Scientific, Singapore, 2002) p 323. 10. P. S. Jessen and I. H. Deutsch, Adv. At. Mol. Opt. Phys. 37,95 (1996)' and references therein. 11. H. Katori, T. Ido, and M. Kuwata-Gonokami, J. Phys. SOC.Jpn. 68,2479 (1999). 12. R. H. Dicke, Phys. Rev. 89, 472 (1953). 13. J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. A 36,R428 (1987). 14. T. Ido and H. Katori, Phys. Rev. Lett. 91, 053001 (2003). 15. H. Katori, M. Takamoto, V. G. Pal'chikov, and V. D. Ovsiannikov, to be published in Phys. Rev. Lett. 16. M. Takamoto and H. Katori, submitted. 17. H. Kluge and H. Sauter, 2. Phys. 270, 295 (1974). 18. V. G. Pal'chikov, Yu. S. Domnin, A. N. Novoselov, J . Opt. B 5,S131 (2003). 19. T. Mukaiyama, H. Katori, T. Ido, Y . Li, and M. Kuwata-Gonokami, Phys. Rev. Lett. 90, 113002 (2003). 20. Irene Courtillot et al., arXiv:physics/O303023. 21. S . V. Karshenboim, Can. J . Phys. 78, 639 (2000).

A CLOCK TRANSITION FOR A FUTURE OPTICAL FREQUENCY STANDARD WITH TRAPPED ATOMS

I. COURTILLOT, A. QUESSADA, R.P. KOVACICH, A. BRUSCH, D. KOLKER, J-J. ZONDY, G.D. ROVERA AND P. LEMONDE BNM-SYRTE, Obseruatoire de P a r i s 61, Avenue de l'obseruatoire, '75014 Paris, France E-mail: pierre.lemonde~obspm.fr

We report the observation of the 5s2 ' S O - 5 s 5 p 3P0 transition in 87Sr. Its frequency is 429 228 004 235 (20) kHz. A resonant laser creates a leak in a magneteoptical trap (MOT): atoms build up in the metastable 3P0 state and escape the trapping process, leading to a detectable decrease in the MOT fluorescence. This line has a natural width of 1 mHz and can be used as a n optical frequency standard using atoms confined in a light shift free dipole trap.

1. Introduction

87Sr probed on the 5s2 IS0 - 5s5p 3P0line at 698 nm seems an ideal system for the realization of a new generation of optical frequency standards with neutral atoms confined in a light shift free dipole trap. Based on a proposal by H. Katori , this new generation would combine the advantages of existing standards using either a single trapped ion or an ensemble of free falling neutrals. The clock transition in 87Sr is only slightly allowed by hyperfine coupling and has a natural linewidth of N 1 mHz '. We report here the first direct observation and frequency measurement of this transition.

'

2. Indirect measurement

We first determined the frequency of the transition by measuring the frequency of the SO- Pl line at 689 nm and the frequency difference between the Pl - 3S1and Po - 3S1transitions at 688 nm and 679 nm respectively (Fig. 1). Optical frequencies were measured relative to a hydrogen maser with a scheme based on a self referenced femtosecond Ti:Sapphire laser 394 with a relative resolution of 3 x for a one second averaging time. The frequency of the's0 - 3P1(F = 9/2) transition was measured with 30

31

5s5p \"."

1"

, Fine StNChUX

5s5p ' P

splitting

clock transition

Figure 1. Left: energy diagram of the relevant 87Sr levels with wavelength and decay rate (s-l). For clarity, the hyperfine structure is not represented ( I = 9/2). Right: o experimental 'So - 3 ~ resonance.

an atomic beam, using a saturated fluorescence technique. The obtained frequency is 434 829 342 950 (100) kHz. The other measurements were performed on a sample of cold atoms collected in a magnetwoptical trap (MOT). The MOT typically contains lo7 atoms at a temperature of 2 mK. While trapped and cycling on the 'SO- 'PI transition, atoms eventually emit a spontaneous photon which brings them to the Dz state and then to the 3P1and 3Pz states (Fig. 1). Atoms in the 3P1state decay back to the ground state and are kept in the trap while atoms in the metastable 3P2 state are lost. We make the fine structure measurement by modifying this escape process. If a laser resonant with one of the hyperfine components of the 3P1- 3S1transition is added to the trap, atoms in the corresponding 3P1state are pumped to the 3P2 and 3P0 metastable states. They escape the trap instead of decaying back to the ground state. This decreases the trapped atom number. If now a 679 nm laser resonant with the 3Po - 3S1 transition is added to this scheme, a dip appears in the 688nm resonance due to coherent population trapping5 (CPT): there exists a coherent superposition of 3P1and 3P0which is not coupled to 3S1. Atoms in this dark state decay back to the ground state in a few 10 p s due to the Pl instability and are kept in the MOT, The fine structure measurement is performed with both lasers locked to resonance. We measured 5 601 338 650 (50) kHz. The 'SO- 3P0 transition is then expected to have a frequency of 429 228 004 300 (110) kHz.

32

3. Direct excitation of the clock transition

We have been able to perform a direct detection with the cold atoms by inducing in the MOT a leak to the 3P0 state with a laser tuned to resonance. The Rabi frequency on the clock transition is 1 kHz with our laser parameters. About of the 2 mK atoms are then expected to be Doppler detuned by less than the Rabi frequency. The lifetime of the trap, however, is two orders of magnitude longer than the duration of a 7r pulse. This leads to a build-up by the same factor of the fraction of atoms escaping the MOT if the transfer rate to 3P0 is constant and if the atoms, once in the 3P0state, actually escape the trapping process. The number of trapped atoms should then decrease by several percent. The Doppler effect induced by gravity is sufficient to fulfill both conditions. The experiment is operated sequentially. We alternate a capture and cooling phase with the blue lasers and a probe phase with the 698 nm laser. During the probe phase atoms are free falling. The frequency sweep caused by acceleration amounts to several times the Rabi frequency per millisecond with the 45" angle formed by the 698nm probe beam and the vertical. Both atoms transferred to the 3P0 state and the corresponding dip in the velocity distribution of the ground state are then rapidly detuned from the excitation laser. With a capture phase of 3 ms duration and a probe phase of 1 ms duration, we still have 3 x lo6 atoms in the MOT and the contrast of the resonance is 1%. In figure 1 is shown the fluorescence of the trapped atoms versus the 698 nm laser detuning from resonance. We locked the laser to the 'SO- 3P0 resonance for 2 hours and found a frequency of 429 228 004 235 (20) kHz. N

4. Conclusion

The observation of the 'SO- 3P0 transition is a first step towards the realization of an optical frequency standard using trapped Sr. With a potential line-Q of 1015 and a reasonable trapped atom number of lo6, ultimate performances are orders of magnitude better than existing devices. A. B. acknowledges a grant from the European Research Training Network CAUAC. BNM-SYRTE is Unit6 Associ6e au CNRS (UMR 8630).

References 1. H. Katori, in Sixth Symposium o n frequency standards and metrology, edited by P. Gill (World Scientific, Singapore, 2002). 2. H.-J. Kluge and H. Sauter, 2. Phys. 270, 295 (1974).

33

3. R. Holzwarth et al., Phys. Rev. Lett. 85, 2264 (2000). 4. D. J. Jones et al., Science 288, 635 (2000). 5. G. Alzetta, A. Gozzini, L. Moi and G. Orriols, Nuovo Cimento B36,5 (1976).

ULTRACOLD ATOMIC STRONTIUM: FROM UNCONVENTIONAL LASER COOLING AND FUTURE OPTICAL STANDARDS TO PHOTON-FREE ANISOTROPIC MANY BODY PHYSICS

T. LOFTUS, X.-Y. XU, T. IDO, M. BOYD, J. L. HALL, A. GALLAGHER, AND J. YE JILA, National Institute of Standards and Technology and University of Colorado, Boulder CO 80309 E-mail: [email protected]. edu We report the first experimental study of sub-Doppler cooling in alkaline earth atoms (87Sr) enabled by the presence of nuclear spin-originated magnetic degeneracy in the atomic ground state. A detailed investigation of system thermodynamics with respect t o trapping beam parameters clearly reveals sub-Doppler temperatures despite the presence of multiple, closely spaced excited-states. This novel result is confirmed by a multi-level theory of the radiative cooling force. In addition, we describe an experimental study of magnetically trapped 3 P state ~ metastable 88Sr, a system that may ultimately provide unique insights into the physics of many-body systems with anisotropic interactions.

1. Single-Stage Sub-Doppler Cooling of s7Sr

The doubly forbidden 87Sr ‘So(F=9/2) - 3 P ~ ( F = 9 / 2 )J = 0 to J = 0 transition (- 1 mHz natural linewidthl) presents unique opportunities for an ultimate optical frequency standard. Realizing this potential, however, requires the stringent separation between external degrees of freedom and internal level structure which is routinely obtained with single trapped ions. As recently demonstrated12this situation can be achieved by cooling neutral atoms into the Lamb-Dicke regime in a “magic wavelength” optical lattice. Given the relatively shallow lattice potential, pre-cooling to ultra-low temperatures is essential. Creating an 87Sr optical standard thus requires a detailed understanding of the cooling dynamics for this unique system. As a step toward this goal, we have performed the first experimental study of sub-Doppler cooling in Alkaline-Earth atoms enabled by nuclear spin originated magnetic degeneracy in the otherwise structure-free ‘SO g r o ~ n d - s t a t e Here, .~ sub-Doppler cooling is achieved in a standard six34

35

beam magneto-optical trap (MOT) despite significant spectral overlap in the excited-state manifold and the MOT magnetic field (see Figure 1). 1.67

.

,

...

.

2.0 2.0,

, I

E" 0.5 10

20

30 40

50

60

Trapping Beam Intensity (niW/an2)

I- 0.0

.

,

I

Doppler Limit

.. ....I, ,,...:;. ...::.

m

L

0

,

.'..'.40 mW/cm

E 1.5

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e

.

-;

..

27mWIm'

-F=9E 0

10

20

30

40

50

60

'So

Tmppirg Beam Detuning6 (MHz)

Figure 1 . Measured 87Sr temperature as a function of (a) trapping beam intensity at a fixed detuning of 6 = -40 MHz and (b) detuning at fixed intensities of 48 mW/cm2 and 27 mW/cmz. Doppler theory predictions (fits to sub-Doppler theory) are shown as dotted (solid) lines. For the lowest intensity in (a), the sample temperature is 300 pK, or the lowest single-step temperature achieved with Alkaline Earth atoms. (c) Cooling transition hyperfine structure. N

This surprising result, in sharp contrast to the cooling dynamics observed in similar systems such as 39K and Li14is confirmed by a fully expanded theory of the radiative cooling force that predicts, due to the large So (F=9/2) ground-state magnetic degeneracy, full-scale sub-Doppler cooling for 87Sr. Moreover, we find that measured values of the MOT damping coefficient ( a ) and spring constant ( K ) are in good agreement with the multi-level theory aside from a global scaling factor of 10. In an effort to fully explore system temperature limits, we have also characterized number and density-related heating and find heating rates for 87Sr that are roughly an order of magnitude larger than the corresponding rates in alkali systems. 2. Magnetic Trapping of Metastable 88Sr

Magnetically trapped 3P2 state metastable Alkaline Earths are expected to display both elastic and inelastic binary collision resonances that arise due to purely long-range molecular bound-states whose origin lies in an interplay between quadrupole interactions and an applied magnetic field.5 Similar interactions, and hence collision resonances, are predicted to occur for polar molecules immersed in electrostatic fields.6 Thus, studies of metastable Sr collision dynamics will likely impact the understanding of a significant and diverse range of physical systems that exhibit anisotropic interactions. To begin these ,measurements, we have studied the lifetime, loading rate, and population of 3P2state magnetic traps loaded via radiative branching

36 from a lS0 - 'PI ssSr MOT.7 Observed loading rates and trap populations scale as expected. More importantly, we find that blackbody radiation can play a key role in the system dynamics, limiting the trap lifetime in the currently employed 360 O C vapor-cell apparatus to 30 ms (see Figure 2). N

0

n,_

8

n ,

-

5.5xlO'cm~'

Depolarization 4d 3D3

Depolarization and State Change 4d 3D2 4d 3D,

1.4~10'cm4

Single Exponential Fit 7 = 32(2)ms

5~ 3p2

(b) L

5P 3P,

Hold Time (ms)

Figure 2. (a) Measured decay of magnetically trapped 3Pz state s8Sr atoms for two different background Sr vapor densities. (b) Blackbody absorption processes that induce the observed loss. Magnetically trapped states are shown in black while absorption (spontaneous emission) events that induce trap loss are shown as gray (black) arrows.

To overcome this limitation and extend our studies of "Sr, we are constructing an UHV atomic beam based apparatus. Additionally, we have recently performed the first study of underdamped two-level atom MOT oscillation dynamics, an experiment that provides important quantitative tests of Doppler cooling theorye8 References 1. H. Kluge and H. Sauter, 2. Phys. 270, 295 (1974). 2. T. Ido and H. Katori Phys. Rev. Lett. 91,053001 (2003); M.Takamoto and H. Katori (in preparation). 3. X.-Y. Xu, T. H. Loftus, J. W. Dunn, C. H. Greene, J. L. Hall, A. Gallagher and J. Ye, Phys. Rev. Lett. 90, 193002 (2003). 4. C. Fort, et al., Eur. Phys. J . D 3, 113 (1998); U. Schunemann, et al., Opt. Commun 158,263 (1998). 5. A. Derevianko, et al., Phys. Rev. Lett. 90 063002 (2002); V.Kokoouline et al., Phys. Rev. Lett. 90,253201 (2003). 6. A. Avdeenkov and J. L. Bohn, Phys. Rev. A 66, 052718 (2002). 7. T. Loftus, et al., Phys. Rev. A 66, 013411 (2002); X.-Y. Xu et al., J . Opt. SOC. Am. B 20, 968 (2003). 8. X.-Y. Xu, T. H. Loftus, M. J. Smith, J. L. Hall, A. Gallagher and J. Ye, Phys. Rev. A 66, 011401(R) (2002).

ULTRACOLD CALCIUM ATOMS FOR OPTICAL CLOCKS AND COLLISIONAL STUDIES

U. STERR, C. DEGENHARDT, H. STOEHR, G. WILPERS, T. BINNEWIES, F. RIEHLE AND J. HELMCKE Physikalisch- Technische Bundesanstalt, Bundesallee 100, 381 16 Braunschweig, Germany E-mail: uwe.sterrQptb.de

CH. LISDAT AND E. TIEMANN Institut fur Quantenoptik, Universitat Hannouer, Welfengarten 1, 30167 Hannover, Germany

1. Introduction

Calcium with its narrow intercombination line 'So - 3P1with a linewidth of 0.3 kHz is a promising candidate for an optical clock based on neutral atoms. The simple level scheme with no hyperfine structure and a single ground state is advantageous for optical clocks and precise collisional studies but it also makes the usual sub-Doppler cooling methods inapplicable. With Doppler cooling on the intercombination line in a quench-cooling scheme, however, recently temperatures of a few microkelvins have been achieved'. 2. Calcium Optical Clock

Using atoms cooled to 3 mK on the resonance line, the transition frequency was measured2 with a fs-optical frequency comb with a relative frequency uncertainty of 2 x At this temperature the uncertainty is ultimately limited by the residual first-order Doppler effect due to the atomic motion in non-ideal wavefronts of the interrogation laser beams. By use of different types of atom interferometers, in principle the leading contributions due to angular misalignment and defocusing of the laser beams can be identified and corrected3. With ultracold atoms also the effect of the higher-order aberrations can be reduced to a few parts in 10l6 due to the 37

38 1.2

i

-20

-1-

1 .o

4HzFWHM

0.8

**-

0.6 0.4

0.2 -0.9

-0.6

-0.3 0.0 0.3 Av (MHz)

0.6

i . ,. , . , . , . , . , . , . I -400 -300 -200 -100 0 100 200 300 400 Av (Hz)

0.9

Figure 1. Ramsey-Bord6 interferences obtained with ultracold calcium atoms.

Figure 2. Beat spectrum between two independent diode lasers systems.

reduced velocity spread. In an asymmetric frequency-dependent four-pulse Bordk atom interferometer with ultracold atoms the Fourier width of the exciting pulse is larger than the Doppler width of the atomic ensemble and hence almost all atoms take part in the interferometry (Fig. 1). With a shelving detection that becomes feasible with ultracold atoms, in principle an instability of the optical clock at the quantum-projection noise limit might be obtained4 that would lead to an instability below in 1 s. As interrogation laser we use an extended cavity diode-laser that is locked to a well isolated reference resonator with a linewidth of a few Hz (Fig. 2). To reach the quantum-projection noise limit, the corresponding phase stability of the laser, however, has still to be improved by more than two orders of magnitude.

0.0

,..-

E VI

-

0.6

0 VI

-?i 1.2

I 66.4

66.6

66.8

67.0

67.2

67.4

A fW z )

Figure 3. Photoassociation spectrum of the rovibrational line of the v' = 69 of the 'C$ state.

I 51.0

.

. 51.5

.

.

52.0

.

. 52.5

. 0

4WW

Figure 4. Quantum-mechanical calculation of the photoassociation signal for different scattering lengths.

39

3. Cold Collisions

To gain a better understanding of cold collisions that may lead to frequency shifts in the optical clock, photoassociation spectroscopy is performed (Fig. 3). The lack of hyperfine structure makes the comparison between theory and experiment simpler than in the case of the alkaline atoms. From the relative height of the J = 1 and J = 3 components (Fig. 4),an improved value for the ground-state scattering length between 50 and 300 Bohr radii ( U O ) could be deduced5. 4. Outlook

Even higher accuracy in an optical clock might be obtained by probing a large number of atoms at the Lamb-Dicke limit in a conservative optical trap. By choosing a “magic” wavelength that shifts both clock-states by the same amount, trapping should be possible without introducing additional shifts6. From preliminary measurements of the ac-Stark shift we have determined this ”magic” wavelength in calcium to be slightly below 800 nm for the intercombination transition. For the even narrower clock transition to the 3P0 state that can be excited in the fermionic isotopes like 43Ca, a “magic” wavelength around 720 nm is calculated. Acknowledgments This work was supported by the Deutsche Forschungsgemeinschaft under SFB 407 and SPP 1116. References 1. T. Binnewies, G. Wilpers, U. Sterr, F. Riehle, J. Helmcke, T. E. Mehlstaubler, E. M. Rase1 and W. Ertmer, Phys. Rev. Lett., 87,123002 (2001). 2. J. Helmcke, G. Wilpers, T. Binnewies, C. Degenhardt, U. Sterr, H. Schnatz and F. Riehle, IEEE Trans. Instmm. Meas., 52, 250 (2003). 3. G. Wilpers, C. Degenhardt, T. Binnewies, A. Chernyshov, F. Riehle, J. Helmcke and U. Sterr, Appl. Phys. B,76,149 (2003). 4. G. Wilpers, T. Binnewies, C. Degenhardt, U. Sterr, J. Helmcke and F. Riehle, Phys. Rev. Lett., 89, 230801 (2002). 5. C. Degenhardt, T. Binnewies, G. Wilpers, U. Sterr, F. Riehle, C. Lisdat and E. Tiemann, Phys. Rev. A , 67,043408 (2003). 6. T. Ido and H. Katori, Phys. Rev. Lett., 91, 053001-1 (2003).

COMPARISON OF TWO SINGLE-ION OPTICAL FREQUENCY STANDARDS AT THE SUB-HERTZ LEVEL CHR. TAMM, T. SCHNEIDER AND E. PEIK Physikalisch-TechnischeBundesanstalt (PTB), Bundesallee 100, 381 16 Braunschweig, Germany E-mail: christian.tann~ptb.de We describe experimental investigations on an optical frequency standard based on a laser cooled 17'Yb+ion confined in a radiofrequency Paul trap. The electric-quadrupole transition from the 'Sln(F=O) ground state to the 'D3n(F=2) state at the wavelength of 436nm is used as the reference transition. The reference transition is probed by a frequency-doubled, frequency-stabilized diode laser and is resolved with a Fourierlimited full halfwidth of approximately 30 Hz. In order to compare two I7'Yb+ standards, separate frequency shift and servo systems are employed to stabilise the probe frequency to the reference transition line centers of two independently stored "'Yb' ions. The present experimental results indicate a relative instability (Allan standard deviation) of the optical frequency difference between the two systems of ~~~(1000 s ) = ~ . O . ~ O - 'and ~ a mean frequency difference of 0.2 Hz. Shifts in the range of several Hertz are observed in the frequency difference if a stationary electric field gradient is superimposed on the radiofrequency trap field. This measurement permits a frst experimental estimate of the electric quadrupole moment of the 'Dsn state of Yb'.

1. Introduction "lYb+ is an attractive candidate for optical frequency standards based on a trapped, laser-cooled single ion because reference transitions with vanishing low-field linear Zeeman frequency shift are available in a level system with relatively simple hyperfine and magnetic sublevel structure.'v2The electricquadrupole transition 2S1,2(F=0)- 2D3n(F=2)of 171Yb+is at a wavelength of 436 nm and has a natural linewidth of 3.1 Hz. The absolute optical frequency of this transition was measured with a total lo fractional uncertainty of l.lO-I4,so that it is now one of the most accurately known atomic transition frequencies in the optical wavelength range.3v4 Here we present experimental results on the high-resolution spectroscopy of the 436 nm reference transition of 17'Yb+and initial results on the comparison of two '"Yb' optical frequency standards. With respect to the statistical uncertainty of the comparison and the ability to resolve small frequency offsets, the results yield an improvement by more than one order of magnitude over previous related work.5 Quadrupole shifts of the atomic transition frequency of the order of a few Hertz, which are introduced in one trap by superimposing a constant component on the confining radiofrequency field, can be clearly resolved. This 40

41

permits a first experimental estimate of the electric quadrupole moment of the 2D3/2state of Yb'. The quadrupole shift caused by electric stray fields is expected to be one of the largest systematic frequency shift effects in optical frequency standards that use ions with alkali-like level systems such as I 7 I Y b + and 199Hg+.6 2. Optical-Excitation Scheme

A scheme of the lowest-lying energy levels of I 7 l Y b + is shown in Fig. 1. For laser cooling, the low-frequency wing of the quasi-cyclic F=l - F=O component of the 2 SiI2resonance transition is excited, and a static magnetic field of approximately 300 pT is applied in order to prevent optical pumping to a nonabsorbing superposition of the magnetic sublevels of the F=l ground state. The natural linewidth of the resonance transition is 21 MHz, which implies a one-dimensional lunetic temperature of 0.6 mK at the Doppler cooling limit. A weak sideband of the cooling radiation provides hyperfine repumping fiom the F=O ground state to the 2P1/2(F=1)level. At the end of each cooling phase, the hyperfine repumping is switched off in order to prepare the ion in the F=O ground state.

Figure 1. Low-lying energy levels of "'Yb' and optical excitation scheme. The main spontaneous decay paths are indicated by dashed lines. Hyperfine splittings are not drawn to scale. The hyperfine splitting frequencies of the S, P, D and [3/2] levels are respectively given by 12.6 GHz, 2.1 GHz, 0.9 GHz, and 2.5 GHz.

The rapid spontaneous decay from the 2P,/2 state to the metastable 2D3/2(F=1)level that occurs during laser cooling is compensated for by coupling this level to the [3/2lIl2(F=1) state, from where the ion readily returns to the ground state. The extremely long-lived 'F712 state, whch is populated at a rate of = 0.3 h-I, is depleted by excitation to the [5/2lSl2level.

42

The F=2 sublevel of the 2D3/2state is not rapidly populated or depleted by the laser cooling excitation. Individual quantum jumps to this state due to excitation of the reference transition can therefore be detected through the interruption of the resonance fluorescence scattering. In the experiments described below, the cooling and reference transitions are excited alternately in measurement cycles of 90 ms duration. During the excitation of the reference transition, the magnetic field is decreased to the microtesla range in order to reduce the quadratic Zeeman frequency shift. When observing the absorption spectrum of the reference transition, the excitation probability to the 2D3/2(F=2)state is registered as a function of the probe laser detuning. In order to operate the system as a frequency standard, both wings of the central resonance of the absorption spectrum are probed alternately, and the probe light frequency is stabilised to the line center according to the difference of the measured excitation probabilities.

3. Experimental Setup The employed ion traps are cylindrically symmetric with a ring electrode diameter of 1.4 mm. Except for the measurement of the quadrupole shft of the 2D3/2level, the applied trap dnve voltage contained no constant component. In this case, the axial and radial secular motion frequencies of a trapped Yb' ion are in the range of 0.7 MHz and 1.4 MHz, respectively. Typical single-ion storage times are in the range of severalmonths. Static electric stray fields in the confinement volume are compensated in three dimensions so that the amplitude of stray-field induced micromotion is smaller than the secular motion amplitude at the Doppler cooling limit. Cooling radiation at 370 nm is generated by frequency doubling the output of an extended-cavity diode laser. Hyperfine repumping radiation is produced by modulating the injection current of this diode laser at a frequency near 14.7 GHz. Extended-cavity diode lasers are also used to generate 935 nm and 639 nm repumping radiation, and light at 871 nm which is frequency doubled in order to produce the 436nm probe radiation. The cooling and repumping radiation is blocked by mechanical shutters during the excitation of the reference transition. In order to stabilise the frequency of the 436nm probe light, a PoundDrever-Hall scheme is used to lock the 871 nm diode laser with a servo bandwidth of 0.5 MHz to a fiber-coupled high-finesse ULE cavity. The cavity is suspended in vacuum by springs of 1 m length for vibration isolation. The cavity temperature is actively stabilised so that the drift of the 436 nm probe frequency is typically mainly determined by the 0.07 Hz/s long-term aging drift of the cavity mate ria^.^

43

The scheme of the frequency comparison experiment is shown in Fig. 2. Both traps use the same cooling laser setup and synchronous timing schemes for cooling, state preparation, and state detection. Using two independent digital servo systems, the error signals resulting from the probing of the atomic resonances are averaged over typically 20 measurement cycles before the detunings between the probe laser frequency and the probe light beams incident on the traps are corrected. In order to minimise servo errors due to the drift of the probe laser frequency, a second-order integrating servo algorithm is used. The servo time constants are in the range of 30s. The differences of the detunings imparted on the probe beams are averaged over time intervals of 1 s and recorded.

Figure 2. Experimental setup for comparison of two "'Yb' frequency standards. AOM: acoustooptic modulators, providing independent frequency shifts between the probe laser and the two ion traps.

4. Spectroscopy of the Reference Transition

Absorption spectra of the 2S1,2(F=0)- 2D3/2(F=2)transition of a single trapped "'Yb' ion are shown in Fig. 3. They were obtained using the setup described in Ref. 7. In Fig. 3, the frequency resolution increases from Fig. 3(a) to (d). For Fig. 3(a) and (b), the linewidth of the probe radiation was increased by whitenoise frequency modulation in order to reduce the number of data points required for the scan. Due to the incoherent optical excitation, the absorption probability here is limited to 0.5 at full saturation. Figure 3(a) shows that the strength of the radial secular motion sidebands is significantly smaller than that of the central recoilless component. This permits the conclusion that the LambDicke condition is well satisfied for the excitation of the reference transition. In Fig. 3(b), the Zeeman structure of the recoilless component is shown for an

44

applied static magnetic field of approximately 1.1 pT. Magnetic fields in the range of 1 pT were also applied in the frequency comparison experiments described below. Figures 3(c) and (d) show the central AmF=O-component of (b) for the case of coherent r-pulse excitation and approximately Fourier-limited resolution. In Fig. 3(d), the maximum absorption probability is reduced relative to Fig. 3(c) because the short-time fluctuations of the probe laser frequency are not negligible relative to the Fourier linewidth limit of 27 Hz.

0.5

0.0 -1 .o

-0.5

0.0

0.5

1 .O MHz

-5.0

-2.5

0.0

2.5

5.0

0.8

,

0.4

-

-120

kHz

I

-60

0

60

120

Hz

Laser Detuning at 435.5 nm Figure 3. Absorption spectra of the 'S1,2(F=O) - 'Dsn (F=2) transition of a single trapped '"Yb+ ion, showing the first-order radial secular-motion sidebands and the central carrier resonance (a); the Zeeman pattern of the carrier resonance (b); and the Amp=O-resonance of (b) in higher resolution (c, d). Each data point corresponds to an average of 20 measurement cycles. The probe pulse length was 1 ms in (a), (b), and (c), and 30 ms in (d). For further details see text.

45

5. Absolute Transition Frequency and Systematic Frequency Shifts Using a femtosecond frequency comb generator, the frequency of the 436 nm 2S1/2(F=0)- 'D3,2(F=2, mF=O) transition of 171Yb+was measured relative to a caesium fountain microwave frequency ~tandard.~ The optical-excitation conditions were identical to those of Fig. 3(d). The measured absolute frequency is I/yb+= 688 358 979 309 312 f 6 Hz. This frequency value includes the sluft of the transition frequency due to isotropic blackbody radiation at an ambient temperature of 300 K. On the basis of computed atomic oscillator strengths, the shift is calculated as -0.4 Hz.'. The total 1 CT measurement uncertainty of f 6 Hz consists of approximately equal statistical and systematic contributions. The dominant source of the systematic measurement uncertainty is given by the electric-quadrupole interaction of the upper level of the reference transition with the gradient of stationary electric stray fields. A maximum stray-field induced quadrupole sluft of the order of 1 Hz is expected for atomic D312and DSl2states.' A non-negligible systematic uncertainty contribution also arises from servo errors due to drifts of the probe laser frequency. The uncertainty contributions of other frequency shifting effects are negligible under the present experimental conditions. The magnetic field applied during the excitation of the reference transition leads to a quadratic Zeeeman shift of only 0.05 Hz. Since the trapped ion is cooled to the Doppler limit, the second-order Doppler and Stark effect shifts caused by the trap field are expected to be in the millihertz range.6

5 h

N

I,o d

-5 -10 0

2000

4000

6000

8000

Time (s) Figure 4(a). Temporal variation of the frequency difference between two probe light fields independently frequency stabilised on the 436 nm reference transitions of two trapped ions. The average frequency difference calculated from this data set is 0.2 Hz. The intervals without data points correspond to times when no frequency correction signal was produced by one of the trapped ions.

46

50

100

200

500

1000

2000

Averaging time 7 (s)

Figure 4(b). Allan standard deviation of the data set shown in Fig. 4(a), normalized to the optical frequency of 688 THz. The dashed line shows the result of a'Monte Carlo simulation of the servo action for the case that the fluctuations of the atomic resonance signals are determined by quantum projection noise.

6. Comparison of Two Traps Figure 4(a) shows the temporal variation of the frequency difference between two independent '"Yb' trap and servo systems, using the experimental setup shown in Fig. 2. The Allan deviation of this data set is shown in Fig. 4(b). The conditions of this measurement were similar to those of Fig. 3(d). Using temporally overlapping probe pulses, the atomic resonance signals were resolved with nearly Fourier-limited linewidths of approximately 30 Hz in both traps. The mean frequency difference of the data shown in Fig. 4(a) is * 0.2 Hz, corresponding to a relative optical frequency offset of 3.10-16. Since is smaller than the Allan deviation for long averaging times (oj,(z) =1.10-" for z 2 800 s), the observed offset is not statistically significant. A change of the drive voltage amplitude of one of the traps by 15% did not cause any significant frequency offset at the 1 Hz level. As shown in Fig. 4(b), the variation of the Allan deviation with the averaging time T is in qualitative agreement with a numerical calculation which simulates the effect of quantum projection noise for the realized experimental conditions. The observed Allan deviation however exceeds the quantum projection noise limit by approximately a factor of two. A possible reason for this excess instability are temporal fluctuations of the probe laser emission spectrum which can lead to fluctuating servo errors. The frequency shifts caused by this effect are not necessarily equal for both servo systems because the probe pulse areas by which the two ions were excited were not exactly matched.

47

7. Quadrupole Shift Measurement

One expects that the interaction of the quadrupole moment of the 'D3l2state of 171 Yb' with a static electric field gradient leads to a shift of the frequency of the 436 nm reference transition. In order to experimentally determine the I 7 l Y b + quadrupole moment, a static field gradient was generated in one of the traps by superimposing a constant (dc) voltage on the radiofrequency trap drive voltage. The orientation of this field gradient is determined by the symmetry axis of the trap. The other trap was operated with a pure r.f. voltage and served as a reference. The result of a corresponding frequency comparison measurement is shown in Fig. 5 .

-5

1I 0

2

4

6

8

10

Field gradient (V/mm*) Figure 5 . Frequency difference between the 436 nm light fields stabilised to two "'Yb+ traps as a function of the dc field gradient generated in one of the traps. The application of a positive voltage to the endcap electrodes increases the optical frequency. The error bars show the statistical measurement uncertainty.

A three-axis magnetic field sensor was used to determine the orientation of the static magnetic field relative to the applied field gradient. Using the formalism described in Ref. 9, we infer a quadrupole moment of 0 = (3.9 k 1.9)ea: for the 2D3/2level of Yb' with e being the electron charge and a. the Bohr radius. The uncertainty of the inferred 0 value is mainly determined by the uncertainty of the measurement of the angle between the magnetic field and the trap axis. References 1. Chr. T a m , D. Engelke and V. Biihner, Phys. Rev. A 61,053405 (2000). 2. S.A. Webster, P. Taylor, M. Roberts, G.P. Barwood and P. Gill, Phys. Rev. A 65, 052501 (2002).

48

3. J. Stenger, Chr. Tamm, N. Haverkamp, S. Weyers and H.R. Telle, Opt. Lett. 26, (2001). 4. T. Quinn, Metrologia 40, 103 (2003). 5. G. Barwood, K. Gao, P. Gill, G. Huang and H.A. Klein, IEEE Trans. IM-50,543 (2001). 6. See, e.g., A. Bauch and H.R. Telle, Rep. Prog. Phys. 65,789 (2002). 7. Chr. Tamm, T. Schneider and E. Peik, in: Proceedings of the 6th Symposiumon Frequency Standards and Metrology, ed. P. Gill (World Scientific, Singapore 2002), p. 369. 8. B. C. Fawcett and M. Wilson, At. Data Nucl. Data Tabl. 47, 241 (1991); J.W. Farley and W.H. Wing,Phys. Rev. A 23,2397 (1981). 9. W.M. Itano, J. Res. NZST105, 829 (2000).

LIMITS ON TEMPORAL VARIATION OF FINE STRUCTURE CONSTANT, QUARK MASSES AND STRONG INTERACTION

V.V. FLAMBAUM School of Physics, The University of New South Wales, Sydney NS W 2052, Australia Theories unifying gravity with other interactions suggest a spatial and temporal variation of the fundamental “constants” in the Universe. A change in the fine structure constant (Y = e2/tic could be detected via shifts in the resonance transition frequencies in quasar absorption systems. We have developed a new approach which improves the sensitivity of this method 30 times. It also provides much better control of systematic errors. We studied three independent samples of data containing 130 absorption systems spread from 2 to 10 billion years after the Big Bang. All three data samples hint that alpha was smaller 7-11 billion years ago. Another very promising method to search for variation of the fundamental constants consists of comparison of different atomic clocks. We performed calculations of the dependence of nuclear magnetic moments on quark masses and obtained limits on the variation of (Y = e 2 / k and (m,/AQco) from recent atomic clock experiments with hyperfine transitions in H, Rb, Cs, Hg+ and an optical transition in Hg+.

1. Introduction

Interest in the temporal and spatial variation of the major constants of physics has recently been revived by astronomical data which seem to suggest a variation of the electromagnetic interaction constant a = e 2 / h at the lop5 level over the time scale 10 billion years. (see ref. 1; a discussion of other limits can be found in the review2 and references therein). To perform measurements of the variation of a we developed a new approach3 which improves the sensitivity to a variation of a by more than an order of magnitude. The relative value of any relativistic corrections to atomic transition frequencies is proportional to a 2 . These corrections can exceed the fine structure interval between the excited levels by an order of magnitude (for example, an s-wave electron does not have spin-orbit splitting but it has the maximal relativistic correction to energy). The relativistic corrections vary very strongly from atom to atom and can have 49

50

opposite signs in different transitions (for example, in s-p and d-p transitions). Thus, any variation of a could be revealed by comparing different transitions in different atoms in cosmic and laboratory spectra. 2. Quasar Absorption Spectra

The above method provides an order of magnitude precision gain compared to measurements of the fine structure interval. Relativistic many-body calculations are used to reveal the dependence of atomic frequencies on a for a range of atomic species observed in quasar absorption spectra It is convenient to present results for the transition frequencies as functions of a2 in the form

’.

w = wo

+ qx,

(1)

%

where x = (“)2 - 1 NN and wo is a laboratory frequency of a particular 010 transition. We stress that the second term contributes only if a deviates from the laboratory value ao. We have performed accurate many-body calculations of the coefficients q for all transitions of astrophysical interest (strong El transitions from the ground state) in Mg, Mg 11, Fe 11, Cr 11, Ni 11, A1 11, A1 111, Si 11, and Zn 11. It is very important that this set of transitions contains three large classes: positive shifters (large positive coefficients q > 1000 cm-l), negative shifters (large negative coefficients q < -1000 cm-l) and anchor lines with small values of q. This gives us excellent control of systematic errors since systematic effects do not “know” about the sign and magnitude of q. Comparison of cosmic frequencies w and laboratory frequencies wo allows us to measure We have studied three independent samples of data containing 130 absorption systems spread over the red shift range 0.2 < z < 3.7. The result’, with raw statistical error, is = (-0.574f0.102) x A very extensive search for possible systematic errors has shown that known systematic effects cannot explain the result. (It is still not completely excluded that the effect may be imitated by very different abundances of isotopes in the past. We have checked that different isotopic abundances for any single element cannot imitate the observed effect. It may be an improbable “conspiracy” of several elements). However, an independent experimental confirmation is needed. The hypothetical unification of all interactions implies that variation of the electromagnetic interaction constant a should be accompanied by a variation of masses and the strong interaction constant. Specific predictions

2.

%

51

need a model. For example, the grand unification model discussed in predicts that the quantum chromodynamic (QCD) scale AQCD(defined as the position of the Landau pole in the logarithm for the running strong coupling constant) is modified as follows

~AQCD -AQCD

6a 34-

a

The variation of quark and electron masses in this model is given by

6m m

-

N

6Q 70-

(3)

Q

This gives an estimate for the variation of the dimensionless ratio

The large coefficients in these expressions are generic for grand unification models, in which modifications come from high energy scales: they appear because the running strong coupling constant and Higgs constants (related to mass) run faster than a. This means that if these models are correct the variation of masses and strong interaction may be easier to detect than the variation of a. Unlike the electroweak forces, for the strong interaction there is generally no direct relation between the coupling constants and observable quantities. Since one can measure only the variation of dimensionless quantities, we want to extract from the measurements the variation of the dimensionless ratio m,/AQcD, where m, is the quark mass (with the dependence on the normalization point removed). A number of limits on the variation of m,/hQcD have been obtained recently from consideration of Big Bang Nucleosynthesis, quasar absorption spectra, and the Oklo natural nuclear reactor which was active about 1.8 billion years ago (see also 5161798

9,10,11,12,13

>.

3. Laboratory Comparison of Atomic Clocks

We consider the limits which follow from laboratory atomic clock comparison. Laboratory limits with a time base of about a year are especially sensitive to oscillatory variation of the fundamental constants. A number of relevant measurements have been performed already and an even larger number have been started or are planned. The increase in precision is very rapid.

52

It has been pointed out by Karshenboim14 that measurements of the ratio of hyperfine structure intervals in different atoms are sensitive to the variation of nuclear magnetic moments. First rough estimates of the dependence of nuclear magnetic moments on m,/AQcD and limits on the time variation of this ratio have been obtained in our paper5. Using H, Cs and Hg+ measurements15i16,we obtained a limit on the variation of m,/AQcD of about 5 . per year. Below we calculate the dependence of nuclear magnetic moments on m,/hQcD and obtain limits from recent atomic clock experiments with hyperfine transitions in H, Rb, Cs, Hg+ and an optical transition in Hgf. It is convenient to assume that the strong interaction scale AQCDdoes not vary, so we will speak about the variation of masses. The hyperfine structure constant can be presented in the following form

The factor in the first bracket is an atomic unit of energy. The second “electromagnetic” bracket determines the dependence on a. An approximate expression for the relativistic correction factor (Casimir factor) for an s-wave electron is the following:

d m - ,

where y = 2 is the nuclear charge. Variation of a leads to the following variation of Fr,l 15:

K=

(Za)2(12y2 - 1) Y2(4Y2 - 1)

More accurate numerical many-body calculation^^^ of the dependence of the hyperfine structure on a have shown that the coefficient K is slightly larger than that given by this formula. For Cs (2=55) K= 0.83 (instead of 0.74), for Rb K=0.34 (instead of 0.29), and for Hg+ K=2.28 (instead of 2.18). The last bracket in eq.(5) contains the dimensionless nuclear magnetic moment p in nuclear magnetons (the nuclear magnetic moment M = p&), electron mass me and proton mass M p . We may also include a small correction due to the finite nuclear size. However, its contribution is insignificant.

53

Recent experiments have measured the time dependence of the ratios of hyperfine structure intervals of Ig9Hg+ and H 15, 133Csand *'Rb l8 and the ratio of the optical frequency in Hg+ and the 133Cshyperfine frequency 19. In the ratio of two hyperfine structure constants for different atoms the time dependence may appear from the ratio of the factors F,,l (depending on a) and the ratio of the nuclear magnetic moments (depending on rn,/AQco). The magnetic moments in the single-particle approximation (one unpaired nucleon) are: P = (9s

+ (2j - l)gd/2

(9)

for j = 1 + 112.

for j = I - 1/2. Here the orbital g-factors are gl = 1for a valence proton and gl = 0 for a valence neutron. The present values of the spin g-factors g9 are g p = 5.586 for a proton and gn = -3.826 for a neutron. They depend on mq/AQco.The light quark masses are only about 1%of the nucleon mass (mq = (mu + m d ) / 2 NN 5 MeV). The nucleon magnetic moment remains finite in the chiral limit of m, = m d = 0. Therefore, one may think that the corrections to gs due to the finite quark masses are very small. However, there is a mechanism which enhances the quark mass contribution: 7r-meson loop corrections to the nucleon magnetic moments which are proportional to the n-meson mass m, = /,; m,=140 MeV is not so small. According to the calculation in Ref. 20 the dependence of the nucleon gfactors on 7r-meson mass m, can be approximated by the following equation N

where a= 1.37/GeV, b= 0.452/GeV2 for the proton and a= 1.85/GeV, b= 0.271/GeVZ for the neutron. This leads to the following estimate: 69, 6% = -0.174-

m4

m,

SP

-69, _

Sm

=-0.0872

- -0.213-

h

T

(12)

= - 0 . 1 0 Sm 72

Sn m, m4 Eqs. (9, 10, 12, 13) give the variation of the nuclear magnetic moments. For the hydrogen nucleus (proton)

54

For lg9Hg we have a valence neutron (no orbital contribution); therefore the result is &P 69, = -0.107-P

Sn

6% m4

For 133Cswe have a valence proton with j=7/2, 1=4 and 6P

6%

- = 0.22P

m7r

6m

=0 . 1 1 2 m4

For 8rRb we have a valence proton with j=3/2, 1=1 and

Deviation of the single-particle values of the nuclear magnetic moments from the measured values is about 30 %. Therefore, we tried to refine the single-particle estimates. If we neglect the spin-orbit interaction the total spin of nucleons is conserved. The magnetic moment of the nucleus changes due to the spin-spin interaction because the valence proton transfers a part of its spin < s, > to the core neutrons (transfer of spin from the valence proton to the core protons does not change the magnetic moment). In this approximation gs = (1 - b)g, bg, for a valence proton (or gs = (1 - b)gn bg, for a valence neutron). We can use the coefficient b as a fitting parameter to reproduce the nuclear magnetic moments exactly. The signs of g p and gn are opposite; therefore a small mixing b 0.1 is enough to eliminate the deviation of the theoretical value from the experimental one. Note also that it follows from eqs. (12, 13) that = $.This produces an additional suppression of the effect of the mixing. This indicates that the actual accuracy of the single-particle approximation for the effect of the spin g-factor variation may be as good as 10 %. Note, however, that here we neglected the variation of the mixing parameter b which is hard to estimate. Now we can estimate the sensitivity of the ratio of the hyperfine transition frequencies to the variation of m,/AQcD. For lg9Hg and hydrogen we have

+

+

-

2

Therefore, the measurement of the ratio of the Hg and hydrogen hyperfine frequencies is practically insensitive to the variation of the masses and the

55 strong interaction. The result of measurement l5 may be presented as a limit on the variation of the parameter 5 = a [ m , / h ~ c ~ ] - ~ ' ~ ' :

I--I51 ddt5 < 3.6 x

l0-l4/yeur

(19)

Other ratios of the hyperfine frequencies are more sensitive to m,/hQcD. For 133C~/87Rb we have

Therefore, the result of the measurement l8 may be presented as a limit on the variation of the parameter x = a0.49[mq/AQcD]0.17: -I d X = (0.2 f 7) x 10-16/yeur

X dt

(21)

Note that if the relation (4) is correct, the variation of X would be dominated by the variation of [ r n q / A ~ c o ]The . relation (4)would give X c( a7 and the limit on the a variation = (0.03 f 1) x 10-16/year . For 133Cs/Hwe have

:%

Therefore, the result of the measurements l6 may be presented as a limit on the variation of the parameter X H = a 0 . 8 3 [ m q / h ~ ~ ~ ] o . z :

(---IX1H dXH < 5.5 x i0-14/year dt If we assume the relation (4),we would have X H c( a8, I;%l < 0.7 x 10-l4/year. The optical clock transition energy E ( H g ) (X=282 nm) in the Hg+ ion can be presented in the following form: m e4 E ( H g ) = const x [ z ] F T e l ( Z a ) h2

(24)

Note that the atomic unit of energy (first bracket) is cancelled out in ratios; therefore, we should not consider its variation. Numerical calculation of the relative variation of E ( H g ) has given 17:

56

Variation of the ratio of the Cs hyperfine splitting A(Cs) to this optical transition energy is equal to

Here we have taken into account that the proton mass M p c( AQCD. The factor 6.0 before 6a appeared from a2FTel in the Cs hyperfine constant (2+0.83) and the a-dependence of E(Hg) (3.2). Therefore, the work l9 gives the limit on the variation of the parameter U = as[me/AQCD][mq/AQCD]'": 1 dU I--+ < 7 x 10-15/year

U dt

If we assume the relation (4), we would have U

c(

10-l6/year. Note that we presented such limits on only since they are strongly model-dependent .

I+%l
1 year span of our data allows us to set bounds on individual elements of k0+, and not only on special linear combinations. Alternatively, following the example of other a u t h o r ~ , ~ ,we ' , ~may analyze our data within the classical kinematical framework by Robertson as well as Mansouri and Sex1 (RMS). Here, one assumes a preferred frame C with a constant velocity of light CO. In a frame moving with a velocity v (usually, the cosmic microwave background is taken for C, so .u = 369 km/s), the speed of light is given by c(v, O)/c, = 1 ( A Bsin' 0) v2/ci, where O is the angle between v' and C: If SR is valid, the coefficients A (boost invariance) and B (isotropy) vanish. In a previous (Kennedy-Thorndike) experiment' comparing a CORE-stabilized laser and an Iodine standard, we determined A = ( 1 . 9 f 2 . 1 ) . (since improvedg t o A = (3.1 f 6.9) . From our MM-experiment we later obtained 6 u / u = (0.73 f 0.48) Hz, which implies a new limit on the isotropy parameter B = (2.2 f 1.5) . lov9, with an uncertainty about 3 times lower than the best previous limit.4 Future versions of the experiment will employ a turntable t o provide active rotation at an optimized rate. The use of fiber coupling and specially designed monolithic CORES are further promising options. Together, this should ultimately lead to an improvement by another two orders of magnitude or more. N

+ +

References 1. H. Muller et al., Phys. Rev. Lett. 91,020401 (2003); Int. J. Mod. Phys. D 11,1101 (2002). 2. H. Muller et al., submitted to Appl. Phys. B (2003). 3. V. A. Kosteleck$ and M. Mewes, Phys. Rev. D 66,056005 (2002).

74 4. 5. 6. 7. 8. 9.

A. Brillet and J.L. Hall, Phys. Rev. Lett. 42, 549 (1979). H. Muller et al., accepted for publication in Opt. Lett. (2003). H. Muller et al., Phys. Rev. D 67,056006 (2003). J. Lipa et al., Phys. Rev. Lett. 90, 060403 (2003). C. Braxmaier et a]., Phys. Rev. Lett. 88, 010401 (2002). P. Wolf et al., Phys. Rev. Lett. 90, 060402 (2003).

UItrafast Spectroscopy

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ULTRA-PRECISE PHASE CONTROL OF SHORT PULSES APPLICATIONS TO NONLINEAR SPECTROSCOPY* W N YE, LISENG CHEN, R. JASON JONES, KEVIN HOLMAN AND DAVID J. JONES JILA, National Institute of Standards and Technology and Universig of Colorado Boulder, Colorado 80309-0440, USA E-mail: Ye@ JILA.colorado.edu Recent progress in precision control of pulse repetition rate and carrier-envelope phase of ultrafast lasers has found a wide range of applications in both precision spectroscopy and ultrafast science. In this contribution we discuss the impact of optical frequency comb to precision molecular spectroscopy, optical standards, nonlinear optics, and sensitive detection.

1. Introduction Precise phase control of femtosecond lasers has become increasingly important as novel applications utilizing the femtosecond laser-based optical comb are developed that require greater levels of precision and higher degrees of control.’ Improved stability is beneficial for both “frequency domain” applications, where the relative phase or “chirp” between comb components is unimportant (e.g., optical frequency metrology), and, perhaps more importantly, “time domain” applications where the pulse shape andor duration is vital, such as in nonlinear optical interactions.’ For both types of applications, minimizing jitter in the pulse train and noise in the carrier-envelope phase is often critical to achieve the desired level of precision. Phase-stabilized mode-locked femtosecond lasers have played a key role in recent advances in optical frequency mea~urement,~’~ carrier-envelope phase stabilizati0n,2.~’~ all-optical atomic clocks,7v8optical frequency synthesizers? coherent pulse synthesis,” and ultra-broad, phase coherent spectral generation.” The capability of absolute optical frequency measurements in the visible and IR spectral regions adds a new meaning to the term of precision molecular spectroscopy. Understanding of molecular structure and dynamics often involves detailed spectral analysis over a broad wavelength range. Such a task can now be accomplished with a desired level of accuracy uniformly across all relevant spectral windows, allowing precise investigations of minute changes in the * This work is supported by ONR, NASA, NIST and NSF. 77

78

molecular structure over a large dynamic range. For example, absolute frequency measurement of vibration overtone transitions and other related resonances (such as hyperfine splitting) will reveal precise information about the molecular potential energy surface and relevant perturbation effects. We have pursued a similar study in iodine molecules, performing high-resolution and hgh-precision measurement of hyperfine interactions of the first excited electronic state ( B ) of I2 over an extensive range of vibrational and rotational quantum numbers towards the dissociation limit. Experimental data demonstrate systematic variations in the hyperfine parameters that confirm calculations based on ab initio molecular potential energy curves and electronic wave functions derived from a separated-atomic basis set. We have accurately determined the statedependent quantitative changes of hyperfine interactions caused by perturbations from other electronic states and identified the respective perturbing states. Our work in I2 near the dissociation limit is also motivated by the desire to improve cell-based portable optical frequency standards.12 Indeed, 12-stabilized lasers have already demonstrated high stability (< 5 x at 1 s averaging time) and have served well for optical atomic clocks.8 For the time domain applications to molecular spectroscopy, stabilization of the “absolute” carrier-envelope phase at a level of tens of milli-radians has been demonstrated and this phase coherence is maintained over an experimental period exceeding many minutes,I3 paving the groundwork for synthesizing electric fields with known amplitude and phase at optical frequencies. Working with two independent femtosecond lasers operating at different wavelength regions, we have synchronized the relative timing between the two pulse trains at the femtosecond level,14and also phase locked the two carrier frequencies, thus establishing phase coherence between the two lasers. By coherently stitchng optical bandwidth together, a “synthesized” pulse has been generated.” With the same pair of Ti:sapphire mode-locked lasers, we have demonstrated widely tunable femtosecond pulse generation in the mid- and far- IR using differencefrequency-generation.l5 The flexibility of this new experimental approach is evidenced by the capability of rapid and programmable switching and modulation of the wavelength and amplitude of the generated IR pulse. A fully developed capability of producing phase-coherent visible and IR pulses over broad spectral bandwidths, coupled with arbitrary control in amplitude and pulse shape, represents the ultimate instrumentation for coherent control of molecular systems. The simultaneous control of timing jitter (repetition rate) and carrierenvelope phase can be used to phase coherently superpose a collection of successive pulses from a mode-locked laser. For example, by stabilizing the two degrees of freedom of a pulse train to an optical cavity acting as a coherent delay, constructive interference of sequential pulses will be built up until a cavity dump is enabled to switch out the “amplified” pulse.I6 Such a passive pulse “amplifier”, along with the synchronization technique we developed for pulse

79

synthesis, has made a strong impact on the field of nonlinear-optics based spectroscopy and imaging of bio-molecular systems, showing significant improvements in experimental sensitivity and spatial r e s o l~ tio n .~ With ~ ~ '* the enhanced detection sensitivity comes the capability of tracking real time biological dynamics. An ultrafast laser locked to a h g h stability cavity is also expected to demonstrate extremely low pulse jitter and carrier-envelope phase noise, which will be particularly attractive for time-domain experiments. In addition, we are exploring the use of pulse-cavity interactions to obtain highly sensitive intracavity spectroscopy (linear and non-linear) with a wide spectral coverage, as well as to enhance nonlinear interaction strengths for h g h efficiency nonlinear optical experiments. 2.

I2Hyperfine Interactions, Optical Frequency Standards and Clocks

The hyperfine structure of I2 rovibrational levels includes four contributions: nuclear electric quadrupole (eqQ),spin-rotation (C), tensorial spin-spin (4,and scalar spin-spin (13)interactions. Agreement between experiment and theory using the four-term effective Hamiltonian is at the lulohertz level for a few selected transitions. For the first excited electronic state B with the 2P3/2+ 2P1,2 dissociation limit, ow goal is to perform a systematic hgh-precision investigation of hyperfine interactions over an extensive range of rovibrational quantum numbers coupled with a large range of internuclear separations. Such a study has allowed us to understand the rovibrational dependence of the hyperfine interactions (as well as the dependence on internuclear distance) based on ab initio molecular potential energy curves and the associated electronic wave functions. Careful analysis of various perturbation effects leads to precise determination of molecular structure over a large dynamic range. Prior studies have concentrated on a few isolated rovibrational levels for the high vibrational states vf = 40 to 82 in the B state.'' For vibrational levels below u' = 43, only functional forms on the state-dependent variations of the hyperfine interactions have been investigated from empirical data.20Combining absolute optical frequency metrology with high-resolution and broad wavelengthcoverage laser spectroscopy, we have measured 80 rovibrational transitions with the upper vibrational levels (from v' = 42 up to u' = 70) stretching from a closely bonded molecular basis to a separated-atomic basis appropriate for the 2P3/2 + 'PI12 dissociation limit, providing kHz-level line accuracies for most hyperfine components. Figure 1 illustrates systematic rovibrational dependences for all four hyperfine parameters. Each solid line is a fit of experimental data for rotational dependence belonging to a single vibrational (vf) level. In general, all hyperfine parameters have monotonic dependence on both rotational and vibrational quantum numbers except for the levels in the vicinity of u' = 57 to 59. However, the v-dependence of eqQBreverses its trend after u' = 60. For the

-

80

sake of figure clarity, the eqQBdata for v' > 60 are not shown. Another important observation is that for levels of v' = 57 - 59 all hyperfiie parameters except for C, bear ab'nonnal J-dependences due to perturbations from a lg state through accidental rotational resonances.

-560 -

8.,

J

J'(J'+l)

J'(J'+ 1)

Figure 1. Rovibrational dependence of the B state hyperfine parameters (a) eq& (b) CB,(c) &, and (d) 6 ~ .Note (b), (c), and (d) are semilog plots and the vertical scale of (c) has been inverted. Each solid line is the J-dependence for each vibrational level (u' indicated in the figure). Experimental data in squares and open circles show abnormal variations of eq@, &, and 6s around u' = 57 and 59.

Combining data from this work and from the literature:' investigations of the hyperfine spectra now cover the majority of the vibrational levels (3 5 u' < 82) in the B state. Therefore, it is now possible and useful to explore the global trend of these hyperfine parameters in the B state. Suppressing the rotational dependence, hyperfine parameters as functions of pure vibrational energy E(v') are found to increase rapidly when molecules approach the dissociation limit,

81

whch is a result of the increasingly strong perturbations from other high-lying electronic states sharing the same dissociation limit with B. While CB'Svariation is smooth over the whole range, eqQB, dB, and 68 all have local irregularities at three positions: u' = 5 where the B": 1, state crosses nearby, around U' = 57 to 59 (see discussions above), and from u' = 76 to 78, due to the lg state.'' To examine these hyperfine parameters in terms of internuclear separation R, the vibrational average of the hyperfine parameters is removed by inverting the expression O(u',J ' ) = (u>t lO(R)(u'y ) , where O(u',J ' ) denotes one of the four hyperfine parameters. Consistent with CB's smooth variation, the interpolation function CB(R) has small residual errors (within f0.02, relative) for the entire range from u' = 3 to 70. On the contrary, the large residual errors in the interpolation of eqQB, dB, and AB for u' 2 56 reflect their abnormal variations observed around u' = 57 and 59, restricting a reliable interpolation only to levels of u' < 56. In the region of R < 5 A, valuable information can be readily extracted from eq& to assist investigation of 12's electronic structure. Udlke the other three hyperfine parameters whose major parts originate from perturbations at nearly all possible values of R, a significant part of eqQB is due to the interaction between the nuclear quadrupole moment (Q) and the local electric field gradient (q(R)) generated by the surrounding charge distribution of a largely B state character. Thus, for R < 5 A, where perturbations from other electronic states are negligible, the vibration-removed interpolation function eq&(R), coupled with a priori dormation on q(R), can be used to determine I2 nuclear quadrupole moment or serve as a benchmark for molecular ab initio calculations of the electronic structure at various values of R. Precision measurements on B-X hyperfine spectra provide an alternative and yet effective way to investigate the potential energy curves (PECs) sharing the same dissociation limit with the B state as well as the associated electronic wave functions. To demonstrate h s , we perform calculations of eqQB,CB, dB, and 8~ based on the available PECs and electronic wave functions derived from a separated-atomic basis set. For both vibrational and rotational dependences, the ab initio calculation results agree very well with the experimental data for u' 2 42 (R-centroid 2 3.9 A). In short, we have extended the range of separatedatomic basis calculations from levels near the dissociation limit to low vibrational levels (u' = 5) and have found very good agreement with the experimental data on both vibrational and rotational dependences. Besides these interesting studies in hyperfine structure, the narrow-linewidth 12 transitions in this wavelength range also provide excellent cell-based optical frequency references for laser stabilization. Frequency-doubled Nd:YAG/'2712at 532 nm has been proved to be one of the best portable optical frequency standards with compact size, reliability, and high stability (< 5 ~ 1 O -atl ~1 s). To reach a higher frequency stability, it is useful to explore I2 transitions at wavelengths below 532 nm, where the natural linewidths decrease at a faster rate

82

than that for the line strengths. We have measured the systematic variation of the I2 transition linewidths within the range of 532 - 498 nm, with the linewidth decreasing by nearly 6 times when the wavelength is changed from 532 nm to near the dissociation limit.” The high SIN results indicate that I2 transitions in the wavelength range of 532 - 501 nm hold great promise for future development of optical frequency standards, especially with the advent of all solid state Yb:YAG lasers. One exciting candidate is the 514.67 nm standard,” with a projected stability < 1 x at 1 s. The Iz-based optical standard has been used to stabilize an entire octave-bandwidth spanning optical frequency comb based on a mode-locked Ti:sapphire laser, thus establishing an optical atomic clock where the RF signal is phase coherently derived from the I2 optical transition.’ With a coherent link established between the femtosecond Tisapphire laser and 1550-nm mode-locked laser sources:* precise time and frequency information can be transferred and disseminated from an optical atomic clock to remote sites via optical telecommunication networks. 3.

Femtosecond Lasers and External Optical Cavities

Understanding the intricate interactions between ultra-short pulses and external passive optical cavities, along with subsequent development of capabilities to efficiently couple and coherently store ultra-short pulses of light inside a h g h finesse optical cavity, will open doors for a variety of exciting experiments. An immediate impact is on precision stabilization of ultrafast lasers.23Similar to the state-of-art stabilization of CW lasers, a cavity-stabilized ultrafast laser is expected to demonstrate superior short-term stability of both the pulse repetition frequency and the carrier-envelope phase. The improved stability is beneficial in particular for time-domain applications where the signal processing bandwidth is necessarily large. Another attractive application lies in broadband and ultrasensitive spectroscopy. The use of high finesse cavities has played a decisive role for enhancing sensitivity and precision in atomic and molecular spectroscopy. We expect a dramatic advancement in the efficiency of intracavity spectroscopy by exploiting the application of ultra-short pulses. In other words, a high detection sensitivity is achievable across the broad spectrum of the pulse simultaneously. Cavity-stabilization techniques for femtosecond lasers allow the comb structure of the probe laser to be precisely matched to the resonance modes of an empty cavity, allowing efficient energy coupling for a spectroscopic probe. Molecular samples introduced inside the high finesse cavity will have a strong impact on the dispersive properties of the cavity. In fact it is this dispersion-related cavity-pulling effect that will aid our sensitive detection process when we analyze the light transmitted through the cavity. Preliminary data on spectrally resolved, time-domain ring down measurement for intracavity loss over the entire femtosecond laser bandwidth are already quite promising.

83

To develop sources for ultrafast nonlinear spectroscopy, a properly designed dispersion compensated cavity housing a nonlinear crystal will provide efficient nonlinear optical frequency conversion of ultrashort optical pulses at spectral regions where no active gain medium exists. Furthermore, by simultaneously locking two independent mode-locked lasers to the same optical cavity, efficient sum and/or difference frequency generation can be produced over a large range of wavelengths. Under a similar motivation, a passive cavity can be used to explore coherent “amplification” of ultra short pulses, with cavity stabilization providing the means to phase coherently superpose a collection of successive pulses from a mode-locked laser. The coherently enhanced pulse stored in the cavity can be switched out using a cavity-dumping element (such as a Bragg cell), resulting in a single phase-coherent “amplified” pulse. The use of a passive cavity also offers the unique ability to effectively amplify pulses at spectral regions where no suitable gain medium exists, such as for the infrared pulses from difference-frequency mixing or the W light from harmonic generation. Unlike actively dumped laser systems, the pulse energy is not limited by the saturation of a gain medium or a saturable absorber needed for mode-loclung. Instead, the linear response of the passive cavity allows the pulse energy to build up inside the cavity until limited by cavity loss and/or dispersive pulse spreading. Therefore storage and amplification of ultra-short pulses in the femtosecond regime requires precise control of the reflected spectral phase of the resonator mirrors as well as the optical loss of the resonator. While the reflected phase and group delay of the mirrors only change the effective length of the resonator, the group delay dispersion (GDD) and higher-order derivatives of the group delay with respect to frequency affect the pulse shape. The net cavity GDD over the bandwidth of the pulse needs to be minimized in order to maintain the shape of the resonant pulse and allow for the coherent addition of energy from subsequent pulses. We have applied the coherent pulse-stacking technique to both picosecond and femtosecond pulses. Initial studies have already demonstrated amplification of picosecond pulses of greater than 30 times at repetition rates of 253 kHz, yielding pulse energies greater than 150 nJ.’* With significant room left for optimization of the cavity finesse (current value of 350, limited by the cavity input-coupling mirror), we expect that amplifications greater than a hundred times are feasible, bringing pulse energies into the pJ range. Whde the use of picosecond pulses allows us to separate out complications arising from intracavity dispersion, for sub-100 femtosecond pulses, dispersive phase shifts in the cavity mirrors becomes an important topic. Preliminary results in enhancing low individual pulse energies for -75 fs pulses illustrate the importance of GDD control. The external enhancement cavity incorporated specially designed negative GDD low-loss mirrors to simultaneously compensate for the Bragg cell’s 3 mm of fused silica and provide a h g h finesse. The input coupling mirror transmission was 0.8 %, with a measured cavity finesse of 440. An intracavity

-

-

84

energy buildup of 163 is expected, leading to single pulse amplifications of approximately 65 for the current setup given the 40% dumping efficiency of our Bragg cell. The negative GDD mirrors were designed to only partially compensate for the total cavity dispersion. The remaining cavity GDD was estimated at +20 to +30 fs’. The excess dispersion results in pulse broadening and a non-uniform filtering of the transmitted pulse spectrum. Experimental results are in good agreement with independent numerical calculations. The observed amplification of only 18 tines is therefore not surprising as the achievable pulse enhancement is limited by the lack of perfect resonance between the femtosecond comb and the external cavity. Controlling the intracavity pressure will allow fine tuning of the net cavity GDD to zero. We thank E. Potma, X.-S. Xie, S. Foreman, I. Thoman, H. Kapteyn, S. Cundiff, T. Fortier, and E. Ippen for fruitful collaborations and J. L. Hall for his support and inspirations. K. Holman is a Hertz Foundation Graduate Fellow. R. J. Jones is a National Research Council Research Associate Fellow.

References 1 . S. T. Cundiff and J. Ye, Rev. Modern Phys. 75,325 (2003). 2. A. Baltuska et al., Nature 421, 6923 (2003). 3. T. Udem et al., Phys. Rev. Lett. 82, 3568 (1999). 4. J. Ye et al., Phys. Rev. Lett. 85, 3797 (2000). 5. D. J. Jones et al., Science 288, 635 (2000). 6. A. Apolonski et al., Phys. Rev. Lett. 85, 740 (2000). 7. S. A. Diddams et al., Science 293, 825 (2001). 8. J. Ye, L.-S. Ma and J. L. Hall, Phys. Rev. Lett. 87,270801 (2001). 9. J. D. Jost, J. L. Hall and J. Ye, Opt. Express 10, 515 (2002). 10. R. K. Shelton et al., Science 293, 1286 (2001). 11. A. Baltuska, T. Fuji and T. Kobayashi, Phys. Rev. Lett. 88, 133901 (2002). 12. W.-Y. Cheng et al., Opt. Lett. 27, 571 (2002). 13. T. M. Fortier et al., Opt. Lett. 27, 1436 (2002). 14. R. K. Shelton et al., Opt. Lett. 27, 3 12 (2002). 15. S. Foreman, D. J. Jones and J. Ye, Opt. Lett. 28,370 (2003). 16. R. J. Jones and J. Ye, Opt. Lett. 27, 1848 (2002). 17. E. Potma et al., Opt. Lett. 27, 1168 (2002). 18. E. Potma et al., Opt. Lett. 28, 1835 (2003). 19. J. ViguC, M. Broyer and J. C. Lehmann, Phys. Rev. Lett. 42, 883 (1979). 20. B. Bodermann, H. Knockel and E. Tiemann, Eur. Phys. J. D 19,31 (2002). 21. R. J. Jones et al., Appl. Phys. B 74, 597 (2002). 22. K. W. Holman et al., Opt. Lett., in press (2003). 23. R. J. Jones and J.-C. Diels, Phys. Rev. Lett. 86,3288 (2001).

OPTIMAL CONTROL OF MOLECULAR FEMTOCHEMISTRY

T. BRIXNER, G. KRAMPERT, P. NIKLAUS AND G. GERBER Physikalisches Institut, Universitat Wiirzburg Am Hubland, 97074 Wzi’rzburg,Germany, Fax: +49-931-888-4906 E-mail: gerberOphysik.uni-wuerzburg.de We describe the method of adaptive femtosecond quantum control. In this technique, ultrashort laser pulses are manipulated in pulse shapers to realize complex temporal intensity and phase profiles of the electric field on a femtosecond time scale. The optimal field parameters are determined in an optimization algorithm which uses direct feedback from experimental observables. Examples of this “closed-loop” scheme discussed here include automated laser pulse compression, gas-phase photodissociation control, selective photoexcitation in liquids, and ultrafast manipulation of the polarization state of light. But also other objectives can be realized, and adaptively shaped laser pulses can be regarded as a very flexible spectroscopic tool.

1. Introduction Since the early days of quantum mechanics there has been a desire to “understand” in detail the general behavior of quantum systems. This quest has been accompanied by the implicit dream not only to be able to observe in a passive way, but in fact also to actively control quantum-mechanical processes. The key question in quantum control is: Can one find external control parameters which guide the temporal evolution of quantummechanical systems in a desired way, even if this evolution is very complex? The theoretical and experimental development of suitable control schemes is a fascinating prospect of modern physics. Immediate applications are found in many different branches of scientific and engineering research such as photochemistry, quantum optics, atomic and molecular physics, biophysics, solid-state physics, telecommunications, quantum computing or quantum cryptography. But in addition to these direct benefits, the successful implementation of quantum control concepts is also likely to provide new insights into the intricacies of the underlying quantum-mechanical dynamics. 85

86

Especially promising in this context is the implementation of adaptive control, with the basic idea of using a “closed-loop” setup in which the difficulties associated with complex quantum-mechanical Hamiltonians are With the help of learning algorithms and experimental circumvented feedback signals, it is possible to achieve automated control over complex systems without the necessity for knowing the underlying potential energy surfaces. Rather than solving Schrodinger’s equation numerically, the quantum system itself is used as a kind of analog computer which “calculates” its response to certain input fields with maximum speed and optimum accuracy. Based on these experimental results, the control-parameter settings are optimized iteratively such that they “adapt” to the needs of the quantum-mechanical system. The required temporal structure of the control fields is tied to the timescale of quantum-mechanical motion. Focusing on chemical reaction dynamics, this timescale is determined by the atomic motions within their molecular frameworks and has been made accessible to experiment by the development of femtosecond laser technology during the last 20 years. Ultrashort light pulses can now be used to follow in real-time the primary events of many chemical-but also physical or biological-processes. Ahmed Zewail has received the Nobel Prize in Chemistry 1999 “for his studies of the transition states of chemical reactions using femtosecond spectroscopy” 2 . The combination of adaptive quantum control with femtosecond laser spectroscopy-adaptive femtosecond quantum control-is a new research field which goes beyond “simple” observation, seeking to control chemical reactions by suitably “shaped” femtosecond light fields. In contrast to the methods used in “conventional” chemistry, this is done on a “microscopic” level-directly in the investigated molecule-by forcing the dynamical e v e lution of quantum wavefunctions into the desired direction. A number of recent review articles and books have treated the subject of quantum control from different perspectives3-’’ Here we first discuss the general scheme of adaptive control experiments and then present a number of selected examples from our laboratory. These examples illustrate the different possibilities for gas-phase, liquid-phase or purely optical experiments using appropriate feedback signals.

’.

87 feedback

fs laser pulse

Figure 1. Experimental setup. A femtosecond pulse shaper (right side) is used to generate phase-modulated laser pulses for closed-loop adaptive quantum control. Suitable experimental feedback signals are processed in a learning algorithm which iteratively improves the applied laser pulse shape until an optimum is reached.

2. Experimental Scheme

The experimental setup is shown in Fig. 1. A femtosecond laser pulse shaper (right side of the figure) is used to impose specific spectral phase modulations. Details of our setup were published previously12. Briefly, the device consists of a zero-dispersion compressor in a 4f-geometry, which is used to spatially disperse and recollimate the femtosecond laser pulse spectrum. Insertion of a liquid-crystal display (LCD) in the Fourier plane of the compressor provides a mechanism for convenient manipulation of the individual wavelength components. By applying voltages, the refractive indices at 128 separate pixels across the laser spectrum can be changed, and upon transmission of the laser beam through the LCD, a frequency-dependent phase is acquired due to the individual pixel voltage values. In this way, an immensely large number of different spectrally phase-modulated femtosecond laser pulses can be produced. The shaped laser pulses are then used in different types of adaptive quantum control experiments wherein experimental feedback signals guide an automated search for optimal electric fields within a learning algorithm. In general the optimal pulse is not known in advance and since the variational space of possible pulse shapes is so huge, scanning the complete parameter space is impossible. In order to overcome this problem, we use a learning loop to find optimized electric fields taking into account the experimental outcome. In an iterative process, the electric laser fields are improved with a computer algorithm until the particular optimization ob-

88

jective is reached and the feedback signals approach the user-specified goal. The global search method we use for this purpose is an evolutionary algcrithm with crossover, cloning, and mutation procedures l 3 , l 4 ? l 5 .The new generation of individuals thus inherit good genetic properties and improve their adaptation to the “environment” so that after cycling through the evolutionary loop for many generations, an optimized laser pulse shape results. In the case of an optical control experiment, second-harmonic generation (SHG) in a thin nonlinear crystal can be used as feedback signal. Control of photodissociation reactions in the gas phase can be carried out by monitoring the photofragment yields in a time-of-flight (TOF) mass spectrometer. Selected mass peaks are then recorded by boxcar averagers, and ratios as well as absolute yields can be optimized within the evolutionary algorithm. With the objective to control excited-state population in the liquid-phase, we have used emission spectroscopy to assign the amount of population which was transferred to the emissive state.

3. Selected Examples

3.1. Automated Pulse Compression In chirped-pulse amplification (CPA) femtosecond laser systems, the generation of bandwidth-limited laser pulses requires the spectral phase function at the output to be flat. With the described method of SHG maximization, it is possible to remove the phase errors and compress the pulses down to their theoretical limit 17313718)19.

before optimization

after optimization

. -200 0 200 time I fs

Figure 2. Automated laser pulse compression. a) The pulse shape before optimization was evaluated from FROG analysis and shows the temporal intensity (solid line) and phase (dashed line). b) experimental setup with pulse shaper, second-harmonic generation (SHG) and detection by a photodiode (PD) as feedback signal. c) The resulting optimized pulse shape displays a much shorter temporal intensity profile (solid line) and essentially chirp-free phase (dashed line).

89 The imperfect laser output without optimization as determined by SHGFROG analysis (FROG = Frequency Resolved Optical Gating) 2o is shown in Fig. 2a. A shoulder contribution is clearly visible in the temporal intensity profile. The experimental setup for automated pulse compression is shown in Fig. 2b, with SHG as feedback signal in the closed-loop evolutionary optimization. The resulting output pulse (Fig. 2c) is temporally shortened and corresponds now to the bandwidth limit. It is even possible to precompensate for the effects of optical elements between laser output and experiment (for example focusing optics such as dispersive microscope objectives). The user simply has to define the feedback signal (SHG) at the spot of the experiment, and the pulses will be optimally short at that location.

3.2. Gas-Phase Control With time-of-flight (TOF) mass spectrometry, it is possible to quantitatively record all photoproducts arising after the interaction of shaped femtoseond laser pulses with isolated molecules in the gas phase. In the first experiments of this kind 21, we have controlled a product branching ratio in the complex organometallic molecule CpFe(CO)2C1, where Cp = C5H5. The two selected separate product channels lead either to the loss of one carbonyl ligand or to almost complete fragmentation where only the F e C l bond remains. As it can be seen in Fig. 3, it is then possible to both maximize (5:l) and minimize (1:l) the ratio CpFeCOCl+/FeCl+ between these two channels as compared to unshaped laser pulses (2.5:l). The optimized electric fields after evolutionary optimization show considerable complexity. A separate analysis 22 revealed that it is not possible to obtain the same or similar results by trivial intensity variation effects, rather the detailed structure is relevant. We have also shown that the optimal electric fields are sensitive with respect to ligand variation 23. In the simpler and more symmetric Fe(CO)5, we have maximized and minimized direct ionization versus complete fragmentation 24. Optimized pulse shapes in that case can be interpreted by comparison with pumpprobe mass spectra, yielding insight into the fragmentation dynamics. It is also possible to simultaneously optimize relative product yields (branching ratios) and absolute yields (efficiencies) by defining a suitable fitness function. However, gas-phase quantum control is not limited to organometallic molecules. Due to the very general implementation in a learning loop, many

90 reactant

products

product yields

5:l

1 ps optimized electric field (maximurn product ratio)

bandwidth-limited laser pulse

2.5:l

1:l

optimized electric field (minimum product ratio)

Figure 3. Quantum control of the CpFeCOCl+/FeCl+ product ratio. Top: relative yields of the two investigated product channels leading to CpFeCOCl+ (black blocks) and FeCl+ (shaded blocks). Bottom: temporal electric fields leading t o a maximum (left) and minimum (right) CpFeCOCl+/FeCl+ ratio, as well as the electric field of a bandwidth-limited laser pulse (middle) leading t o an intermediate branching ratio.

classes of molecular photodissociation processes can be controlled. For example, we have shown bond-selective photochemistry in CH2ClBr 25 and in lactic acid 2 6 . In atomic Ca we have optimized double ionization, finding phase-shaped (and not bandwidth-limited) laser pulses to be optimal for this high-order nonlinear process 27. Meanwhile, molecular photodissociation control was also accomplished in other groups 28,29.

3.3. Liquid-Phase Control Probably the most intriguing initial motivation for quantum control was selective photochemistry in the liquid phase by which macroscopic amounts of chemical substances could be synthesized. While this dream has still not been realized experimentally, a number of breakthroughs toward achieving this goal have been achieved. For example, the possibility of cleaving specifically selected bonds within large molecules has been demonstrated in the gas phase (see above). Our first approach to liquid-phase quantum control dealt with selective photoexcitation, monitoring a photophysical (rather than photochemical) observable. We have shown how light pulses can be optimized such that they selectively transfer electronic population within one specific complex dye molecule in solution 30. This could be used for selective photoexcitation within mixtures of molecules. For this, we have investigated the dye molecule DCM and the organometallic complex [Ru(dpb)~] ( P F s ) 2 , where dpb = 4,4’-diphenyl-2,2’bipyridin, both dissolved in methanol. After excitation with at least two

91

photons at 800 nm and the possibility for additional interaction of the excited species with the electric field of the shaped femtosecond laser pulse, the amount of excited-state population is recorded by monitoring the spontaneous emission signal from each of the two molecules. The objective here was to excite DCM while not exciting (or at least reducing the excitation of) [R~(dpb)g]’+31.

h/nm

energy/pJ

~“/104fs2

h/nm

generation

Figure 4. Control of liquid-phase molecular excitation. a) The relative DCM/[Ru(dpb)3I2f linear absorption ratio (solid line) is shown as a function of wavelength. The second-order power spectrum (dotted line) of a bandwidth-limited laser pulse is shown t o illustrate possible two-photon transition frequencies. The DCM/[Ru(dpb)3I2+ emission ratio is plotted b) for varying pulse energies of unshaped laser pulses, c) for varying second-order spectral phase (i.e., linear chirp), d) for scanning a symmetric and rectangular window of 5 nm width over the laser spectrum, and e) for many-parameter phase-shaping as a function of generation number within the evolutionary algorithm.

Selective excitation can be achieved with many-parameter adaptive quantum control in which all 128 LCD pixels are optimized independently. It is shown in Fig. 4e how the emission ratio evolves as a function of generation number within the evolutionary algorithm so that finally a 50% increase is observed. Thus it is possible to selectively excite one specific molecular species even within mixtures of molecules with identical absorption profiles. It should be emphasized that control is possible in the presence of complex solutesolvent interactions. The feedback signal rises significantly above the level given by the (unsuccessful) single-parameter schemes (indicated by the dashed line). The failure of the single-parameter schemes indicates that the control mechanism cannot be based on the initial excitation step (which is identical for the two molecules), but exploits the differences in the dynamical wavepacket evolutions on excited-state potential energy surfaces. 3.4. Polarization Control Molecules are three-dimensional objects. It would therefore be fascinating if not only scalar but also vectorial properties of light fields could be ex-

92

ploited to guide dynamical evolution. We recently developed the technique of femtosecond polarization pulse shaping by which the polarization state of light (i.e., its ellipticity and orientation) as well as the intensity and light oscillation frequency can be varied as functions of time within a single femThis is done within a frequency-domain pulse tosecond laser pulse shaper which contains two LCD layers instead of just one. 32133t34.

Figure 5 . Quasi-three-dimensional electric field representation for a polarization-shaped femtosecond laser pulse. Time evolves from left to right, and electric field amplitudes are indicated by the sizes of the corresponding ellipses. The momentary frequency can be indicated by colors or grey-shading, and the shadows represent the amplitude envelopes of component Ei (bottom) and component Ez (top) separately.

An example for a complex but experimentally produced and characterized laser pulse is shown in Fig. 5 . The electric field evolves as a function of time from left to right, and the tip of the electric field vector spirals on the surface of this quasi-three-dimensional object. It is seen that many different elliptical, linear or circular light states are reached in a transient fashion within the same light pulse. This is not a “random” light burst, but a completely coherent light field with a total duration of several picoseconds. Polarization femtosecond pulse shaping can be considered a novel spectroscopic technique, because the temporal as well as three-dimensional spatial properties of quantum wavefunctions can be addressed and controlled. But there are also fundamentally new prospects which rely on the vectorial manipulation of light-matter interaction (for example, reaching enantiomer selectivity in quantum control The door to many experimental possibilities has just been opened. 35336137.

93 4. Conclusion

With adaptive quantum control, it is possible to manipulate the dynamics of complex quantum systems. While applications of the closed-loop learning scheme to molecular systems were first demonstrated in a populationtransfer experiment by Bardeen et al. 38, our group was then the first to realize automated quantum control of photodissociation reactions in complex molecules ”. Other early examples of the adaptive scheme include the excitation of different vibrational modes in a molecular liquid 39 and the control of vibrational dynamics in a four-wave mixing experiment 40. But optimal quantum control as described here is not limited to molecular systems. Femtosecond laser pulse shapers and learning loops have been used for automated pulse compression as described above and optimized generation of arbitrary laser pulse shapes control of two-photon transitions in atoms shaping of Rydberg wavepackets 4 5 , optimization of high-harmonic generation 46, and control of ultrafast semiconductor nonlinearities 47. Shaped electric fields have also been suggested to be of use in the context of laser cooling 48. Recent developments include furthermore the transfer of adaptive control methods to achieve selective photoexcitation in the liquid phase 31, control over energy transfer even in a biological system 49, and the optimization of Raman-type nonlinear spectroscopy and microscopy One challenge for the future is certainly still the realization of reaction control in liquids where specific bonds are broken and others are formed. This could lead to applications in pharmaceutical industry, in connection with the synthesis of expensive substances. But there are many more fundamental questions still to be explored in quantum control. Novel technological developments such as field polarization shaping or spatiotemporal pulse shaping can stimulate new types of experiments. Although the initial dream of microscopic chemical reaction control is still at the heart of this research field, many other ideas have been realized already. 41142,

43144,

50751.

52153

Acknowledgments We would like to thank our coworkers, A. Assion, M. Bergt, N. H. Damrauer, C. Dietl, B. Kiefer, V. Seyfried, and M. Strehle, for their dedicated efforts. We gratefully acknowledge financial support from the European Coherent Control Network (COCOMO): HPRN-CT-1999-00129, the German-Israeli Cooperation in Ultrafast Laser Technologies (GILCULT): FKZ-13N7966, and the “Fonds der chemischen Industrie.” T . B. thanks

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the German Science Foundation for a n “Emmy-Noether” Fellowship.

References 1. 2. 3. 4. 5.

6. 7.

8.

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

R. S. Judson and H. Rabitz, Phys. Rev. Lett. 68,1500 (1992). in Nobel Price in Chemistry 1999 (The Nobel Foundation, 1999). W. S. Warren, H. Rabitz, and M. Dahleh, Science 259,1581 (1993). R. J. Gordon and S. A. Rice, Annu. Rev. Phys. Chem. 48,601 (1997). S. A. Rice and M. Zhao, Optical Control of Molecular Dynamics (Wiley, New York, 2000). H. Rabitz et al., Science 288,824 (2000). M. Shapiro and P. Brumer, in Advances in Atomic, Molecular, and Optical Physics, Vol. 42 edited by B. Bederson and H. Walther (Academic Press, London, 1999), pp. 287-345. T. Brixner, N. H. Damrauer, and G. Gerber, in Advances in Atomic, Molecular, and Optical Physics, Vol. 46 edited by B. Bederson and H. Walther (Academic Press, London, 2001), pp. 1-54. D. J. Tannor, Introduction to Quantum Mechanics: A Time Dependent Perspective (University Science Press, Sausalito, 2002). M. Shapiro and P. Brumer, Principles of Quantum Control of Molecular Processes (Wiley, New York, 2003). T. Brixner and G. Gerber, Chem. Phys. Chem. 4,418 (2003). A. M. Weiner, Rev. Sci. Instrum. 71,1929 (2000). T. Baumert et al., Appl. Phys. B 65,779 (1997). D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, 1993). H.-P. Schwefel, Evolution and Optimum Seeking (Wiley, New York, 1995). B. A. Mamyrin, Int. J . Mass Spectrom. Ion Processes 131,1 (1993). D. Yelin, D. Meshulach, and Y. Silberberg, Opt. Lett. 22, 1793 (1997). A. Efimov et al., Opt. Lett. 23,1915 (1998). D. Zeidler et al., Appl. Phys. B 70, S125 (2000). R. Trebino et al., Rev. Sci. Instrum. 68,3277 (1997). A. Assion et al., Science 282,919 (1998). T. Brixner, B. Kiefer, and G. Gerber, Chem. Phys. 267,241 (2001). M. Bergt et al., J. Organomet. Chem. 661 199 (2002). M. Bergt et al., J . Phys. Chem. A 103,10381 (1999). N. H. Damrauer et al., Eur. Phys. J. D 20,71 (2002). T. Brixner et al., J . Mod. Opt. 50, 539 (2003). E. Papastathopoulos, M. Strehle, and G. Gerber, in preparation. S. Vajda et al., Eur. Phys. J . D 16,161 (2001). R. J. Levis, G. M. Menkir, and H. Rabitz, Science 292,709 (2001). T. Brixner et al., J . Chem. Phys. 118,3692 (2003). T. Brixner et al., Nature 414,57 (2001). T. Brixner and G. Gerber, Opt. Lett. 26,557 (2001). T. Brixner et al., Appl. Phys. B 74,S133 (2002). T. Brixner et al., J. Opt. SOC.Am. B 20,878 (2003).

95 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53.

M. Shapiro and P. Brumer, J . Chem. Phys. 95, 8658 (1991). Y. Fujimura et al., Chem. Phys. Lett. 306,1 (1999). M. Shapiro, E. Fkishman, and P. Brumer, Phys. Rev. Lett. 84,1669 (2000). C. J. Bardeen et al., Chem. Phys. Lett. 280, 151 (1997). T. C. Weinacht et al., J. Phys. Chem. A 103,10166 (1999). T. Hornung, R. Meier, and M. Motzkus, Chem. Phys. Lett. 326,445 (2000). D. Meshulach et al., J. Opt. SOC.Am. B 15, 1615 (1998). T . Brixner et al., Appl. Phys. B 70,S119 (2000). D. Meshulach and Y . Silberberg, Nature 396,239 (1998). T. Hornung et al., Appl. Phys. B 71,277 (2000). T. C. Weinacht, J . Ahn, and P. H. Bucksbaum, Nature 397,233 (1999). R. Bartels et al., Nature 406,164 (2000). J. Kunde et al., Appl. Phys. Lett. 77, 924 (2000). D. J. Tannor and A. Bartana, J . Phys. Chem. A 103,10359 (1999). J. L. Herek et al., Nature 417,533 (2002). D. Oron et al., Phys. Rev. Lett. 8 8 , 063004 (2002). N. Dudovich, D. Oron, and Y . Silberberg, Nature 418,512 (2002). M. M. Wefers, K. A. Nelson, and A. M. Weiner, Opt. Lett. 21,746 (1996). T . Feurer et al., Opt. Lett. 27,652 (2002).

SPECTRALLY RESOLVED FEMTOSECOND 2-COLOUR 3-PULSE PHOTON ECHOES FOR STUDIES OF MOLECULAR DYNAMICS L.V. DAO, C.N. LINCOLN, R.M. L O W AND P. HANNAFORD Centre for Atom Optics and Ultrafast Spectroscopy Swinbume University of Technology PO Box 218, Hawthorn, Victoria 3122, Australia E-mail: dvlap @swin.edu.au We report on the use of spectrally resolved femtosecond 2-colour 3-pulse photon echoes as a potentially powerful multidimensional technique for studying molecular dynamics in ground and excited electronic states of complex molecules, including biomolecules.

Multidimensional femtosecond nonlinear techniques are finding increasing application in studies of molecular dynamics and energy and charge transport in molecular systems.' Techniques with two independent time delays, such as 3-pulse photon echo peak-shift? have yielded information about molecular dynamics in a range of systems. New techniques such as 2-D Fourier transform3 or spectrally-resolved photon echoes4may give additional information by tracing the third-order polarization P(3)with more than two degrees of freedom. We report on the use of spectrally resolved femtosecond 2-colour 3-pulse photon echoes (PE) as a potentially powerful multidimensional technique for studying molecular dynamics.435 With this technique four degrees of freedom can be independently controlled to provide detailed information about the dynamics of the molecule. The spectrally resolved PE signals reflect the temporal evolution of P(3)on a femtosecond time scale, while the use of two independent pulse time delays, t l 2 and tZ3, reveal the dynamics of the coherence and population ensembles, respectively, and allow separation of contributions from homogeneous and inhomogeneous broadening. Furthermore, the wave-lengths of the pump and probe pulses can be independently selected to drive particular quantum pathways and to study the dynamics of vibrational relaxation and vibrational coherences in both the excited and ground states of the molecule. In our PE experiment the sample is illuminated by two femtosecond pump pulses with wave vectors kl, kz and wavelength and a probe pulse with wavevector k3 and wavelength approbe.The laser pulses generate a third-order polarization P"'(t, t12, tZ3)which is dependent on the time ordering and the frequencies of the femtosecond pulses. The femtosecond laser system consists of a Tixapphire laser and regenerative amplifier pumping two independent optical parametric amplifiers that provide pulses of around 100 fs duration over a

sump

96

97 RhB: Pump 550 nm Probe 565 nm

R h 6 Pump 565 nm Probe 550 nm

RhlOl: Pump 560 nm Probe 575 nm

Figure 1. Two-colour three-pulse photon echo spectra versus population time tz3 (a, d, g) and coherence time tlz (b, c, e, f, h, i) for 104M RhB (a-f) and RhlOl (g-i) in methanol at fixed values of the other time delay. Insets show contour plots of the PE spectra, with the straight lines indicating the centre probe laser wavelength (vertical lines) and zero population time (horizontal lines).

broad range of wavelengths (250-2OOOnm). The PE signal is detected in the phase-matching direction k4 = k3 + k2 - k, and analysed in a spectrometer with CCD detector. Multidimensional PE spectra are obtained by scanning over the coherence time t12or the population time tz3for various fixed values of the other time delay. Figure 1 shows 2-colour 3-pulse PE spectra recorded for the dye molecules Rhodamine B (W)and Rhodamine 101 (RhlOl) in methanol. The absorption maxima for W and RhlOl occur at 550 nm and 575 nm, respectively. When &,mp < &,be (Fig. 1 a-c, g-i) we observe a delayed enhancement in the PE signal on the red side when scanning the population time t23 (Fig. 1 a, g) and an increase in the PE signal on the red side at longer population times when scanning the coherence time tI2(Fig. 1 b, c, h, i). In this case the wavelengths and pulse sequence favour quantum pathways leading to generation of coherence and population in the excited electronic state of the molecule. Relaxation and

98

coherence transfer between vibrational levels in the excited state lead to the delayed red shft in the PE spectrum. When APu, > & , b e (Fig. 1 d-f) the wavelengths are favourable for generating population and coherence in high vibrational levels of the ground electronic state. The PE signal is enhanced on the blue side with increasing population time, reflecting the vibrational dynamics in the ground state. The PE spectra for RhlOl are broader than for RhB and show more structure (Fig. 1, insets), reflecting the differing structure of the molecules. RhlOl is a rigid, fairly planar molecule with a stable excited state comprising a range of vibrational levels and (for dpUmp ' ~ . To obtain the number of atoms in the condensate and in the thermal cloud, and the temperature, we make an absorption image after a 20 ms time of flight. The number of atoms, obtained to within 20%, is calibrated from a measurement of the temperature of condensation T,. The temperN

118

ature T is extracted from a two-component fit to the absorption images, yielding the temperature of the thermal cloud fitted by an ideal Bose function with zero chemical potential. In fact, the temperature obtained at the end of the evaporation ramp can be chosen a priori to within 20nK by controlling the final trap depth (final rf frequency, as compared to the one totally emptying the trap, with is checked every five ramps) to a precision of 2 kHz. This provides a high reproducibility which is a real asset for these experiments. As in Shvarchuck et al.13, we observe strong shape oscillations at the formation of the condensatelg, despite a slow evaporation (less than 100 kHz s-l) across T,. We then hold the condensate for a time of typically 7 seconds in the presence of an rf shield, to damp the axial oscillations enough that they do not affect the Bragg spectra (see below).

4. Measurement of the Spatial Coherence Function by

Bragg Spectroscopy The most direct way to measure the spatial coherence function relies on atom interferometry. It turns out, however, that this method demands a high level of shot to shot stability of the interferometerz1, and we have decided to first use a complementary method, that directly yields the momentum distribution (momentum spectrum) P ( p z ) ,which is nothing else than the Fourier transform of C(s). This is analogous to the two complementary methods of “Good Old Spectroscopy”, where one can either measure the temporal correlation function of the light (the so called “Fourier-transform spectroscopy”), or directly obtain the frequency spectrum. It is well known that the second type of method is much easier to implement, because some physical phenomena (such as dispersion in dielectric media) or “simple” devices (such as diffraction gratings) are able to separate frequency components whose weight can be directly measured. The methods yielding directly the spectrum were in fact the only ones that were used for centuries, until stable and reliable enough interferometers became available. In our problem of Atom Optics, it turns out that we also have a method that directly yields the momentum spectrum: this is “Bragg S p e c t r o s c ~ p y ” ~Our ~ ~ momentum ~. distribution measurement is based on four-photon velocity-selective Bragg diffraction. In this process, atoms are extracted out of the condensate by interaction with a moving standing wave, formed by two counter-propagating laser beams with a relative de-

119

tuning 6. In a wave picture, this can be interpreted as a (second order) Bragg diffraction of the atomic matter waves off the grating formed by the light standing wave, and it is therefore resonant only for a given atomic de Broglie wavelength, or equivalently for a given value of the atomic momentum. The number of extracted atoms is therefore proportional to the density of probability of that particular value of the momentum. Writing the Bragg condition of diffraction off a thick grating, one finds that the momentum component resonantly diffracted out of the condensate depends on the velocity of the light standing wave, and therefore on the detuning 6, according to

P,

= M ( 6 - S W R ) / ( ~ ,~ L )

(1)

with W R = h k 2 / ( 2 M ) ,M the atomic mass, and k~ = 2n/X (A = 780.02nm). By varying the detuning 6 between the counter-propagating laser beams that make the moving standing wave, and measuring the fraction of diffracted atoms versus 6, one can build the momentum distribution spectrum. The stability of the differential frequency 6 determines the spectral resolution, and must be as good as possible. The optical setup is as follows. A single laser beam is spatially filtered by a fiber optic, separated into two arms with orthogonal polarizations, frequency shifted by two independent 80 MHz acoustc-optic modulators, and recombined. The modulators are driven by two synthesizers stable to better than 1Hz over the typical acquisition time of a spectrum. The overlapping, recombined beams are then sent through the vacuum cell, parallel (to within lmrad) to the long axis of the trap, and retro-reflected to obtain two standing waves with orthogonal polarizations, moving in opposite directions. To check the differential frequency stability, we have measured the beat note between the two counter-propagating beams forming a standing wave. The average over ten beat notes had a half-width at half-maximum (HWHM) of 216(10) Hz for a 2ms pulse23. 5. Axial Bragg Spectrum of an Elongated Condensate The following experimental procedure is used to acquire a momentum spectrum. At the end of forced evaporative cooling, the radio frequency knife is held fixed for about 7s to allow the cloud to relax to equilibrium (see above). The magnetic trap is then switched off abruptly, in roughly loops, and the cloud expands for 2ms before the Bragg lasers are pulsed on for

120

2 ms. The lasers are tuned 6.6 GHz below resonance to avoid Rayleigh scattering, and the laser intensities (about 2 mW/cmz) are adjusted to keep the diffraction efficiency below 20 %. We take the Bragg spectrum after expansion rather than in the trap to overcome two severe problems. First, in the trapped condensate, the mean free path (about 10 pm) is much smaller than its axial size of 260 pm, so that fast Bragg-diffracted atoms would scatter against the cloud at restz5. Second, the inhomogeneous mean field broadening6 would be of the order of 300Hz, i.e., larger than our instrumental resolution. By contrast, after 2ms of free expansion, the peak density has dropped by two orders of magnitude26,and both effects become negligible. One may wonder whether this 2ms expansion doesn’t affect the momentum distribution we want to study. In fact, the phase fluctuations do not significantly evolve in 2ms, since the typical timescale for their complete conversion into density ripples varies from 400 ms to 15 s for the range of temperatures we explorez7. Also, the mean field energy is released almost entirely in the radial direction, because of the large aspect ratio of the trapz6, and contributes only about 50Hz of Doppler broadening in the axial direction. The only perturbation due to the trap release seems to be small axial velocity shifts (around 100 pm s-l) attributed to stray magnetic gradients that merely displace the spectra centers. After application of the Bragg pulses, we wait for a further 20ms to let the diffracted atoms separate from the parent condensate, and take an absorption image. Diffraction efficiency is defined as the fraction of atoms in the secondary cloud. We repeat this complete sequence typically five times at each detuning, and we average the diffraction efficiency for this particular value of the detuning. Repeating this process at various values of the detuning (typically 15), we plot the diffraction efficiency versus 6, and obtain an “elementary spectrum”. As shown on Fig. la, an elementary spectrum at a given temperature still shows a lot of noise. To average out this noise, we have taken many elementary spectra at the same temperature (between 10 and 40), but during different runs (on different days). We have also varied the hold time (of about 7 s, see above) over a range of 125ms, to average out possible residual shape oscillations. We then take all the spectra corresponding to the same temperature, and we reduce them to the same surface, background, and center, and superpose them2*. Figure l b shows the result of such a data processing, at a temperature well above Tb,i.e., for a case where the broadening due to phase fluc-

121

b

0

I

0

1

2

3

Figure 1. Bragg spectrum of an elongated BEC a t T = 261(13) nK, corresponding to

TIT+ = 20(2). (a) Elementary spectrum, i.e., diffraction efficiency vs. relative detuning of the Bragg lasers corresponding to 75 diffraction efficiency measurements made at the same temperature, but at various detunings (see text). Measurements at the same detuning (typically 5 ) are averaged. (b) Averaged spectrum, obtained by averaging 12 (recentered and rescaled, see text) elementary spectra. A typical statistical error bar is shown. This spectrum is the superposition of 12 “elementary spectra”, as described in the text. The solid line is a Lorentzian fit, giving a half-width at half-maximum (HWHM) of 316 (10) Hz. (c) and (d) show, respectively, the (folded) residuals of a Lorentzian and of a Gaussian fit to the above spectrum.

tuations is expected to dominate over the instrumental resolution. The width (316 (10) Hz, HWHM) is definitely larger than the resolution (less than 216 (10) Hz, HWHM). Moreover, it is clear by simple visual inspection, and confirmed by examining the residuals of fits (see Fig. l c and Id) that the profile shape is definitely closer to a Lorentzian than to a Gaussian. Such a shape is in contrast to the gaussian-like profile expected for a pure condensate6>z0,and it is characteristic of large 1D phase fluctuationsz7, which result in a nearly exponential decay of the correlation function. 6. Results. Comparison with Theory

Bragg spectra have been taken at various temperatures between T+and T,. Using a Lorentzian fit, we extract a measured half-width AUMfor each temperature, and plot it (Fig. 2a) vs. A Y ~a ,parameter convenient to compare with the theory. That parameter can be directly expressed as a function of the number of atoms and the temperature, and theory predicts that the ideal spectrum half-width should be proportional to A u ~with , a multiplying factor a , depending on the density profile, of the order of 1 (a= 0.67 in our c a ~ e ) ’ ~ In , ~ fact, . since our spectral resolution is limited, the measured spectrum width AVMresults from a convolution of the ideal Lorentzian profile with the resolution function, that we assume to be a Gaussian, of half-width W G . The measured spectrum is then expected to be a Voigt profile, whose shape can hardly be distinguished from a Lorentzian in our range

122

+

+

of parameters, but with a width (HWHM) aAv4/2 d w i (aAv4)2/4. We use that expression to fit the data of Fig. 2a, taking a and W G as free parameters. We find WG = 176 (6) Hz, and a = 0.64 (5)(5). The first uncertainty quoted for a is the standard deviation of the fit value. The second results from calibration uncertainties in the magnification of the imaging system and in the total atom number, which do not affect W G . We note first that the fitted value of W G is slightly smaller than 216 Hz, as it should be23. The agreement of the measured value of a with the theoretical value 0.67, to within the 15 % experimental uncertainty, confirms quantitatively the temperature dependence of the momentum width predicted in Ref. 10.

0.10

250

0.06

200

6

AV4 (Ha

10

16

20

26

Jo

T/T+

Figure 2. Momentum distribution width and coherence length for TIT,+, > 1. (a) Halfwidths at half-maximum AVM of the experimental Bragg spectra versus the parameter Av,+, (see text). The solid line is a fit assuming a Voigt profile with a constant Gaussian resolution function. (b) Coherence length &(divided by the condensate half size L) vs. temperature (divided by T+).The coherence length is obtained after deconvolution from the resolution function.

Conclusion We have shown that the momentum spectrum of an elongated condensate at a temperature smaller than T, but larger than T4 has a Lorentzian shape, in contrast with the Gaussian shape of a “normal” condensate. That shape, as well as the measured broadening, agree quantitatively with the predictions for a phase fluctuating condensate. The coherence lengths corresponding to the spectrum widths are smaller than the condensate axial size (Fig. 2b). It would be interesting to also measure the coherence length for temperatures approaching T4, to study how coherence develops over the whole condensate. Since this corresponds to smaller and smaller momentum widths, the method presented here is not well adapted, and an interferometric measurement directly yielding the spatial correlation function at large separations would be a method of choice. Such measurements require a large path dif-

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ference of the interferometer, and the stability of the interferometer is a crucial issue. We are currently working in that direction.

Acknowledgments This work is supported by CNRS, Ministbre de la Recherche, DGA, EU (Cold Quantum Gases network; ACQUIRE collaboration), INTAS (project

01-0855). References 1. F. Dalfovo et al., Rev. Mod. Phys. 71,436 (1999). 2. E. A. Cornell and C. E. Wieman Rev. Mod. Phys. 74,875 (2002); W. Ketterle Rev. Mod. Phys. 74, 1131 (2002). 3. R. Simon et al., Phys. Rev. Lett. 91,010405 (2003). 4. F. Gerbier et al., cond-mat/0307188. 5. E. W. Hagley et al., Phys. Rev. Lett. 83,3112 (1999). 6. J. Stenger et al., Phys. Rev. Lett. 82, 4569 (1999). 7. I. Bloch et al., Nature 403,166 (2000). 8. D. S. Petrov et al., Phys. Rev. Lett. 8 5 , 3745 (2000). 9. J. 0. Andersen et al., Phys. Rev. Lett. 88, 070407 (2002); D. L. Luxat and A. Griffin, cond-mat/0212103; C. Mora and Y. Castin cond-mat/0212523. 10. D. S. Petrov et al., Phys. Rev. Lett. 87,050404 (2001). 11. S. Dettmer et al., Phys. Rev. Lett. 87,160406 (2001); D. Hellweg et al., A p p l . Phys. B 73,781 (2001). 12. H. Kreutzmann et al., cond-mat/0201348. 13. I. Shvarchuck et al., Phys. Rev. Lett. 89, 270404 (2002). 14. B. Desruelle et al., Phys. Rev. A 60,R1759 (1999). 15. V. Boyer, These de doctorat, Institut d’OptiqueUniversit6 Paris VI (2000). 16. A. Gorlitz et al., Phys. Rev. Lett. 87,130402 (2001). 17. F. Schreck et al., Phys. Rev. Lett. 87,080403 (2001). 18. M. Greiner et al., Phys. Rev. Lett. 87 , 160405 (2001). 19. F. Gerbier et al., cond-mat/0210206. 20. F. Zambelli et al., Phys. Rev. A 61,063608 (2000). 21. Note however that recently the Hanover group has succeeded in such a measurement, and obtained results analogous to ours: D. Hellweg et al., Phys. Rev. Lett. 91,010406 (2003). 22. J. Steinhauer et al., Phys. Rev. Lett. 88, 120407 (2002). 23. Since supplementary mirrors must be added to perform this measurement, the result is an upper limit to the linewidth of 6. 24. S. Richard, These de l’universit6 d’Orsay (2003) 25. A. P. Chikkatur et al., Phys. Rev. Lett. 85, 483 (2000). 26. Y. Castin and R. Dum, Phys. Rev. A 55, R18 (1997); Yu. Kagan et al., Phys. Rev. A 55, R18 (1997). 27. F. Gerbier et al., Phys. Rev. A 67,051602(R) (2003).

EXPERIMENTAL STUDY OF A BOSE GAS IN ONE DIMENSION *

W. D. PHILLIPS, J. H. HUCKANS, B. LABURTHE TOLRA, K. M. O'HARA, J. V. PORTO, S. L. ROLSTON National Institute of Standards and Technology Gaithersburg, MD, USA M. ANDERLINI NFM, Dipartimento di Fisica, Universitii d i Pasa Via Buonarroti 2, I-56127, Pisa, Italy

There has recently been considerable interest in reduced-dimensionality systems involving trapped Bose gases. For example, a gas that would be weakly interacting in 3D can be strongly correlated in 1D permitting the study of phenomena beyond mean field theory. Understanding the coherence properties of 1D systems may be important for atom interferometers using atomic waveguides. This short, qualitative paper provides an extended abstract of our presentation at ICOLS 03 which described our recent studies of 1D Bose gases. Here we report on both the observation of a strong reduction in the three-body recombination rate in 1D gases due to their correlation properties,' and the observation of a 1D Mott transition when these 1D gases are loaded into a 1D optical lattice.' Our approach for studying 1D quantum degenerate Bose gases is to load a Bose-Einstein condensate (BEC) into a sufficiently deep 2D optical lattice. Such a lattice is formed by intersecting two 1D optical lattices at the position of the BEC, which creates a regular 2D array of one-dimensional tubes separated by distances on the order of the optical wavelength. The lattice provides strong confinement transverse to the tubes (along the directions lying in the plane of the laser beams) and weak confinement along the tubes (perpendicular to the plane of the laser beams). In the case where the transverse confinement is strong enough that the tunnelling of atoms between the tubes is negligible, the tubes are an array of independent 1D *This work was supported in part by ARDA and NASA.

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125

systems. The tubes are 1D in the sense that the energy associated with transverse motion, tiwl, is large compared to all other energies, i.e., the temperature, the longitudinal energy tiw,, and the atom-atom interaction energy 47rh2n3~a/m,where m is the atomic mass, n 3 is ~ the 3D density and a is the scattering length. We investigate this system experimentally using a nearly pure condensate of up to 5 x lo5 87Rb atoms, produced in a magnetostatic trap. By slowly applying a 2D optical lattice, we transfer the BEC into the array of tubes producing up to 200 atoms in the central tube. The behavior of the 1D gas is characterized by the dimensionlessparameter y = 2a/(a:nl~), where as = (fi/mwl)1/9 is the transverse harmonic oscillator length and nlD is the linear atomic density in the tube. y > 1 is the Tonks-Girardeau regime where atoms cannot pass each other along the tube. In this latter situation the bosonic atoms behave like fermions; in ) higher correlation particular the pair correlation function g2(q - ~ 2 and functions go to zero as ( 2 1 - 2 2 ) goes to zero. Note the counter-intuitive behavior wherein a low density corresponds to the strongly correlated regime, in contrast to the 3D case. These predictions for the behavior of a 1D gas are discussed in Refs. 3 and 4. We have investigated the appearance of fermionic behavior by comparing the rate of three-body loss in a 3D and a 1D gas. In the 3D gas, held in the magnetostatic trap, we measure the atom number N as a function of time and fit N ( t )to a functional form that includes one-body loss and threebody loss to extract a rate coefficient for the three-body loss. (Two-body loss is assumed to be negligible for 87Rb atoms.5@)We perform the same measurement for atoms held in a 2D lattice (i.e., a bundle of 1D tubes). We find that the three-body rate-coefficient is significantly smaller in the 1D case (see Fig. l),meaning that for the same atomic density, the three-body collision rate is significantly lower. We interpret this to mean that the probability that three particles are close to each other is much smaller in the 1D case, indicating the appearance of fermionic correlations. In our case, y is approximately unity, representing the crossover from Thomas-Fermi to Tonks-Girardeau behavior. We have also investigated the occurrence of a Mott insulator transition in the 1D tubes. When a BEC is adiabatically loaded into a sufficiently deep optical lattice it goes into the Mott insulator state, the ground state of the system. Adiabaticity requires that the loading be slow with respect to the time scales for band excitation in the lattice and for excitation in

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Figure 1. Decay of the number of atoms for the 3D BEC (squares) and for the atoms loaded into the 2D optical lattice (full circles). Although the effective density of the gas loaded into the 2D lattice is more than three times larger than the 3D BEC density, the three-body recombination process is smaller, which is interpreted as the effect of local correlations in the 1D case (or “fermionization”). The inset shows the same data for the first few seconds. The dashed line is an exponential fit at late times t o the 1D data. At short times, the three-body recombination is taken into account (solid line). The dot-dashed line is a fit of 1-body and 3-body decay to the 3D data, assuming the BEC is described by a Thomas-Fermi distribution.

the overall trapping potential, as well as the time-scale corresponding to the on-site interaction energy. In the Mott state, atom number fluctuations from site to site are reduced (number squeezing) and tunneling is inhibited by the on-site interaction energy.’ The number squeezing is accompanied by a large uncertainty in the phase of the atomic wavefunction from site to site and results in the disappearance of atomic diffraction when the atoms are suddenly released from the lattice. This has been observed in a linear array of 2D planes of atoms (created by a 1D optical lattice)8 and in a 3D l a t t i ~ eHere . ~ ~we ~ have ~ observed similar behavior in the array of 1D tubes

127 ,7

..

1.0 -

I...-.

I #

e,

380

400

420

440

Time (ms)

Figure 2. Signature of a 1D Mott transition. The 1D tubes of gas are loaded into an additional 1D optical lattice. The diffraction pattern, well resolved for low lattice depth (a), is no longer resolved when the 1D lattice is deep enough (b), whereas the diffraction pattern is restored when the lattice is again reduced to a shallower depth (c).

created by a 2D lattice. We first turn on the 2D lattice, establishing the array of 1D tubes. Then we slowly turn on a 1D lattice along the tubes, testing for phase coherence by looking for an atom diffraction pattern at various 1D lattice depths. When the turn-on time is sufficiently long, we see that the diffraction pattern disappears for a sufficiently deep 1D lattice and that it reappears when the 1D lattice is slowly turned off (see Fig. 2). This behavior is qualitatively the same as has been observed in the case of the 3D Mott transition, and we interpret this to suggest the occurrence of a similar transition in our 1D case. References 1. B.Laburthe Tolra, K. M. O’Hara, J. H. Huckans, J. V. Porto, S. L. Rolston, and W. D. Phillips, in preparation. 2. B. Laburthe Tolra, K. M. O’Hara, J. H. Huckans, M. Anderlini, J. V. Porto, S. L. Rolston, and W. D. Phillips, to be published in J. Phys. IV fiance. 3. V. Dunjko, V. Lorent, and M. Olshanii, Phys. Rev. Lett. 86,5413 (2001). 4. D. S. Petrov, G. V. Shlyapnikov, and J. T. M Walraven, Phys. Rev. Lett. 8 5 ,

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3745 (2000). 5. E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 79, 337 (1997). 6. J . Soding, D. GuBry-Odelin, P. Desbiolles, F. Chevy, H. Inamori and J. Dalibard, Appl. Phys. B 69,257 (1999). 7. D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. 81,3108 (1998). 8. C. Orzel, A. K. Tuchman, M. L. Fenselau, M. Yasuda, and M. A. Kasevich, Science 291,2386 (2001). 9. M. Greiner, 0. Mandel, T. Esslinger, T. W. Hansch and I. Bloch, Nature 415, 39 (2002). 10. J. V. Porto, S. Rolston, B. Laburthe Tolra, C. J. Williams, and W. D. Phillips, Phil. Pans. R. SOC.Lond. A 361,1417 (2003).

QUANTUM DEGENERATE BOSONS AND FERMIONS IN A 1D OPTICAL LATTICE

C. FORT, G. MODUGNO, F.S. CATALIOTTI, J. CATANI, E. D E MIRANDES, L. FALLANI, F . FERLAINO, M. MODUGNO, H. O T T , G. ROATI AND M. INGUSCIO LENS and Dipartimento d i Fisica, Universitci d i Firenze, and INFM Vza Nello Carrara 1, I-50019 Sesto Fiorentino, Italy

We review our recent experiments on ultracold and quantum degenerate samples of bosonic and fermionic atoms in 1D optical lattices. We study the expansion and the collective excitations of a Bos+Einstein condensate in the lattice, showing how the dynamics is governed by a tunable effective mass. We then study the dipolar oscillations of a Fermi gas along the lattice, which reveal band-structure effects. The comparison with the dynamics of ultracold bosons allows us to identify a strong effect of atomic collisions in the lattice.

1. Introduction

The combination of ultracold atoms and periodic potentials created by light has opened up the possibility of investigating fundamental phenomena in relatively simple and “clean” experiments. Atoms in optical lattices with adjustable strength can mimic a variety of different crystals, allowing the study of phenomena such as band transport’, superfluid flow’ and insulating phases3. The nonlinearities induced by the periodic potentials can also result in dynamical instabilities or bright solitons. Furthermore, cold atoms in optical lattices have been indicated as promising for the implementation of quantum computing, or for the observation of fermionic superfluidity. The possibility of preparing quantum degenerate samples of both bosons4 and fermions5 in optical lattices allows one to study the implication of the different quantum statistics in a periodic potential. In this work we review our recent experiments on the dynamics of bosons and fermions moving in the presence of a 1D optical lattice, which show the interplay between band-structure effects and interactions between particles.

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130 2. Expansion of a Bose-Einstein Condensate in the Lattice

From solid state physics it is well known that in the presence of an infinite periodic potential the energy spectrum of a free particle is modified and a band structure arises6. In the rest frame of the lattice the eigenenergies of the system are En(q),where q is the quasi-momentum and n the band index. According to band theory, the velocity in the n-th band is vn = E 1 a E n / a q and the effective mass is m* = h2(d2En/aq2)-1. Thanks to its very narrow momentum spread a BEC is a very good candidate to probe the band structure produced by an optical lattice. The h / R where R is the physical momentum spread of a condensate is Ap size of the BEC4. This must be compared to the periodicity introduced in the momentum space by the lattice given by 2qB = 2hik~,where k L = 2n/A is the modulus of the laser beam wave-vector (A being the wavelength of the laser creating the periodic potential). Typically R 100pm and in the optical domain A < 1pm, so that A p / 2 q ~ lop3. N

N

N

4 1.5

lattice ofl lattice on

.3

0.0 0.0

0.5

Figure 1. (a) Velocity of the condensate in the rest frame of the optical lattice in the first two energy bands as a function of the quasimomentum q (in units of qB = W W B = 4 ~ ) . The continuous line is the corresponding curve given by the band theory. (b) Effective mass of the condensate moving in the optical lattice. Data points are extracted from the measured velocity and the solid line is the corresponding theoretical prediction. The data are obtained with an optical potential depth Vopt = 1 . 3 E ~ (ER = h2/2mX2).

In a first experiment we have investigated the dynamics of a 87Rb cigarshaped BEC released from the harmonic magnetic trap and expanding in a moving optical lattice7, aligned along the long axis of the condensate. The lattice velocity has been varied by changing the frequency difference of

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two counterpropagating laser beams that interfere and produce a moving standing wave. After an initial expansion of 1 ms, we adiabatically switch on the moving optical lattice and let the condensate continue its expansion in this environment. From the center-of-mass position of the BEC moving in the lattice, we extract the velocity in the rest frame of the lattice as a function of the quasi-momentum in the first and second energy band. The results are shown in Fig.1 together with the values of the effective mass derived by evaluating the derivative dvldq from the finite increment between two consecutive data points of Fig. la.

lloi 100

90

80 70

60 0.0

0.5

1 .o

1.5

Quasimomentum (q~) Figure 2. Axial and radial dimensions of the condensate after an expansion of 13 ms in . experimental points (filled and open circles) show the optical lattice (V,,t = 2 . 9 E ~ ) The the Thomas-Fermi radii of the cloud extracted from a 2D fit of the density distribution. The dotted lines show the dimensions of the expanded condensate in the absence of the optical lattice. The continuous lines are theoretical calculations obtained from a 1D effective model 7,9.

By varying the velocity of the optical lattice we can adiabatically load the condensate in the periodic structure with different quasi-momenta q; thus we can choose the value of the effective mass from positive to negative. The effective mass enters into the diffusive kinetic term of the Schrodinger equation determining the expansion of the condensate. In particular, a negative effective mass is equivalent to a time reversal, so that an initial expanding condensate loaded into an optical lattice with negative effective mass is observed to contract7is. In Fig. 2 we show the measured radial and

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axial radii of a condensate expanding inside the optical lattice. Varying the velocity of the lattice, or equivalently the quasi-momentum q of the condensate in the rest frame of the lattice, we range from a slower axial expansion to a contraction ( q 5 q B ) or a faster expansion ( q 2 q B ) , that we attribute, respectively, to m* > m, m* < 0 and 0 < m* < m. In Fig. 2 we also report the radial dimension of the BEC. We observe that the radial expansion is enhanced in the region of axial contraction ( q 5 q B , m* < 0) where the nonlinear coupling between axial and radial direction is expected to increase7.

Figure 3. Measured dipole (a) and quadrupole (b) mode frequencies of the condensate moving in the harmonic magnetic potential in the presence of an optical lattice. The frequencies are reported as a function of the optical potential depth (in units of the recoil energy ER). The dashed lines correspond to the dipole and quadrupole frequencies in the pure harmonic trap while the continuous lines have been obtained by renormalizing the frequencies with the effective mass introduced by the periodic potential.

The modified dynamics of a BEC moving in an optical lattice can be evidenced also by studying the low-lying collective modes of a trapped condensate in the combined potential created by adding to the magnetic harmonic trap the periodic potential created by a standing wavelo. As predicted by the theory", the hydrodynamic equations of superfluids for a weakly interacting Bose gas can be generalized to include the effects of a periodic potential. The generalization can be done through the introduction of a renormalized interaction coupling constant and the effective mass. As a consequence, the frequencies of the modes characterized by a motion along the optical lattice axis are down shifted as the optical potential depth is increased and the effective mass at the center of the first band increases. In

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Fig. 3 we report the dipole and quadrupole mode frequencies measured as a function of the optical lattice depth. The experimental data are compared with the prediction (continuous line) based on the effective mass calculation. Also in this case the constant effective mass of the band theory is adequate to describe the dynamics of the condensate in the presence of the periodical potential created by the optical lattice.

3. Dipolar Oscillations of a Fermi Gas in the Lattice In a second experimental apparatus we have studied the dynamics of a Fermi gas of 40K atoms in a 1D optical lattice. The picture is now completely different, because the momentum spread of the atomic sample is comparable with the Bragg momentum q ~ More . precisely, for our typical experimental conditions, the Fermi energy is just comparable with the recoil energy5, E F M ER. This clearly complicates the description of all the observable phenomena, but the system is certainly worth a deep investigation, because fermions in periodic potentials are at the basis of very fundamental phenomena, like the electron transport in a crystal. In a first series of

A

-30

0

50

100

150

200

Time (ms) Figure 4. Dipolar oscillations of a Fermi gas in the magnetic trap alone (triangles), and in presence of a lattice with U = ~ E R(circles). The continuous lines are fits to a damped sinusoidal oscillation.

experiments we have studied the transport of a Fermi gas of identical (spinpolarized) particles along the lattice. Since the atoms are not interacting at low temperature, we have performed the experiment by keeping the atoms

134

in the magnetic potential, and by inducing a sloshing motion along the lattice, aligned along the weak axis of the trap, as in the experiments with BEC. As shown in Fig.4, in general we observe a damped dipolar oscil26 24 h

8 22

Figure 5 . Measured dipole frequency of the Fermi gas in the lattice. The continuous line is calculated from the effective mass of the fermions in the lattice at the band center. The good agreement with the experiment indicates that only the atoms oscillating a t the bottom of the band are contributing to the macroscopic motion.

lation of the cloud, which proceeds at a frequency smaller than the one in the bare magnetic potential, and with a displaced center-of-oscillations. All these features can be explained with a simple model of non-interacting particles moving in the magnetic potential with the usual band-like velocity dispersion given by the lattice. Using the semiclassical approximation6, it is indeed possible to study the trajectory of individual particles in the combined potential. We find that particles with sufficiently low energy perform harmonic oscillations at an effective frequency that scales with l/m, as observed for the BEC. As the particle's energy increases towards the nonparabolic part of the band, the oscillation gets distorted, and finally the atoms start to perform Bloch oscillations on one side of the magnetic potential. The two latter phenomena are clearly responsible for the decay of the oscillation, which is actually due to the dephasing of the motion of the independent particles, and for the offset that we observe in the experiment. Both the decay rate and the offset increase as the lattice strength is increased (we have explored the range V&=0-8 E R ) , due to an increasing

135 nonlinearity of the bands. Although a quantitative analysis is still underway, the consistency between the observation and the simple band model is confirmed by the quite good agreement of the measured oscillation frequency with theory, as shown in Fig.5. We have also studied the role of thermal excitation in the dynamics of the Fermi gas, and found that the temperature does not change the picture as long as k B T < E F , i.e., as long as the energy distribution is not strongly modified and most of the atoms are confined to the first band. On the contrary, for k B T > ER more atoms are thermally excited to the non-parabolic part of the bands, resulting in a larger damping of the dipolar oscillations. 4. Role of Collisions in the Transport of Bosons and

Fermions To check for the role of interactions between particles, we have performed a second series of experiments using a mixture of either two spin states of fermions or spin-polarized fermions and bosons12, or even with bosons alone. We find that elastic collisions between particles, that are allowed

0

100

200

300

400

500

Time (ms) Figure 6 . Dipolar oscillation in presence of a shallow lattice, with Vopt = 0 . 6 E ~for , a pure Fermi gas (circles) and for a Fermi gas in presence of a Rb thermal cloud at T=150nK (triangles). The strong damping in the latter case is due to boson-fermion collisions in the lattice. Due to the presence of boson-boson collisions, under the same conditions the bosonic cloud is very strongly damped even in the absence of the Fermi gas.

in all these cases, results in a very strong damping of the dipolar motion,

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as shown in Fig. 6. We explain this as a consequence of the non-parabolic dispersion of the energy bands, which implies a not strict conservation of the pair momentum in a binary atomic collision (only the quasi-momentum must be conserved in the lattice6). On the other hand, momentum conservation is just what ensures undamped oscillations of cold atom clouds in a parabolic magnetic potential. We also observe a slow decay of the offset from the trap center, as a consequence of the collisional decay of the atoms performing Bloch oscillations on one side of the potential. This kind of system is interesting for two reasons. On the one hand one can speculate whether the lattice could be used as 6, very efficient detector for elastic collisions, to be possibly used to study the transition from a strongly collisional two-species Fermi gas to a superfluid state. On the other hand, a more detailed study of Bloch oscillations in the combined potential in various collisional regimes might allow one to understand the role of collisions in the macroscopic transport along a perfect periodic potential.

Acknowledgments We acknowledge contributions by M. Zawada and useful discussions with the BEC theory group in Trento. This work was supported by MIUR, by EC under contract HPIUCTl999-00111, and by INFM, PRA “Photonmatter”.

References 1. B. P. Anderson and M. A. Kasevich, Science 281,1686 (1998). 2. F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi and M. Inguscio, Science 293,843 (2001). 3. M. Greiner, 0. Mandel, T. Esslinger, T. W. Hansch and I. Bloch, Nature (London), 415,39 (2002). 4. J. Stenger et al., Phys. Rev. Lett. 82,4569 (1999). 5. G. Modugno, F. Ferlaino, R. Heidemann, G. Roati and M. Inguscio, Phys. Rev. A 68,011601(R) (2003). 6. N. Ashcroft and N. Mermin, Solid State Physics (Saunders, Philadelphia, 1976). 7. L. Fallani, F.S. Cataliotti, J. Catani, C. Fort, M. Modugno, M. Zawada and M. Inguscio, cond-mat/0303626. 8. B. Eiermann et al. Phys. Rev. Lett. 91,060402 (2003). 9. P. Massignan and M. Modugno, Phys. Rev. A 67,023614 (2003). 10. C. Fort, F.S. Cataliotti, L. Fallani, F. Ferlaino, P. Maddaloni and M. Inguscio, Phys. Rev. Lett. 90,140405 (2003). 11. M. Kramer, L. Pitaevskii and S. Stringari, Phys. Rev. Lett. 88 180404 (2002). 12. G. Roati, F. Riboli, G. Modugno and M. Inguscio, Phys. Rev. Lett. 89, 150403 (2002).

DYNAMICS OF A HIGHLY-DEGENERATE, STRONGLY-INTERACTING FERMI GAS

J. E. THOMAS, S. L. HEMMER, J. M. KINAST, A. V. TURLAPOV, AND M. E. GEHM Physics Department, Duke University, Durham, NC, 27708-0305, USA E-mail: j e t Ophy. duke. edu

K. M. O’HARA National Institute of Standards a n d Technology, Gaithersburg, MD

We use all-optical methods to produce a highly-degenerate, Fermi gas of atoms near a Feshbach resonance, where strong interactions are predicted. In this regime, the zero-energy scattering length is larger than the interparticle spacing, and both the mean field energy and the collision rate take on universal forms as a consequence of unitarity and many-body interactions. We observe strong interactions in the highly anisotropic expansion of the gas upon release from the trap and discuss the interpretation of the data in terms of collisional and superfluid hydrodynamics in the universal limit.

1. Introduction

Strongly interacting Fermi systems are expected to exhibit universal behavior In atomic gases, such strong forces can be produced in the vicinity of a Feshbach resonance, where a bound molecular state in a closed exit channel is magnetically tuned into coincidence with the total energy of a pair of colliding particles 2 . In this case, the zero energy scattering length a s , which characterizes the interactions at low temperature, can be tuned through f o o . For very large values of lasl, the important properties of the system (e.g., the effective mean field potential, the collision rate, the superfluid transition temperature) are predicted to lose their dependence on the magnitude and sign of a s , and instead become proportional to the Fermi energy with different universal proportionality constants In the vicinity of a Feshbach resonance, a two-component Fermi gas of atoms is predicted to exhibit a superfluid transition at a particularly high 137

138

temperature, Tc = V C T Fwhere , TF is the Fermi temperature at which degeneracy occurs. Below Tc,the atomic system exhibits pairing analogous to a superconductor. For an ordinary metallic superconductor, the Fermi temperature corresponds to an energy of 1 electron volt or TF N 1040K and qc N so that the transition occurs at a few Kelvin. The best high-temperature superconductors achieve vc N l o p 2 , and superconduct at temperatures N 100 O K . By contrast, the atomic systems are predicted to achieve qc = 0.25 - 0.5 Although the Fermi temperature for the gas is of the order of 10 pK, the atomic system is analogous to a metallic superconductor with a transition temperature of several thousand OK! In the following, we first review briefly the basic properties of the gas in the strongly interacting regime, and then describe the theory of hydrodynamic expansion in the collisional and collisionless superfluid regimes. The predictions are compared with our recent experiments in the last section. 39435.

2. Universal Interactions

In a degenerate Fermi gas, the interparticle spacing L sets the scale of the local Fermi wavevector, k~ N 1/L. This is a simple consequence of the Pauli principle which requires no overlap between atomic wave packets with the same spin. The local density n of a gas comprising a 50-50 mixture of twospin components each of density n/2 is related to kF by

n ( x )= k;(x)/(37r2).

(1)

When the effective range R of the collision potential is small compared to the interparticle spacing L N ',k , but the magnitude of the zero-energy scattering length lasl is large compared to the interparticle spacing, i.e., R > 1 since w~ = w z ,wy >> w,, and the gas expands much faster in the transverse (I) directions than the axial. The maximum local Fermi energy occurs at the center of the trap, where E F ( O ) = ICBTF = For our trap, w, = 27r x (230 f 20) Hz, w, = wy = 27r x (6625 f 50) Hz, and N = 1.6 x lo5. Then, TF is of the order of 8 pK in the experiments. In this case, y N 27r x 8 kHz. At low , blocking can suppress the collision rate temperatures, T < 0 . 1 5 T ~ Pauli in the trap 16, but the gas can become collisional after release 17. In a noninteracting atomic Fermi gas, the momentum distribution depends only on the kinetic energy p2/(2M). Hence, for long expansion times, the cloud assumes a spherical shape, reflecting the initial isotropic momentum distribution.

141

For a gas in the collisionally hydrodynamic regime, the pressure gradient and the trap potential determine the stream velocity u ( x ,t ) ,

The density n(x,t ) and the stream velocity obey the continuity equation, dn V . (nu)= 0.

at

+

We assume also that the gas is irrotational, and

vxu=o.

(15)

For a normal fluid at nearly zero temperature, the pressure P is given in the local density approximation by 2 P = -(1+ P N ) E F ( X ,t )n(x,t ) . (16) 5 Here, the first term is just the Fermi pressure, while the second arises from the mean field potential in the unitarity limit, as described above. The parameter / 3 ~is the ratio of the mean field energy to the kinetic energy (or the ratio of the mean field contribution to the chemical potential to the local Fermi energy) for a normal fluid. Since E F ( X , t ) 0; n2/3,we have P 0; n5/3and V P / n 0; VEF(n),where EF(n)is given by Eq. 2. From Eq. 13 and Eq. 15, one easily obtains

In equilibrium for the trapped gas, the pressure forces are balanced by the confining potential, and the right hand side of Eq. 17 vanishes with u = 0. Assuming that the trapping potential is abruptly extinguished, the ~ below. gas expands according to Eq. 17 for U T = ~0 as~discussed In the collisionless superfluid case, we assume that the force is given by the gradient of the local chemical potential, p(x) 14,

M

-u+u.vu

(:t

)

= -Vp(x).

Eq. 18 assumes a pair correlation energy A which is large compared to the oscillation frequencies in the trap. A is a substantial fraction of E F , so that A l w , is of the order of (3XN)'/3 >> 1 for the conditions of our trap where N 2 1.6 x lo5 and X = wz/w:, = 0.035. The local chemical potential takes the form of Eq. 7,

P(x)= (1+ P S ) E F ( X ) + U T r a p ( X ) .

(19)

142

Here, we assume that the parameter Ps may be different from that of a normal fluid. Using Eq. 19 in Eq. 18 yields

We see that Eq. 20 and Eq. 17 differ only in the parameter P, i.e., ,Bsand P N , respectively. In both cases, c F ( n ) c( n2I3and the equations of motion allow an exact solution in terms of a scale transformation, where 2 = z/b,(t) and u,(x, t ) = Zb,(t) 14. The one dimensional density distribution of the expanding gas is then

n(x,t) = The scale factors obey simple equations

where bi(0) = 1 and &(O)

= 0,

14J8

which are independent of

P,

for i = z,y,t.

4. Experiment

A highly degenerate, two-component Fermi gas of 6Li is produced by forced evaporation near a Feshbach resonance l9 in our COa laser trap. This is accomplished by subjecting the gas to a 910 G field while lowering the trap laser intensity over 3.5 sec 13. After evaporative cooling, the trap is adiabatically recompressed to full depth and the gas is released from the trap and imaged at various times in the 910 G field to observe the anisotropy. The C 0 2 laser power is extinguished in less than 1 ps with a rejection ratio of 2 x of the maximum value. A CCD camera images the gas with a magnification of 4.9 f 0.15. The cloud widths a,(t) and a z ( t )are determined from the measured column density i i ( z ,t)by integrating over t and z, respectively, and fitting zero temperature Thomas-Fermi distributions, Eq. 11, for each direction. Fig. 1 shows the extrapolated initial transverse dimension a,(O) obtained both from the measured a,(t), i.e, a,(O) = a,(t)/b,(t) and from the measured a,@), i.e., a,(O) = Xa,(t)/b,(t), where X = w , / w l . Ideal hydrodynamic scaling would produce identical horizontal lines with the same value of a,(()). In the region where both the axial and transverse plots are nearly horizontal, we obtain a,(O) = 3.58 f 0.02pm. This is in excellent

143

agreement with the value of gz(0) = 3.6 f 0.1pm obtained for the transverse zero-temperature Fermi radius based on the measured atom number and trap frequencies for p = 0. Including a temperature correction by using a finite temperature Thomas-Fermi distribution for the fits, we find that the zero temperature radii are smaller by approximately 7%, yielding p = 0.26 f 0.07 12.

0.0

0.5

1.o

1.5

2.0

Expansion Time (ms) Figure 1. Extrapolated initial width az(0)obtained by scaling the measured transverse and axial dimensions: The solid curve shows the results for the transverse direction ( ( ~ ~ (=0 o ) z ( t ) / b z ( t ) )which , appear as a nearly horizontal line beyond 0.1 ms. The 0 )Xo, (t)/bz(t))which exhibit more deviation dotted curve shows the axial data ( ( ~ ~ ( = from perfect hydrodynamic scaling at long times than the transverse data, due t o residual thermal energy. The curves are added to guide the eye. For ideal hydrodynamic scaling, both sets of data should be overlapping horizontal lines.

5. Conclusions

Figure 1shows that the expansion data are well fitted by a scale transformation which arises from hydrodynamics. However, the current experiments cannot distinguish between superfluid and collisional behavior. Since the pressure gradient is much larger in the transverse direction than the axial, nearly all of the available energy is expended by expansion in the transverse direction. The remaining energy in the axial direction is then very small, producing a slower than ballistic scaling for the axial widths. We find that the transverse data closely matches the hydrodynamic scaling with no free parameters, except at the shortest times, where the transverse radius of

144

the cloud is too small to resolve and diffraction of the light from the cloud produces spurious results for the widths. T h e axial data fit the expected hydrodynamic scaling to 21 1 ms. Beyond this time, the axial widths grow faster than expected, suggesting residual thermal energy in the axial direction 20. Investigation of hydrodynamics both in the trap and during expansion, as a function of magnetic field and temperature, will be topics of future experiments.

Acknowledgments

This research is supported by the Chemical Sciences, Geosciences and Biosciences Division of the Office of Basic Energy Sciences, Office of Science, U. S. Department of Energy, the physics divisions of the National Science Foundation and t h e Army Research Office, and the Fundamental Physics in Microgravity Research program of the National Aeronautics and Space Administration. References 1. H. Heiselberg, Phys. Rev. A 63,043606 (2001). 2. See E. Tiesinga, B. J. Verhaar and H. T. C. Stoof, Phys. Rev. A 47, 4114 (1993) and references therein. 3. M. Holland, et al., Phys. Rev. Lett. 87, 120406 (2001). 4. E. Timmermans, et al., Phys. Lett. A 285, 228 (2001). 5. Y . Ohashi and A. Griffin, Phys. Rev. Lett. 89, 130402 (2002). 6. D. A. Butts and D. S. Rokhsar, Phys. Rev. A 55, 4346 (1997). 7. This argument was suggested to us by R. Furnstahl, Ohio State University, private communication. 8. J. V. Steele, e-print nucl-th/0010066 (2000). 9. G. A. Baker, Jr., Phys. Rev. C 60, 054311 (1999). 10. J. Carlson, et al., e-print cond-mat/0303094 (2003). 11. This method of deriving the equation of state, as well as an independent derivation of the trap Fermi radii in terms of p, were provided by S. Stringari, private communication. 12. M. E. Gehm, et al., Phys. Rev. A 68, 011401(R) (2003). 13. K. M. O’Hara, et al., Science 298, 2179 (2002). 14. C. Menotti, et al., Phys. Rev. Lett. 89, 250402 (2002). 15. L. Pitaevski and S. Stringari, Science 298, 2144 (2002). 16. M. E. Gehm, et al., e-print cond-mat/0304633 (2003). 17. Henning Heiselberg, Jason Ho, Wolfgang Ketterle and Jook Walraven, private communication. 18. Y . Kagan, et al., Phys. Rev. A 55, 18 (1997). 19. K. M. O’Hara, et al., Phys. Rev. A 66,041401(R) (2002). 20. This idea was suggested to us by J. Walraven, private communication.

SPECTROSCOPY OF STRONGLY CORRELATED COLD ATOMS

A. J. DALEY, P. 0. FEDICHEV, P. RABL AND P. ZOLLER Institute for Theoretical Physics, University of Innsbruck A-6020 Innsbruck, Austria A. RECATI Dipartimento d i Fisica, Universitd d i Dento and INFM I-38050 Povo, Italy

J. I. CIRAC Max-Planck Institut f u r Quantenoptik, 0-85748 Garching, Germany J. VON DELFT AND W. ZWERGER Sektion Physilc, Universitat Munchen, 0-80333 Munchen, Germany. We study theoretically the dynamics and spectroscopy of cold atoms in optical potentials. Specific examples include atomic quantum dots, i.e., atoms stored in a tight optical trap coupled to a superfluid reservoir via laser interactions, and the high fidelity loading of optical lattices with a precise number of atoms per lattice site by a novel filtering scheme.

1. Introduction

A new frontier of atomic physics is the study of cold atoms as a strongly interacting many body system'. In this regime atomic experiments may help to increase understanding of several physical phenomena that have been predicted or observed in solid state systems. They may also allow one to test and make new predictions in the field of quantum statistical mechanics, and even to investigate important concepts in quantum information such as multiparticle entanglement. Here we will summarize briefly two recent examples, including: (i) the dynamics and spectroscopy of atomic quantum dots (corresponding to atoms stored in tight optical or magnetic 145

146

Figure 1. Schematic setup of an atomic quantum dot coupled to a superfluid atomic reservoir. The Bose-liquid of atoms in state a is confined in a shallow trap V a ( x ) .The atom in state b is localized in a tightly confining potential v b ( X ) . Atoms in state a and b are coupled via a Raman transition with effective Rabi frequency 0. A large on-site interaction Ubb > 0 allows only a single atom in the dot.

traps) coupled to a superfluid reservoir via laser interactions12 and (ii) the high fidelity loading of optical lattices with a precise number of atoms per lattice site by a novel filtering ~ c h e m e . ~

2. Atomic Quantum Dots Coupled to BEC Reservoirs

A focused laser beam superimposed on a trap holding an atomic BoseEinstein condensate (BEC) allows the formation of an atomic quantum dot (AQD), i.e., a tight optical trap where the laser provides the coupling to a reservoir of Bose-condensed atom^.^^^ This configuration can be created by spin-dependent optical potentials,' where atoms in the dot and the reservoir correspond to different internal atomic states connected by Raman transitions (see Fig. 1). Atoms loaded in the AQD will repel each other due to collisional interactions. In the limit of strong repulsion, obtained, for example, by tuning near a Feshbach resonance, a collisional blockade regime can be realized where either one or no atom occupies the dot, but higher occupations are exc1uded.l The AQD coupled to a superfluid reservoir can be modeled by a spin-112 system describing the occupation/ non-occupation of the dot by a single atom in state ( b ) coupled to a bath of harmonic ascillators, which corresponds to the phonon modes of the superfluid in level ). .1 It can then

147

be shown that the Hamiltonian for the dot is 1 H = hv, IqlbAb, 2gabfi(O)dz hR(a+e-id(0) h.c.),

c

+

+

+

(1)

9

where we use Pauli matrix notation for the pseudo-spin-1/2. The first term in Eq. (1) is the Hamiltonian of the phonon bath with phonon operators b,. The second term describes the collisional interaction of the atom in the dot with the density fluctuations of the superfluid, Pa(x)= Pa ft(x), at the position of the dot x = 0. Finally, the last term describes Raman transitions with Rabi frequency R between) . 1 to ( b ) ,which couple to the phase fluctuations of the reservoir &x). In terms of the phonon operators b, with wave vector q we have

+

1/2

=

eiq'x(b, - b-,), t

4

where pa is the superfluid density, v, is the sound velocity, m is the atomic mass and V is the sample volume. A unitary transformation allows us to rewrite this Hamiltonian in the form of a spin-boson Hamiltonian,

where the phonon-bath coupling A, = lmhqv~/2VpaI1/2 (gabpalmu: - 1) has contributions from both the collisional interaction and the laser coupling. This interference between the coupling to the density and phase fluctuations is the key feature of the AQD dynamics, and results in a controZZabZe spin-1/2-phonon coupling. In particular, a complete decoupling of the dot from the dissipative environment can be achieved, thus realizing a perfectly coherent two-level system which can be considered as an analogue of the "stable charge qubit" of condensed matter systems. The spin-1/2-phonon coupling is characterized by the effective density of states J(w)=

C A ; ~ ( W- w,)

= 2awS,

(5)

4

where a is the dissipation strength due to the spin-phonon coupling and D = s the dimension of the superfluid reservoir. In the standard

148

termin~logy,~ s = 1 and s > 1 correspond to the ohmic and superohmic cases, respectively. The spin-boson model exhibits very rich dynamics which has been the subject of a significant number of theoretical papers5. Thus an experimental realization in terms of an AQD, leading to a spin-boson model with controllable parameters, is of particular interest. This is especially true for the case of ohmic dissipation corresponding to a 1D (Luttinger) superfluid, where the system exhibits damped Rabi oscillations in the regime 0 1. The latter two regimes are separated by a dissipative phase transition. Furthermore, the tunability of A, and thus a results in a novel spectroscopic tool to measure the dynamics of 1D superfluids, all the way from the weakly interacting Bogoliubov case to the Tonks gas limit.

i,

3. Defect Suppressed Atomic Crystals

The possibility to realize Bose-Hubbard models with very precise control over the system parameters using BECs loaded in optical lattices has opened many possibilities for the study of strongly correlated phenomena and for the implementation of quantum gates. However, the experimental realization of many of these ideas also requires that the system be prepared in a well defined pure initial state with high precision. The loading of a BEC in a Mott Insulator (MI) phase, which in principle corresponds to having a definite number of atoms on each lattice site, has already been observed.6 However, non-ideal conditions will always result in defects in that phase (i.e., missing atoms and overloaded sites). We propose a coherent filtering scheme which, beginning with an uncertain number of atoms on each site, provides a method to transfer a definite number of atoms on each site into a different internal state, and so load a new lattice of atoms with an exact number of particles per site. This process dramatically reduces the site occupation number defects, and can be extended, under experimentally reasonable conditions, to allow the production of doped or pattern loaded initial states with almost unit fidelity (i.e., atomic crystals). We consider a system of bosons loaded into an optical lattice such that they do not tunnel between neighboring sites. The atoms are in a particular internal state, la), and have onsite interaction strength U,. We then couple the atoms into a second internal state, Ib), (which is trapped by a second lattice potential) via an off-resonant Raman transition with Rabi frequency

149

f@), which is detuned from state Ib) by 6 ( t ) . If we denote the onsite interaction between particles in state Ib) as u b and the onsite interaction between particles in different internal states as U a b , then we obtain the Hamiltonian (for ti = 1)

where ti and 6 are the annihilation operators for particles in states la) and ~ b )respectively, , and fi, = titti, f i b = 6t6. If we write the state at each site as )n,,nb), where n, and n b are the number of particles in the states la) and Ib), respectively, then the initial state at a given site is IN,O), where N E 1 , 2 , . . . ,Nmax.We can then transfer exactly one particle to state Ib) if we vary the detuning 6 from some initial value 6i to some final value 6f so that the system evolves along the appropriate avoided crossing in the energy eigenvalues, undergoing an adiabatic passage from IN, 0) + ( N - 1,l), and does not evolve along any other such avoided crossings. 6i and 6f must be chosen so that the process works simultaneously for all values of N . Once the system is in state IN - 1, I), we can turn off the lattice trapping state la), leaving a pure state with exactly one atom per lattice site. Because the relative locations of the avoided crossings in the energy eigenvalues are determined by the values of u a / u b and U a b / U b , this procedure will only work for particular choices of these parameters. The parameter ranges for which appropriate values of 6i and 6f can be chosen in the adiabatic limit are shown in Fig. 2a for N,,, = 4. Larger values of N,, result in a more restrictive allowed parameter range. Substantially different values of U, and u b are required, which is possible using either spin-dependent lattices, where U, and u b are independently controlled via the different lattice shape for atoms in each internal state, or Feshbach resonances, near which the scattering length a, is different for atoms in different internal states as we tune an external magnetic field. U a b can also be independently controlled by slightly displacing the lattices trapping internal states la) and ]b). We define the fidelity of the final state at a single lattice site to be the probability that the site contains exactly one atom, i.e., F(,= (112&11), where 2i)b = Tra2ir, and 2i) is the final density operator for the site. Similarly, an initial fidelity for a state with filling factor N , may be defined as Fa = ( N , , O l z i r 0 l N a , O ) , where GOis the initial density operator for the

150 A

3

a>

2 1

Figure 2. a) The parameter range for which appropriate values of 6i and 65 can be found when Nmax = 4, in the adiabatic limit (light shading) and from numerical simulations with a smoothed rectangular pulse which rises and falls with a sin2(t) shape, T = 1OOU;' and max Q = 0.3Ub, giving a transfer error E < 1% (dark shading). b) Initial and final state fidelities, 3aand 3 b r as a function of temperature, for an initial MI state described at each site using the Grand Canonical Ensemble. The inset shows values of 1 - Fa (solid line) and 1 - 3 b for Fa= 2 (dashed) and Fa= 3 (dotted) on a logarithmic scale.

site. The total fidelity for M sites is 3TM. Fidelities less than one arise from two sources. Firstly, in practice the system must evolve along the avoided crossings in a time which is well within the decoherence time of the optical lattice. Numerical calculations with realistic pulse shapes and time scales show that this error can be made of the order of (which corresponds to one defect in lo4 sites) even without optimization of n(t) and 8(t). The parameter range in which this is possible is only slightly reduced (see Fig. 2a). The second source of fidelities smaller than one is that this scheme cannot correct occupation number defects in which no atoms are present Thus, to obtain the highest possible at a particular lattice site in state ). .1 fidelities we should start in a MI phase with an average filling factor of 2 or more particles per site, where the probability of zero occupation is small. Modeling particle number fluctuations in a MI phase at temperature T using a Grand Canonical Ensemble, we find that the error in the initial state is exponentially suppressed with exponent fli by the filtering scheme (see Fig. 2b). For example, in the case of zero transfer error, an initial state = 2 and a defect at every tenth lattice site, 3a= 0.9, T / U a = 0.17, with results in a final state with 1 - 3 b 3 x i.e., less than one defect in every three hundred thousand lattice sites. This scheme can be easily extended to multiple atoms by a different choice of 6f.It can also be used to characterize defects in current MI states

ma

ma

N

151

by measuring occupation number distributions if bi and 6s are chosen so that the transfer only occurs for particular values of N . Also, the addition of a superlattice to the lattice trapping state Ib) will site-dependently shift the energy of that state, preventing atoms in particular sites from coupling to the Raman transition, and allowing the loading of high fidelity spatial patterns of atoms (atomic crystals). The scheme is also applicable to the production of high fidelity initial states in fermion systems (including doped states and the loading of composite objects), and when combined with moving optical lattices, the techniques involved can be used t o measure interesting two-body correlation functions. These include correlation functions giving us information about long range order and pair correlation lengths in systems of fermions with two spin species. One example application of these ideas t o strongly correlated fermion systems is a scheme to produce and characterize a BCS state from fermions in an optical lattice with a precisely chosen filling f a ~ t o r . ~ 4. Summary

In summary, we have discussed two examples which illustrate the emerging interface between quantum degenerate atomic gases (in optical and magnetic traps) and condensed matter physics of strongly correlated systems. First, we have shown that an AQD coupled to superfluid reservoirs leads to spin-boson model with tunable parameters. For 1D reservoirs this provides a route to study dissipative phase transitions with cold atoms, and a spectroscopy of bosonic Luttinger liquids. Second, we have discussed a coherent filtering scheme which allows the production of a wide range of high-fidelity atomic crystals in systems of bosons and fermions in an optical lattice through techniques which can also be used to measure two-body correlation functions. These schemes should greatly enhance the application of atoms in optical lattices t o the study of strongly correlated condensed matter systems and to quantum computation.

References 1. J. I. Cirac and P. Zoller, Science 301 176 (2003). 2. A. Recati, P.O. Fedichev, W. Zwerger, J. von Delft and P. Zoller, unpublished. 3. P. Rabl, A. J. Daley, P. 0. Fedichev, J. I. Cirac and P. Zoller, condmat/0304026 and to appear in Phys. Rev. Lett. 4. R.B. Diener, B.Wu, M.G. Raizen and Q. Niu, Phys. Rev. Lett. 89, 070401

(2002). 5 . A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg and

152

W. Zwerger, Rev. Mod. Phys. 59, 1 (1987); U. Weiss, Quantum Dissipative Systems, World Scientific 1999. 6. M. Greiner, 0. Mandel, T. W. Hansch and I. Bloch, Nature 415,39 (2002); M. Greiner, 0. Mandel, T. Esslinger, T.W. H ansch and I. Bloch, Nature 419, 51 (2002).

STRONG CORRELATION EFFECTS IN COLD ATOMIC GASES B. PAREDES, G. METALIDIS, V. MURG AND J. I. CIRAC Max-Planck Institut fur Quantenoptik Hans-Kopfermann Str. 1, D-85748Garching, Germany C. TEJEDOR Universidad Autnoma de Madrid, Canto Blanw, Spain We analyze several regimes in which atoms in optical lattices display strongcorrelation effects, including Tonks-Girardeau behavior, bosonic Cooper pairing, Kondo effect and magnetism.

1

Introduction

An atomic gas at very low temperature can display a rich variety of quantum phenomena which strongly depend on the role played by the interatomic interactions. In an ideal gas the atoms tend to accumulate in the ground state of the potential which confines them. For weak interacting gases the atoms teud to accumulate in a state Ip) which is no longer the ground state of the confining potential but of an effective potential which includes the atom-atom interactions. In both cases, the , a singleparticle many-body state IS) can be approximated as I@) N I P ) @ ~ i.e., wavefunction can be used to describe the collective atomic behavior. For sufficiently strong interactions this is no longer true, and the gas must be described in terms of highly correlated (entangled) states. These entangled states display very exotic phenomena which are of great interest in other fields of physics. The exquisite control in experiments with cold atoms offers new possibilities to study all these interesting phenomena. In order to observe strong correlations, the typical interaction energy between two atoms must exceed the typical single particle (kinetic and/or potential) energies. This can be achieved, for example, by storing the atoms in an optical lattice [see Fig. l(a)], which provides a periodic potential for the atoms. At sufficiently low energies the atoms are confined to the first Bloch band, and we can consider a single level in each lattice site (minimum of the periodic potential). Atoms can move from one site to the next one by tunneling across the potential barrier, a process which is characterized by an energy t. On the other hand, to locate two atoms in the same site costs an energy which we will denote by U . Strong correlation effects will appear for U > t. In fact, in a remarkable experiment, these effects have been observedl following the ideas proposed in Ref. 2. This particular system offers new and exciting possibilities since the parameters t and U can be easily tuned by varying the optical potential. Moreover, one can store bosonic orland fermionic atoms with several internal states,,and use lasers or magnetic fields to induce new 153

154

Figure 1. Parameters for: (a) Bosonic atoms in an optical lattice; (b) Bosonic atoms with two internal states; (c) Fermionic atoms in a superlattice with two internal states.

physical phenomena. In this contribution we will propose and analyze several possibilities to observe strong-correlation phenomena in this system in the limit U >> t. We will consider bosons and fermions with one or two (relevant) internal levels. Some of the results presented here have been reported in Refs. 3 and 4. 2

Spinless Bosons in an Optical Lattice

We consider N bosonic atoms stored in a lattice with M sites. The Hamiltonian describing this situation is M

H=-t

a!aj+UxnS,

i=l

where ai annihilates an atom at the i-th lattice site and ni = aiai is the particle number at that site. If N = vM (v = 1 , 2 , . . .) the ground state of the system in the regime U >> t is given by the Mott-insulator state, where there are exactly v atoms in each site. The reason for this is that the energy cost of having a site with v 1 atoms (and one with v - 1) is very high (equal to U ) , and thus this cannot be the minimal energy (ground) state. The situation gets more interesting when v is not integer. In that case, there still cannot be one site with double occupation, since the energy cost would be too high. In some sense, the bosons behave like fermions which cannot be in the same state. One has to be nevertheless careful with this statement since still the bosonic states have to be symmetric with respect to the atomic exchange. In any case, several of the properties of the ground state can be understood in terms of non-interacting fermions. In a 1D lattice one can make the previous qualitative argument more rigorous. Let us consider first the case v < 1. In this case, one can use the JordanWigner transformation to reexpress the bosonic operators a! in terms of fictitious

+

155 0.1

. . . . v=1 .oo ' - v=o.90 -V=0.50 - -V=O.lO '

0.08

x

c

-0

5

k

10

15

Figure 2. Quasimomentum distribution of strongly interacting bosons in a 1D lattice for 200 sites and u = 0.1,0.5,0.9,1. Note that for u = 1 we have the whole Bloch band filled, which corresponds to the Mott phase.

fermionic operators c,t as follows:

Thus, we can reexpress all atomic observables in terms of fermionic operators. There will be some which will keep the form (i.e., will have the same functional dependence on the c's as they had on the a's) but there will be some for which this dependence will be different. For example, the particle number at site i remains the same (ni= afai = clci), whereas the momentum distribution changes. In particular, the Hamiltonian becomes H = -tC,i,j, cfcj, i.e, the one corresponding to free fermions. Thus, all spectral properties (density of states, heat capacity, etc) of the bosonic atoms will coincide with those of the free fermions. However, other measurable quantities will be different. In particular, the diffraction pattern when the bosons are released acquires a form which is completely different to the FermiDirac distribution, as shown in Fig. 2. In fact, one can easily show that what one obtains in the case u < 1 is a discretized version of the Tonks-Girardeau gas 6, with correlations (afai+a) A-l12 for A --* 00. Thus, the Tonks-Girardeau gas can be observed with atoms in an optical lattice as follows: (i) raise the barrier along the 5 and y directions in order to ensure that there is not tunneling in those directions and that one has a 1D situation; (ii) raise slowly the barrier along the z direction until U > t. At this point, one should start seeing the typical difraction pattern of the Tonks-Girardeau gas, which should become more pronounced as U gets larger. For v > 1 one can also determine the properties of the ground state, as well as when one includes a harmonic potential confining the atoms. These results will be presented elsewhere '.

-

156

3

Bosons with Two Internal Levels in an Optical Lattice

The fact that strongly interacting bosons may behave, in some aspects, like free fermions is very intriguing. However, it would be more intriguing if one could have that bosons behave as interacting fermions, since in that case one would be able to observe Cooper pairing (and other typically fermionic behavior) with bosons. This is possible if now we consider bosons with two internal states (pseudospins, 0 1) and the interactions are appropriate. Let us denote by U (V) the interaction energy corresponding to two atoms in the same site and in the same (different) internal state [see Fig. l(b)]. The corresponding Hamiltonian can be written as

=r,

,u

i,u

i

where we have used the obvious notation for the bosonic operators. In the limit U >> IVl,t two atoms with the same internal state will not share the same site, i.e., in some sense they behave like fermions. If in addition V < 0, atoms with different spin will tend to pair, and thus we would expect to see some Cooper pairing behavior. Again, in 1D one can make the above argument rigorous by resorting to an extension of the Jordan-Wigner transformation including the internal levels:

Under this transformation, Hamiltonian (3) becomes the familiar Fermi-Hubbard one which has been thoroughly studied in the literature ’. Also, density-density as well as spin-spin correlation functions remain unchanged, and thus some fermionic behavior can be observed by measuring the corresponding observables. In particular, for V < 0 one can detect the formation of pairs by measuring (&+a&),where

Si= nz.T - n .2 1 .

The physics of the ground state in the case of V < 0 can be understood by using a variational formulation, similar to the one used in BCS theory ’. We write

where sgn denotes the sign function. The variational function cp represents the wavefunction corresponding to the relative coordinate A of a Cooper pair. If it is narrow it indicates that both atoms are basically in the same site, whereas if it is broad it indicates that they are more delocalized. In Fig. 3 we have plotted this function. As expected, for increasing values of IVI the Cooper pairs tend to localize more due to the attractive interaction. 4

Fermions in a Superlattice

So far, we have dealt with strongly correlated bosons in optical lattices. Given the experimental progress in cooling fermionic atoms lo it is expected that very

157 1 -0.9

10l2 0.:

c

-2

-1

2

Figure 3. I’plz as a function of A for the variational wavefunction given in the text, N = 6, t = 0.1Uand different values of V/U.

soon they will be loaded in optical lattices to observe a large variety of interesting phenomena, ranging from the superfluid phase transition to the simulation of high-Tc superconductivity states ll. Another intriguing possibility is to confine the fermions in a superlattice, which is formed by several laser standing waves of different frequencies (which is achieved, e.g., by tilting the propagation direction of the lasers) 12. The parameters that enter the problem are illustrated in Fig. l(c) ‘. We will call “impurities” the superlattice sites with lower potential and simply “sites” the other sites. As before, tunnelling between sites is described by t , whereas tunnelling between an impurity and a site is characterized by w. Interactions at the impurity have energy U , whereas interactions at the sites have energy V . The potential energy difference between the sites and the impurities is denoted by A. The Hamiltonian in this case has the Anderson Lattice Hamiltonian (ALH) l3 form

,u

Here, .

fsu

annihilates an atom in internal state

S

LT

at the impurity s, and n!u =

fimfsu. The ALH has been extensively studied in the literature 14, and it is known

to capture the physics of a variety of strongly correlated phenomena, from the Kondo effect to RKKY magnetism 15. Typical condensed matter systems described by Anderson models are metallic or intermetallic compounds with a low concentration of magnetic impurities. The usual scenario is then that of impurities located far from each other, each of them coupled to a continuum of delocalized electrons. In our case ‘, we have a quite different situation and thus cold atoms allow us to study a very peculiar regime

158

which has not been described so far. We have an array of impurities connected through small islands with a discretized set of levels (for simplicity we will consider here one dimensional superlattices). In order to reduce the parameter space, we will concentrate on the limit where U , A >> t, v and V = 0. This regime is naturally achieved in the superlattices, since at the impurities the density is larger and therefore the interactions are stronger. In fact, one can allow more freedom in the values of the parameters if one uses four internal levels instead of two and stimulates the tunnelling with lasers '. In this regime, one can adiabatically eliminate the states in which there are none or two atoms in an impurity, t o obtain a Kondo lattice Hamiltonian 16,

3 xu,,, fiu

where SL = T,,,! fsa, with r a vector of Pauli matrices. The operators Sg are defined in the obvious way. Here, a new parameter is defined J = 2v2/A which describes the exchangelike interaction between the atoms in each impurity with the atoms sitting in neighboring sites. In the following we will analyze two regimes depending on the ratio t/J. 4.1

Strong Coupling Limit

For J > t/L the Kondo effect dominates the physics of the problem. Since tunneling is very small, the impurities are basically disconnected from each other and we can treat each impurity independently (interacting only with a small number of sites). The exchange energy is minimized by forming singlets between the impurity and the neighboring site. If t increases slightly, the impurity tends t o delocalize among several sites. A more quantitative analysis can be made by using a generalization of the variational wave function of Varma and Yafet l7 in which a singlet is created between the impurity and the different lattice sites. One obtains a Kondo temperature of the order of J . For a large number of sites one recovers the Kondo effect, although the temperature displays the typical non-analytic behavior with the effective interaction parameter J . When the size of the singlets becomes comparable t o the impurity separation (which occurs in the weak coupling limit), this approximation is no longer valid and the impurities start to interact with each other.

4.2

Weak Coupling Limit

A very different situation corresponds t o the regime in which the spacing between energy levels in the conducting islands ( w 2t sin(r/L)) becomes much larger than the Kondo temperature ( w J ) . Within this limit atomic orbital degrees of freedom are completely frozen, with excitations above the Fermi level in each of the islands taking part of the problem only as virtual states. Performing adiabatic elimination of these excitations in Hamiltonian ( 7 ) , we obtain an effective Hamiltonian for the spin degrees of freedom. As an interesting feature the resulting Hamiltonian depends on the parity of the number of particles per conducting island, N,. This even-odd effect is a clear manifestation of the finite size of the conducting islands.

159

N,=even. The Fermi level of each island is occupied by two atoms. In this case the only spin degrees of freedom correspond to atoms localized in supersites. An effective spin-spin interaction between neighboring supersites appears, mediated by the Fermi sea in between them. To second order perturbation theory in J ,

+

+

where J Z f = sin(kp) sin(kp(L - 1 ) )sin(kp Ak) sin((kp A k ) ( L - l ) ) ,and Ak = T / L . We see that the Kondo effect disappears and magnetism is induced for localized atoms, the ground state being antiferromagnetic or ferromagnetic depending on both L and k p . We note that due to the characteristic topology imprinted by the superlattice, Heisenberg (and not RKKY) magnetism is induced. N,=odd. The Fermi level of each island is occupied by one atom, whose spin comes into play. The effective Hamiltonian is in this case H = J 3 f C , S i . S ~ F , where S t F = C,,,, ra,,~AslcFoAskFa,, t and J$f = 4 J / L . The Kondo effect remains in this case and magnetism is not induced. The ground state consists in this case of singlets formed by each localized atom and the atoms at the Fermi level in neighboring islands. The Kondo temperature is T = 2JeKff.

a

4.3 Numerical Results To illustrate the predictions above we have numerically diagonalized Hamiltonian ( 6 ) for a small 1D superlattice. In Fig. 4 we plot the spin-spin correlation functions ( S f . Se) (spatial correlation of a fixed f-spin with the rest of the sites in the chain, l ) for the exact ground state. We consider different cases: a) L = 4, N, = 4 [Fig. 4(a)]. There is a clear smooth transition from local Kondo singlet formation to magnetism of localized spins. For small values of t / J each localized f-spin is antiferromagnetically correlated with its next neighboring sites (forming a singlet with them). As t / J increases correlations of each impurity with its neighboring islands disappear, at the same time that correlations between next impurities are induced. The transition (arrow in figure inset) takes place around TK N 2 A ( t / J N 3.25) as predicted. As stated by Hamiltonian (8) impurities are antiferromagnetically coupled ( J Z f = - J 2 / 1 6 t ) . b) L = 4, N, = 3 [Fig. 4(b)]. The Kondo effect appears in this case as t / J increases. Fig. 4 shows how singlet-type correlations become more and more extended along the conducting islands next to each impurity, whereas neighboring impurities remain uncorrelated. Delocalization of the singlet becomes more evident in momentum space, where a resonance, the Kondo resonance, appears at the Fermi level. The Kondo temperature is always of the order of J , reaching the limiting value T 2 J Z f = 2J for J ~ 2 ~ should stabilize to a lattice in equilibrium, the mechanisms of their formation and stabilization are still not fully understood. Stirring of a condensate described by the Gross-Pitaevskii equation produces vortices, but in a turbulent state, rather than a lattice-however, inclusion of dissipation is sufficient to give stabilization to a lattice. Vortex lattices have also been produced by growing a condensate from a rotating vapour cloud, and this process can clearly only be described by an adaptation of the kind of kinetic theory needed to describe the growth of a condensate from a vapour. It is also believed that even in the growth of a condensate from a non-rotating cloud some vortices can be produced. This paper gives a sketch of our work on this subject. The reader should read it in conjunction with several movies which are available on the world wide web1.

2. Quantized Vortices If we write a condensate wavefunction in terms of phase O(x, t ) and density n(x,t ) in the form $(x,t ) = , / m e x p [ i O ( x , t ) ] then , the phase is directly related to the velocity by v(x,t ) = AVO(x, t )which leads to the conclusion that *This work is supported by the Marsden fund under contract number PVT202

171

172

Figure 1. a) An ideal vortex lattice; b) Vortex lattices created in the MIT laboratory5

Vxv(x, t ) = 0 if 0 exists. However, when +(x, t ) --t 0 the phase becomes ill defined and this corresponds to a quantized vortex such that

f

h v(x) * dl = N m

Vortex lattice: When we have many vortices each moves in the velocity field of all the others. There are stable vortex lattice solutions, rotating rigidly at angular velocity R, and quantized so that 2m)OIA= h, where A is the area occupied by each vortex in the lattice. The spacing of the vortices is independent of the density of the condensate, so that in a trapped Bose-Einstein condensate, in spite of the strongly varying density, we expect to see this regular lattice. Experimentally vortex lattices were first produced in ENS Paris 6,7,MIT8v5, JILA’ and Oxford1’. 3. Mechanisms of Vortex Formation

Stir a condensate with an anisotropic potential: Theoretical analyses of stirring have been made by Fetter and Svidzinskyl’, Dalfovo and Stringari12, Anglin13, Muryshev and Fedichev14. In these analyses the perturber mixes the condensate ground state and excited condensate states, and a perturbative calculation is used to obtain a critical rotational speed at which an instability leads to growth of the vortex state. However, in a complete non-perturbative calculation for a localized rotating stirrer done by us15 using a Gross-Pitaevskii equation description of the stirred condensate, we found a kind of coherent “Rabi cycling”. The condensate exhibits a vortex cycling from infinity to the central regions of the condensate. No stable lattice is ever formed using the Gross-Pitaevskii equation, and it is clear that some

173

kind of dissipative process is required to allow the vortices to settle into a lattice.

m m 1 Grow a condensate from a rotating vapor: The Boulder groupg developed a technique to grow a condensate from a rapidly rotating vapour, and in this way created vortex lattices. The mechanism for this must be a modification of condensate growth theory16 for a rotating frame of reference, and indeed may be the fundamental process in vortex lattice stabilization.

4. Vortex Growth Equation In a recent paper17 we introduced dissipation semiphenomenologically. We considered the transfer of atoms between the condensate and a rotating thermal cloud with temperature T , chemical potential p, and angular velocity a and used a modified version of our phenomenological growth equation18 (an adaptation of quantum kinetic theorylg) in a rotating frame for the growth of the condensate wavefunction. The dissipative term came from collisions between atoms in a thermal cloud in which one of the colliding atoms enters the condensate after the collision. The simpl$ed vortex growth equation in the rotating frame is

Vortex growth equation for a rotating trap: Using a stirrer amounts to rotating a non-axially-symmetric trap, and in the frame rotating with the trap potential, in which there is also a thermal cloud rotating at angular velocity a,and a nonaxially-symmetric trap potential rotating with angular velocity fl, we found the vortex growth equation became

If fl # a the distribution of the thermal cloud is not truly a thermodynamic equilibrium distribution, and the equation is only approximate. We conjectured that the production of vortex lattices by stirring a condensate involves the excitation of a thermal cloud (or thermal quasiparticles) whose interaction with the condensate provides the same kind of stabilization. Our equations are superficially very similar to Tsubota et aLZ0but in fact are very different as shown in our paper17.

174

1-

Simulation results for a rotating vapour cloud: We solve (2) with an initial condition in which we "seed" with a superposition of angular momentum states with 1 = 1 to 30, centered at the Thomas-Fermi radius with amplitude 2x To begin, an imperfect ring of 19 vortices arrives from infinity to just outside the Thomas-Fermi radius. The ring shrinks further, and several vortices are shed. A dominant ring of 16 vortices passes through the Thomas-Fermi radius into the interior of the condensate. An instability ensues and the vortices distribute themselves quasi-uniformly over the condensate. During the process the condensate expands and picks up angular velocity. Over a long period, further vortices leave (and for larger values of a may enter) the dense region, until finally a regular lattice of 12 vortices rotating at angular velocity a in the lab frame remains The process is initially one of gain into peripheral Bogolyubov excitations with angular momentum 1, and energy eigenvalues cn,l relative to p c . The initial condensate wavefunction takes the form for angular momentum hl, and energy tiw, E Aal, - €0,1, N

Here J ~ ( T is ) the initial rotationally symmetric condensate wavefunction, 4 is the azimuthal angle u ( T ) , V(T) are obtained by solving the Bogolyubov-de Gennes equations. The essential behavior can be seen by neglecting W(T) and in this case this means that $ has 1, interference zeroes-that is vortices. Vortices all occur at the same radius, initially at infinity. The long distance behavior of U ( T ) is less rapid than that of C ~ ( T so ) that the ring shrinks as the excitation grows. Including all values of 1 gives rise to an imperfectly circular ring of x 1, vortices. We have verified this by decomposing the condensate into angular momentum occupation values Pl-the spatial particle density for each 1 very accurately matches the prediction.

5. Influence of Noise and Fluctuations-Primordial Vortices As shown by Davis et a1.21, vortices are created during condensate growth, as in Fig. 2. Do these primordial vortices survive to become the vortices which form the lattice? How are they to be included in a description based on the Gross-Pitaevskii equation? We can do this using the stochastic Gross-Pitaevskii equation which we have now formulated in a rather rigorous formulation22. The stochastic Gross-Pitaevskii equation: This is a development Quantum Kinetic theorylg and the phenomenological description of 18, in which great care

175

Figure.2 Vortices produced in a simulation of condensate growthz1 for a) Low temperature, b) Moderate temperature, c) High temperature. is taken to separate the subspaces of the non-condensate band (thermalized particles, treated kinetically, with energies > ER) from the condensate band (lower lying excitations with energy < ER which merit a full quantum-mechanical description). We are able to derive the stochastic Gross-Pitaevskii equation in which the condensate band amplitude is explicitly restricted to the condensate band subspace. It has similarly restricted noise terms which arise from: a) Growth from and loss to the thermal cloud; b) Scattering from the thermal cloud. Noise generates primordial vortices where the density is low. The basic description is in terms of the projected Gross-Pitaevskii operator:

Here PC is the projector into the condensate band, which can be written in terms of non-interacting trap eigenfunctions Y,(x)as

Pc(x,x’)=

c

Yn(x)Yn(x’).

(6)

E n <ER

This projector is difficult to implement efficiently, and we are at present developing efficient methods based on wavelet expansions. The stochastic GrossPitaevskii equation can then be written (in the Stratonovich formz3)

( S ) da(x,t ) = -z&a(x, t ) dt E,

-Pc

{

1

kBT

{

*}

- a*(x,t)&a(x,t ) - a(x,t ) (&a(x, t ) )

dt

176

The first line represents the basic condensate band evolution, the second the growth/loss process of atoms from the non-condensate band entering and leaving the condensate band, and the last two lines represent scattering processes between the two bands. The nonzero correlations of the noises from the last two processes are

Growth Scattering

dW;(x, t)dWG(x’,t ) = ~ G ‘ ( x ) ~-( x’) x dt,

(8)

~ W M (t)dWM(x’, X, t) = 2M 6 ( x - x ’ ) d t .

(9)

GV

6. Scenarios for Growth of Vortex Lattices We have so far made only a preliminary exploration of possibilities. However we have seen the following three kinds of phenomena: A: Noise has little effect-vortices come in from infinity as in the simple vortex growth equation scenario. B. Primordial vortices cause the condensate and the lattice to grow simultaneously. C. Some compromise between A and B;-primordial vortices are to some extent . pushed away by the growing condensate.

VJ pZil

7. Conclusions and Outlook The simulations appear to show that the stabilization of vortex lattice is independent of noise, while the initiation of the lattice is very dependent on the noise. The simulations have been carried out so far: a) Without full implementation of the projector; b) With only the growth noise term; c) Only in two dimensions, for which it is not easy to get a definitive idea of what size of the noise should represent a physical three dimensional-but almost two dimensional-situation, such as a rapidly rotating condensate almost in the form of a pancake. Full implementation of the projected stochastic Gross-Pitaevskii equation in three dimensions will be needed for definitive conclusion, and we believe this is a realistic ambition.

References Movies for this paper can bedownloaded from http://www2.vuw.ac.nz/staff/crispin~g~dinernCOLS2003.html. P. Nozieres and D. Pines, Theory of Quantum Liquids (Part 2 ) (Addison-Wesley, Reading, Massachusetts, 1999). E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics Part 2 (Butterworth-Heineman,

Oxford, 1980,1986,1991).

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4. R. J. Donnelly, Quantized Vortices in Helium N (Cambridge U. P., Cambridge, 1991). 5. J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, Science 292,476 (2001). 6. K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys. Rev. Lett. 84, 806 (2000). 7. K. W. Madison, F. Chevy, and J. Dalibard, Phys. Rev. Lett. 86,4443 (2001). 8. C. Raman el al., Phys. Rev. Lett. 87,210402 (2001). 9. P. C. Haljan, 1. Coddington,P. Engels, andE. A. Comell, Phys. Rev. Lett. 87,210403 (2001). 10. E. Hodby et al., Phys. Rev. Lett. 88,010405 (2001). 11. A. L. Fetter, Phys. Rev. A 64,063608 (2001). 12. F. Dalfovo and S . Stringari, Phys. Rev. A. 63,011601 (2000). 13. J. R. Anglin, Phys. Rev. Lett. 87,240401 (2001). 14. A. E. Muryshev and P. 0. Fedichev,cond-mat 0106462 (2001). 15. B. M. Caradoc-Davies,R. J. Ballagh, and K. Bumett, Phys. Rev. Lett. 83,895 (1999). 16. M. J. Davis and C. W. Gardiner, J. Phys. B 35,733 (2002). 17. A. A. Penckwitt, R. J. Ballagh, and C. W. Gardiner, Phys. Rev. Lett. 89, 260402 (2002). 18. C. W. Gardiner, J. R. Anglin, and T.I. A. Fudge, J. Phys. B 35, 1555 (2002). 19. C. W. Gardiner and P. Zoller, Phys. Rev. A 58,536 (1998). 20. M. Tsubota, K. Kasamatsu, and M. Ueda, Phys. Rev. A 65,023603 (2002). 21. M. J. Davis, S. A. Morgan, and K. Bumett, Phys. Rev. A 66,053618 (2002). 22. C. W. Gardiner and M. J. Davis, The Stochastic Gross-Pitaevskiiequation 11, condmatl0308044,(2003). 23. C. W.Gardiner, A Handbook of Stochastic Methods, 2nd ed. (Springer, Berlin, Heidelberg, 1985).

A STORAGE RING FOR BOSE-EINSTEIN CONDENSATES

C. S. GARVIE, E. RIIS AND A. S. ARNOLD Deptrtment of Physics, University of Strathclyde 107 Rottenrow, Glasgow G4 ONG, UK

A vertically-oriented magnetic storage ring for cold atoms has been demonstrated and a Bose-Einstein condensate has been formed in a localised section of the torus. The unique magnetic coil design can be utilised as a magneto-optical trap, a IoffePritchard magnetic trap or a storage ring. The ring has a large atom number and magnetic trap lifetime compared to other rings. Additionally, its large (locm) diameter and high level of optical access should facilitate atom interferometry.

We use a double magneto-optical trap (MOT) system,’ in which the low pressure MOT contains around NLP = 1 x lo9 87Rb atoms. Atoms can then be loaded into a Ioffe-Pritchard magnetic trap with axial and radial trap frequencies v, = 10Hz and v, = 300Hz, respectively. Four circular coils form a toroidal quadrupole field.2 The geometric centre of the coils has a diameter of 10 cm. Each coil consists of two water-cooled loops carrying a current of 500 A. Additional pinch coils at the top of the storage ring enable the formation of MOT and Ioffe-Pritchard field configurations in a section of the torus. This unique coil design allows cold atoms or condensates to be formed inside the storage ring.2 A one-dimensional classical Monte Carlo simulation of cold atoms in the storage ring is sufficient for describing their time-dependent angular distribution (Fig. 1). The angular motion of the atoms is determined using the rigid pendulum equation: O”(t)= g sin(e)/r. Gaussian initial angular velocity and angular position distributions are chosen, with standard deviations determined from the atomic temperature (a, = d m / ~ and - )the Ioffe-Pritchard trap parameters (PO= d m / ( 2 7 r v 2 r ) )respectively. , The first storage rings contained lo6 atoms with lifetimes of less than half a Our ring (Fig. 2) contains more than lo8 atoms with a lifetime of 40s. We have recently heard of a ferromagnetic storage ring for cold atoms5 which contains up to lo7 atoms, with a lifetime of a few seconds. 178

179

1

2 time (s)

3

Figure 1. Monte Carlo simulation of the time-dependent atomic distribution as a function of angle around the ring for 0.5 mK atoms released from a Ioffe-Pritchard trap at the top (0 = 0) of the vertically-oriented storage ring. The black lines at 0 = f0.07rad mark the extent of the experimental viewing region.

Figure 2. Cold atoms with a temperature of 1mK confined in the storage ring. They disappear after 200ms and reappear after 600ms. Each absorption image is 6.4mm wide, and the spots on either side of the trapped atoms are due to imperfections in the CCD camera.

By evaporatively cooling atoms in the storage ring's Ioffe-Pritchard magnetic trap configuration, Bose-Einstein condensates (BECs') were created in a localised section of the Strathclyde ring in June 2003 (Fig. 3). The condensate fraction contains up to N = 5 x lo5 atoms. To date, experiments utilising the full extent of our storage ring have only been performed with cold atoms (Fig. 2). However we intend to launch condensates around the ring in the near future.

180

Figure 3. The Bose-Einstein phase transition at Strathclyde. The absorption images are taken after a 23ms drop time, for final RF evaporation frequencies 750,745 and 740 kHz. Each image is 500pm square.

As condensates are formed inside and a t the top of our verticallyoriented storage ring, we first intend to simply release condensates into the ring by lowering the current in the pinch coils. The one-dimensional expansion of the condensate in this case is likely t o lead to the formation of phase fluctuation^.^ At a later stage we intend to use Bragg scattering' to launch very cold, coherent pulses of condensate in both directions around the storage ring. With this approach it is more likely that phase coherence will be maintained until the two condensate wavepackets interfere at the top of the storage ring. This would enable the creation of a high-precision large-area Sagnac atom interferometer.2

References 1. K. Gibble, S. Chang and R. Legere, Phys. Rev. Lett. 75, 2666 (1995); C . J. Myatt et al., Opt. Lett. 21, 290 (1996). 2. A. S. Arnold and E. Riis, J . Mod. Opt. 49, 959 (2002). 3. F. M. H. Crompvoets, et al., Nature 411, 174 (2001). 4. J. A. Sauer, et al., Phys. Rev. Lett. 87, 027041 (2001). 5. W. Rooijakkers, M. Vengalattore, R. Conroy and M. Prentiss,

http://www.eps.org/aps/meet/DAMOP03/baps/abs/S390008.html 6. GSU BEC website: http://becOl.phy.gasou.edu/bec.htrnl/ 7. D. S. Petrov, G. V. Shlyapnikov and J . T. M. Walraven, Phys. Rev. Lett. 87, 050404 (2001); S. Dettmer et al., Phys. Rev. Lett. 87, 160406 (2001). 8. J. Stenger et al., Phys. Rev. Lett. 82, 4569 (1999); M. Kozuma et al., Phys.

Rev. Lett. 82, 871 (1999).

BRAGG SPECTROSCOPY O F AN ACCELERATING BOSE-EINSTEIN CONDENSATE

K. J. CHALLIS, R. GEURSEN, R. J. BALLAGH AND A. C. WILSON Department of Physics, University of Otago, PO Box 56, Dunedin, New Zealand E-mail: kchallisC3physics.otago.ac.n.z We investigate experimentally and theoretically Bragg spectroscopy of a condensate undergoing circular motion. The experimental Bragg spectra demonstrate additional structure and width compared to that expected of a condensate in a static harmonic trap. We have developed a simple momentum-space model which allows us to readily calculate and interpret the Bragg spectra.

Bragg spectroscopy has been established as a tool capable of measuring condensate properties with spectroscopic precision’. In particular, measurement of the relative scattered population for a range of Bragg frequencies allows the momentum distribution of stationary condensates to be inferred’. In the present paper we investigate Bragg spectroscopy of a condensate in a Time-averaged Orbiting Potential (TOP) trap3. An experimental study of the Bragg spectra reveals a significantly altered Bragg spectrum due to the nonstationary momentum distribution of the condensate. In addition to a full theoretical treatment, we present a “two-state” momentum-space model which provides a simple physical interpretation of the observed spectra. In a TOP trap the rotating bias field causes the condensate to execute a circular micro-motion which can be described analytically. An experimental study of the effect of this micro-motion on Bragg spectra has been carried out4, and in Fig. 1 we plot a typical experimental Bragg spectrum as ( x ) . This differs dramatically from the predicted spectra for a condensate in a harmonic trap equivalent to the static form of the time-averaged TOP trap, shown as (- -). The full theoretical description of Bragg spectroscopy of a condensate in a TOP trap is given by the Gross-Pitaevskii equation (GPE) with a time dependent trap term. We plot in Fig. 1 a solid line corresponding to 181

182

, Frequencydifference,S/2n (kHz)

Figure 1. Experimental ( X ) and theoretical (-) Bragg spectra of a condensate accelerating due to the dynamic T O P trap. For comparison the expected Bragg spectrum for a stationary condensate is also shown (- -). The radial trapping frequency and bias field rotation frequency are 27r x 18 Hz and 27~x 2:78 kHz, respectively. The tangential micro-motion velocity is 30% of the two-photon recoil velocity.

the theoretical Bragg spectrum for a condensate with no meanfield interactions. This linear theory can be seen to quantitatively predict the overall spectral width and scattered fractions of the experimental spectrum. Additional numerical simulations (not presented here) show that the effect of the nonlinearity is to increase the scattered fraction and narrow each spectral resonance. Both one and two dimensional calculations yield the same Bragg spectrum. These numerical calculations of the full GPE require intensive computation for each point of the spectral curve and provide little physical insight. We can obtain a simpler, approximate, treatment by transforming into the momentum frame where the condensate is stationary and applying the two momentum group approximation of Blakie e t d 5 .This yields two coupled ordinary differential equations for the momentum wavefunctions of the zeroth and first Bragg orders6, which can be readily solved to give the Bragg spectra shown as (- -) in Fig. 2. For comparison, the results of the full GPE treatment are reproduced from Fig. 1, as a solid line. In principle our momentum-space model will also treat non-circular accelerations provided that the transformation into the condensate rest frame is linear. In the “two-state” approximation the effects of the trap are entirely represented by the circular motion of the condensate. Thus, it is possible to extract an effective light potential for a condensate in an equivalent, trap-

183

Frequency difference,WZn (kHz)

Figure 2. The numerical simulation of the full GPE from Fig. 1 (-) and the corresponding result of the “two-state” momentum-space model (- -). The vertical lines indicate the Bessel function resonances.

less system. This provides a rigorous basis for the interpretation that in the condensate rest frame the optical lattice appears frequency modulated7. An approximate relative weight of each Bragg spectral resonance can be immediately inferred from a Bessel function expansion of the effective potential, where the resonances are separated by the bias field rotation frequency, as shown by the vertical lines in Fig. 2.

Acknowledgments The authors thank C. W. Gardiner for helpful discussions. This work was supported by Marsden Grants U00508 and 02-PVT-004.

References 1. J. Stenger et al., Phys. Rev. Lett. 82, 4569 (1999). 2. P. B. Blakie et al., Phys. Rev. A 6 5 , 033602 (2002). 3. J. H. Muller et al., Phys. Rev. Lett. 85, 4454 (2000). 4. R. Geursen et al., cond-mat/0307224, to appear in Phys. Rev. A. 5. P. B. Blakie and R. J. Ballagh, J . Phys. B: At. Mol. Opt. Phys. 33, 3961 (2000). 6. K. J. Challis and R. J. Ballagh, in preparation. 7. M. Cristiani et al., Phys. Rev. A 6 5 , 063612 (2002).

DISPERSION MANAGEMENT AND BRIGHT GAP SOLITONS FOR ATOMIC MATTER WAVES

B. EIERMANN, TH. ANKER, M. ALBIEZ, M. TAGLIEBER, P. TREUTLEIN* AND M. K. OBERTHALER Fachbereich Physik and Center for Junior Research Fellows Universitat Konstanz, Fach M696, 78457 Konstanz, Germany E-mail: so1itonOvortex.physik.uni-konstanz.de

We demonstrate the control of the spreading of matter wave packets utilizing periodic potentials. We also report on the first demonstration of bright atomic gap solitons. The matter wave packets are realized by Bose-Einstein condensates of *?Rbin an optical dipole potential acting as a one-dimensional waveguide. A weak optical lattice is used t o control the dispersion of the matter waves during the propagation of the wave packets.

The broadening of particle wave packets due to the free space dispersion is one of the most prominent quantum phenomena discussed in almost every textbook of quantum mechanics. Dispersion management utilizing periodic potentials allows to control the linear dynamics. Non-spreading wave packets - solitons - are one of the most prominent effects in nonlinear physics. Solitons can only be formed if nonlinear and linear dynamics compensate each other. Therefore bright solitons for 87Rbi.e., repulsive interaction, can only be realized for anomalous dispersion. 1. Dispersion Management

Since the realization of Bose-Einstein condensates of dilute gases wave packet dynamics can be observed in real space on a macroscopic scale. The approach of using periodic potentials to experimentally engineer the dispersion relation is known from nonlinear optics' and is likewise applicable to atomic matter waves2. Generally the matter wave dispersion in a *Present address: Max-Planck-Institut fur Quantenoptik und Sektion Physik der Ludwig-Maximilians-Universitat, 80799 Miinchen, Germany

184

185

periodic potential can be adjusted in magnitude and sign by changing the depth of the potential and the mean quasimomentum of the initial wave packet. In our experiment the anomalous matter wave dispersion has been realized by an adiabatic acceleration of the periodic potential up to the recoil velocity. This allows the preparation of the atomic wave packet at the Brillouin zone edge of the corresponding band structure leading to an effective negative mass. Figure 1 shows the results of an experiment in which the propagation of an atomic wave packet is studied in the normal (Fig. l b ) and anomalous (Fig. lc) dispersion regime. Figure Id reveals that the consecutive realization of normal and anomalous dispersion allows to reverse the spreading of a wave packet. c) anomalous

a ) initial wave packet

.. . . . lispersion

m=l.25 q

d) dispersioi

Figure 1. Controlling the dispersion of an atomic wave packet in a waveguide using a periodic potential. Shown are absorption images of atomic wave packet averaged over four realizations and the corresponding density distributions n(x,t) along the waveguide. (a) Initial wave packet. (b), (c) Images taken after an propagation time of t=26ms for different dispersion regimes with different effective masses as indicated. (d) Wave packet subjected to dispersion management: an initial stage of expansion for t=17ms with normal dispersion is followed by propagation with anomalous dispersion for t=9 ms. The broadening in the normal dispersion regime has been reversed by anomalous dispersion.

For the wave packet dynamics in a quasi one-dimensional situation many new effects are expected due to the interplay between nonlinearity resulting from atom-atom interaction and dispersion. In particular, non-spreading wave packets such as gap solitons and self trapped states are predicted. 2. Atomic Gap Solitons

Dark solitons for repulsively interacting Bose-Einstein condensates have been demonstrated 5, while bright solitons have been observed by switching the interaction from repulsive to attractive using a Feshbach resonance 6 .

186

We demonstrate for the first time the generation of bright solitons for repulsive interaction. Solitonic solutions in periodic potentials are also known as gap solitons. As shown in Sect. 1dispersion management allows the realization of an effective negative mass corresponding to anomalous dispersion which is necessary t o compensate for the repulsive interaction. We have observed wave packets of 5 p m width with 300 atoms, which do not spread over a time interval of 60 ms. This corresponds to a lifetime of 10 times the soliton period. Systematic measurements of the atom number as a function of the effective negative mass reveal the expected scaling for bright gap solitons.

Figure 2. Experimental observation of a bright atomic soliton for repulsive interaction after 25 ms of propagation. It shows the soliton and the radiated atoms not fulfilling the soliton condition. The soliton packet contains 300 atoms and has a width of 5 pm.

Acknowledgments We wish t o thank J. Mlynek for his generous support, A. Sizmann and B. Brezger for many stimulating discussions. This work was supported by Deutsche Forschungsgemeinschaft, Emmy Noether Program, and by the European Union, Contract No. HPRN-CT-2000-00125.

References 1. G.P. Agrawal, Applications of Nonlinear Fiber Optics (Academic Press, San Diego, 2001). 2. B. Eiermann et al., Phys. Rev. Lett. 91,060402 (2003). L.Fallani et al., condmat/0303626 (2003). 3. M. Steel and W. Zhang, cond-mat/9810284 (1998). P. Meystre, Atom Optics (Springer Verlag, New York, 2001) p 205, and references therein. 4. A. Trombettoni and A. Smerzi, Phys. Rev. Lett. 86,2353 (2001). 5. S. Burger et al., Phys. Rev. Lett. 83,5198 (1999). J. Denschlag et al., Science 287,97 (2000). 6. L. Khaykovich et al., Science 296, 1290 (2002). K.E. Strecker et al., Nature 417,150 (2002).

ALL-OPTICAL REALIZATION OF AN ATOM LASER BASED ON FIELD-INSENSITIVE BOSEEINSTEIN CONDENSATES

G. CENNINI, G. RITT, C. GECKELER AND M. WEITZ Physikalisches Institut der Universitat Tubingen Auf der Morgenstelle 14, 72076 Tubingen, Germany A rubidium Bose-Einstein condensate is produced by direct evaporative cooling of atoms in a single beam C02-laser dipole trap. When a magnetic field gradient is applied, the trap geometry allows the formation of the condensate in field insensitive m F = 0 states only. This suppresses fluctuations of the chemical potential caused by stray magnetic fields. We have extracted condensed atoms into a collimated, monoenergetic beam to realize a novel atom laser.

Soon after the realization of Bose-Einstein condensation (BEC) by evaporative cooling in magnetic traps, condensed atoms have been extracted to form a coherent monochromatic beam. This source is commonly referred to as an atom laser'. In these experiments, a major experimental issue was to circumvent the influence of stray magnetic fields, which cause a shift of the chemical potential of order % 67 nK/mG. On the other hand, e. g., atomic precision spectroscopy has demonstrated the benefit of using magnetic field insensitive mF = 0 states. Atoms in such states cannot be kept in magnetic traps. However, optical dipole traps can confine atoms in all spin states. Chapman et al. have demonstrated that a Bose-Einstein condensate can be created by direct evaporative cooling in a crossed beam optical dipole trap2. We report here on an all-optical realization of an atom laser3. Our experiment is based on direct Bose condensation of rubidium atoms in a tightly focused COz-laser optical dipole trap. The use of a single running wave allows a very stable operation of the condensate production. Previous experiments creating quantum degeneracy in dipole traps have either required the use of a more alignment-sensitive crossed dipole trap geometry2 or Feshbach resonances4i5to enhance the collisional rate. In our experiment, due to the weak confinement along the trapping beam axes, already 187

188

a moderate magnetic field gradient removes atoms in field-sensitive states during evaporative cooling, and allows a stable trapping of atoms only in field-insensitive mF = 0 states. The remaining magnetic field sensitivity of the chemical potential of the generated condensate is 0.014 pK/(mG)2, and determined only by the quadratic Zeeman effect. An alternative way to realize a field-insensitive BEC is t o use atoms with spin singlet ground states, as reported by Takahashi et al. in these Proceedings6. Finally, we have extracted condensed atoms from the trap by smoothly ramping down the trapping potential to form a well collimated atom laser beam. Our experiment proceeds as follows. In a vacuum chamber, a magnetooptical trap (MOT) initially captures atoms of the isotope s7Rb. In a 30s loading time, typically 6 x lo7 atoms are collected. Far off-resonant radiation near X = 10.6pm for quasi-static dipole trapping of atoms is generated by a CO2-laser. The emitted mid-infrared radiation is coupled into the vacuum chamber and focused to a beam waist of 27pm. This trapping beam is oriented horizontally and the focus overlaps with the MOT. Following the MOT loading, a 60 ms-long dark-MOT phase is applied during which the atomic cloud is compressed and atoms are pumped into the lower hyperfine state. Subsequently, the near-resonant cooling light is extinguished and the MOT magnetic field is switched off. By this time, the atoms are confined in the quasistatic dipole trapping potential alone. An analysis of the trapped atomic cloud is performed by extinguishing the dipole trapping beam at the end of an experimental cycle and employing absorption imaging. The number of trapped atoms in the optical dipole trap after the darkMOT phase is 4 x lo6. After an initially looms-long plain evaporation phase, the atoms have relaxed to a clean Maxwell-Boltzmann distribution with a measured temperature of 140 pK. The atomic density at this time is n N 1.2 x 1013cm-3, and we derive the value nX& N 1.2 x Although this result is below best values obtained in quasistatic crossed dipole trap configurations or optical lattices2i7, the obtained high collisional rate of 6.2 kHz encourages further, forced evaporative cooling. To cool the atoms to quantum degeneracy, we have smoothly lowered the power of the trapping beam power. In an evaporative cooling time of 7s, the trapping beam power is reduced from an initial value of 28 W to a final value of 200mW. In initial measurements, the MOT quadrupole field was switched off after the dark-MOT phase. Figure l a shows an image recorded directly after shutting off the trap laser, i.e., with no free expansion phase. The shown trapped cloud is cigar shaped and strongly elongated

189

Figure 1. Free expansion of a Bose-Einstein condensate (field of view 240 pm x 240 pm). Images taken with (a) no free expansion, (b) 8ms and (c) 15ms, respectively, of free expansion.

along the weakly confining beam axis. Figures l b and l c give time of flight images recorded after a 8 ms and 15ms, respectively, long free expansion phase. We interpret the observed inversion of the aspect ratio during free expansion as clear evidence for BEC. This condensate has a spinor nature. The distribution of Zeeman components is analyzed by applying a SternGerlach magnetic field gradient during the detection phase.

Figure 2. Shadow images of Bose-Einstein condensates after a 15 ms free expansion time (field of view 380pm x 380pm). (a) Stern-Gerlach magnetic field applied only during the free expansion phase. (b) Stern-Gerlach field activated also during the evaporative cooling phase.

Figure 2a shows a typical measurement, where a separation into clouds with different spin projections is clearly visible. The so-produced condensates contain typically 12,000 atoms distributed among the m F = - 1 , O , 1 Zeeman components of the F = 1 ground state. In other experiments, the MOT quadrupole field was left on throughout the experimental cycle. The atoms then condense in the mF = 0 state alone. Figure 2b shows an image of an almost pure Bose-Einstein condensate of 7,000 atoms in the F = 1 , m = ~ 0 component of the electronic ground state. Atoms in field sensitive spin projections are removed during the final stage of the evaporation by the magnetic field gradient. To realize an all-optical atom laser, we produce a field insensitive condensate as described and then further reduce the COz-laser beam power in a 100ms long linear ramp. The condensate is outcoupled once the dipole trap-

190

Figure 3. Shadow images of atoms extracted from the dipole trap by lowering the trapping potential: (a) thermal cloud (b) for a Bose-Einstein condensate, where a coherent atom laser beam is formed. The field of view comprises 0.28 mm x 0.5 mm.

ping potential fails to support the atomic cloud against gravity. Figure 3 shows typical images of the emitted beam, where the atomic temperature was lowered from clearly above T, (a) to below T, (b). In the latter image, an intense, well directed atom laser beam is visible. The momentum spread is limited by the uncertainty principle only. As fluctuations of the magnetic field cause a shift of the chemical potential via the second order Zeeman effect only, no magnetic shielding is required for this measurement even in a magnetically noisy environment. The output stability of the present scheme is limited only by the intensity fluctuations of the trapping laser. We foresee that the technical simplicity of the demonstrated method t o produce Bose-Einstein condensates opens up new applications in basic and applied sciences. Further, we expect the the demonstrated atom laser is awaiting fascinating applications in atom optics experiments, such as new atom interferometric measurements of gravitation and rotation.

References 1. M. 0. Mewes et al., Phys. Rev. Lett. 78,582 (1997); M. R. Andrews et al., Science 282, 1686 (1998); E. W. Hagley et al., Science 283, 1706 (1998); I. Bloch et al., Phys. Rev. Lett. 8 2 , 3008 (1999). 2. M. D. Barrett, J. A. Sauer and M. S. Chapman, Phys. Rev. Lett. 87, 010404 3. 4. 5. 6. 7.

(2001). G. Cennini, G. Ritt, C. Geckeler and M. Weitz, cond-mat/0307620. T. Weber et al., Science 299, 232 (2003). S. R. Granade et al., Phys. Rev. Lett. 88, 120405 (2002). Y. Takahashi et al., in these Proceedings. S. F'riebel et al., Appl. Phys. B 67,669 (1998).

DYNAMICAL EFFECTS OF BACK-COUPLING ON AN ATOM LASER

N. P. ROBINS, J. E. LYE, C. S. FLETCHER, S. A. HAINE, J. DUGUE C. BREME, J. J. HOPE AND J. D. CLOSE Department of Physics, Faculty of Science, The Australian National University Canberra, A C T 0200, Australia E-mail: niclc.robinsOanu. edu.au Atom lasers have been demonstrated in both pulsed and semi-continuous modes. The dynamics of these devices is predicted to be dramatically different to the optical laser due to fundamental differences between the two types of boson fields; however to date it is the similarities that have been emphasized. Here we show how the atom laser dynamics are strongly affected by coupling of the output beam back to the condensate.

Typically, research in the field of Bose-Einstein condensation (BEC) relates to studies of fundamental properties, such as collective modes or to the response of the BEC to external stimuli 3. In contrast, the atom laser is one of the most promising technologies to arise from BEC, being the direct atomic equivalent of the optical laser 4 . Mewes et al.5 demonstrated the first atom laser, based on the application of pulsed radio-frequency (RF) fields to induce controlled spin flips from magnetically trapped to untrapped states of a Bose-Einstein condensate. Later it was shown by Hagley et a1.6 that a pulsed Raman outcoupling could be used to achieve a quasi-continuous multi-state atomic beam. Bloch et al.7 achieved continuous RF outcoupling for up to 100 ms, producing a single state atom laser beam, and showed that this beam could be coherently manipulated in direct analogy to the optical laser '. These experiments, and others have conclusively demonstrated that the outcoupling process can be made coherent. They have also demonstrated a number of other similarities between the optical and atom lasers such as high flux and low divergence of the output beam. In all these works, the similarities between the two systems have been emphasized. However, ultimately it is the differences such as mass and wavelength that will lead to applications of the atom laser. In moving continuously from a low to high pulse repetition rate, we have 9310,

191

192

observed the effect of back-coupling on the dynamics of the atom laser. In order to study the ‘crossover’ regime, we produce a F = 2 , m ~ = 2 87Rb condensate, typically consisting of about 50,000 atoms, via evaporation in a water-cooled QUIC magnetic trap l 1 (for this experiment our trap was set to up = 253 H z , u, = 20 H z and Bo = 1G). After evaporative cooling: the BEC is left to equilibriate both thermally and motionally for 100 ms. We then apply a TTL initialization signal to a pulse generator which is used to drive the TTL input of an RF signal generator set in gated burst mode. The RF pulses are amplified and radiated perpendicular to the magnetic bias field of the trap through a single loop of 22 mm radius, approximately 18 mm from the BEC.

Figure 1. Pulsed atom laser dynamics, showing the crossover dynamics. The applied radio-frequency pulses are varied: (a) 4 pulses, (b) 5 pulses, (c) 6 pulses, (d) 7 pulses, (e) 10 pulses.

We calibrated the system by measuring the number of trapped and untrapped atoms after the application of a single RF pulse of varying amplitude. For the experiment described hereafter, we selected an R F amplitude that coupled only a very small fraction of atoms into the untrapped states while still giving a large enough number in the mp = 0 state for good signal to noise ratio. For one, two, three and four RF pulses, we observe predictable outcoupling from the atom laser system. Figure l a is indicative of this behavior, where four RF pulses (separation 2 ms) have been applied to the BEC and we see four mF = 0 atom pulses in the position expected from gravity. In Figure l b five RF pulses (separation 1.6 ms) have been applied, and we observe five atomic wave-packets again in the expected positions. However,

193

we note that in the later 3 pulses, there is a significant blurring between the atomic pulses. This effect is not due to interference between the wavepackets which are quite separate and distinct. Rather, atoms appear to be leaking from the condensate region between the outcoupling RF pulses. In Figure l c , six RF pulses were applied (separation 1.2 ms); however only five atomic pulses were observed with the first atomic pulse being entirely absent. At a higher pulse repetition rate the output is further distorted from the ideal (Fig. Id, separation 1ms). In Fig. l e (pulse separation 800 pus) the atomic beam is longer than expected from pure gravitational acceleration, indicating that the anti-trapped mF states are playing a significant role in the dynamics. This highly repeatable effect can be understood by considering that the R F outcoupling is coherent, and hence atoms can not only be coupled out of the magnetically trapped condensate states but also into them. Backcoupling results when the pulse spacing is of the same order as the time it takes an outcoupled pulse to leave the resonance, leading t o atoms being coupled back into the condensate and to interference effects between fields.

References 1. M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell, Science 269, 198 (1995); K. B. Davis, M. -0.Mewes, M. R. Andrews,

N. J. van Druten, D. S. Durfee, D. M. Kurn and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995); C. C. Bradley, C. A. Sackett, J. J. Tollett and R. G. Hulet, Phys. Rev. Lett. 75, 1687 (1995). 2. M.-0. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, C. G. Townsend and W. Ketterle Phys. Rev. Lett. 77, 988 (1996). 3. A. E. Leanhardt, Y . Shin, D. Kielpinski, D. E. Pritchard and W. Ketterle, Phys. Rev. Lett. 90, 140403 (2003). 4. S. L. Rolston and W. D. Phillips, Nature 416,219 (2002); W. Ketterle, Physics Today 52, 30 (1999). 5. M.-0. Mewes, M. R. Andrews, D. M. Kurn, D. S. Durfee, C. G. Townsend and W. Ketterle Phys. Rev. Lett. 78, 582 (1997). 6. E. W. Hagley et al., Science 283, 1706 (1999). 7. I. Bloch et al., Phys. Rev. Lett 82, 3008 (1999). 8. I. Bloch et al. Phys. Rev. Lett. 87, 030401-1 (2001); M. Kohl et al., Phys. Rev. Lett. 87, 160404 (2001); M. Kohl et. al. Phys. Rev. A 65, 021606 (2002). 9. B. P. Anderson and M. A. Kasevich, Science 282, 1686 (1998). 10. J. L. Martin, C. R. McKenzie, N. R. Thomas, D. M. Warrington and A. C. Wilson, J. Phys. B: At. Mol. Opt. Phys. 33, 3919 (2000). 11. T. Esslinger et al., Phys. Rev. A . 58, 2664 (1998).

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Cold Molecules and Cold Collisions

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PHOTOASSOCIATION SPECTROSCOPY OF ULTRACOLD METASTABLE HELIUM ATOMS: NUMERICAL ANALYSIS

M. LEDUC, M. PORTIER, J. LEONARD AND M. WALHOUT Labomtoire Kastler Brossel, Ecole Nonnale Supirieure, 24 rue Lhornond, 75231 Paris, France E-mail: [email protected]

F. MASNOU-SEEUWS AND K. WILNER Laboratoire Aim6 Cotton, CNRS, Bt. 505, Campus d 'Orsay, 91405b Orsay Cedex E-mail: masnouQ1ac.u-ps.ud.fr

A. MOSK University of Twente P . 0.Box 21 7, 7500AE Enschede, The Netherlands E-mail: [email protected] We present a mapped Fourier grid method to analyse the results of a photoassociation spectroscopy experiment with metastable helium atoms in the z3S1 state. The ultracold atoms, close to the BEC transition, are excited with a laser that is red detuned near the 23S1-23Po atomic transition. Photoassociation produces giant dimers of average size up to 900 a0 in a shallow, purely long range molecuThe positions of five bound vibrational states in this potential lar potential .0; are measured using a calorimetric detection. The multichannel calculations of the spectrum make use of a grid with adaptive coordinate. The precision of the measurements and of the calculations is comparable and the agreement is very good.

1. Introduction

Laser-induced photo-association (PA) spectroscopy is an appropriate method to derive information about the wave functions and the collisional properties of laser cooled atoms. It has provided a rich array of results for alkali atoms' and allowed tests of calculations of molecular states as well as measurement of lifetimes in excited states of atoms with good precision. 197

198

In the case of Bose Einstein Condensation (BEC), it has also led to good estimates of the s-wave scattering length'. The case of metastable helium in the 23S1 state is distinctive in that it carries a large internal energy of 20 eV, which does not prevent BEC3. The first PA experiments with metastable helium were performed in a magnetooptical trap4. Our group at ENS set up a PA experiment in different and more favorable conditions of temperature and density by use of an evaporatively cooled spin polarized atomic cloud. In a recent letter5 we reported the observation of giant helium dimers in ultracold metastable helium atoms with the PA laser detuned to the red of the 23S1 - 23P0 atomic line. The five observed bound molecular states belong to a shallow 0; purely long-range potential well (see Figure 1). We here provide the theoretical background for the calculation of the long range part of the z3S - 23P molecular interaction potentials. We compute the bound states in the 0; potential using the mapped Fourier grid method6, where the mapping procedure7 is adapted to the 0: potential.

1083 nm ' 1 i5 0 '

2

"

0

I

5 00

-2x1o4 -3x1O4 0

"

I

1

O H F ' -iXio4j

"

I

'

'

1000

+ 23s,

23s,

T

'

1I

v

15

30

500

1000

'

Internuclear distance R (a,) Figure 1. Principle of a photoassociation experiment with metastable 4He in the 23S1 state. The 0: potential is purely long range. Note the different scales for the upper and lower potentials.

199

2. Measurements of Molecular Bound States in the 0:

Potential The atomic cloud is maintained in a magnetostatic trap of the IoffePritchard type and evaporated down to a few pK just above the critical temperature T,for condensation, in the range 2 to 30 pK. The bias field Bo of the trap can be varied between 0.1 up to 10 Gauss, which changes the compression of the cold gas and thus its density n and critical temperature T,. For an average value of 4 Gauss for Bo, T, is of order 2.5 pK and n of order 1013 cme3. The PA beam is sent perpendicular to the axial direction of the cloud while keeping the magnetic trap on. The laser frequency, detuned slightly red from the atomic 23S1 - 23P0line, is stabilized on this DOatomic transition with a precision of 0.3 MHz. To reach four of the five measurable levels (v = 1- 4), the detuning bv is adjusted with a set of 2 to 3 acousto-optical modulators. For reaching the v = 0 state at larger detuning, the laser frequency is compared to that of the DZ transition using a Fabry-Perot cavity; the resulting precision (10 MHz) is less good. The duration and intensity of the PA pulse are typically of order a few ms and a few 10pW/cm2. When the PA laser is resonant with a molecular transition, there is an increase in the temperature of the trapped cloud. We use absorption imaging to detect the temperature increase as a function of PA frequency. One explanation for this heating is the decay of the molecular bound state into pairs of “hot” atoms, some of them remaining in the trap and depositing energy in the cloud by elastic collisions with the other atoms, after a few hundreds of ms of thermalization between the PA and detection pulses. The values of the energy hb of the molecular bound states are deduced from the measured bv detuning of the PA laser, corrected by 2pBo 3 k ~ T for shifts related to the magnetic field in the trap and the finite temperature of the cloud’. Other types of shifts, such as light shifts, mean field shifts and recoil shifts are smaller than the experimental precision of order 0.5 MHz and therefore negligible. Experimental results5 are reported in the first column of Table 1.

+

3. Hamiltonian of the 25 - 2P Molecule

Let us consider the system resulting from the two helium atoms 23S and 2 3 P . The Schrodinger equation for this system can.be written in the following way:

200

fit*=)

= ('f'n

+ f i o + fifs+ O(R))lQa)

(1)

the nonrelativistic Hamilwhere fn is the kinetic energy of the nuclei, tonian of the atoms (the ground level of energies), H f s the fine structure of atom 23P, O(R)the energy of electrostatic interaction between the two atoms at distance R , the long range part of which is dominated by the dipole-dipole interaction term in C3/R3. Given that the relative motion of the nuclei is characterized by the angular momentum = J'the kinetic energy of the nuclei can be written as in equation 2:

XI

where p is the reduced mass of the system, J' the total angular moment of the system, conserved during the collision, and the total electronic angular momentum. Hc is the Coriolis coupling term between the angular momentum of the electrons and the ro-vibrational movement of the molecule. The fine structure Hamiltoniang can be written as:

where L' and are, respectively, the total electronic orbital angular momentum and the total spin of the atoms. CY and p are deduced from spectroscopic measurements of the 23P fine structure splittinglO. Let us first consider the basis set in which the dipole-dipole interaction is diagonal (Hund's case (a)). The projection of on the molecular axis (A) is a good quantum number in this basis set and the states are labelled 2~9+1Ah 9,u, where g,u refer to the g e m d e , ungerade eigenstates of parity, respectively. In our case only the ungerude states are accessible as one starts from the gerude 'CCg+state. In the first approximation, one should consider the following terms in calculating the dipole-dipole interaction: +2C3/R3 for 'CC,+, -2C3/R3 for 'C$ and 5C$, -C3/R3 for 31-Iu, +C3/R3 for 'II, and 'nu. The C3 coefficient can be expressed as a function of r, the width of the 2s - 2P atomic line and of k, the wave number, according to C3 = (3/4)Ii17/k3. At large distance one needs to take into account the propagation time of the electromagnetic field, as we deal with giant molecules of size R of order 50 nm, which is not negligible compared with

201

the laser wavelength X of 1083 nm. Taking into account the retardation effects, one finds1' the following corrections for states 11 and C :

C3c

-+

C3n

-+

+ R/Xsin(R/X)) x C, (cos(R/X) + R/Xsin(R/X) - (R/X)'COS(R/X)) x Cs

(cos(R/X)

(4)

If now one considers the fine structure Hamiltonian H f , , it is not diagonal in the basis set of Hund's case (a). It is diagonal in the basis set of Hund's case (c) which is labelled R:,g, where R is the projection of J , on the molecular axis. Actually, fijs couples the Hund's case (a) eigenstates of U with each other inside the R = 0 subspace, which leads to the purely long range potential 02. 4. Calculation of Molecular Potentials Using the Mapped

Fourier Grid Method We write a general wavefunction of the form:

where the @p are the electronic states of the Hund's case (a) basis set, which is diabatic (independent of R), UP and Y p are the radial and angular parts of Qa. We solve the radial Schrodinger equation using equation 1. It gives:

-(@Pa

1%

s I%+P

(R)

(6)

We decided to solve the problem in its most general case, for maximum precision in view of the comparison with the experimental results. We perform a multichannel calculation, which allows us to take into account the coupling between different potentials and the predissociation which occurs when a bound state is embedded in a continuum. For this we follow the mapped Fourier grid method. We use the projection of the wavefunction on the sine (particle-in-a-box)basis, which ultimately requires solving a matrix problem6. One considers a representation by discrete variables on a grid of length L made of N points. In the phase space (R, P ) , this grid is a rectangle of size L * Pmax, where P k a x / 2 p = E,,,, the maximum

202 energy considered in the calculation. The characteristic length is the local de Broglie wavelength h / P , which strongly varies over the L distance for the 02 potential. R is chosen to vary between two hard walls, one at a short distance of 25 ao, the other one at a large distance of 5000 ao. In order to avoid the implementation of a grid with a very narrow step, the mapping procedure introduces an adaptive coordinate x( R) subjected to the variation of the local de Broglie wavelength as a function of the internuclear distance R: the grid spacing is not fixed but adapted to the variations of the potential7. The minimum number of grid points is fixed by dividing the rectangle ( X J J , ) , where p , is the momentum associated with 2 , in elementary cells of area h and putting one grid point per cell, so that the stepsize becomes Ax = h/2pmal, that is to say at least of the order of one point per node of the wave function. The Jacobian of this change of variables is set up as in Ref. 6 and calculated from a C3/R3 potential. 5. Results

Calculations have been performed using for the C3 coefficient the value: C3 = 6.4102 atomic unit, as quoted in Ref. 12. Three levels of approximation have been applied: 0

0

0

0

approximation (A): in the subspace 0; of dimension 4, one takes into account the dipole-dipole interaction terms, the fine structure terms and the main rotation term. We neglect the Coriolis coupling, this is to say we ignore the ro-vibrational couplings. approximation (B): the Coriolis couplings included in the 0: subspace of dimension 4 are taken into account. approximation (C): one takes into account the O:, 0; and 1, supspaces, forming a 19 x 19 matrix, in which all the rotational couplings for J=l are considered. approximation ( C ’ ) :calculations are the same as in approximation (C), but a different value of the C3 coefficient has been used: C3 = 6.405, as used in Ref. 8 from Ref. 13.

The different theoretical results and the experimental values are shown in Table 1. We varied the position of the hard wall at large distance from 5000 to 10 000 a0 and found a difference smaller than 0.01 MHz. Here 5000 a0 is used. We also varied the grid step in the (x,p,) phase space within a factor of 2 and found no significant difference. The value taken for Em,, is chosen to be 1 MHz above the dissociation limit of the 23S1-23P~

203

potentials. Table 1. Energies of the bound states in the Oj molecular potential for 7=1, expressed in MHz. The first column gives the experimental results for the vibrational levels from u=l to v=4. The next columns give the theoretical results for three levels of approximation (see text). Experiment

v=5 v=4 v=3 v=2 v=l

v=0

-18.2(5) -79.6(5) -253.3(5) -648.5(5) -1430(10)

(A) -3.70 -22.23 -89.40 -272.62 -682.00 -1469.33

(B) -2.49 -18.12 -79.39 -252.75 -647.96 -1417.26

(C) -2.59 -18.31 -79.68 -253.17 -648.52 -1417.92

(C') -2.58 -18.27 -79.57 -252.97 -648.26 -1417.72

One notices the rather large difference between the results of approximations (A) and (B), which indicates that the Born Oppenheimer approximation gives inaccurate values for the energies. The comparison between approximations (B) and (C) shows that the vibration-induced couplings between electronic and nuclear degrees of freedom introduce tiny differences significant at the level of precision of the experiment. The calculations made in approximation (C) are more precise than in approximations (A) and (B), as the basis of considered electronic states is larger. One must thus compare the experimental results (column 1) to those of approximation (C) (column 4). One notices that the agreement between experiment and theory is remarkably good for u=l to the u=4 vibrational levels, for which the energy was measured with a precision of 0.5 MHz. It is also satisfactory for the v=Q level, the position of which was measured with much less precision. The grid method calculation has allowed us to check how these results depend on the value of the parameter Ca related to the lifetime of the 23P atomic level. The comparison between calculations (C) and (C'), where Cy, differs by one part in 1000, shows that the variations of the results are below the sensitivity of the experiment. We estimate that the assumed Cy, value is confirmed to within a few 10~3. 6. Conclusion

The photoassociation method allowed us to measure the energy of the vibrational levels of giant metastable helium dimers in the shallow purely long range 0^ molecular potential. The same energies could also be calculated using the mapped Fourier grid method with a precision comparable to the

204

experimental precision. T h e agreement is excellent. T h e agreement is also very good with another theoretical analysis of the data performed at ENS, using a less general method *. T h e preprint by Venturi et a1 l 2 recently brought t o our attention gives the results of a multichannel calculation. They are in very good agreement with t he present experimental data and with the mapped Fourier grid method computations described here.

References 1. see e.g. review articles W.C. Stwalley, H. Wang, J. Mol. Spec. 195, 194 (1999), J. Weiner, V. S. Bagnato, S. Zilio, P. S. Julienne, Rev. Mod. Phys. 71, 1 (1999), F. Masnou-Seeuws, P. Pillet, Advances in Atomic, Molecular, and Optical Physics 47,53 (2001), and references therein. 2. E. R. I. Abraham, W. I. McAlexander, C.A. Sackett, R. G. Hulet, Phys. Rev. Lett. 74,1315 (1995); J. R. Gardner, R. A. Cline, J. D. Miller, D. J. Heinzen, H. M. J. M. Boesten, B. J. Verhaar, Phys. Rev. Lett. 74,3764 (1995). 3. F. Pereira Dos Santos, J. LBonard, Junmin Wang, C. J. Barrelet, F. Perales, E. Rasel, C. S. Unnikrishnan, M. Leduc, C. Cohen-Tannoudji, Phys. Rev. Lett. 86,3459 (2001); A. Robert, 0. Sirjean, A. Browaeys, J. Poupard, S. Nowak, D. Boiron, C. I. Westbrook, A. Aspect, Sci. Mag. 292,463 (2001). 4. N. Herschbach, P. J. J. Tol, W. Vassen, W. Hogervorst, G. Woestenenk, J.W. Thomsen, P. van der Straten, A. Niehaus, Phys. Rev. Lett. 84,1874, (2000). 5. J. LBonard, M. Walhout, A. P. Mosk, T. Mueller, M. Leduc, C. CohenTannoudji, Phys. Rev. Lett. 91,073203-1 (2003). 6. K . Willner, 0. Dulieu, F. Masnou-Seeuws, J . Chem. Phys., in press. 7. V. Kokoouline, 0. Dulieu, R. Kosloff, F. Masnou-Seeuws, J . Chem. Phys. 110,9865 (1999). 8. J. LBonard, M. Walhout, A. P. Mosk, P. van der Straten, M. Leduc, C. Cohen-Tannoudji, submitted to Phys. Rev. A . 9. H. A. Bethe, E. E. Salpeter, “Quantum mechanics of one- and two-electron atoms”, Springer-Verlag (1957). 10. M.C. George, L.D. Lombardi, E.A. Hessels, Phys. Rev. Lett. 87, 173002, (2001), and references therein; J. Castillega, D. Livingston, A. Sanders, D. Shiner, Phys. Rev. Lett. 84,4321, (2000), and references therein. 11. E.I. Dashevskaya, A.I. Voronin, E.E. Nikitin, Can. J . Phys. 47,1237 (1969); W. J. Meath, J . Chem. Phys. 48,227 (1968). 12. V. Venturi, P. J. Leo, E. Tiesinga, C. J. Williams and I. B. Whittingham, Phys. Rev. A 68,022706-1 (2003). 13. G. W. F. Drake in Atomic, Molecular and Optical Physics Handbook, edited by G. W. F. Drake, AIP Press, Chap.11 (1996).

PRODUCTION OF LONG-LIVED ULTRACOLD LIZ MOLECULES FROM A FERMI GAS

J. CUBIZOLLES, T. BOURDEL, S. J. J. M. F. KOKKELMANS AND C. SALOMON Laboratoire Kastler Brossel, Ecole Normale Supe'rieure 24 rue Lhomond, 75291 Paris 05, fiance

G. V. SHLYAPNIKOV FOM Institute AMOLF, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands, and Russian Research Center Kurchatov Institute, Kurchatov Square, 123182 Moscow, Russia We create weakly-bound Liz molecules from a degenerate two-component Fermi gas by sweeping a magnetic field across a Feshbach resonance. The atom-molecule transfer efficiency can reach 85 % and is studied as a function of magnetic field and initial temperature. The bosonic molecules remain trapped for 0.5s and their temperature is within a factor of two of the Bose-Einstein condensation temperature. A thermodynamical model reproduces qualitatively the experimental findings.

Feshbach resonances constitute a unique tool to tune the microscopic interactions in ultracold bosonic and fermionic gases 1 ,2 ,3 . By varying an external magnetic field, one can adjust the magnitude and sign of the scattering length, leading, for bosons, to collapse of Bose-Einstein condensates for negative scattering length 4 , bright soliton formation and coherent oscillations between an atomic condensate and molecules 7, For fermions with attractive interaction, it is expected that the superfluid transition temperature is enhanced near a Feshbach resonance We have used the Feshbach resonance between the two Zeeman sublevels of the hyperfine ground state of 6Li, 11/2, -1/2) and 11/2,1/2) at a magnetic field of 810G (Fig. 1) to produce in a reversible manner ultracold and trapped molecules from a quantum degenerate 6Li gas 13. The production efficiency exceeds 80% and the observed molecule lifetime reaches half a second. The phase-space density of these bosonic molecules is on the order 516

819110,11,12.

205

206 of one, the highest value reported so far. Our work paves the way to Bose-Einstein condensation of molecules and to the study of the crossover between the regime of molecular BEC and the regime of superfluid BCS pairing in Fermionic systems Our method is illustrated in Fig. 1 and was suggested for Bose gases in Refs. 14-16. It consists in scanning over a Feshbach resonance from the region of attractive interaction (scattering length a < 0) to region 2 in Fig. 1, where a is large and positive and where a weakly bound molecular state exists with energy Eb = -h2/ma2 (where m is the atomic mass). Using a similar method with fermionic 40K atoms, the JILA group recently reported molecule production with a lifetime of 1ms, atom to molecule conversion efficiency of 50% and direct measurement of the molecular binding energy 17. Molecules have also recently been produced from 87Rb and 133Cs condensates and also from a cold 133Cscloud ". 819910111912.

18119

-E S

u

a

200 100

O

-100 -200 0

0.5

1

1.5

Magnetic Field [kG]

2

) Figure 1. Calculated scattering length a versus magnetic field for the 6Li [ F , r n ~= 11/2,1/2), 11/2, -1/2) mixture near the 810Gauss Feshbach resonance.

Experimentally, using the setup described in Ref. 21, we prepare N1 = Nt + N J = 1.5 lo5 atoms in a 50(10)% mixture of 11/2, -1/2) and 11/2,1/2) in a crossed Nd:YAG dipole trap at B = 1060 G where a = -150 nm is large and negative (position 1 in Fig. 1). By evaporation in the optical trap from the initial conditions wx/27r = 2.9(3) kHz, w,/27r = 5.9(5) kHz and w,/2n = 6.5(6) kHz, the temperature T of the gas mixture can be tuned between 0.2 and 0 . 5 T ~ where , TF is the Fermi temperature defined by TF = E (3Nl)'/'/k~ and Z = ( W ~ W ~ W , ) ~ / ~ . The Fermi gas quantum degeneracy is measured through analysis of absorption images after abrupt (20 ps) switch-off of the magnetic field and

207

time of flight expansion of the cloud. Free 6Li atoms are detected at zero magnetic field, for which a z 0, using laser light tuned to the D2 line, position 4 in Fig. 1. From position 1, we sweep the magnetic field in 50 ms to position 2 where a is large and positive, the region where a weakly-bound molecular state exists. Absorption images indeed reveal that the number of atoms Nz as counted after B switch-off to position 4, as explained above, has dramatically decreased to =20% of the initial number N1. After leaving the system at position 2 for a variable wait time t , between 0 and 5s, we sweep the magnetic field back to position 1 in 50ms and count the number of atoms, N3, after this round-trip through resonance. Surprisingly, when position 2 is at 689 Gauss and t , = 0, we find N3 N NI, indicating that no loss has occurred in 100 ms. We are thus led to conclude that: (i) at position 2,80% of the atoms were in a molecular bound state which is not detected using light resonant with the atoms a t position 4, (ii) the atom-to-molecule formation process is reversible. (N3 - N2)/2 represents thus the number of molecules at position 2.

versus magFigure 2. a: Fraction of atoms in molecular bound state, ( N 3 - NZ)/N3, netic field a t position 2 in Fig. 1. b: Corresponding atom temperature at position 2 (squares) and 3 (circles). Solid (dotted) line corresponds to the non-interacting (interacting) thermodynamical model described in the text.

The relative fraction of atoms bound in a molecule, (N3 - Nz)/N3, and the temperature of the atoms at position 2 and 3 are plotted in Fig.2 as a function of the magnetic field at position 2. Initially, we have Nl = 8. lo4 atoms at 1060 G at a temperature of T = 4.7pK with TF = 11pK. One observes that very few molecules are detected above 0.77 kG.When increasing the binding energy by decreasing the magnetic field, the fraction of molecules increases up to SO%, because the molecular state becomes energetically favorable. Below 650 G, losses become important and the

208

molecular fraction decreases. A key parameter in our detection scheme is the switch off time ( 2 0 ~ s ) of the Feshbach magnetic field from position 2 because molecules will dissociate in this process if the relative rate of change of the binding energy dEb/Ebdt is greater than the oscillation pulsation &/A. a = hdEb/Eb2dt then governs the dissociation of molecules. If a >> 1,they dissociate and appear as free atoms in the absorption image. Our detection scheme is unable to prove molecule formation. On the contrary, for a Nz is the signature of molecule formation. Since for B = 700 G one has cx N 1,we attribute the decrease of the molecular fraction above 720 G in Fig. 2a to the crossover between the adiabatic and non adiabatic regimes. The temperature of the atoms at position 2 and 3 is plotted in Fig. 2b. The increase with decreasing B we observe is expected since, during the molecule formation process, the released binding energy must be converted into kinetic energy. Morever, sweeping back the magnetic field to the initial position, the temperature returns close to its initial value, demonstrating the reversibility of the molecular formation process.

20.6 0.4

50.2

0

20

40

60

80

Trap Depth [ pK]

100

Figure 3. F’raction of atoms bound in a molecule as a function of the depth of the dipole trap for B = 689 G , a = 78 nm, Eb = 12 pK.

Interestingly, the efficiency of molecule formation strongly depends on the trap depth and reaches values up to 80 % in Fig. 3, the highest efficiency reported so far. The trap depth is slowly (800 ms) reduced at B = 1060 G, before the magnetic field sweep through resonance, decreasing the temperature and Fermi degeneracy by evaporation and adiabatic cooling. At 689 G, a = 78nm, E b / k B = 12pK. Reducing the trap depth, T and TF become small compared to E b and the molecular fraction increases towards unity. Indeed increases with decreasing T , resulting in a higher occupancy

209

of the bound state. Our results can be understood within a thermodynamical model 13~22 assuming equilibrium between the atoms and molecules during the magnetic field sweep since the 3-body recombination rate to the molecular state and all collision rates are large compared to the resonance’s scanning rate. 23324

Figure 4. Decay of Liz molecules in the optical trap for two values of the magnetic field: ( a ) B = 689 G, a = 78 nm, (b) B = 636 G, a = 35 nm. Note the difference in time scales. Solid lines are fits with 2-body and l-body processes giving initial time constants of 500ms in (a) and 20ms in (b).

What is the lifetime and degeneracy parameter of the produced molecular cloud, two crucial quantities for the formation of a molecular BEC? The decay of the molecules in the optical trap is found by measuring the quantity (N3 - N2)/2 as a function of the wait time t , at position 2 before returning to 3 for the detection process. Fig.4 presents the number of remaining molecules as a function oft, in a trap with mean frequency 5/27r = 1.6(2) kHz. In Fig. 4 a and b, the initial temperatures at position 1 are the same: 1.1pK = 0 . 3 T ~ .Fig 4a corresponds to B = 689G, a = 78nm, E b / k ~= 12pK, whereas Fig.4b corresponds to B = 636G, a = 35nm, and E b l k ~= 60pK. The decays are well fitted by 2-body plus l-body loss processes with strikingly different initial lifetimes of N 500ms in (a) (689 G) and 20ms in (b), whereas the l-body decay exceeds 4 seconds. A strong decrease in the lifetime of molecules with decreasing a is expected assuming that they undergo collisional relaxation to deep bound states 24. However, the observed factor of 25 in lifetimes is surprisingly large. Let us now evaluate the phase space density of the trapped molecules. In Fig. 2, at 689 G, 1.8 lo4 molecules are confined with 3.3 lo4 atoms at Tmol = Tat = 6.7pK. The critical temperature for molecule condensation is then 3.5 pK giving T / T c N 2 . In fact, for all data between 675 G and 750G, the phase space density of the molecules is not far from the

210

condensation point. Since, the lifetime of the molecules, 500ms, is long in this region, it should be possible to evaporate the molecules further to reach the Bose-Einstein condensation threshold. In summary, we have produced long-lived and trapped Liz dimers. The atom-molecule conversion efficiency can approach unity when the Fermi quantum degeneracy is strong. The lifetime of the trapped molecules strongly depends on the scattering length. Current research concentrates on evaporation towards a molecular BEC. Prospects for producing superfluid Fermi mixtures and for investigating the transition between molecular condensates and superfluid Fermi gases are promising We are grateful to D. S. Petrov, S.Stringari, Y .Castin, R. Combescot and J. Dalibard for useful discussions. 11112,24925.

Acknowledgments This work was supported by CNRS, College de France and R6gion Ile de France. Laboratoire Kastler Brossel is Unit6 de Recherche de 1’Ecole Normale SupCrieure et de l’Universit6 Pierre et Marie Curie, associ6e au CNRS. S.J.J.M.F.K acknowledges a Marie Curie grant from the E.U. Contract number MCIF-2002-00968

References 1. H.Feshbach, Ann. Phys. (N.Y.) 5,357 (1958); 19,287 (1962). See for instance, “Ultracold Matter” in Nature Insight 416 (2002).

2. 3. 4. 5. 6. 7. 8.

S.Inouye, et al., Nature 392,151 (1998). E. A. Donley, et al., Nature 412,295 (2001). L. Khaykovich, et al., Science 296,1290 (2002). K.E. Strecker, et al., Nature 417,150 (2002). E. A. Donley, et al., Nature 417,529 (2002). P. Nozihres and S. Schmitt-Rink, J . Low temp. Phys. 59,195 (1985). 9. C. A. R. S B de Melo, M.Randeria, and J. R.Engelbrecht, Phys. Rev. Lett. 71, 3202 (1993). 10. M.Holland, et al., Phys. Rev. Lett. 87,120406 (2001). 11. Y. Ohashi, and A. Griffin, Phys. Rev. Lett. 89,130402 (2002). 12. J. N. Milstein, S. J. J. M. F. Kokkelmans, and M. J. Holland, .Phys. Rev. A 66, 043604 (2002). 13. J. Cubizolles et al.,cond-mat/0308018. 14. E. Timmermans, et al., cond-mat/9805323. 15. F. A. van Abeelen and B. J. Verhaar, Phys. Rev. Lett. 83, 1550 (1999). 16. F. H. Mies, E. Tiesinga, and P. S. Julienne, Phys. Rev. A 61,022721 (2000). 17. C. A. Regal, Nature 424,47 (2003). 18. J. Herbig, et al., 10.1126/science.1088876 (2003).

21 1

S. Durr, et al., cond-mat/0307440. C. Chin, et al., Phys. Rev. Lett. 90, 033201 (2003). T. Bourdel, et al., Phys. Rev. Lett. 91, 020402 (2003). S. J. J. M. F. Kokkelmans, G. V. Shlyapnikov and C. Salomon, cond-mat/0308384. 23. D. S. Petrov, Phys. Rev. A 67,010703(R) (2003). 24. D. S. Petrov et al., cond-mat/0309010. 25. L. D. Carr, Y .Castin, and G. V. Shlyapnikov, to be published. 19. 20. 21. 22.

FESHBACH RESONANCES IN DILUTE QUANTUM GASES

M. J. HOLLAND JILA 440 UCB, University of Colorado Boulder, CO 80309-0&0 USA Feshbach resonances allow the interactions in a quantum degenerate gas to be controlled at a microscopic level. In general terms this manifests as an ability to tune the scattering length from large to small values and from positive (repulsive interactions) to negative (attractive interactions). However, near the resonance, the magnitude of the scattering length becomes large resulting in a break-down of the simple mean-field theories used to describe dilute gases. This requires a fully quantum field theory of the resonance superfluid to be developed. This theoretical formulation is the subject of this article.

1. Introduction

There has been an explosive growth of interest in the physics of BoseEinstein condensation in dilute atomic gases since the first demonstrations were made seven years ago l. One of the major current theoretical challenges in this field is the development of a many-body theory to describe the situation in which resonances dominate the individual binary interactions. This topic connects with many of the future research goals of the field; e.g., the production of molecular condensates, the demonstration of high temperature superfluidity in fermion gases, and the investigation of a broad range of strongly-correlated phenomena with analogy to strongly-correlated electron systems in condensed matter physics. At first glance, it is remarkable that the collective phenomena of atomic gases could make any contact with strongly-correlated physics or hightemperature superfluidity. The gases are extremely dilute - the range of the interatomic potential is typically two to three orders of magnitude smaller than the spacing between particles. This means that the interactions are generally considered to be weak and that perturbation theory at low order can provide an accurate description of the quantum kinetic evolution 2 , 3 . Such considerations, however, are incomplete since they neglect the complex internal structure of the atoms. This means that collisions are intrinsi212

213 cally multichannel scattering events. Such a system can support scattering resonances (Feshbach resonances) which result in the atoms spending an extraordinary amount of time in close proximity 4 . This implies the possibility of generating a wide variety of collective phenomena in dilute gases which require the interactions to be strong. The presence of strong interactions, however, poses a complex theoretical question; how does one systematically develop a many-body theory of atomic gases in which resonant scattering events are correctly accounted for? 2. What is Meant by Resonant Interactions

w

Internuclear Separation

-

Figure 1. Characteristic interatomic potential. The dashed line shows the position of the threshold at zero scattering energy.

When two ground state alkali atoms collide, the scattering process is described by an interatomic potential of the characteristic form illustrated in Fig. 1. At short range the overlap of the electronic orbitals leads to a strong repulsion, and at long range the mutual polarization leads to a characteristic van der Waals attraction. What is important in considering the implications of this characteristic form on the properties of a quantum degenerate gas is the separation of scales. The scattering energy in the millikelvin to nanokelvin region is much less than the potential depth which may be hundreds of Kelvin. The spatial range of the potential, on the order of a few nanometers, is minuscule in comparison to the characteristic interparticle spacing which may be several micrometers. In general, these considerations allow a contact description of the two-body scattering to be an accurate formulation. The enormous height of the centrifugal barrier for channels with nonzero angular momentum, in comparison to relevant scattering energies, imply that for the most part only s-waves need to be considered. The low energy collisions are then usually encapsulated by the

214

s-wave scattering length, a. This is evident in the Gross-Pitaevskii equation for example;

where the only scattering parameter which appears is a constant T-matrix (transition matrix). This parameter is completely determined by a and the mass m according to T = 4nh2a/m. This nonlinear Schrodinger equation has been remarkably successful in the quantitative description of BoseEinstein condensates in atomic gases. Note that at the Gross-Pitaevskii level of the field theory, the many-body state is assumed to be completely factorisable into self-consistent single-particle orbitals 4(z). The nonresonant picture is obviously incomplete since it is possible to tune the scattering length through infinity by application of a Feshbach resonance, as illustrated in Fig. 2. A closed channel potential, typically corresponding to a distinct spin configuration, can support bound states with energies in close proximity to the scattering threshold. The difference in the magnetic moments of the open and closed channels allow the detuning vo to be varied by application of an external magnetic field 5 . When the bound state crosses threshold, the scattering length passes from positive infinity to negative infinity.

Internuclear Separation

-

Figure 2. Illustration of a Feshbach resonance. A bound state of a closed potential is in close proximity (with detuning YO) to the scattering threshold (dashed line).

In this situation intrinsic assumptions made in deriving the GrossPitaevskii equation fail and the formulation of the field theory must be reconsidered. In close proximity to a resonance, the T-matrix is not constant and becomes strongly dependent on the scattering energy. The Tmatrix may even acquire a substantial imaginary component as shown in Fig. 3. The assumption that the fields factorize into single-particle orbitals

215

Scattering Energy (W)

Figure 3. Real (solid) and imaginary (dashed) parts of the 2'-matrix in length units (a0 is the Bohr radius) for scattering of two 40Katoms. The detuning value is shown by the dash-dot line in the inset. The inset illustrates the behavior of the scattering length as the magnetic field is varied. The dotted (vertical) line shows a typical maximum scattering energy.

is no longer valid and quantum correlations must be included. The behavior of the scattering length is shown in the inset of Fig. 3. There are three parameters required to encapsulate all the relevant two-body physics associated with a Feshbach resonance, including the bound state physics of the coupled channels problem and the scattering phase shifts at finite energy. These three parameters are: (1) the background scattering length Ubg, given by the asymptotic value of the scattering length at large detuning from the resonance, ( 2 ) 6 ,characterizing the coupling matrix element between the open and closed channel potentials (i.e. the width of the resonance), and (3) the detuning, VO.

The behavior of the scattering length, for example, is given by

Based on these ideas we have recently developed a field theory formulated on the three microscopic parameters. The following sections summarize some of the results of the studies based on this approach.

216

3. Time-Dependent Fields; Dynamic Cooper Pairing

The resonance field theory has been quantitatively compared with a recent experiment involving a Feshbach resonance in bosonic 85Rb. In direct analogy with a Ramsey fringe experiment in atomic physics, a timedependent magnetic field was used to create two intervals of strong coupling (i.e., near resonance) separated by a waiting interval tevolvewhich could be varied in duration. In the strong coupling regions the scattering length was positive and many thousands of Bohr radii in magnitude (the highest values of the dimensionless parameter nu3, where n is the density and a is the scattering length, approach unity). This is typically considered to be a very challenging regime to explore theoretically. After applying the pulse sequence the experimenters observed three components; a residual condensate, a burst of atoms at higher energy with distinct spatial profile, and a missing fraction. As the time interval tevolve was varied they observed oscillations in the proportion of atoms in each component. Since the microscopic parameters are known to high precision for 85Rb, we were able to model the dynamic experiment completely with no adjustable parameters. We calculated evolution trajectories for each value of tevolve by solving the time-dependent Hartree-Fock-Bogoliubov equations appropriately renormalized to remove the ultraviolet divergences which would otherwise arise. We thereby predicted occupations in the atomic condensate, molecular condensate, and noncondensate components (Fig. 4). The agreement with the experimental observation is striking allowing the observed components to be unambiguously identified. Some of the properties which are correctly accounted for include the fringe frequency at the one percent level (which is a direct measure of the binding energy of the molecule in the 100 kHz scale), and the visibility and position of the fringes for each component. A phase offset in the oscillations is noted in the experimental data and reproduced in the theory. An important aspect of the theory is the anomalous density which plays a major role in the calculated trajectories. In the case of f e m i o n s , the Hartree-Fock-Bogoliubov dynamics are straightforward to calculate in an analogous way. However, in that case there is no atomic condensate, and the generation of an anomalous pairing field is usually referred to as the formation of Cooper pairs. 798

217 Theory

9'0

r;

;o

i

t MI

Experiment

3.0

3.5

10

15

20

25

30

35

40

Lvfw

Figure 4. Ramsey fringe oscillations between the atomic condensate (theory-solid line) and the atomic noncondensate (theory-dashed line). These two components sum to the total number of recovered atoms (theory-squares), which excludes the molecular component. The experimental plot illustrates the directly analogous quantities for comparison.

4. Resonance Superfluidity in Fermion Gases

There is currently a significant experimental effort to achieve superfluidity in a dilute Fermi gas by utilizing Feshbach resonance interactions. The direct application of the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity to a dilute Fermi alkali gas lo in this situation is incomplete because it involves the treatment of the interatomic interactions by the scattering length a. The requirement to have a high critical temperature in close proximity to the Fermi temperature in order for the superfluid transition to be achievable experimentally means that the theory cannot involve just the scattering length and must account for the resonance state explicitly. We previously developed a theory of the critical temperature for the formation of the fermion superfluid state which takes this into account 11, We studied the behavior of the critical temperature in the crossover region in a path integral formulation We were able to consider the important role of fluctuations in the resonance regime (by performing the path integral calculation in the beyond saddle-point approximation) and utilized the same renormalization procedure described above for the Ramsey fringe studies. The characteristic dependence of the critical temperature on detuning is shown in Fig. 5. Note that these calculations interpolate between 12313914315.

218

the BEC limit at large negative detuning, and the BCS superfluidity limit at large positive detuning.

Figure 5 . Ratio of the critical temperature Tcto the Fermi temperature TF as a function of the magnetic field detuning v across a Feshbach resonance for 40K (solid line). At large negative detuning the transition is characterized by strongly bound composite pairs which Bose condense. At large positive detuning the prediction of the Bardeen-CooperSchrieffer theory emerges (dashed line).

One of the reasons the superfluid fermion gas, if realized, would be interesting is the connection with preformed Cooper pairing and the formation of a pseudogap. A major effort has been put into understanding the normal phase of high-Tc cuprate superconductors due to a growing belief that it is this phase that holds the key to understanding the exotic behavior of these materials 16717. One reason for this belief is the formation of a ‘pseudogap’, which is a strong suppression of weak excitations (as opposed to a ‘gap’ which causes a complete suppression), at temperatures far above T,. One possible explanation for this pseudogap phase requires the onset of local superconducting correlations l8 at a temperature T’, which only become coherent when the temperature drops below T,. Our Feshbach resonance model is an ideal system in which to study the pseudogap phase since it allows us to move well within this regime by varying the strength of the resonant interactions, an easily controlled experimental parameter. For many reasons atomic gases appear to offer an ideal system in which to study the pseudogap phase since they potentially may allow a much simpler description to be developed than their condensed matter analogs. They are not encumbered by a number of complicating mechanisms, such

219

as strong Coulomb interactions, d-wave pairing, interlayer tunneling, a n d so on, though these are interesting in their own right. I n addition the microscopic physics behind the interparticle interactions a r e well understood and parametrized at an extremely precise level.

Acknowledgments Supported by t h e National Science Foundation and by the Department of Energy

References 1. M. H. Anderson, et al., Science 269,198 (1995); K. B. Davis, et al., Phys Rev. Lett. 75,3969 (1995); C. C. Bradley, et al., Phys. Rev. Lett. 75,1687 (1995); Phys. Rev. Lett. 79,1170(E)(1997). 2. R. Walser, J. Williams, J. Cooper, and M. Holland, Phys. Rev. A 59,3878 (1999). 3. R. Walser, J. Cooper, and M. Holland, Phys. Rev. A 63,013607 (2000). 4. H. Feshbach, Ann. Phys. 5, 357 (1958); ibid., Ann. Phys. 19,287 (1962); H. Feshbach, Theoretical Nuclear Physics, (Wiley, New York, 1992). 5. E. Tiesinga, B. J. Verhaar, and H. T. C. Stoof, Phys. Rev. A 47,4114 (1993). 6. S.J.J.M.F. Kokkelmans, et al., Phys. Rev. A 65,053617 (2002). 7. S.J.J.M.F. Kokkelmans and M. J. Holland, Phys. Rev. Lett. 89,180401 (2002). 8. E.A. Donley, N.R. Claussen, S.T. Thompson, and C.E. Wieman, Nature 417, 529 (2002). 9. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108,1175 (1957); J. R. Schrieffer, Theory of Superconductivity, Perseus Books, Reading, MA, (1999). 10. A. G. Leggett, J. Phys. (Paris) C7, 19 (1980); M. Houbiers and H. T. C. Stoof, Phys. Rev. A 59, 1556 (1999); G. Bruun, Y . Castin, R. Dum, Eur. Phys. J. D 7, 433 (1999); H.Heiselberg, C. J. Pethick, H. Smith, and L. Viverit, Phys. Rev. Lett. 85,2418 (2000). 11. M. Holland, S.J.J.M.F. Kokkelmans, M.L. Chiofalo, and R. Walser, Phys. Rev. Lett. 87,120406 (2001). 12. P. Nozibres and S. Schmitt-Rink, J. Low Temp. Phys. 59,195 (1982). 13. See M. Randeria in Bose-Einstein condensation, ed. by A. Griffin, D.W. Snoke and S. Stringari, Cambridge Uni. Press, Cambridge (1995). 14. Y . Ohashi and A. Griffin, Phys. Rev. Lett. 89,130402 (2002). 15. J.N. Milstein, S.J.J.M.F. Kokkelmans, and M.J. Holland, Phys. Rev. A 66, 043604 (2002). 16. M. Randeria, Varenna Lectures, (1997), available on xxx.lanl.gov as condmat/9710223. 17. T. Timusk and B. Statt, Reports Prog. Phys. 62,61 (1999). 18. V. Emery and S.A. Kivelson, Nature (London) 374,434 (1995); V. Emery and S.A. Kivelson, Phys. Rev. Lett. 74,3253 (1995).

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Atom Optics and Interferometry

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COLD ATOMS NEAR METALLIC AND DIELECTRIC SURFACES

M. P. A. JONES, C. J. VALE: D. SAHAGUN, B. V. HALL, B. E. SAUER, AND E. A. HINDS Blackett Laboratory, Imperial College, London SW7 2B W, UK E-mail: ed.hindsimperial.ac. uk C. C. EBERLEIN Department of Physics and Astronomy, Sussex University Falmer, BN1 9QH, UK

K. FURUSAWA AND D. RICHARDSON Optoelectronics Research Centre, Southampton University Southampton, SO17 lBJ, UK

We describe experiments in which ultracold rubidium atoms held by a microscopic magnetic trap near a room temperature metal wire are used to study the magnetic fields above the wire. By studying the density distribution of the atoms we detect a weak static magnetic field component AB, parallel to the wire. This field is proportional t o the current in the wire and is approximately periodic along the wire with period X = 230pm. We find that the decrease of this field with distance from the centre of the wire is well described by the Bessel function Kl(27ry/X), as one would expect for the far field of a transversely oscillating current within the wire. We have also studied the lifetime for the atoms t o remain in the microtrap over distances down to 27pm from the surface of the metal. We observe the loss of atoms from the microtrap due to spin flips. These are induced by radio-frequency thermal fluctuations of the magnetic field near the surface, as predicted but not previously observed.

1. Introduction

The ability to control cold atom clouds in microscopic magnetic traps1y2i3 and w a v e g u i d e ~ has ~ ? ~created >~ the new field of miniaturized atom optics7l8. With the use of atom chips it becomes possible to control cold atoms on *Current address: Dept. of Physics, University of Queensland, Brisbane 4072, Australia.

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the pm length scale and to anticipate the construction of integrated atom interferometersg*lOJ1.Ultimately there is the possibility of controlling arrays of individual atoms for use in quantum information p r o ~ e s s i n g l ~ > ' ~ . For these applications it is important to avoid fluctuating or inhomogeneous perturbations, which can destroy the quantum coherences. Clouds within 100 pm of a current-carrying wire and cooled below a few pK have recently revealed three surface-related decoherence effects. First, the clouds break into fragments along the length of the wire as a result of a corrugated trapping p ~ t e n t i a l ' ~ ? 'The ~ . corrugations are caused by a small spatially alternating magnetic field AB, parallel to the wire16, which is due to a small transverse component of the current. The second effect is heating of the c l o ~ d due ~ ~to~ audio ' ~ frequency technical noise. Finally, trapped atoms are l o ~ t ' ~ 9through '~ spin flips induced by radio frequency noise. This has two distinct origins. In addition to technical noise on the wire currents, there is a more fundamental source associated with the thermal fluctuations of the magnetic field. Atoms trapped some tens of pm above a metal interact with the thermally fluctuating near field of the surface, whose spectrum is very different from the blackbody spectrum. Recent c a l c ~ l a t i o n shave ~ ~ shown that spin flips induced by this near field can cause atoms to be ejected from a magnetic trap in less than a second. In this paper we investigate two of these surface effects. Firstly we present our study of the anomalous magnetic field component that leads to fragmentationlsil9. Secondly we describe the first experimental measurements18 of the coupling between trapped atoms and the thermal fluctuations of the surface, which in the presence of technical effects has not previously been a c c e ~ s i b l e ' ~ ~ ~ ~ .

2. Preparation of Cold Atoms

The arrangement of wires used to form our microtrap is shown in Fig. 1. The main wire is a 500pm diameter guide wire. In cross section it has a 370 pm diameter copper core, surrounded by an aluminium layer 55 pm thick with a 10pm thick ceramic outer coating. This wire is glued into the 200 pm deep channel formed by a glass substrate and two glass cover slips. The cover slips are gold coated to form a mirror. The loading of the microtrap is described e l s e ~ h e r e The ~ ~ end ~ ~ result ~. is a microtrap 225 pm above the top of the wire that contains 2 x lo7 87Rb atoms in the IF, m) = 12,2) state. The atoms are then evaporatively cooled over 6 s to approximately 6 p K . They are brought closer to the surface by

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Figure 1. Construction of the magnetic microtrap (not to scale). A current along z and a bias field along z together form a long thin trap parallel to the guide wire. The ends of the trap are closed by currents in the transverse wires.

a smooth reduction of the current in the guide wire during the last 1 s of evaporation. We probe the cloud using resonant absorption imaging. The absorption yields an image of the column density of the atoms, viewed in the y - z plane. Integrating this over y gives the probability distribution of atoms along the *direction, p ( z ) . Assuming that the gas is in thermal equilibrium at temperature T , p ( z ) is related to the confining potential U ( z ) by the Boltzmann factor exp(-U(z)/lcT). In the absence of AB,, the potential V ( z ) is harmonic with an axial frequency of 2627Hz over the range of heights studied. Fitting a parabola to - lnp(z) then determines the temperature T . For example, the top curve in figure 2 taken 97 pm above the surface of the wire gives T = 5.8 pK.

3. Fragmentation When the cloud is brought closer to the surface, p ( z ) develops additional structure, as can be seen in Fig. 2. The difference between -1np(z) and the fitted harmonic potential is the potential of interest AB,/kT, and since we know T from the harmonic fit, we obtain AB,. In this way we have measured how AB, varies along z at the 8 different heights h of the cloud above the wire. AB, is proportional to the current in the guide wire16, so to compare AB, at different heights we scale the result at each height to the current used. Figure 3 is a map of AB,(y,z) scaled to a fixed 3.7A in the guide wire. We see AB, undergoing 2 full periods of oscillation along the wire. The phase of the oscillation is fixed with respect to the wire and did not change over several months. The field has many more oscillations along the wire, with an average wavelength X of 230 f 10 pm, but in this particular experiment the atoms are confined to these two periods

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Position along wire (pm) Figure 2. Distribution of atoms p ( z ) along the length of the trap for a variety of heights above the surface of the wire. The curves show the probability of finding atoms within a range 6.z = 5 pm and are all normalized to unit total probability. They are successively offset by 0.01 for the sake of clarity.

by the trap potential. It would seem that the current follows an oscillatory or helical path along the wire with wavelength A. In this case, if the distance y from the current to the atoms is much greater than the transverse excursions of the current, the decay of the field component AB, with distance y should be well approximated by the modified Bessel function Kl(Icy) exp(-Icy)/&, where Ic = 27r/X. The amplitude of the AB, oscillations a t each height h above the wire are plotted in Fig. 4. The solid line is a least-squares fit to the function aK1(27r(h 6)/X), in which a, 6, and X are fitting parameters. The best fit has a x2 of 4.8 for 5 degrees of freedom and gives the results X = 217 f 10 pm and 6 = 251 f 12 pm. The close agreement between this value of X and the value of X = 230 f lOpm obtained directly from the images strongly suggests that our model is correct. The value of S is equal to the wire radius indicating that the centre of the oscillating current coincides with the centre of the wire. N

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Figure 3. Anomalous magnetic field A B , as a function of position for a current of 3.7 A in the guide wire.

0

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Height h above wire (pm) Figure 4. Magnetic fields versus height h above a wire carrying 3 . 7 A . Data points: amplitude of the anomalous magnetic field variation A B , . Solid line: best fit of the modified Bessel function aK1(2n(h 6)/X). This has X = 217 f 1 0 p m and 6 = 251 f 12pm. Dashed line: The (usual) azimuthal field referred t o the auxiliary ordinate on the right.

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4. Thermal Fluctuations

We probe the coupling between the atoms and the near field of the surface of the wire by studying how the lifetime of the atoms in the magnetic trap depends on the distance from the surface. Far from the wire, the lifetime 100s. As the atoms are brought close to the surface, the lifetime is decreases dramatically, as shown in Fig. 5. The increased loss rate is due to spin flip transitions 12’2) -+ ( 2 , l )+ (2’0)driven by thermal fluctuations of the magnetic field close to the wire. The resonant frequency fo for this spin flip transition is $ p ~ B o / hwhere , p~ is the Bohr magneton, h is Planck’s constant, and Bo is the magnetic field at the center of the trap. Figure 5 shows two series of measurements using two different values of the spin-flip frequency: fo = 1.8(1)MHz (open circles), and fo = 560(10) kHz (filled squares). In each series, the lifetime exhibits a strong dependence on the distance from the surface, decreasing by an order of magnitude as the distance is reduced from 80pm to 30pm. At a given height, atoms with a lower spin flip frequency have a shorter lifetime. N

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Height (pm) Figure 5. Lifetime of trapped atoms versus distance from A1 surface of the wire. Filled squares (open circles): measurements with spin flip frequency fo = 560kHz (fo = l.8MHz). Solid (dashed) lines: calculated lifetimes above a thick slab of c o g per (aluminum) for these two spin flip frequencies. Dotted line: expected scaling for technical noise.

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The solid (dashed) curves in figure 5 show calculated thermal spin-flip lifetimes T for rubidium atoms in the 12,2) state above a thick plane slab of Cu (Al) with fo = 560 kHz and 1.8 MHz. In calculating these curves, we have found that the analytical expression given in equation 23 of Henkel et al.17 is in error by a factor of 2-3 over this range. We therefore integrated their equation 22 numerically to obtain the results shown21. Figure 5 shows lifetimes limited only at the level of the thermal spin flips: the lifetime has not been unexpectedly cut short by some other mechanism. This is in itself a significant experimental result. Beyond that, our results provide a preliminary test of the thermal spin flip theory. To the best of our knowledge, the theory has only been worked out for the case of a plane s1ab17, and since our experiment involves atoms above a cylindrical wire, we cannot expect the points in Fig. 5 to lie exactly on the lines. However, the atomsurface distance is less than half the radius of the wire, so it is reasonable to expect some correspondence between this theory and our experiment , and indeed that is seen in Fig. 5. At the lower frequency, fo = 560 kHz, the skin depth of the aluminum is llOpml whereas the thickness of the aluminum outer layer is 55pm. We therefore expect that fluctuations in the copper core will contribute to the decay rate and that the lifetime will be somewhere between the two lowest theory curves. At the higher frequency, the skin depth is only 61pm, and the lifetime should be closer to the a h minum value shown in the topmost curve. The agreement between theory and experiment is remarkably good given the differences in geometry of the two. This result has been achieved after taking considerable care to control technical noise. If there were any residual rf current in the guide wire, the resulting field would vary as 1 / ~giving , a lifetime proportional to r 2 , as demonstrated elsewhere2'. This power law is indicated by the dotted line in Fig. 5, arbitrarily placed to pass through one of our data points. Clearly the height dependence we observe is much stronger than that. Spin flips induced by rf pickup cause a loss rate proportional to the rf power at the spin flip frequency fo. This leads to a constant ratio of lifetimes, not the height-dependent ratio that we observe.

5. Conclusions

We have used a cloud of ultracold atoms to study the magnetic fields above a current carrying wire. On approaching the wire, the cloud breaks into fragments due to an anomalous magnetic field parallel to the wire. Along

230 the wire, this field oscillates with a period of 230f10 pm and with increasing radius it decays according to a modified Bessel function K1. This is characteristic of the field produced by a periodic transverse current distribution in the wire. We found a characteristic decay length of 217 f 10 pm, in good agreement with the observed period and we have shown that the anomalous current is centred on the middle of the wire. The lifetime for the atoms to remain in the trap is also strongly reduced as the atoms approach the wire. This is due to spin-flips induced by the fundamental thermal fluctuations of the magnetic field above the wire. Both these effects have important implications for the coherent manipulation of cold atoms above metallic surfaces.

Acknowledgments This work was supported by the UK EPSRC, and by the FASTNET and Cold Quantum Gases networks of the EU.

References 1. J. D. Weinstein and K. G. Libbrecht, Phys. Rev. A 52, 4004 (1995). 2. V. Vuletic et al., Phys. Rev. Lett. 80,1634 (1998). 3. J. Fortdgh et al., Phys. Rev. Lett. 81,5310 (1998). 4. D. Miiller et al., Phys. Rev. Lett. 83,5194 (1999). 5. N. H. Dekker et al., Phys. Rev. Lett. 84,1124 (2000). 6. M. Key et al., Phys. Rev. Lett. 84,1371 (2000). 7. E. A. Hinds and I. G. Hughes, J. Phys. D 32, R119 (1999). 8. R. Folman et al., Adv. Atom. Mol. Opt. Phys. 48,263 (2002). 9. E. A. Hinds, C. J. Vale and M. G. Boshier, Phys. Rev. Lett. 86,1462 (2001). 10. W. Hansel et al., Phys. Rev. A 64,063607 (2001). 11. E. Anderson et al., Phys. Rev. Lett. 88,100401 (2002). 12. T. Calarco et al., Phys. Rev. A . 61,022304 (2000). 13. P. Horak et al., Phys. Rev. A 67,043806 (2003) 14. J. Fortdgh et al., Phys. Rev. A 66,041604(R) (2002). 15. A. E. Leanhardt et al., Phys. Rev. Lett. 89,040401 (2002). 16. S. Kraft et al., J . Phys. B 35, 469 (2002). 17. C. Henkel, S. Potting and M. Wilkens, Appl. Phys. B 69,379 (1999). 18. M. P. A. Jones et al., Phys. Rev. Lett. 91,080401 (2003). 19. M. P. A. Jones et al., cond-mat/0308434 20. A. E. Leanhardt et al., Phys. Rev. Lett. 90,100404 (2003). 21. We are indebted to P. K. Rekdal for calculating the integrals.

COHERENT ATOMIC STATES IN MICROTRAPS

PH. TREUTLEIN, P. HOMMELHOFF, T. w. HANSCH AND J. REICHEL Max-Planck-Institut f u r Quantenoptik and Sektion Physik der Ludwig-Maximilians- Universitat Schellingstr. 4, 0-80799 Munchen, Germany jakob.reiche1Ophysik. uni-muenchen.de http://www. mpq. mpg. de/ -jar We have created coherent superpositions of two 87Rb ground state sublevels in a magnetic microchip trap. Coherence times between the levels, measured by Ramsey spectroscopy, exceeded 2 s and showed no adverse influence of the roomtemperature chip surface, even for atom-surface distances in the 10pm range. These are encouraging results in view of quantum information processing on the chip. We also discuss prospects for a chip-based trapped atom clock.

1. Introduction Chip-based microtraps are an exciting tool for employing atomic coherence in novel ways. They have already proven their usefulness as compact, rugged and cheap BEC sources1i2. Experiments under way in a growing number of laboratories are aimed towards practical applications of coherent atomic ensembles, such as interferometric sensors3, and towards quantum information processing (QIP)475. As a first step towards these ambitious goals, recent investigations have led to a better understanding of atom-surface interactions at micrometric distance^^^^^^; up-to-date results are documented in other contributions in this volume. Like atom clocks, many of the cited applications require long-lived atomic coherences. In quantum computing based on controlled collision^,^^^ for example, the qubit information is encoded in two ground-state sublevels of a single neutral atom or a small ensemble. Coherence between these states must be preserved over many gate times, where the gate time is limited by the trap oscillation period. Thus, depending on the strength of the trap confinement, coherence times from a few milliseconds to many hundreds of milliseconds are required. Whereas coherence lifetimes of several seconds have been demonstrated with 87Rb atoms in a conventional magnetic traplo, it is not clear a priori whether comparable coherence times can 231

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be achieved in a chip trap, where the proximity of the room-temperature chip surface can induce strong dissipative interactions’’. We have used Ramsey spectroscopy to measure the lifetime of coherence between the IF = 1,m = -1) and the IF = 2, m = 1)state of s7Rb in a chip trap for atom-surface distances down to 4pm. In this first experiment employing internal-state superpositions in a chip trap, we found coherence lifetimes of more than 2s - i.e., similar to those obtained in conventional trapslo - down to the smallest atom-surface distances. This means that the favourable coherence properties of this state pair are fully preserved in the presence of the chip surface, which is an extremely encouraging result in view of integrated QIP and interferometry applications. In the following, we briefly describe the experimental setup and procedure, present the coherence measurements, and conclude with a short discussion of the prospects for a chip-based atomic clock. A more comprehensive study, including systematic variation of the trap-surface distance and a measurement of the Allan variance, is currently under way. N

2. Experiment

The experimental setup and procedure is similar to our earlier experiments’, except that the atoms are optically pumped to the IF = 1 , m = -1) state immediately before the magnetic trap is switched on. Thermal atoms are used in all measurements. After an evaporative cooling phase in a strongly compressed trap, the trapping potential is adiabatically transformed into a more relaxed “measurement trap”, with trap frequencies 370 Hz transversely and 60 Hz longitudinally. Evaporative cooling was adjusted so that the atom cloud in this measurement trap had a temperature of N 500nK and a density of N 4 x 10l2cmP3. Because the coherence time is strongly dependent on density and on the bias field Bo in the center of the trap (see below), great care is taken to achieve the same density and the same Bo = 3.23 G regardless of trap-surface distance. For every coherence measurement, we have checked the density and the longitudinal and transverse temperature of the cloud, verified the calculated trap frequencies by parametric excitation, and calibrated Bo spectroscopically. A two-photon, microwave and radiofrequency transition is used to coherently transfer atoms to the IF = 2,m = 1) state. The microwave is set to a fixed frequency of 6.835GHz, so that it is detuned by 1.2MHz below the target state. This frequency is produced by a DRO (dielectric resonator oscillator), which is phase-locked to a synthesizer at 1/100 of its frequency. The DRO signal is amplified to 15W and directed into the glass N

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cell containing the chip trap using a simple WR137 coaxial-to-waveguide transition as an antenna. The adjustable radiofrequency is the same that is used for R F cooling and has a power of 1W. Both sources are referenced in to an ultra-stable, oven-controlled quartz oscillator (stability 1 x 1s, and better than 2 x 10-l' per day). The effective two-photon Rabi frequency is between 200Hz and 1kHz, depending on the position of the microwave antenna. Generally speaking, all effects that produce a differential energy shift of the two states can limit their coherence time if they vary temporally, or spatially across the atom cloud. The two states used here experience the same first-order Zeeman shift at a magnetic field of Bo = 3.23 G, and only a quadratic shift of 6v = 431 Hz/G2 x ( B - B o )remains. ~ Thus, although each level is Zeeman shifted by 10 kHz over the extent of a 500 nK cloud, the differential shift of the two levels across the cloud is less than 1Hz". Collisional interactions within the cloud are another important source of differential shift; however, this effect is mitigated in 87Rb by the fact that the relevant scattering lengths all have similar values. At our densities in the middle 10" cm-3 range, collisions contribute on the order of 1...2 Hz to the differential shiftlo. By the same token, any atom-surface interaction that shifts the energy levels will only limit the coherence time through the dzfferential shift that it creates. This makes this state pair particularly suitable for experiments that require long coherence times. N

'(s atoms Figure 1. Layer structure of the trapping chip (not to scale).

We have performed Ramsey spectroscopy both in the time and frequency domain, and for various trap-surface distances. The surface of our chip is a 200nm silver layer, which acts as a mirror for the MOT beams" and is separated from the conductors by a 30pm layer of vacuum-compatible epoxy glue (Fig. 1). Figure 2 shows a Ramsey oscillation for a trap-surface distance of 10pm. A fit to the data with an exponentially decaying envelope yields a decay time constant of 2.4s. This value agrees with the N

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maximum coherence time that we expect for the density and temperature of the atom cloud in our experiment. It is also very similar to the value measured in Boulder" for the same density. Thus, even at this relatively small distance, no effect associated with the surface is apparent within this coherence time limit. We have also measured at various larger distances, and found very similar values. a

10000 -j

. , . , . , . , . , . ,< 0

100 200 300 400 500 600 700 800 900 10001100120013001400

Time between n/2 pulses [ms] Figure 2. Ramsey oscillation in a magnetically trapped thermal atom cloud close to a surface. Squares indicate the number of atoms detected in the IF = 2) state after the second pulse, as a function of time between pulses. The coherence time, determined from a fit (continuous line), is 2.4s in this example. The atom cloud has a temperature lOpm from the chip surface. Trap frequencies are 370Hz of 650nK and is located transversely and 60 Hz longitudinally. In separate measurements, the trap lifetime was measured t o be 10s with all atoms in the IF = 1 , m = -1) state, 1 . 5 s with all atoms in the IF = 2, m = 1) state, and 2.9s after a single 7r/2 pulse.

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3. Discussion In our experiments, the atoms are exposed to a layered metal and dielectric surface. Interactions may be stronger close to a purely metallic surface". Note, however, that many envisaged chip experiments will work under conditions similar to, or even less critical than those used here. This would be the case, in particular, for an on-chip trapped-atom clock. Indeed, with the level of stability achieved in these measurements, it is natural to consider the usefulness of this scheme for metrology. Let us briefly compare the sources of fluctuations in this experiment to those of an atomic fountain clock. First of all, the differential Zeeman shift here is of the same order as that of the m = 0 -+ m = 0 transition used in atomic clocks. The main difference is that the thermal motion of the atoms broadens the transition, because the atoms move in the inhomogeneous field of the trap. Thus, it

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is advantageous to adiabatically reduce the trapping frequencies as much as possible, in order to reduce the temperature. This will also reduce the density, and it should be possible to attain the same low densities that are used in fountain clocks. Furthermore, chip traps can also be made more box-like (something that is diffult to achieve with conventional coils), leading to a more uniform density distribution, and thus to a smaller frequency spread across the cloud. The special case of a trap with frequencies (230,230,7)Hz has been analyzed quantitively in Ref. 10, and a relative precision of 1 x has been predicted for this case. However, the relatively high transverse trapping frequency leads to a quite restrictive atom number vs. density compromise, and it is clear that shallower t r a p s will allow better precision. Although it is unlikely that the performance of a fountain clock can be approached, even with a shallow trap, a trapped-atom clock can still have interesting application when it is realized on a chip. The short evaporation time’ improves the duty cycle, and most importantly, the “physics package” can be confined to the size of the chip. We will discuss this aspect in more detail in a future publication. In conclusion, these first and very encouraging measurements of atomic internal-state coherences close to a microchip shed new light on the feasibility of collisional phase gates for QIP. They also suggest that a compact, chip-based atomic clock with a fractional stability in the range may be a realistic goal. Acknowledgements We thank Thomas Udem and Marcus Zimmermann for providing us with the quartz reference oscillator. This work was supported in part the by the European Union’s IST programme under contract IST-2001-38863 (ACQP). References 1. W. Hansel, P. Hommelhoff, T. W. Hansch and J. Reichel, Nature 413,498 (2001). 2. H. Ott, J. Fortagh, G. Schlotterbeck, A. Grossmann and C. Zimmermann, Phys. Rev. Lett. 87,230401 (2001). 3. M. A. Kasevich, Science 298, 1363 (2002). 4. T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer, J. I. Cirac and P. Zoller, Phys. Rev. A 61,022304 (2000). 5. H.-J. Briegel, T. Calarco, D. Jaksch, J. I. Cirac and P. Zoller, J . Mod. Opt.

47,415 (2000). 6. J . Fortagh, H. Ott, A. Kraft, A. Giinther and C. Zimmermann, Phys. Rev. A 66,041604 (2002).

236 7. A. E. Leanhardt, Y . Shin, A. P. Chikkatur, D. Kielpinski, W. Ketterle and D. E. Pritchard, Phys. Rev. Lett. 90, 100404 (2003). 8. M. P. A. Jones, C. J. Vale, D. Sahagun, B. V. Hall and E. A. Hinds, Phys. Rev. Lett. 91, 080401 (2003). 9. D. Jaksch, H.-J. Briegel, J. I. Cirac, C. W. Gardiner and P. Zoller, Phys. Rev. Lett. 82, 1975 (1999). 10. D. M. Harber, H. J. Lewandowski, J. M. McGuirk and E. A. Cornell, Phys. Rev. A 66, 053616 (2002). 11. C. Henkel, S. Potting and M. Wilkens, Appl. Phys. B 69, 379 (1999). 12. J. Reichel, W. Hansel and T. W. Hansch, Phys. Rev. Lett. 83, 3398 (1999).

ATOM OPTICS WITH MICROTRAPS AND ATOM CHIPS; ASSEMBLING TOOLS FOR QUANTUM INFORMATION PROCESSING

L. FEENSTRA, K. BRUGGER, R. FOLMAN, S. GROTH, A. KASPER, P. KRUGER, X. LUO, S. SCHNEIDER, S. WILDERMUTH AND J. SCHMIEDMAYER Physikalisches Institut der Universitat Heidelberg Philosophenweg 12 69120 Heidelberg, Germany E-mail: [email protected] Magnetic microtraps and atom chips have proven to be small-scale, reliable and flexible tools to prepare ultra-cold and degenerate atom clouds as sources for various atom-optical experiments. We present a short overview of several possibilities of the devices, including the first results of using electrostatic interaction to realise traps.

Wire traps, based on the superposition of the magnetic fields of a current carrying wire and a homogeneous magnetic bias field oriented perpendicular to it, have proven t o be able to capture large numbers of atoms, both thermal and Bose-Einstein condensed. Surface mounted wire traps and atom chips enable extremely tight traps in various geometries while combining mechanical stability and robustness with low power consumption. They also allow the combination with optical elements, such as microresonators and cavities. These properties make the devices ideally suited for the realisation of various proposals on quantum information processing with neutral atoms '. A comprehensive review on trapping and manipulation of atoms with magnetic micro-traps can be found in '. The heart of the experiment is the atom chip. Since our microfabricated chip traps are usually too shallow and too small t o capture a sufficient number of atoms to allow evaporative cooling to a large BEC, a robust copper structure is embedded in an ultra-high vacuum (UHV)-compatible ceramic directly behind the chip in the atom chip holder assembly (see Fig. 1). The copper structure can be operated with U- and Z-shaped currents (generat112v3,4

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Figure 1. The atom chip is mounted directly against a ceramic carrier, housing a copper device for the initial, larger traps. The structure is used for U- and Z-shaped currents for quadrupole fields and Ioffe-Pritchard traps. Right: The mounted atom chip. The structure is used upside down t o allow time of flight studies.

ing a quadrupole-like or a Ioffe-Pritchard-type field, respectively) of up to 60 A for over a minute without degrading the vacuum. It thus enables prolonged trapping up to several mm away from the chip surface, facilitating the loading of large atom clouds The experimental cycle starts with loading a mirror-MOT in a UHV chamber, either from a vapour cell MOT via a continuous push beam or by pulsed operation of Rb-dispensers or a Li oven. The magnetic quadrupole field for the MOT is initially supplied by external coils, later replaced by that of a U-trap (U-MOT); these configurations can be operated with the same optical setup enabling perfect transfer. Compressing the U-MOT pulls the atoms closer to the chip and automatically aligns the cloud for the subsequent magnetic trap. After a brief molasses and optical pumping phase we can either load the chip directly for experiments with small ensembles of thermal atoms, or we can first load the “copper trap” to pre-cool the atoms before transferring them to the chip. Because the quadrupole-like field region of a U-trap is relatively small, its overlap with the MOT-laser beams is insufficient for loading a large MOT, so that additional quadrupole coils remain indispensable. However, an optimised U-structure, in which the base wire is replaced by a thin, wide slab, can generate a sufficiently large quadrupole-like field to enable loading > lo8 atoms in a U-mirror-MOT in a few seconds 11, which proved to be sufficient for Bose-Einstein Condensation. Since this improved UMOT removes the need for a separate set of quadrupole coils, it reduces the complexity and power consumption of the setup considerably. It also allows a much better optical access to the atoms. The described system allows Bose-Einstein Condensation of s7Rb atoms in the copper structure without accessing the chip itself 637. BECs of 617.

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Figure 2. Creating a BEC and loading it in the atom chip. Far left: Atoms in the final trap. Middle left to right: Growth of a BEC of 3 x lo5 atoms upon lowering the temperature (final RF-frequencies 800 kHz, 650 kHz, 630 kHz) (all images after 16 ms expansion). Far right: BEC after transport to the chip (and 16 ms expansion). Each image displays N 1.8 x 2.3 mmz. The chip surface is indicated with the black line.

3 x lo5 atoms in the IF = 2 , mF = 2) state can be made up t o about 250 pm from the chip surface, and subsequently can be loaded onto a Z-wire trap on the chip (base wire is 2 mm long, 200 pm wide) in typically 100 ms. The trapping parameters of a wire trap depend on the vector addition of all fields present. For instance, exciting the condensate in the copper trap to the lowest quadrupole mode shows that the current density distribution in the copper structure cannot be neglected for correct modelling of the trap frequencies. Similar finite size effects are found when approaching trap wires closer than 2 - 3 times their width, and near junctions Changing the angle between the wire and the bias field by rotating the direction of the wire on the chip therefore also provides a means to alter the trap. This effect renders it difficult to transport atoms to arbitrary sites on the chip. However, one can realise a guide that is independent of the direction in a certain plane using two parallel wires with counterpropagating currents and a bias field perpendicular to the plane of the wires. This technique allows the guiding of atoms in every direction on the chip, as is shown in Fig. 3, where a cloud of thermal Li atoms expands along a 11417J2,13.

Figure 3. Two parallel, counterpropagating currents with a bias field perpendicular to the chip-plane allow a 360' rotating atom guide, as shown here with thermal Li atoms expanding on an inward spiralling path. Images: 10 ms and 50 ms after loading into the guide.

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Figure 4. A Ioffe-Pritchard type trap made from two partially counterpropagating Zwires a t differentheight levels. &,= 2.0 A, IcU= 8.4 AA10.7 G, Natoms= 3 x 106,T N 10 fiK.

spiral-shaped guide 14. The double layered setup of both chip and copper structure can be used to have the one layer generating a bias field for the other, as shown in Fig. 4. Here the currents in the two Z-structure counterpropagate in their bases, so that the fields subtract for radial trapping 15, but they are copropagating in the arms so that the axial confinement is increased, slightly raising the trap bottom. Thus the fields combine to form a robust Ioffe-Pritchard trap without needing coils. By combining a magnetic potential with a spatially varying electrostatic field, the attractive electrostatic interaction can counteract the repulsive magnetic barrier to the trapping wire, effectively creating a potential well in the guide 16. In this manner a homogeneous magnetic guide can be transformed into a chain of traps, of which each one can be addressed independently (see Fig. 5 ) . Applying time-varying voltages to diagonally opposite pads allows shifting, splitting and combining trapped clouds. By increasing the electric field strength the magnetic barrier can be overcome altogether, causing atom loss from the trap due to collisions with

Figure 5. Combined electric and magnetic traps: charging especially designed pads to 300 V next to a magnetic guide transforms it to a series of localised traps. (A) microscope image of the Chip (the arrow indicates the trap wire). (B) charging all six pads (each pad is 200 pm wide). (C) charging pads 1, 2, 3 and 5 (left to right).

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the chip surface. Since the magnetic interaction is mF-state dependent and the electric interaction is not, the combined force can be used to open a trap that is filled with atoms in different magnetic hyperfine states just so far that the weaker-held atoms are lost to the surface while the magnetically tighter-held atoms remain trapped. In this way a state filter can be realised. In conclusion, the above discussion indicates a number of techniques that allow atom chips to become the method of choice when aiming to control neutral atoms for experiments on atom optics and quantum information processing.

Acknowledgments This work was supported by the EU-projects ACQUIRE, ACQP, and FASTNet, and the Deutsche Forschungsgemeinschaft (grants Schm1599/11 and Schm1599/2-1). The chips were made at the Weizman Institute of Science, Israel, as part of the ACQUIRE-collaboration. L.F. acknowledges support of the Alexander von Humboldt Foundation.

References 1. R. Folman et al., Adv. At. Mol. Opt. Phys. 48, 263 (2002). 2. R. Folman et al., Phys. Rev. Lett. 84, 4749 (2000). 3. D. Cassettari et al., Phys. Rev. Lett. 85, 5483, (2000). 4. J. Reichel, Appl. Phys. B 74, 469 (2002). 5. J. Schmiedmayer, R. Folman, and T. Calarco, J . Mod. Opt. 49, 1375 (2002). 6. S. Schneider et al., Phys. Rev. A 67,023612 (2003). 7. A. Kasper et al., J. Opt. B 5 , S143 (2003). 8. J. Reichel, W. Hansel, and T.W. Hansch, Phys. Rev. Lett. 83, 3398 (1999). 9. K.I. Lee et al., Opt. Lett. 21,1177 (1996). 10. T. Pfau and J. Mlynek., In Proceedings of the European Quantum Electronics Conference, p. 33, editor K.. Burnett, Optical Society of America, Washington, DC, USA (1996). 11. S. Wildermuth et al., In preparation. 12. A.E. Leanhardt et al., Phys. Rev. Lett. 89, 040401 (2002). 13. P. Schwindt. PhD thesis, University of Colorado, USA, (2003). 14. X. Luo et al.,, In preparation. 15. J. FortAgh et al. Appl. Phys. B 70, 701 (2000). 16. P. Kriiger et al., arXiv:quant-ph/O306111.

ON-CHIP LABORATORY FOR BOSE-EINSTEIN CONDENSATION J. FORTAGH, H. om, s.KRAFT, A. GONTHER, c.TRUCK, c. ZIMMERMANN Physikalisches Institut der Universitat Tubingen Auf der Morgenstelle 14, 0-72076 Tubingen http://www.pit.physik.uni-tuebingen.de/zimmennann/

Our recent experimental work has been devoted to three different aspects of the physics of magnetic microtraps. Firstly, we have characterized an anomalous magnetic field component, which appears on the surface of a current carrying copper conductor. Thereby, clouds of ultracold thermal and condensed atoms were used as sensors for weak magnetic fields. Next, we investigated the dynamics of Bose-Einstein condensates while they were moving within the microtrap. We identified a shape oscillation of the condensate, which is driven by the center of mass motion if moving in an anharmonic potential. Our latest development is a dual-layer microstructure with a multiplicity of microtraps. On the chip, three-dimensional positioning and excitationless transport of condensates are possible.

1. Introduction The application of degenerate quantum gases for quantum technology has motivated several theoretical and experimental investigations into the spatial and temporal manipulation of matter waves. Thereby, remarkable progress became apparent in loading of Bose-Einstein condensates into microscopic magnetic traps. The alluring feature of microtraps is the possibility of constructing complex potential systems and of accurate spatial and temporal control. In our experiment, different mictrotaps are located in an ultrahigh vacuum chamber at lo-" mbar. After collecting 3 x lo8 "Rb atoms in a six-beam magneto-optical trap, the magnetically confined atoms (F=2, mF=2 hyperfine ground state) are adiabatically transferred into the microtrap in which the cloud is cooled to condensation by forced rf-evaporation. We reach condensation with lo6 atoms at 1 pK temperature in a cigar shaped trap with a radial and axial frequency of w 2 n x 800 s-l and 0,=2nx 14 s-', respectively. The trap frequencies can be independently modified during the experimental run. The microstructure is mounted upside down in order to get time of flight images if the trapping potential is switched off. The magnetic setup surrounding the microtrap and the magnetic hardware for the magneto-optical trap are placed into

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the vacuum chamber. The compact design' leads to high long term stability, and automatic data acquisition over night is thus made possible. 2.

Anomalous Magnetic Field

One focus of our work is directed on surface microscopy with matter waves. A condensate moved close to a surface is able to detect weak potential modulations due to electric or magnetic fields. By imaging the density distribution of the cloud with absorption imaging, the magnitude and the spatial profile of the confining potential can be determined. In t h s way, we have detected an unexpected potential modulation along copper wires carrying a current necessary for generating the mi~rotrap.~ It arises from a periodic magnetic field component whch is oriented parallel to the conductor axis4 The anomalous field is induced by the electric current and its strength changes if the current is modified. By inverting the current in the conductor, the orientation of the anomalous field component likewise changes. The experiment in Fig. la shows the fragmentation of a'thermal cloud which has been released for free expansion into a waveguide. The waveguide was

0

500

1000

1500

2000

Position (pm)

Figure I . a) Density distribution of a thermal cloud (1 pK) in a waveguide after 100 ms free expansion. The axial potential was ramped to zero within 400 ms. b) Density distribution by inverted offset field Bo. Bo was inverted within the last millisecond while the axial confinement was ramped to zero.

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generated by the circular magnetic field of a microscopic copper conductor (30 pm width, 2 pm height, 24 mm length), an additional bias field perpendicular to the conductor axis and an offset field Bo parallel to the conductor. For comparison, Fig. Ib shows the absorption image after the offset field Bo has been inverted. Because the axial magnetic potential of a trap is given by the modulus of the sum of all magnetic fields oriented along this axis, the change in the sign of Bo changes potential minima into maxima and vice versa if and only if the modulation is given by an axial magnetic field component. The current dependence of the anomalous field becomes apparent if one inverts the current in the conductor. As a result, the same potential structure observations can be made as for an inverted offset field. Thus, the relative orientation of the current flow and offset field Bo determine the position of the potential minima and maxima. The modulation appears here with a pronounced periodicity of 260 pm and we do not observe any shift of its position if the current in the conductor is changed. Similar results have been found in experiments with different conductor geometries; even a copper wire with 90 pm diameter shows the same effects. We derive the absolute value of the anomalous field component from the measurement of the chemical potential of Bose-Einstein condensates which are split due to the influence of the surface field. By positioning the center of the harmonic trap over a maximum of the surface field, a symmetrical double-well potential is generated in which the atom cloud is cooled to condensation. The potential barrier U in the double-well potential is then estimated from the chemical potential of the condensate at the separation point in two equal parts. This result gives a lower limit for the surface field which is 1.5 mG at 100 pm distance to the surface of a copper wire (90 pm diameter), driven by a current of 1 A. Figure 2 shows trap-surface distances for different current intensities where a potential barrier of kB x 200 nK was measured. The origin of the longitudinal

-k

wire, 9Op diameter

I

v

58

100

-

-

.,-0 al

?m

50-

c

u) ._ U

0,1

current (A)

1

Figure 2. Distance-current pairs for Il/ks200 nK corresponding to an axial field amplitude of at least 1.5 mG.

245

0.0

0.1

0.2

0.3

0.4

0.5

0.6

time (s) Figure 3. Center of mass motion for two different initial displacements (A: 0.45 mm, C: 0.79 mm) of the condensate.

field component is still unexplained, such that further data on different conductor geometries and on other metals will be necessary to answer the question to what extent this anomalous field may influence applications of microtraps. 3.

Dynamics of Condensates in Waveguides

With the manipulation of Bose-Einstein condensates in microscopic traps, it is an important question, how the condensate behaves while moving inside the trap. For this reason, we have studied the oscillation of the condensate in an elongated trap for different initial displacements. The radial confiiement was kept constant by 04=27t x 110 s-' and the oscillation took place along the long axis characterized by a basic frequency of w,=2n: x 8 s-'. The center of mass motion was detected periodically by using absorption imaging and 15 ms time of flight (Fig. 3). For small amplitudes, the motion is described by a single sine function . which frequency can be determined with a high accuracy of d A o ~ 2 0 0 0 0 For larger amplitudes, the motion becomes anharmonic and the potential shape can be reconstructed up to the fifth harmonic by fitting the data points. The anharmonicity of the potential is also observed in the shape oscillation of the condensate. While having a center of mass motion described by the second graph in Fig. 3, the ratio of the axial and radial size of the condensate oscillates by a factor of 30! The Fourier-spectra of the aspect ratios show the fundamental frequency w, with its harmonics, the eigenfrequency of the condensate (5l2)'" w, and mixed frequencies due to the nonlinear character or the Gross-Pitaevskii e q ~ a t i o n .For ~ large oscillation amplitudes and long oscillation times, the theory even predicts a chaotic behavior.6 Similar effects are expected through the manipulation of condensates in microtraps which put serious restrictions on

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.1,5

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4.5 0.0

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-1,O

4.5

0.0

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-1.5

-1.0

4.5 0.0

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paslion (mm)

Figure 4. Shape oscillation of the condensate for three initial displacements in the trap. A: 0.45 mm, B: 0.63 mm, C: 0.79 mm. The absorption images show the position and shape of the condensate in 10 ms cycle, after 15 ms time of flight.

atom-optical applications. For use in atom optics, the density of atoms has to be reduced such that the interaction energy is smaller than the radial level spacing of the trap. Then, the condensate is confined to the oscillator ground state and behaves like an ideal wavepacket with quadratic dispersion relation (quasi-one dimensional regime). The physics of such an atom laser beam is the topic of current investigations.* 4.

On-chip Laboratory for Bose-Einstein Condensation

We have developed a dual-layer atom-chip where matter waves can be manipulated in a versatile manner by the magnetic field generated by the chip only (Fig. 5 ) . Ultracold thermal atoms and Bose-Einstein condensates are adiabatically transported in a waveguide placed along an array of micro traps. The transport over 2 cm is done without changing the trap frequencies and without excitation of the condensate. Neighboring traps can exchange atoms directly, and more distant traps are connected via the waveguide. Thus, a three dimensional manipulation of the condensate on the chip surface is realized. With the construction of a set of model potentials on the chip, we realized an experimental system with which different aspects of the physics and technology of matter waves can be studied during the same run.Figure 6 demonstrates the operation of the conveyor belt with a thermal cloud. The atoms are transported

247

Figure 5 . Dual-layer atom-chip. The waveguide in the center of the chip connects an array of model potentials. The Bose-Einstein condensate is moved adiabatically between the micro traps.

P

time [s] Figure 6 . Forth and back transport of thermal atoms over a total of 16 transport distances each 650 pm long. The data points display the center of the cloud. The theoretical line was calculated based on the slopes of driving currents.

over more than one centimeter and the experimental data are well described by the simulation. The simulation also predicts an extremely smooth transportation with a variation of the trap frequencies. The new chip provides gold conductors with widths between 30 and 100 pm at different geometries and offers promising opportunities to study the anomalous magnetic field of gold conductors.

References 1. 2. 3. 4. 5. 6.

H. Ott et al., Phys. Rev. Lett. 89, 230401 (2001). J. Fortigh et al., Appl. Phys. B 76, 157 (2003). J. Fortagh et al., Phys. Rev. A 66,041604(R) (2002). S. Kraft et al., J. Phys. B: At. Mol. Opt. Phys. 35, L 469 (2002). H. Ott et al., Phys. Rev. Lett. 91,040402 (2003). H. Ott et al., J. Phys. B: At. Mol. Opt. Phys. 36,2817 (2003).

ATOM OPTICS AND QUANTUM INFORMATION PROCESSING WITH ATOMS IN OPTICAL MICRO-STRUCTURES

M. VOLK, T. MUTHER, F. SCHARNBERG: A. LENGWENUS, R. DUMKE, W. ERTMER, AND G. BIRKL Institut fur Quantenoptik, Universitat Hannover, Welfengarten 1, D-30167 Hannover, Germany

We experimentally demonstrate interferometer-type guiding structures for neutral atoms and novel structures for the realization of registers of atomic qubits based on dipole potentials created by micro-fabricated optical systems.

State-of-the-art technology in micro- and nano-fabrication can be combined with the quantum optical techniques of laser cooling, laser trapping, and quantum control to open a new gateway for integrated matter wave optics and quantum information processing with atomic systems. Several groups have applied micro-fabricated magnetic or current carrying structures for this purpose l . In our work, micro-fabricated optical systems have been used to create light fields that allow one to trap and guide neutral atoms as a result of the optical dipole force experienced by the atoms 2 . These approaches open the possibility to scale, parallelize, and miniaturize systems for quantum information processing and atom optics in fundamental research and application.

Integrated Atom Optics Because of the high intrinsic sensitivity, atom interferometers have to be built in a robust way to be applicable under a wide range of environmental conditions. A new approach to meet this challenge lies in the development of miniaturized and integrated atom optical setups based on micro-fabricated guiding structures. Using micro-fabricated current carrying wires, several Permanent address: Swinburne University of Technology, Melbourne, Australia

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configurations for atom guides and beam splitters have been realized’. The demonstration of a setup suitable as a guided-atom interferometer remains an important goal. In our work, we have achieved the experimental implementation of atom guides, beam splitters, and structures for atom interferometers based on micro-fabricated optical elements 3 . We demonstrate the guiding of neutral atoms along the focal lines created by arrays of micro-fabricated cylindrical lenses making use of optical dipole potentials (Fig. 1 (top left)). A key advantage of optical micro-structures is the possibility to image the focal plane into the vacuum chamber. Thus micro-optical elements can be placed outside the vacuum and the light fields of several elements can be combined. By superimposing two of these arrays under a variable relative angle, we realize X-shape beam splitters as well as interferometer-type configurations like Mach-Zehnder (Fig. 1 (top right)) or Michelson-type structures.

Figure 1. Interferometer-type structures for guided atoms based on dipole potentials created by microfabricated optical systems.

Figure 1 (bottom) shows the propagation of atoms through a MachZehnder-type structure for guided atoms. We load one of the input ports with atoms from a single dipole trap. The atom sample propagates to the first beam splitter and splits into two paths. At the next intersections these split into a total of four paths. Two of the paths recombine at the the fourth intersection. This Mach-Zehnder-type structure has an enclosed area of 0.3 mm2 with a total required array including the loading and detection stages of below 1 mm2. It presents the first experimental demonstration of a

250

structure suitable for atom interferometry based on atom guides. Numerical simulations show that for typical experimental conditions coherent splitting of atom waves and matter-wave interference at the output beam splitter can be achieved '. As an important result, a variation in the relative phase between the two paths of a Mach-Zehnder-type structure (e.g., by inserting a variable dip in the guiding potential as phase shifter) results in a complementary periodic variation of the atom number in the two final output ports, thus clearly predicting the existence of interference fringes. Specific to optical guiding structures is the possibility to use the internal atomic structure for the splitting process. For guided atoms, state-selective splitting can be achieved by applying a state-selective optical potential in a small section of one output port shortly after the beam splitter. We demonstrated a state-selective guided-atom beam splitter for 85Rb by employing an additional laser field which is red-detuned for atoms in the 5 S l p ( F = 2 ) hyperfine ground state and blue-detuned for atoms in the 5&12(F = 3 ) hyperfine ground state.

Quantum Information Processing Quantum information processing has emerged as one of the most active fields of research in physics in recent years. Fundamental issues as well as the question of practical implementation are investigated. Among the broad range of possible approaches, important progress has been obtained with atom physical schemes. We have demonstrated the first experimental implementation of microfabricated optical systems for quantum computing purposes with atoms 5 . Using two-dimensional arrays of spherical, diffractive microlenses, we can trap neutral atoms in one- and two-dimensional arrays of far-detuned dipole traps (Fig. 2 (a)). More than 80 traps hold 85Rb atoms in Fig. 2 (b). Each trap can act as a memory site for quantum information encoded in the two hyperfine ground states of the atoms. Thus, the arrays can serve as registers of atomic qubits. In addition to its scalability, our approach is especially suited to fulfill an important requirement for the physical implementation of quantum: information processing, namely, the ability to selectively address, initialize, and read out individual qubits. The large lateral separation of 125 pm between the dipole traps enables us to selectively address the individual traps in a straightforward fashion. We demonstrate this by focusing a near-resonant laser beam onto one of the dipole traps, thus removing atoms from the

251 addressed trap. As can be seen in Figure 2 (c), no atoms are left at the addressed site, while the atoms at the adjacent sites remain unaffected. We could also demonstrate the site-specific and state-selective initialization and readout of atomic quantum states (Fig. 2 (d)). Here, we illuminate a one-dimensional atom array with light only resonant with the 5Sl/z(F = 3) -+ 5P3/2(F' = 4) transition during detection. Since the atoms are almost exclusively in the lower hyperfine ground state 5 S I l 2 ( F= 2) after the loading phase (first row), they do not scatter the detection light unless we actively pump them into the upper hyperfine ground state 5S1/2(F= 3) during the time the atoms are stored in the dipole traps, i.e., prior to the detection phase. This has been done for one (second row) or, alternatively, all (third row) of the trap sites.

Figure 2. Micro-optical realization of a 2D array of dipole traps for quantum information processing

It is also possible to actively control the distance between individual traps in our system, if smaller or adjustable distances are required, e.g., for quantum gate operations and the entanglement of atoms via atom-atom interactions. This can be accomplished by illuminating one microlens array with two beams at slightly different angles, which results in two interleaved sets of arrays of trapped atoms. The separation only depends on the angle between the two laser beams and can easily be changed, especially to smaller values. By reducing the relative angle to zero, overlapping traps are created. Several schemes for quantum gates based on the direct interaction of

252 neutral atoms have been proposed theoretically (see a list of references in Refs. 5 and 6). We could show that the requirements for the implemenIn addition to that, tation of these gates can be fulfilled by our system new proposals for the implementation of quantum gates, particularly suitable for our setup, were developed together with the group of Prof. M. Lewenstein.'

'.

Acknowledgments

We thank F. Buchkremer, K. Eckert, H. Kreutzmann, M. Lewenstein, J. Mompart, and A. Sanpera for many stimulating discussions. This work is supported by the SFB 407 and the Schwerpunktprogramm 'Quanteninformationsverarbeitung' of the Deutsche Forschungsgemeinschaft and by the project ACQP of the European Commission.

References 1. For an overview see R. Folman, P. Kruger, J. Schmiedmayer, J . Denschlag, and C. Henkel, Advances in Atomic, Molecular and Optical Physics 48, 263,

B. Bederson and H. Walther (editors), Elsevier Science (2002) and references therein. 2. G. Birkl, F.B.J. Buchkremer, R. Dumke, and W. Ertmer, Opt. Comm. 191, 67 (2001). 3. R. Dumke, T. Muther, M. Volk, G. Birkl, and W. Ertmer, Phys. Rev. Lett. 89, 220402 (2002). 4. H. Kreutzmann, U. Poulsen, M. Lewenstein, R. Dumke, W. Ertmer, G. Birkl,

and A. Sanpera, in preparation. 5 . R. Dumke, M. Volk, T. Muther, F.B.J. Buchkremer, G. Birkl, and W. Ertmer, Phys. Rev. Lett. 89, 097903 (2002). 6. K. Eckert, J. Mompart, X. X. Yi, J. Schliemann, D. BruB, G. Birkl, and M. Lewenstein, Phys. Rev. A 66, 042317 (2002); J. Mompart, K. Eckert, W. Ertmer, G. Birkl, and M. Lewenstein, Phys. Rev. Lett. 90, 147901 (2002).

A CONTROLLABLE DIFFRACTION GRATING FOR MATTER WAVES

HILMAR OBERST, SHIGENORI KASASHIMA AND F U J I 0 SHIMIZU Institute for Laser Science, University of Electro-Communications Chofu-shi, Tokyo 182-0025 Japan E-mail: [email protected] VICTOR I. BALYKIN Institute of Spectroscopy, Russian Academy of Science Troitsk 142190, Moscow Region, Russian Federation We report on the experimental realization of an atom optical device, that allows scanning and coherent splitting of an atomic beam. We used a timemodulated evanescent wave field above a glass surface t o diffract a continuous beam of metastable neon atoms at grazing incidence. The diffraction angles and efficiencies were controlled by the frequency and form of the light intensity modulation, respectively. With an optimized shape, obtained from a numerical simulation, diffraction efficiencies into a single order of more than 50% were achieved.

We report on the experimental realization of a diffraction grating for atomic waves, that makes it possible to control the diffraction angles and efficiencies by an electronic signal. We diffracted an atomic beam at grazing incidence from a glass surface by using a time-modulated evanescent wave. Our technique was the spatial version of the work by Steane et al. ', in which they generated frequency sidebands of n x hf with a sinusoidally modulated evanescent light of frequency f in a perpendicularly reflected atomic beam. The modulation of the reflecting potential leads to the absorption of energy quanta perpendicular to the surface and to a splitting of the atomic beam into diffraction orders that are separated in space. The effect can also be interpreted as a dynamically created diffraction grating for atoms moving parallel to the mirror plane, with an adjustable grating period d = v / f , where v is the incident velocity. The shape of modulation can be used to control the diffraction efficiencies, similar to a blazed grating in optics. This technique has been used in the standing optical wave case2. The experimental setup is shown in Fig. 1. Metastable neon atoms were 253

254 releasing laser at 598 nm -.

....... .......... ... . ..... .........,,.........

0 MOT of lssneon atoms

Ip

I

I 1s3neon atoms

.,...,..,.,.

118 cm I -

MCP ' detector

to computer

Figure 1. Experimental setup.

trapped and cooled in a magneto-optical trap using the transition at 640 nm, and continuously released by optically pumping to the 1 s g ['PO]state with a laser at 598nm, which was focused into the cloud of trapped atoms. The atoms started a free fall and had a velocity of 3ms-' with a spread of less than 1%,when they arrived at the modulated mirror. The angle of incidence was around 21 mrad. The atom mirror was realized by total internal reflection of an intense, blue-detuned laser beam in a BK7 glass prism. We used the optical dipole transition at 627nm and worked with a detuning of about +2GHz. The laser passed through an acousto-optical modulator, and by modulating the rf-intensity of the AOM driver, we created the desired light intensity modulations. The atoms were detected by a microchannel plate detector, which allowed us to determine the position of each single incident atom. In order to calculate the diffraction pattern we numerically solved the one-dimensional, time-dependent Schrodinger equation, describing the reflection of a Gaussian wave packet from the time-modulated potential. In this calculation we also included the van der Waals surface potential3. We optimized the shape of modulation in order to transfer a maximum amount of atoms into one of the first order beams. The arbitrary shape of modulation was described by a Fourier series, and the optimum Fourier coefficients

255

a 0 2 C

0 In

c

C

8 E

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z

A

=-30 kWz

A 5

6

z

7

0

mm from center

9

Figure 2. (a) The measured distribution of atom counts on the detector, when the wave form was optimized to concentrate atoms into one diffraction order. The strong peak is the first order beam. (b) The corresponding wave form for each modulation frequency.

determined by fitting the calculated diffraction efficiencies to the chosen ideal distribution. The optimized wave forms for each modulation frequency are shown in figure 2(b) and the measured distributions of atom counts on the detector are shown in figure 2(a). The reflectivity is more than 80% and more than 50% of the incident atoms are transferred into the first order beam over the whole scanning range of about 8 mrad. The technique might, for example, be applied in atom lithography to scan an atomic beam or in atom interferometers as a beam splitter or switch. The method can easily be adapted to different atomic species and velocity ranges. This work was partly supported by Grant in Aid for Scientific Research (No. 11216202) from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

References 1. A. Steane, P. Szriftgiser, P. Desbiolles and J. Dalibard, Phys. Rev. Lett. 74, 4972 (1995). 2. S. Bernet et al., Phys. Rev. A 62,023606 (2000). 3. F. Shimizu, Phys. Rev. Lett. 86, 987 (2001).

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Cavity QED

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CAVITY QED BY THE NUMBERS

H. J. KIMBLE, A. BOCA, A. D. BOOZER, W. P. BOWEN, J. R. BUCK, C. W. CHOU, L.-M. DUAN, A. KUZMICH AND J. MCKEEVER Norman Bridge Laboratory of Physics 12-33 California Institute of Technology Pasadena, C A 91125, USA E-mail: [email protected] Observations of cooling and trapping of N = 1,2,3,... atoms inside a small optical cavity are described. The atom-cavity system operates in a regime of strong coupling for which single photons are sufficient to saturate the atomic response. New theoretical protocols for the efficient engineering of multi-atom entanglement within the setting of cavity QED are described. By trapping a single atom within the cavity mode, a oneatom laser is experimentally realized in a regime of strong coupling. Beyond the setting of cavity QED, quantum correlations have been observed for photon pairs emitted from an atomic ensemble and with a programmable time offset.

1. Introduction

A long-standing ambition in the field of cavity quantum electrodynamics (QED) has been to trap single atoms inside high-Q cavities in a regime of strong coupling In this regime, the Rabi frequency 290 for a single quantum of excitation exceeds the decay rates (7,K ) for the atom and cavity mode, respectively, resulting in critical atom and photon numbers (No,no) > ( I C , ~ ) ) , resulting in critical photon and atom numbers no 3 y2/(29;) N 0.013, NO 3 21~y/9:N 0.084. The upper level F‘ = 3‘ is pumped by the external drive 0 3 , while effective decay of the lower level F = 4 takes place via the combination of the drive 0 4 and decay 7 3 4 , 4 --+ 4‘ -+ 3. In essential character this system is analogous to a Raman scheme with pumping 3 -+ 3’, lasing 3’ -+ 4, and decay 4 -+ 3. We have made measurements of intracavity photon number versus the pump intensity 103l2 that exhibit “thresholdless” behavior, and infer that the output flux from the cavity mode exceeds that from atomic fluorescence by more than tenfold We have also reported observations of the second-order intensity correlation function g(2)( T ) demonstrating that our one-atom laser generates manifestly quantum (i.e., nonclassical) light that exhibits both photon antibunching g(2)(0) < g ( 2 ) ( ~and ) sub-Poissonian photon statistics g(2)(0) < 1. 314.

2i6.

394.

5. Quantum Information Processing with Atomic

Ensembles Beyond the setting of cavity QED, there is a very different avenue for the realization of scalable long distance quantum communication and the distribution of entanglement over quantum networks, following the lead of Ref. 10. As an enabling step in the implementation of the protocol of DLCZ, we have reported observations of quantum correlations for photon pairs generated in the collective emission from an atomic ensemble The nonclassical character of the fields is evidenced by the violation of a Cauchy-Schwarz inequality for the two fields (1, 2). As compared to prior investigations of nonclassical correlations for photon pairs produced in atomic cascades and in parametric down conversion, our experiment is distinct in that the

264 correlated (1, 2) photons a r e separated by a programmable time interval At, with At N 400 nsec in our initial experiments.

6. Acknowledgements This work was supported by t h e National Science Foundation, by the Caltech MURI Center for Quantum Networks, by the Office of Naval Research, and by the NSF Institute for Quantum Information.

References 1. H. J. Kimble, Physica Scripta T76,127 (1998). 2. J. McKeever, J. R. Buck, A. D. Boozer, A. Kuzmich, H.-C.Nager1, D. M. Stamper-Kurn and H. J. Kimble, Phys. Rev. Lett. 90, 133602 (2003); available as quant-ph/0211013. 3. J. McKeever, A. Boca,A. D. Boozer, J. R. Buck and H. J. Kimble, Nature 425, 268 (2003). 4. A. D. Boozer, A. Boca, J. R. Buck, J. McKeever and H. J. Kimble, submitted t o Phys. Rev. A (2003); preprint available at http://lanl.arXiv.org/archive/quant-ph. 5. J. Ye, C. J. Hood, T. Lynn, H. Mabuchi, D. W. Vernooy and H. J. Kimble, IEEE Transactions on Instrumentation and Measurement 48, 608 (1999). 6. J. Ye, D. W. Vernooy and H. J. Kimble, Phys. Rev. Lett. 83, 4987 (1999). 7. H. J. Kimble, Phys. Rev. A 90, 249801 (2003); available as quant-ph/0210032. 8. L.-M. Duan, A. Kuzmich and H. J. Kimble, Phys. Rev. A 67,032305 (2003); available as quant-ph/0208051. 9. L.-M. Duan and H. J. Kimble, Phys. Rev. Lett. 90, 253601 (2003); available as quant-ph/0301164. 10. L.-M. Duan, M. Lukin, J. I. Cirac and P. Zoller, Nature 414, 413 (2001); herein refered to as DLCZ. 11. A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan and H. J. Kimble, Nature 423, 731 (2003).

MANIPULATING MESOSCOPIC FIELDS WITH A SINGLE ATOM IN A CAVITY S. HAROCHE', A. AUFFEVES, P. MAIOLI, T. MEUNIER, S. GLEYZES, G. NOGUES, M. BRUNE AND J.M. RAIMOND Laboratoire Kastler Brossel, De'partement de Physique de 1'Ecole Normale Supe'rieure 24 rue Lhomond, F-75231 Paris Cedex 05 E-mail: [email protected] r

Fields made of several tens of photons are manipulated with a single atom in a high-Q cavity. At resonance, the atom-field coupling splits the field into two components whose phases differ by an angle inversely proportional to the square root of the average photon number. The field and the atomic dipole are phase-entangled. These effects are directly related to the collapse and revivals of the quantum Rabi oscillation of the atomic populations. They are manifestations of photon graininess which vanish at the classical limit. The Schrtidinger cat states of the field generated in this way are much larger than the ones obtained previously. Their study opens the way to new investigations of the quantum-classicalboundary.

1. Atom Interacting with a Field in a Cavity: from Vacuum to Classical Fields via the Mesoscopic Regime

In cavity quantum electrodynamics (CQED) experiments, a single atom can have a large effect on a field stored in a cavity. The presence in the cavity of a two level atom (with internal energy states e, g) changes the frequency of the field by an amount depending upon the atom-field detuning, the atom's internal state and its position in the field mode.' Various manifestations of this effect have been observed in optical and microwave CQED experiments. In the optical domain, changes in the transmission of the cavity field have been detectedzv3and used' to analyze the atom's motion through the cavity mode. In the microwave domain, the dispersive phase shift produced by an atom on the field has been utilized to generate Schrodinger cat like states of the field.4 Conversely, the effect of the field on the phase of the atomic dipole has been used to count photons in a nondestructive way.5

* Also Coll2ge de France, 1 1 Place Marcelin Berthelot, F-75231 Paris Cedex, France.

265

266

The influence of a single atom is in fact dependent on the "size" of the field. At the vacuum field limit, the cavity coupling to a single atom is very dramatic. If the atom is initially injected in an empty cavity in the upper level e, the system starts oscillating between the atom-field states le,O> and lg, 1> representing the atom in levels e and g, in the presence of 0 and 1 photon, respectively.6 This quantum Rabi oscillation occurs at the vacuum Rabi frequency R proportional to the zero-point fluctuations in the cavity. It results in a periodic entanglement between the atom and the field and strongly modifies the state of the latter. The coupling to a single atom is, on the other hand, negligible for coherent fields made of very large photon numbers (classical limit). The interaction between an atom and a coherent field with an average photon number is characterized by the Rabi frequency The classical limit corresponds to a situation where t h s frequency assumes a finite value R c, while Q tends to 0 and n to infiiity. This can be realized by increasing the size of the cavity (thus diluting the zero-point fluctuations) and increasing the photon number accordingly, while keeping the atom-field coupling finite. At this limit, the field is unperturbed by the atom which undergoes at resonance a reversible oscillation at frequency R c between its levels e and g, Between the vacuum field case corresponding to a large field perturbation and the classical regime, for whch the field remains unaffected, lies the regime of mesoscopic fields made of large but finte numbers of photons. In such fields, a single atom can, as we show here, produce an observable perturbation, provided the coupling is allowed to last long enough.

n

2.

Qfi.

Coupling a Single Atom with a Mesoscopic Coherent Field

Let us consider an atom initially in level e, resonantly coupled with a coherent 2 field with a real amplitude a (average photon number n = a ). The system's state at time t in the interaction picture can, to a good approximation, be expressed as:'

where the field states la'(t)) and atomic states Iv;(q)are:

267

and the probability to detect at time t the atom in level g is given by:

These equations veld a synthetic view of the evolution of a mesoscopic field coupled to a single atom. Equation (1) shows that the atom and the field get entangled. The field evolves into two coherent components rotating in opposite directions in phase space (Eq. (2)). These components are correlated to two different atomic state superpositions corresponding to mean atomic dipoles whose phase also rotates in opposite directions (Eq. (3)). Note that the field components have their phases not only shfted but also spread out. The factor whose exponent is quadratic in n in Eq. (2) is responsible for the phase spreading. The Rabi oscillation appears as a quantum interference effect between the amplitudes associated with the atomic state superpositions (Eq. (4)). Due to the splitting of the field into different states, the amplitude of the Rabi oscillation is predicted to collapse and revive periodically.* The collapse is a direct consequence of complementarity. When the field is split into two orthogonal components with different phases, it stores information about these atomic superpositions, thus destroying the interference between them. The Rabi oscillation revives when this information is erased, as the two components of the field periodically merge together. These equations reveal also the existence of two very different time scales for the system's evolution. The Rabi oscillation corresponds to a fast evolution whose frequency, remains finite at the - classical limit. The phase drift of the field and atomic dipole occurs at a 4 n times slower frequency R/4& which vanishes at the classical limit. In fact, the formulae given by Eqs. (1) to (3) are obtained by replacing in the exact atom-field state expressions the radicals by J;;+ 142&) -and by developing then all functions of n around n up to second order in n - n . This procedure takes into account the discrete nature of the field photon number. The atom-field entanglement and the slow

CIA,

268

field and atom phase diffusion thus appear as effects of photon graininess which disappear for classical fields. Although they have been extensively studied the~retically,',~ these effects have not been up to now experimentally investigated in the mesoscopic field domain. Evidence of the collapse and revival of the Rabi oscillation has been obtained in CQED experiments"*" with fields containing no more than about 2 or 3 photons (similar collapse and revival experiments have been realized with ion traps,12with about 3 ion vibration quanta replacing the photons). The CQED experiments focused on the atomic evolution in a regime where only a few photons were involved and did not study the evolution of the field or its entanglement with the atom in the mesoscopic regime. We describe here the first ob~ervation'~ of the action of a single atom undergoing Rabi oscillation on a field made of several tens of photons. The phase splitting of the field into two components rotating in opposite directions in phase space is observed by a homodyne method and very good agreement between experiment and theory is found. This experiment demonstrates that a single atom leaves its imprint on a mesoscopic field, opening the way to applications in CQED.

3.

The Setup and the Experimental Procedure

Our set-up is sketched in Fig. 1 (a) and described in details in Ref. 6 . Rubidium atoms coming from an oven 0 and velocity selected by laser optical pumping are prepared in box B in the circular Rydberg state with principal quantum number 51 (level e ) or 50 (level g ) by a combination of laser and radiofrequency excitations. The atomic preparation is pulsed, so that the position of each atom is known along the beam within k 1 mm. The atoms cross, one at a time, the

Position

Figure 1. (a) Sketch of experimental set up. (b) Timing of homodyne field phase measurement.

269

cavity C sustaining a Gaussian field mode (waist w = 6 m)exactly resonant with the e +g transition at 5 1.1 GHz. The cavity, cooled to 0.6 K, is made up of two superconducting niobium mirrors. The Rabi frequency is R = 3.10' s-'. The atom and field relaxation times are T, = 30ms and Tc = 0.85 ms corresponding to firc= 250, llfilling the strong coupling regime condition. The atomic velocity is chosen from two values, v, = 335 m s" and v b = 200 m s'' corresponding to effective atom-cavity interaction times t, = h c w/v,= 32 ps and th = 3 ps. The atoms are detected after C by a field ionization detector D (quantum efficiency 70%) which discriminates e and g. The experimental procedure follows the time sequence shown in the spacetime diagram of Fig. l(b). The preparation box B, cavity C and detector D are represented from left to right by vertical bands. The cavity is initially empty (vacuum state). A first atom A / , prepared in e or g with velocity v, or vh, is sent across the set-up (lower diagonal line). Before this atom reaches C a coherent field F I is injected (lower horizontal line). The atom then crosses C. As soon as it exits the mode, a probe field F2 (upper horizontal line) is injected, with the same amplitude as F I and a relative phase #in.The F I and F2 fields add in C and their sum is read out by a second (probe) atom prepared in g, which reaches C just after the injection of F2 (upper diagonal line). We realize in this way, by detecting A2, a homodyne phase sensitive dete~tion'~ of the coherent field FI modified by its interaction with A , . The probe atom absorbs the final field in C, ending up in e with a large probability when # is such that there is one photon or more in C. By repeating the sequence many times, we reconstruct a signal Sg(#) equal to the probability versus # that A2 remains in g. This signal exhibits peaks revealing the final phase pattern of F, after its interaction with A , . This signal is observed either in coincidence with the detection of Al or without A , atom (signal providing a phase reference). 4.

Splitting the Phase of a Mesoscopic Field with a Single Atom

Typical homodyne signals are shown in Fig. 2. The open circles represent &(#) when no A l atom is sent, with 29 photons on average in C. The black 111 squares correspond to the signal when A l has interacted during t , = 32 ps with a coherent field having = 36 photons on average. In both cases, the solid lines are Gaussian fits to the data. The splitting of the phase of a mesoscopic field by a single atom is clearly observed, the phase distribution exhibiting two well resolved peaks. We have recorded similar signals with atom-field interaction time th. We have also varied the average photon numbers (between 15 and 36). These numbers were independently determined by measuring the light shift induced by the field on an auxiliary Rydberg atom. The phase splittings measured in these experiments are shown in Fig. 3, where we have plotted the position of the experimental peaks in Sg(# versus the dimensionless parameter

n

270 Qt/4&. The points are experimental, with their error bars. The dotted line corresponds to the phases predicted by Eq. (1) (linear variations). The solid line results from a numerical simulation solving the field equation of motion and talung into account all known experimental imperfections, including cavity damping. The maximurn phase splitting observed for = 15 and tb=52 ps is 90".

n

"'1 0,4!

-200

'

I

-150

I

-100

,

I

-50

,

I

0

I

50

*

8

100

,

I

150

*

i00

4 (degrees)

Figure 2. Field phase distribution $(I$) for n = 36 and t, =32 p (black squares). The open circles give the phase distribution when no A1 atom is sent (yielding the phase origin). The points are experimental and the lines Gaussian fits.

Figure 3. Phases of the two field components versus Rt/4&. theoretical, as explained in text.

Dotted and solid line are

27 1

5.

Atom-Field Entanglement and Schrodinger Cat States of the Field

The splitting of the mesoscopic field into two components with different phases indicates that a Schrodinger cat state superposition is generated. Each component of the field is, according to Eq. (l), correlated to a different atomic state, superposition of e and g . We have checked this by preparing separately the Ip;(o)) and lp;(o)) atomic superpositions with a combination of Rabi and electric field pulses and verified that the field coupled to the atom then evolves or instead of the two components observed into a single component

l a +)

la-),

when the atom is initially prepared in le), superposition of lq;(o)) and lu);(o)>. This experiment, described in Ref. 13, reveals the correlation between the atomic and field states, a necessary condition to demonstrate atom-field entanglement. The coherent nature of the atom-field superposition must also be proven. A proof of coherence would be obtained by observing the Rabi oscillation revival, which occurs only if a quantum phase relationship is maintained between the two terms of the state superposition in Eq. (1). The revival time, t = 4n&/R, is of the order of 200 ps for = 30 and its observation would require atoms much slower than the ones available in our set-up. Instead of waiting for the revival to occur spontaneously, we use a trick to induce it at an earlier time. After the collapse of the Rabi oscillation, we apply to the atom, at a time T, an electric field pulse whose effect is to switch the signs of the quantum amplitudes associated with e and g. According to Eq. (I), this pulse suddenly exchanges the atomic states correlated with the laf> and la-> field components. The atomfield evolution resumes afterwards, reversing the sign of rotation of the field components. At time 2T, the two field states are back in phase and the Rabi oscillation revives. This stimulated revival, analogous to a spin echo, is a signature of the coherence of the atom-field state superposition produced in the cavity at time T. This experiment will be described in a forthcoming paper. The square of the distance in phase space between the field components, d’ = 4 n s i n Z ( ~ / 4 & ) ,is a measure of the mesoscopic character of this superposition. In the range of n values we have explored, d-’ is nearly constant versus n , equal to about 20 for t, = 32 ps and to about 40 for th = 53 ps. These figures are much larger than the separations obtained in the Schrodinger cat states realized in OUT earlier CQED experiments (d’ IS). The larger cat states obtained here are due to a hlgher-Q cavity and to the use of the resonant atom-field interaction whch achieves a larger phase splitting than the dispersive coupling used previously. The theoretical decoherence time of the final cat state, 2Tc Id’, is about 43 ps for t b = 52 ps. The superposition then loses its coherence as fast as it is generated. The situation is better for the smaller cat states prepared faster (to = 32 ps), which have a decoherence time of about 85 ps.

n

272

These experiments show that a single atom leaves a quantum imprint on a mesoscopic field made of tens of photons. As the size of the field increases, it takes a longer time for the atom to get entangled with the field. At some point, this time is so long that decoherence sets in before the quantum superposition can be prepared. The system then becomes classical. The limit between the quantum and classical regimes ultimately depends upon our ability to protect the system from decoherence. By using better cavities and letting the atom interact longer with the field, we hope to be able to push the quantum classical boundary further and to prepare larger mesoscopic field superpositions involving hundreds of photons.

Acknowledgments We acknowledge support of the European Community, of the Japan Science and Technology corporation (International Cooperative Research Project: Quantum Entanglement). Laboratoire Kastler Brossel is a laboratory of University Pierre and Marie Curie and ENS, associated with CNRS (UMR 8552).

References 1. 2. 3. 4. 5.

6. 7. 8. 9. 10. 11. 12. 13. 14.

S. Haroche and J-M. Raimond, in Cavity Quantum Electrodynamics, editor, P. Berman (Academic Press, 1994). C.J. Hood, T.W. Lynn, A.C. Doherty, A.S. Papkins and J. Kimble, Science 287, 1447 (2000). T. Fisher, P. Maunz, P.W.H. Pinkse, T. Puppe and G. Rempe, Phys. Rev. Lett. 88, 163002 (2002). M. Brune et al., Phys. Rev. Lett. 77,4887 (1996). G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J-M. Raimond and S. Haroche, Nature 400,239 (1999). J-M. Raimond, M. Brune and S. Haroche, Rev. Mod. Phys. 73, 565 (2001). J. Gea-Banacloche, Phys. Rev. A 44,5913 (1991). J. Eberly, N. Narozhny, J. Sanchez-Mondragon, Phys. Rev. Lett. 44, 1323 (1980). V. Buzek and P. Knight, in Progress in Optics XXXIV, Vol34, Elsevier (1992). G. Rempe, H. Walther and N. Klein, Phys. Rev. Lett. 58, 353 (1987). M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J-M. Raimond and S. Haroche, Phys. Rev. Lett. 76, 1800 (1996). D.M. Meekhof, C. Monroe, B.E. King, W.M. Itano and D.J. Wineland, Phys. Rev. Lett. 76, 1796 (1996). A. Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J-M. Raimond and S. Haroche, to be published (2003). P. Bertet, A. Auffeves, P. Maioli, S . Osnaghi, T. Meunier, M. Brune, J-M. Raimond and S. Haroche, Phys. Rev. Lett. 89,200402 (2002).

VACUUM-FIELD MECHANICAL ACTION ON A SINGLE ION JURGEN ESCHNER, PAVEL BUSHEV, ALEX WILSON, FERDINAND SCHMIDTKALER, CHRISTOPH BECHER, CHRISTOPH RAAB AND RAINER BLA'lT Institut fur Experirnentalphysik, Universitat Innsbruck 6020 Innsbruck, Austria Email Juergen. [email protected] In an experiment with a single trapped barium ion we demonstrate that a laser-excited atom whose surrounding electromagnetic vacuum field is modified by a distant mirror experiences vacuum-induced forces which notably change its trapping conditions.

The mechanical action of radiation has long been known. Being the basis for laser cooling and optical trapping, it is one of the most important tools for quantum optical experiments with atoms. The trapping of atoms in few-photon fields of high-finesse optical cavities has been experimentally demonstrated'. It is also known that the electromagnetic vacuum, with no photons present that an atom could absorb, may still exert a force on the atom and, in principle, trap it2. We demonstrate this effect with a single barium ion whose surrounding vacuum field is modified by a single mirror at 25 cm distance. 493 nm mifror

on pezo

-

Vacuum wffldow

Da2

SW2-

Ba+ ion in Count rate vs. mirror distance

Sideband in FFT spectium

Figure 1. Two lasers at 493 nm and 650 nm excite a single trapped '"Ba+ ion continuously. Lens L1 (f-number 1.1, wave front distortion < 2 5 ) and a retro-reflecting mirror (25 cm from trap) are arranged so they image the ion onto itself, thus leading to high-contrast interference fringes in the PMT signal as the ion-mirror distance is vaned. The mirror reflects only 493 nm light. The 493 nm photons detected by PMT are counted in 0.1 s intervals. The intensity modulation of the PMT signal (motional sideband) due to the ion's trap oscillation at 1.02 MHz is detected with an FFT spectrum analyzer.

273

274

In our experiment (Fig. l), a single '38Ba+ion is held in a Paul trap and continuously excited and cooled by near-resonant lasers. A collimating lens and a distant mirror are placed such that part of the ion's resonance fluorescence is retro-reflected, thus leading to inhibited and enhanced spontaneous emission3,as well as to an energy shift of the excited level, W ( z ) ,which is a function of the ion-mirror distance, z, see Ref. 4:

SU = -h-

E r

sin(2kz) (1) 2 Here r is the decay rate of the upper level and k = 2n7'A is the wave vector of the light with wavelength A. Ths potential creates different mechanical effects: around sin(2kz) = 0, a force is exerted on the ion which points either towards or away from the mirror; around sin(2kz) = *l, a binding (-1) or anti-binding (+1) potential is formed. Since the vacuum field couples only to the excited level, the forces are scaled by the probability, P,, for the atom to be in the excited Pl12 state. The maximum force at sin(2kz) = 0 is therefore calculated as P,Ak&T. The bindindanti-binding potential at sin(2z) = *1 is characterized by the oscillation frequency which an otherwise force-free atom would have in the respective potential well, q,uc = (2Pe&rtrk2/m)'",with atomic mass m. For an ion with trap frequency wrap,m/(z) adds to the trapping potential and changes the trap frequency by a small amount 6wru,,(z) which is well approximated by P, el? #.k2 aWtrup(Z) sin(2kz) (2) wtrap

We measure this variation of the trap frequency with < 10 Hz error, by positioning the ion with run-accuracy relative to the mirror and detecting the intensity modulation in the scattered light due the ion's oscillation in the combined potential of trap and vacuum. Figure 2a shows two spectra which were recorded directly one after the other with the ion positioned on the midpoints of a positive and negative slope of the interference signal (count rate versus ion-mirror distance). The shift is clearly visible and amounts to 3 10 Hz in this case. The value is within the range expected from Eq. 2, which predicts up to 350 Hz, talung values P, =5%, E -2%, A = 493 nm, and r = 2n x 15.4 MHz. We emphasize that no changes are made to the setup between recording the two spectra, apart from translating the distant mirror by 2 4 . A typical measurement of trap frequency vs. time, alternating between the two slopes, is shown in Fig. 2b. Whle the trap frequency itself varies due to slow drifts of the trap drive intensity and due to thermal effects, a constant difference is observed between the values on the two slopes. Depending on details of the experiment such as the settings of the lasers, their directions, and the fine alignment of the back-reflecting mirror, we observe values between 50

275

-70

-

m

s

._ C -72 -

t

U

5 -74-

z

-ae -76-

H

0-l -781017

1018

1019

1020

Frequency (ktlz)

1021

O

l 0

j

,

, 5

,

, . , . , . , . , I 10

I5

20

25

30

Time of measurement (mm)

Figure 2. (a) Signal on the spectrum analyser for the ion positionedgn the positive (right curve) and negative slope (left curve) of the interference signal. The centre frequency of a Lorentzian fit to the data is taken as the trap frequency. The broadening of the lines is a consequence of the ongoing laser cooling. The size of the Lorentzian curve above the Poissonian noise level serves as a measure of the amplitude of the trap oscillation. (b) Trap frequency measured on the positive (full circles) and negative slope (open circles) of the interference signal versus measurement time.

and 350 Hz, all within the range expected from Eq. (2). It is important to note that we always find the higher trap frequency on the positive slope of the interference fringes, in agreement with the theoretical prediction4. While the data in Fig. 2 were taken at the midpoints of the interference h g e s , Fig. 3 shows the dependence of the trap frequency on the ion-mirror distance. It has been obtained by stabilizing the ion's position to a certain average signal on the PMT. The result agrees well with the expected sinusoidal behaviour. The calculated level shift and change of decay rate of the PI12 state are also shown. By varying the laser power, we also measured the trap

Figure 3. Measured trap frequency (data points) at variation of ion-mirror distance. (The total ionmirror distance is about 25 cm.) The solid line is a sinusoidal fit. This data set showed particularly small drifts of the trap frequency. The dotted line is the frequency shift of the PL,2state, and the dashed line is the modification of its decay rate; these curves are calculated from the fit using the relations given in the text.

276 frequency shift vs. the mean occupation probability of the P112state, P,, and found the linear dependence expected from Eq. (2).

References 1.

P. W. H. Pinkse et al., Nature 404, 365 (2000); C. J. Hood et al., Science 287, 1447 (2000).

2. 3. 4.

S. Haroche et al., Europhys. Lett. 14, 19 (1991); C. Schon and J. 1. Cirac, Phys. Rev. A 67,043813 (2003). J. Eschner et al., Nature 413,495 (2001). U. Dorner and P. Zoller, Phys. Rev. A 66,023816 (2002).

COUPLING OF ATOMS, SURFACES AND FIELDS IN DIELECTRIC NANOCAVITIES G. DUTIER, I. HAMDI, P.C.S. SEGUNDO", A. YAROVITSKIb, S . SALTIEL', M.-P. GORZA, M. FICHET, D. BLOCH AND M. DUCLOY Laboratoire de Physique des Lasers, UMR7538 du CNRS Universite' Paris 13, F-93430 Villetaneuse, France E-mail: [email protected]

D. SARKISYAN, A. PAPOYAN AND T. VARZHAPETYAN Institute for Physical Research, Armenian Academy of Sciences Ashtarak-2, 378410 Armenia The high sensitivity of Laser Spectroscopy has made possible the exploration of atomic resonances in newly designed "nanomemc" vapour cells, whose thickness lies in the 201000 nm range. In addition to the observation of the optical analogue of coherent Dicke microwave narrowing that provides a linear sub-Doppler spectral response, it offers ways to explore the long-range atom-surface van der Waals interaction with spatial resolution, and to probe an unexplored range of distances. We are also developing an analysis of the influence over the atom-surface interaction of the near-field thermal excitation of the vacuum, as induced through the vicinity of a "hot" surface.

1. Optical Spectroscopy: from Thin Cell of Dilute Vapour to Nanocell When a cell of dilute vapour is thin enough to make the "mean" free path anisotropic (atoms flying from wall to wall), the transient build-up of the resonant interaction with light is responsible for a specific enhancement of the response of the slowest atoms. This has provided the principle of a novel method for Doppler-free spectroscopy, which is applicable to a variety of situations: velocity-dependent optical pumping, linear absorption, two-photon transitions, etc. (see Ref. 1 and references therein). However, in these experiments, the cell thickness (10-1000 pm) remained much larger than the optical wavelength A. For this reason, the amplitude of the specific sub-Doppler signature was limited, so that the preferred mode of observation uses a frequency modulation (FM). Also, imperfect parallelism of the windows made a

Permanent address : Universidade Federal de Paraiba, JoZo Pessoa, Brazil. Permanent address : Lebedev Institute of Physics, Moscow, Russia. Permanent address : University of Sofia, Bulgaria. 277

278

interferometric effects not observable, and the atom-surface interaction was neglected. With the recent fabrication of extremely thin cells (ETC) of dilute vapour,* whose thickness typically varies from 20 rim to 1 pm, new prospects are opened for sub-Doppler spectroscopy, with the additional possibility of detecting atom-surface interaction effects in an unexplored range of distances. With ETC, we have extended to the optical domain3 the original observation by Romer and Dicke4 in the microwave domain of a coherent spectroscopic narrowing in a gas sample with a hckness 1 2 . Let us recall that when an atom leaves the wall, it is suddenly excited, precessing in phase with the electromagnetic field at the wall position. Because of the atomic motion, this excitation gets out-of-phase with the local driving field, the phase mismatch finally reaching kL (where L is the cell thickness and k the wave number). Hence, one predicts a periodical revival of the Dicke narrowing, that we succeeded in observing for a cell thickness close to 3iu2.3 In the elementary case of a a2 cell and with the addition of a FM technique, the signal narrows down to a width essentially related to the homogeneous width, with no reduction in the peakamplitude if the FM amplitude is comparable to the optical width. Owing to its observation in the linear regime of atom-light interaction, such a sub-Doppler narrowing may be particularly appealing to obtain frequency references with weak molecular transitions that are not suitable for saturated absorption spectroscopy. As a major difference with spectroscopy previously developed in a thin cell, the probed atoms are as close as those usually probed (i.e., 1 2 7 4 in selective reflection (SR) spectroscopy (see, e.g., Ref. 5 and references therein), so that the long-range atom-surface interaction is susceptible to induce observable effects. Note also that to realise an ETC, the parallelism of the windows must be excellent (e.g., deviation < 1 pm over a 10 mm transverse extension), unavoidably generating Fabry-PCrot (FP) type effects. On the one hand, this has provided an efficient method to measure the local cell thickness, e.g., through the measurement of the reflection coefficient of the cell at various wavelengths (see Ref. 3), and on the other hand, this implies that any spectroscopic observation in the ETC is actually the result of an intricate mixing of transmission and reflection.6

-

2. Observing Atom-Surface Van Der Waals Interaction at Small Distances The ETC structure confines the atom in a distance range where the surface interaction, notably the long-range van der Waals (vW) atom-surface interaction with its z - ~dependence (where z is the atom-surface distance), can induce large

279 effects. In a first step, we have performed a systematic analysis of the transmission spectrum of the Cs D, resonance line ( A = 894 nm). When the local cell thickness is very small (- 50-100 nm, accurately controlled to f 5 m),we have notably demonstrated a lineshape distortion and red-shift (up to 200 MHz) that largely exceeds those observed with SR spectroscopy. The observation includes either the direct transmission mode (most often in a differential detection scheme to enhance the sensitivity) or the FM mode, of notable interest for larger thickness when the overall VW shft remains small. The spectral lineshapes are in good agreement with theoretical modelling, which takes into account an integration over atomic trajectories of the van der Waals potential described, in an electrostatic model, as the interaction with the multiple images induced in the two reflecting walls. Alternately, at these small distances, the reflection spectra, recorded over a low level of non-resonant reflection as due to FP interferences, offer a competitive possibility to observe the VW shft (see Fig. 1).

. .. ...

.. .. .. *.


13 and algorithms7. In our experiment, we first create a Bell state by applying laser pulses to ion 1 and 2 on the blue sideband and on the carrier transition. To produce = l/fi(lS, D)fID,S))we use the pulse sequence U*, = the Bell state @(n, fn/2) . Rz(n, n/2) . R?(n/2, -n/2) applied to the IS, S) state. The first pulse Rf(n/2, -n/2) entangles the motional and the internal degrees

292

Time (p)

+

Figure 3. Right: Joint probabilities for the ions prepared initially in I S D , S). The controlled-NOT gate operation is performed according to the table of laser pulses. The data points represent the probability for the ion string to be in the state indicated on the right-hand side by the corresponding CCD image. The measurement procedure is the same as in Fig. 2. Left: CCD images of the fluorescence of the two-ion crystal as measured in different logic basis states: ISS), [SO),IDS), and [ D O ) . The ion distance is 5.3 fim.

._

s o

a

-0.5

90

180

270

I

360

Phase I$ (deg)

+

Figure 4. The quantum register is initially prepared in the superposition ( S D , D ) . After the control-NOT gate operation we obtain an entangled output state. To analyze the entanglement, we apply after the gate operation to both ions n/2-pulses on the carrier transition, with a phase 4, and measure the parity P = Pss PDD- (PSD P D S )as a function of the phase. The quantum nature of the gate operation is proved by observing oscillations with cos 24, whereas a non-entangled state would yield a variation with cos 4 only. The observed visibility is 0.54(3).

+

+

of freedom. The next two pulses R ; ( ~ T~,/ 2 ) . R 2 ( 7~r ,/ 2map ) the motional degree of freedom onto the internal state of ion 2 . Appending another Tpulse on the carrier transition frequency, R ~ ( TO),, to the sequence UQ* produces the state @&. The pulse sequence takes less than 200 ps. Our tomographic method consists of individual single bit rotations, followed by a projective measurement. For the analysis of the data, we employ

293 a maximum likelihood estimation of the density matrix following the procedure as s ~ g g e s t e d ' ~and > ~implemented ~ in experiments with pairs of entangled photons16. As an example, Fig. 5 shows the reconstructed density matrix p of one out of four Bell states. To monitor the evolution of these

Figure 5 .

Real and imaginary part of the density matrix p a , that approximates fidelity is F= ( @ + ( p a + 1 9 + ) = 0.91.

l/t/z(\S,S + D , D)). The measured

entangled states in time we introduce a waiting interval before the state tomography. We expect that Bell states of the type @@ = IS, D) ei0(D,S) are immune against collective dephasing due to fluctuations of the qubit energy levels or the laser freq~ency'~.However, a magnetic field gradient that gives rise to different Zeeman shifts on qubits 1 and 2 leads to a linear time evolution of the relative phase eiP between the IS,D) and the component of the Q* states. We plot the maximum overlap F, = r n a s { ( @ , ~ l pIQ'p)} ~ , for the reconstructed density matricies as a function of time, see Fig. 6. A coherence time of a few ms is reached. In some measurements, the lifetime of the @* states exceeds 20 ms. Finally, we specify the entanglement of the four Bell states, using the entanglement of f ~ r m a t i o n ' ~and , find E(Q-)=0.79(4), E(@+)=0.75(5),E(@+)=0.76(4) and E(@- ) =O. 72(5).

+

10,s)

5. Conclusion

On the way towards a scalable quantum processor18, we have shown singlequbit rotations and a universal two-qubit gate operation. Entangled quantum states of two qubits have been fully characterized by their density matrix. We have found coherence timedg which exceed by far the time which is used for the entanglement operation.

294 1

0.9

0 0.8 -

$ 0.7

0.6

0.5

0

1

2

3

4 Time (ins)

Figure 6. Decay of the measured fidelity is F, part of the density matrix after 2 ms and 8 ms.

5

6

7

8

= r n a ~ { ( Q ~ l p ~ ~ I ’ l ”Inset: p ) } . Real

Acknowledgments We thank I. L. Chuang, Media Lab, MIT, Boston, for his important contributions. This work is supported by the Austrian ‘Fonds zur Forderung der wissenschaftlichen Forschung’, by the European Commission, and by the ‘Institut fur Quanteninformation GmbH’.

References 1. J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995). M. Sasura, and V. Buzek, J . Mod. Opt. 49, 1593 (2002). F. Schmidt-Kaler, et al., Nature 422, 408 (2003). H. Rohde, et al., J. Opt. B 3, S34 (2001).

2. 3. 4. 5. 6. 7. 8. 9.

C. Roos, et al., Phys. Rev. Lett. 83,4713 (1999). C. Monroe, et al., Phys. Rev. Lett. 75, 4714 (1995). S. Gulde, et al., Nature 421, 48 (2003). C. A. Sackett et al., Nature 404, 256 (2000). D. T. Smithey, M. Beck, M. G. Raymer and A. Faridani, Phys. Rev. Lett. 70, 1244 (1993). 10. D. Leibfried et al., Phys. Rev. Lett. 77, 4281 (1996). 11. Ch. Kurtsiefer, T. Pfau and J. Mlynek, Nature 386, 150 (1997). 12. I. L. Chuang, N. Gershenfeld, M. G. Kubinec and D. Leung, Proc. R. Soc. London A 454, 447 (1998). 13. D. Leibfried, et al., Nature 422, 408 (2003). 14. Z. Hradil, Phys. Rev. A 5 5 , R1561 (1997). 15. K. Banaszek, G. M. D’Ariano, M. G. A. Paris and M. F. Sacchi, Phys. Rev. A 61, 010304 (1999). 16. D. F. V. James, P. G. Kwiat, W. J. Munro and A. G. White, Phys. Rev. A 64, 052312 (2001). 17. W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). 18. Quantum Information Science and Technology Roadmapping Project (ARDA), http://qist .lanl.gov/ 19. F. Schmidt-Kaler, et al., J. Phys. B: At. Mol. Opt. Phys. 36, 623 (2003).

BUILDING BLOCKS FOR A SCALABLE QUANTUM INFORMATION PROCESSOR BASED ON TRAPPED IONS

D. LEIBFRIED, M. D. BARRETT, A. BEN KISH, J. BRITTON, J. CHIAVERINI, B. DEMARCO, w. M. ITANO, B. JELENKOVIC, J. D. JOST, C. LANGER, D. LWCAS, V. MEYER, T. ROSENBAND, M. A. ROWE, T. SCHAETZ AND D. J . WINELAND National Institute of Standards and Technology, 325 Broadway, Boulder, CO 80305, USA E-mail: dilQboulder.nist.gov We describe the underlying concept and experimental demonstration of the basic building blocks of a scalable quantum information processor architecture using trapped ion-qubits. The trap structure is divided into many subregions. In each several ion-qubits can be trapped in complete isolation from all the other ion-qubits in the system. In a particular subregion, ion-qubits can either be stored as memory or subjected to individual rotations or multi-qubit gates. The ion-qubits are guided through the array by appropriately switching control electrode potentials. Excess energy that is gained through the motion of ion-qubits in the array or other heating mechanisms can be removed by sympathetic cooling of the ion-qubits with another ion species. The proposed architecture can be used in a highly parallel fashion, an important prerequisite for fault-tolerant quantum computation.

1. Basic Concept 1.1. Original Cimc/Zoller Architecture

Quantum information processing with trapped ion-qubits was first proposed by J. I. Cirac and P. Zoller in 1995l. The original architecture consisted of a string of ions lined up in a linear quadrupole ion-trap. In each ion two long-lived electronic levels are used to implement a qubit. For all gate manipulations the ion-qubits are individually addressed with focussed laser beams. Single qubit rotations are performed with laser pulses exciting resonant transitions between the internal levels of the ion-qubit in question. Two-qubit gates use one normal mode of vibration of the ion string as a means to couple the possibly distant partners in the gate'. To this end all normal modes of vibration should be cooled close to the ground state before the algorithm starts. 295

296

This proposal stimulated a new field in trapped ion research, but it soon became clear that it is difficult to scale the original architecture to more than a few ions. In particular, confining a linear string of thousands of ions in one trap would lead to unrealistically high voltages on the endcap electrodes. Also, for a fixed value of the lowest normal mode frequency, the distance between neighboring ions decreases as the ion number increases. Keeping this distance above the diffraction limit of the addressing laser beam requires rather low motional frequencies for a large string. This is in conflict with other requirements, for example, ground state cooling, which has only been demonstrated at higher motional frequencies. Low motional frequency also limits gate speeds for the computation (In the original proposal the gate rate has to be below the lowest motional frequency). Finally, from a practical point of view, the emergence of 3N normal modes for N ions plus their sum and difference frequencies leads to an increasingly crowded excitation spectrum where components are difficult to identify and off-resonant coupling to parasitic transitions is hard to avoid.

1.2. Multiplexed Trap Architecture

In 1998 we proposed a multiplexed trap architecture2i3 that might alleviate the problems described above and is modular, so scaling to higher qubit numbers seems to be more feasible. Other schemes have also been proposed The basic idea is to expand the orignal architecture to an array of many independently controllable subtraps that hold the ion-qubits in certain configurations at each stage of the algorithm (see Fig. 1). Qubits that do not partake in given step of the algorithm are stored in memory regions. To execute a gate on certain qubits, they are separated from other ions in the memory regions and shifted into a “processor” region. Moving ion-qubits around does not lead to decoherence in the computational Hilbert-space spanned by the qubits since the motion is only used for coupling two qubits during the gate. Once the gate is completed, the motion factors out and no entanglement with the computational Hilbert-space is left. Since the electrostatic forces controlling the ions in the array do not couple to the internal qubit states, the computation is not affected by the movement. The only relevant phases are brought about by the fact that the ions could be illuminated at different spatial positions in the array by the laser beams, but these phases depend only on the respective positions and not on the exact trajectory the ion took from one place in the array to another. Through movement in the array, the ion-qubits may gain some

297 to additional memory or processor units

0

I o

qubit-ion cooling ion

Figure 1. Multiplexed trap architecture. An array of independently controllable subtraps holds the ion-qubits. Qubits that are not involved in a given step are held in a memory region (m). Before performing a gate on a certain pair of qubits, they are shifted into a “processor” unit (p), and sympathetically recooled with another ion species. Singlebit rotations or ancilla readouts can be performed in any region of the array as long as the qubit in question is sufficiently isolated from the remaining qubits (for example in

b)).

excess motional energy, but they can be sympathetically recooled close t o the ground state with another ion species (a “refrigerator” ion) before the next gate is applied. Due to the strong Coulomb-coupling one can simultaneously laser cool all ions trapped in the same subtrap. The cooling laser interacting with the “refrigerator” ion can be far detuned from all qubit transitions and therefore cause negligible phase and spin-flip errors of the qubits, even if they are directly illuminated with the cooling light. Sympathetic recooling will also remove motional energy that the ion-qubits might acquire due to other heating mechanisms in the trap6. The heating time constant is estimated to be on the order of one quantum per second at our typical motional frequencies for the most benign source, blackbody radiation of the room temperature experimental apparatus2. In the small traps used in our experiments we empirically find a much larger heating rate with time constants on the order of several milliseconds6. With sympathetic recooling, an even larger heating rate would not limit computation time, so algorithms that run for much longer times could be implemented. In this case the time available for a computation would be limited only by the decoherence of the internal qubit states. The lifetime of hyperfine

298

ground states is extremely long (many years), so the memory decoherence of qubits composed of such states is primarily due to phase errors induced by external perturbations, e.g. magnetic field fluctuations. In a carefully controlled environment decoherence timescales could theoretically be on the order of many days with experimentally demonstrated lower limits of several minutes (see, e.g., Ref. 7). A final advantage of the array architecture is the ability to read out some qubits without perturbing other qubits (for example, ancillae within a round of error correction). Since the readout of an ion-qubit usually requires approximately lo4 scattered photons and the fractional solid angle subtended by the scattering cross section of a neighboring ion is on the order of X2/(87r2d2)2 (assuming X N 313 nm is the wavelength of the scattered light and d 5 5 pm is the inter-ion distance) it is not possible to perform many ancilla readouts without compromising neighboring ionqubits in a string through rescattered photons. In the array architecture these readouts might instead be performed in areas that are sufficiently spatially isolated from the remaining qubits. All steps described above can be done in a highly parallel fashion, an important prerequisite for efficient error correction. Scaling to many, possibly thousands of ions is technically challenging, but seems possible without fundamental problems. To demonstrate the feasibility of this architecture one has to experimentally demonstrate the following basic building blocks: (1) Build trap arrays: Arrays containing several independent sub-traps

and eventually also crossings and/or “T”-junctions must be constructed in a precise and reproducible fashion. The dimensions and techniques used must be scalable to large arrays that are able to hold and control thousands of ions. (2) Move ions: Ion movement in a trap array must be reliable and repeatable without gaining too much excess energy. The typical timescales of these movements should not substantially limit the speed of the algorithm. (3) Ability to separate and recombine ions: To execute quantum logic gates, certain ions have to be picked out of the memory regions reliably and combined with another ion-qubit and the refrigeratorion in the processor unit. The typical timescale of these processes should also not substantially limit the speed of the algorithm. (4) Sympathetic recooling: Excess kinetic energy of the ion-qubits, brought about by their movement in the array, by external heat-

299

ing mechanisms, or by the recoil suffered in ancilla-readout steps, can be removed by sympathetic laser cooling with a second ion species (the refrigerator ion). Re-cooling must leave the ions sufficiently close to the motional ground state and should not disturb the qubit. (5) Robust single-qubit and two-qubit gates: For extended algorithms it will be necessary to reach gate fidelities on the order of 0.9999. All single bit and two-qubit gate mechanisms considered should reach this fidelity with technically realistic improvements and still be compatible with the other features of the architecture, for example, the presence of a “refrigerator” ion during gate operation. The experiments performed so far at NIST to implement these basic building blocks will be briefly described in the next section. 2. Experimental Demonstrations

2.1. Building h

p Arrays

In the last two years we have built and characterized three multi-zone traps, two of them with the ability to load and hold ’Be+ and 24Mg+ simultaneously. The sub-traps are aligned along one common axis and the traps had 3, 5 and 6 trapping zones respectively. The two larger arrays had a dedicated loading zone to shield the other zones from plating with neutral Be and Mg from the ovens. We were able to cool Be+ to the ground state in all three traps and observed heating rates of 1 quantum in 10 ms at 2.9 MHz axial trap frequency for the 3-zone trap’, 1 quantum in 1 ms at 4.5 MHz for the 5-zone trap, and 1 quantum in 5 ms at 4.1 MHz for the 6-zone trap. 2.2. Moving Ions

We transferred a single ion between two traps 1.2 mm apart by continuously changing the potentials on five pairs of control electrodes We initially prepared the ion in the motional ground state. Using numerical solutions for our trap geometry, trap potentials were designed so that during the transfer all motional frequencies would be held constant. After a hold period in the target trap the transfer process was reversed. Following the transfer back to the starting trap, we measured the average gain in occupation number of the axial motional mode. For a transfer time of 28 p s and a trap frequency of 2.9 MHz, about half a quantum on average was gained

’.

300

from the transfer. From a numerical integration of the classical equations of motion, we expected that the ion should gain an amount of energy equal to one motional quantum for a 30 p s transfer duration. This estimate indicates approximately when the transfers are no longer adiabatic and agrees reasonably well with our observations. Overall, this transfer process is robust in that we have not observed any ion loss due to transfer. We also verified that the coherence of the internal qubit-states was not affected by the transfer’.

2.3. Separating I o n s We separated two ions from a common trap-well into two separate wells 300 pm apart by continuously changing the potentials on five pairs of control electrodes. For separation, the common trap potential is relaxed, so the Coulomb repulsion drives the ions apart until a distance is reached where an external electrostatic “wedge” can be ramped up between the two ions to separate them into independent sub-traps. At one point during this process the external potential is essentially flat, leading to rather small motional frequencies of the ions. At this point the ions are most susceptible to external heating and to field errors due to imperfect trap geometry and stray charges. It is therefore essential to keep this minimum frequency as high as possible. Ideally one would like to have a very sharp wedge, but that is only possible if the trap electrodes are small and close to the ion. In the 6-zone trap we included a pair of electrodes with a width of 100 pm and about 140 pm distance from the ion. Ramping the potential on this electrode pair creates the electrostatic wedge that separates the ions into two adjacent zones over two electrode pairs 200 pm wide. In preliminary experiments in this array we were able to separate two ions in about 2 ms reliably. The time varying potentials on all electrodes were designed to be in the adiabatic regime with a minimum oscillation frequency of about 350 kHz. A heating measurement after separation yielded that the motional energy increased by less than 10 quanta during the separation. Current efforts are devoted to increasing the speed and reducing the heating during separation. 2.4. Sympathetic Recooling

Sympathetic recooling could be a crucial step to extend the capability of quantum information processing with trapped ions to timescales much longer than the time constant for motional heating in the ion traps. It

301

can also serve to remove the excess kinetic energy of ion-qubits that might be produced by their movement in the array or by the recoil suffered in ancilla-readout steps. Recooling must leave the ions sufficiently close to the motional ground state so that quantum gates are not limited in fidelity by the fluctuations of the thermal state produced by cooling. For the gate mechanism used in the two-ion experiment described below, 99% ground state cooling on all modes will lead to a fidelity of about 0.9999. Sympathetic cooling has previously been demonstrated using "refrigerator" ions that are the same as the qubit ionsg or an isotope of the qubit ions". In order to gain higher immunity from decoherence caused by stray cooling light, we have chosen a different ion species for the refrigerator ion. In our sympathetic cooling experiment we trapped and Doppler cooled one gBe+ and one 24Mgs ion in one trap, with 2.0 MHz and 4.1 MHz axial normal mode frequencies. We then cooled either the Mg+ or the Be+ ion close to the ground state using interspersed red-sideband Raman pulses and resonant repumping pulses". Finally the average occupation number A was determined by comparing red and blue sideband strengths of both modes on the Be+ ion. When cooling on the Mg+ ion we were technically limited by our Raman-detuning and achieved f i = 0.19(6) and A = 0.52(7) on the two normal modes, respectively. This result will be improved in the future by implementing higher Raman-detuning. Cooling the two ions through the Be+ ion (where the detuning is large) yielded a limit of A = 0.03(2) and A = 0.04(3)11.

2.5. Robust One and Two-qubit Gates

Single qubit rotations can be executed in any region of the array where the selected qubit is sufficiently isolated from all other qubits. For one-qubit spin-flip gates we use stimulated Raman transitions where the two laser beams have a parallel wave-vector. This makes these rotations highly immune to the motional state of the ions and to any common fluctuations of the beam path of the two Raman beams. We have demonstrated nrotations with a lower fidelity limit of 0.99. Higher fidelity seems possible with improved intensity stability of the Raman-beams (currently,fluctuating by about 1% ) and by reducing the magnetic field sensitivity of our qubit states. Beryllium ions offer a qubit transition that is first-order magnetic field independent at about 120 G. Working at this field should reduce the sensitivity to fluctuating magnetic fields by at least two orders of magnitude. One ion-phase gates (2-rotations) need not be executed by

302

laser pulses; they can be incorporated by adjusting the phase of subsequent spin-flip gates. We recently demonstrated a two-qubit geometric phase gate utilizing a state-dependent dipole force. In our implementation, we coherently excited the motion of two ion-qubits along a closed path in motional phase space if they were in different internal states, while they were not excited if they were in the same state12. For different internal states the total state of the ions picked up a phase proportional to the phase-space area circumscribed, leading to the following truth table (I 1) and I T) denote the qubit logical states):

I Tt)

-+

I Tt)

*

The gate is universal and can be converted into a n-phase gate or a CNOTgate with single bit rotations for 4 = n/2. Starting with the state I 11) and sandwiching this gate between n/2 and 3x12 pulses applied to both ions, we were able to produce maximally entangled states of the form I$) = 1/fi(I 11) il TT)) with a fidelity of 0.97. Under the reasonable assumptions that the error of the gate operating on I i t ) is of equal magnitude to the one on I TL) and much larger than the errors on I 11) and I TT) (which are not excited by the gate pulse), the fidelity of producing the maximally entangled state can be shown to be equal to the gate fidelity. Individual ion addressing is not required during this gate and the accumulated phase depends only on the path area, not on the exact starting state distribution, path shape, orientation in phase space, or the time it takes to traverse the closed path. Thus within the Lamb-Dicke regime, ground state cooling is not required for accurate gate operations. The main sources of gate error in our experiment are fluctuations in the trap frequency and fluctuations in the Raman-beam intensity, both roughly at the 1% level, and a spontaneous emission probability of about 2.2% for each gate operation. If frequency drift and intensity errors could be reduced to order and spontaneous emission suppressed (i.e., by using a different ion specie^'^), the expected gate fidelity is on the order 0.9999. In the future, with a refrigerator-ion present in the processor trap, the normal-mode amplitudes of each ion will be different, making it technically more difficult to obtain equal laser beam couplings, as required in

+

303

the S ~ r e n s e n / M ~ l m gate14. er Equal coupling is not required for a general geometric phase gate since the extra phases on each qubit can be absorbed into previous or subsequent single-qubit rotations.

3. Conclusions and Outlook In the last two years, all basic building blocks for a scalable architecture of a quantum information processor with trapped ion-qubits have been individually experimentally demonstrated. Although it will be a nontrivial technological challenge, no fundamental problems seem to prohibit sealing to many qubits. It also appears technically feasible to reach the fault tolerant level with the demonstrated one- and two-qubit gates. Therefore trapped ion-qubits remain a promising candidate for the implementation of large-scale quantum information processing.

Acknowledgments The work described in this paper was supported by ARDA/NSA and NIST.

References 1. J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995). 2. D. J. Wineland et al., J . Res. Nat. Znst. Stand. Technol. 103,259 (1998). 3. D. Kielpinski, C. Monroe, and D. J. Wineland, Nature 417, 709 (2002). 4. R. G. DeVoe, Phys. Rev. A 58,910 (1998). 5. J. I. Cirac and P.Zoller,Nature 404, 579 ( 2000). 6. Q. A. Turchette et al., Phys. Rev. A 61, 063418-1 (2000). 7. J. Bollinger et al., ZEEE Trans. Znstrum. Meas. 40, 126 (1991). 8. M. A. Rowe et al., Quantum Inf. Comput. 2, 257 (2002). 9. H. Rohde et al., J . Opt. B 3,34 ( 2001). 10. B. B. Blinov et al., Phys. Rev. A 65,040304 (2002). 11. M. D. Barrett et al., submitted to Phys. Rev. A ; quant-ph/0307088(2003). 12. D. Leibfried et al., Nature 422, 414 (2003). 13. D. J. Wineland et al., Phil. Trans. R . SOC.Lond. A 361,1349 (2003). 14. A. Sgrensen, and K. Malmer, Phys. Rev. Lett. 82,1971 (1999).

CONTROLLED TRANSPORT OF SINGLE NEUTRAL ATOM QUBITS

D. SCHRADER, S. KUHR, W. ALT, Y. MIROSHNYCHENKO, I. DOTSENKO, W. ROSENFELD, M. KHUDAVERDYAN, V. GOMER, A. RAUSCHENBEUTEL AND D. MESCHEDE Institut f u r Angewandte Physik, Universitat Bonn, Wegelerstr. 8, 0-53115 Bonn, Germany E-mail: schraderOiap.uni- bonn.de We have prepared and detected quantum coherences of trapped cesium atoms with long dephasing times. Controlled transport by an “optical conveyor belt” over macroscopic distances preserves the atomic coherence with slight reduction of coherence time. The dominating dephasing effects are experimentally identified and found to be of technical rather than fundamental nature.

1. Introduction

Obtaining full control of all internal and external degrees of freedom of individual microscopic particles is the goal of intense experimental efforts. The long lived internal states of ions and neutral atoms are excellent candidates for quantum bits (qubits), in which information is stored in a coherent superposition of two quantum states. The engineered construction of quantum systems of two or more particles can be implemented by transporting selected qubits into an interaction zone. For this purpose we demonstrate the quantum state preparation and transportation of neutral atoms in a dipole trap which exhibit coherence times of up to -200 ms 2. Preparation of Neutral Atom Qubits

We trap cesium atoms in a standing wave dipole trap (A = 1064nm) with a potential depth of Uo = 1mK, loaded from a high-gradient magneto-optical trap (MOT) The MOT is also used to determine the exact number of trapped atoms by observing their fluorescence. The single-atom transfer efficiency between the two traps is better than 95%. By shifting the standing wave pattern along the direction of beam propagation, we can transport the 213.

304

atoms over millimeter-scale distances. This is realized by mutually detuning the frequencies of the dipole trap laser beams with acousto-optical modulators. Additionally, we use microwave radiation at u h f s / 2 r = 9.2 GHz to coherently drive the I F = 4, m F = 0 ) 4 I F = 3, m F = 0 ) clock transition of the Ci2S1/2 ground state with Rabi frequencies of 52127~= 10 kHz. The initial state is prepared by optically pumping the atom into I F = 4, m F = 0 ). A state selective detection method, which discriminates between the atomic hyperfine states, is implemented by exposing the atom to a a+-polarized laser, resonant with the F = 4 +F‘ = 5 cycling transition. It pushes any atom in F = 4 out of the dipole trap, whereas an atom in F = 3 remains trapped The number of atoms remaining in the dipole trap is then determined by transferring them back into the MOT. Coherent superpositions of qubit states are prepared by driving the transition between the I F = 4, m F = 0 ) and the I F = 3, m F = 0 ) hyperfine states with a resonant 7r/2 microwave pulse. After a second pulse of variable delay (“Ramsey spectroscopy”) we detect the atomic state. Using this technique, we measured transverse relaxation times on the order of 1-30 ms depending on the depth of the trapping potential. This dephasing is caused by an inhomogeneous distribution of light shifts due to the temperature distribution of the atoms in the dipole trap 4. In analogy to NMR-techniques, we reversed the inhomogeneous dephasing of the atomic spins by means of a r-pulse in between the n/2-pulses. We observed a pronounced spin echo for pulse delays of up to 300 ms. We found that the decay of the echo visibility is due to the pointing instabilities of our dipole trap laser. Any change of the relative position of the two interfering laser beams changes the interference contrast and hence the light shift, causing an irreversible dephasing of the atomic coherence.

’.

3. Quantum State Transport

Finally, we demonstrate the controlled quantum state transportation of neutral atoms. We show that the atomic coherence persists while moving the atoms back and forth over macroscopic distances by shifting the standing wave dipole trap. For this purpose, we essentially perform a spin echo measurement, with the addition that the atoms are transported between the microwave pulses. The sequence is visualized in Fig. l(a). The corresponding spin echo signal is also shown in Fig. 1, together with a reference signal without transportation (b). The spin echo prevails if we transport the atom between the microwave pulses, however with slightly reduced vis-

306

ibility, see Fig. l(c). This can be attributed to an energy gain of the atoms due to abrupt acceleration of the potential during transport 3 .

d2pulse

=: 2

c)

time'

1'

0 4 . . 30 35

, . 40 45

. ,

50

withbansport

time of second rr/2pulse [ma]

Figure 1. (a) Quantum state transportation. An atom prepared in a superposition of hyperfine states is displaced by lmm. After transporting the atom back to its initial position, the state superposition is analyzed by means of a second rl2-pulse. (b) Spin echo without transport; ( c ) Spin echo including transport by 1 mm. The lines in (b) and (c) are fits according to the analytic model presented in Ref. 1.

4. Outlook

Our experiment opens the route to the realization of a "quantum shift register". Most recently] we have realized first steps to individually address the atoms in a magnetic gradient field. The possibility of coherently transporting quantum states should allow us to let atoms interact at a location different from the preparation and read out. More specifically, our experiments aim at the deterministic transport of two or more atoms into an optical high finesse resonator] where they could controllably interact via photon exchange.

Acknowledgments We have received support from the Deutsche Forschungsgemeinschaft and the state of Nordrhein-Westfalen.

References 1. 2. 3. 4.

S. K u h r et al., quant-ph/0304081, submitted t o Phys. Rev. Lett. S . K u h r e t al., Science 293,278 (2001). D. Schrader et al., A p p l . Phys. B 73,819 (2001). W. Alt et al., Phys. Rev. A 67,033403 (2003).

FERRETING OUT THE FLUFFY BUNNIES: ENTANGLEMENT CONSTRAINED BY GENERALIZED SUPERSELECTION RULES

HOWARD M. WISEMAN Centre for Quantum Computer Technology, Centre for Quantum Dynamics School of Science, Grifith University, Brisbane, Queensland 4 1 11 Australia E-mail: H. WisemanQgrifith.edu. au STEPHEN D. BARTLETT Department of Physics, Macquarie University Sydney, New South Wales 21 09, Australia. JOHN A. VACCARO Division of Physics and Astronomy, University of Hertfordshire, Hatfield AL10 9AB, UK. Entanglement is a resource central t o quantum information (QI). In particular, entanglement shared between two distant parties allows them t o do certain tasks that would otherwise be impossible. In this context, we study the effect on the available entanglement of physical restrictions on the local operations that can be performed by the two parties. We enforce these physical restrictions by generalized superselection rules (SSRS), which we define to be associated with a given group of physical transformations. Specifically the generalized SSR is that the local operations must be covariant with respect to that group. Then we operationally define the entanglement constrained by a SSR, and show that it may be far below that expected on the basis of a n a h e (or “fluffy bunny”) calculation. We consider two examples. The first is a particle number SSR. Using this we show that for a twomode BEC (with Alice owning mode A and Bob mode B), the useful entanglement shared by Alice and Bob is identically zero. The second, a SSR associated with the symmetric group, is applicable t o ensemble QI processing such as in liquid-NMR. We prove that even for an ensemble comprising many pairs of qubits, with each pair described by a pure Bell state, the entanglement per pair constrained by this SSR goes to zero for a large ensemble.

1. Introduction

Entanglement is profoundly important in quantum information (QI) and has been much studied in recent years. Surprisingly, it is still a contro307

308

versial topic, even for pure states. For example, consider a Bose-Einstein condensate (BEC) containing N atoms, suddenly split into two modes (say two internal states, or two wells). Sprrenson, Duan, Cirac and Zoller claimed that after some evolution (inter-mode cycling and intra-mode collisions), there would be entanglement between the particles. They showed that this would be useful for precision measurement, exactly as in non-condensed spin-squeezing 3. Hines, McKenzie and Milburn 4 , however, criticized this characterization of entanglement, saying that “the decomposition of the system . . . into subsystems made up of individual bosons is not physically realizable, due to the indistinguishibility of the bosons within the condensates.” The power of entanglement is seen most strikingly in its form as a nonlocal resource, allowing two distant parties to do certain tasks that would otherwise be impossible. In this context, one can understand the criticism by Hines et al. of the formalism of Sprrenson et al. As an alternative, Hines et al. propose calculating the entanglement between the two modes, which is maximal immediately after the split. However, we argue, this entanglement can only be accessed by the two parties if they violate fundamental conservation laws. Thus, in both of the above formalisms, the calculated entanglement is useless as a non-local resource for &I tasks. In this context, both are examples of what Burnett has called puffy bunny entanglement ‘. In this work we show how to calculate entanglement between two distant parties in the presence of physical restrictions on the local operations they can perform. The first step is to give an operational definition of entanglement in the presence of physical constraints (Sect. 2). The next is to define generalized superselection rules (SSRs) as a way to formulate a wide variety of constraints (Sect. 3). For such constraints an expression for the entanglement can then be derived and, for pure states, simply evaluated (Sect. 4). The two main examples we have considered are the SSR associated with particle number conservation (Sect. 5 ) and the SSR associated with particle permutation covariance (Sect. 6 ) . We conclude in Sect. 7 with a summary and a discussion of future investigations.

‘

‘

2. Operational Definition of Bipartite Entanglement

Say two distant parties, Alice and Bob, share some quantum system S which for simplicity we will for the moment assume to be in a pure state [ $ J A B ) . Bipartite entanglement is a resource that enables Alice and Bob to do certain quantum tasks, such as teleportation independent of the

‘,

309

medium that holds that entanglement. We thus assume that, in addition to S , Alice and Bob each have a conventional quantum register, initially pure. We now define the entanglement in S to be the maximal amount of entanglement which Alice and Bob can produce between their quantum registers by physically allowed local operations 0. With no physical constraints, the entanglement would simply be

WI~CIAB))

PA),

(1)

s(P)= -n[PlogzP], and P A = nB[I1CIAB)(~ABI]/(1CIABI1CIAB).Here n~denotes the trace over Bob's Hilbert space, and for convenience we are allowing for unnormalized states. With physical constraints, it may be impossible to transfer this entanglement to the registers, so the constrained entanglement will be less than E ( ~ + A B )A ) . simple example to which we will return is the one-particle state 1 0 ~ ) I l ~ 1) 1 ~ ) l O ~ )This . appears to contain 1 ebit of entanglement. But to transfer this into the conventional register would require a SWAP gate Such a gate between this system and a conventional register is physically forbidden, because it could act as follows:

+

'.

Creating such a superposition, of the vacuum and one particle, would violate fundamental conservation laws, such as those for charge, lepton number et cetera, and hence is forbidden by superselection rules (SSRs). 3. Physical Constraints as Generalized SSRs

We define a SSR to be a restriction (fundamental or practical) on the allowed local operations on a system. It is not, in our view, a restriction on its allowed states. Note that "operations" includes measurements as well as unitaries More particularly, we define a SSR to be a rule associated with a group G of local physical transformations g. The rule, which we denote the G-SSR, is that operations must be G-covariant. Here we define an operation 0 to be G-covariant if

where T ( g ) is a unitary representation of the group G. Traditionally8, one talks of a SSR for an operator, rather than a SSR associated with a group of transformations. For example, a SSR for local charge Q means that it is forbidden to create a superposition of states with different Q-values. This can be derived from the conservation of global charge, an assumption that the initial state of the Universe had definite

310

charge, plus the fact that global charge is the sum of local charges. However, as we will see, other SSRs may be practical rather than fundamental constraints. Note that it is pointless to talk of a SSR for global charge because that is a conserved quantity ’. Local charge is a non-conserved quantity, so a SSR for it is meaningful. The terminology “SSR for or “Q-SSR” is compatible with our above definition if it is read as “SSR associated with the Lie group G generated by Q.” That is, the operations that cannot create local charge superpositions satisfy V p and V< E R, O[e-ztQpezcQ] = e-icQIOp]eacQ.If a SSR associated with G is in force, then all experimental predictions are unchanged if a state p is replaced by the state f ’ ( g ) p f ’ t ( g ) for any g E G. The most mixed state (that is, the state containing no irrelevant information) with which p is physically equivalent is

4”

We call this the G-invariant state, as Vg E G , ?(g) [Gp]pt( 9 ) = Gp. 4. Entanglement Constrained by a G-SSR

Having precisely defined SSRs, we can now generalize (and specialize) our operational definition of entanglement to

EG-ssR(P~;) = “0”” ED( n s y s [O(P”,”;; 8 e::)]).

(5)

The generalization is that we have allowed for a mixed system state p y i . As a consequence the entanglement is not uniquely defined, so we have to specify the entanglement measure. Since we are interested in how much useful entanglement ends up in the registers, the entanglement of distillation ED is a natural choice lo. The initial register state : : Q is still pure and separable. The specialization of our previous definition is that the physical restrictions are those enforced by a generalized SSR. That is, the operations 0 are G-covariant local operations. We can now prove the following 7 : Theorem: The SSR can be enforced by removing all irrelevant information from p by the decoherence process p + Gp. That is,

EG-SSR(PAB) = ED(GPAB) .

(6)

5. Example: Particle Number SSR

As for charge, global conservation laws and suitable initial conditions lead to a SSR for local particle number N , which is however not conserved.

31 1

Because they are not subject to a number SSR, quanta of excitation such as photons or excitons are not particles by our meaning of the word. But electrons, protons, and Rubidium atoms in a specified electronic state are. Note that the global conservation law need not be for particle number; the total number of Rb atoms in the universe is not conserved. However there are conservation laws for lepton number, baryon number and so on that ensure that there is a Rb atom number SSR. In this case the the entanglement constrained by the c-SSR is E , Q - S S R ( P A B ) = ED(NPAB)

where the operation N destroys coherence between eigenspaces of ing different local particle number n:

(7)

fin,hav-

n

Here, “local” could mean either Alice’s or Bob’s; it makes no difference. Consider a simple pure-state example. We use the notation of separating the occupation numbers of Alice’s mode(s) from those of Bob’s mode(s) by a comma, as in In~,nng). For a particle in a mode split between Alice and Bob, the state is = 10,l) 11’0):a superposition, apparently with one e-bit of entanglement. But the equivalent invariant state is N (I$J)($JI) = (0,1)(0,1( ( l , O ) ( l , O l , an unentangled mixture. For general pure states (which we here assume to be normalized), the entanglement constrained by the N-SSR is

+

+

Some pure state examples are given in Table 1. Table 1. Entanglement of various states according to the measure of Hines et aL4, Smenson et aL2 and the present work.

312

We now discuss some properties of EfidSsRillustrated by these examples. The first is super-additivity 6: Efi-SSR(\$)

(814)) 2 E f i - S S R ( \ @ ) )

+ Efi-SSR(\4))'

(10)

All standard measures of entanglement are sub-additive l o . One could attribute this anomaly to the fact that for identical particles one /$) is not truly independent of another 14). The second property is asymptotic recovery of standard entanglement '. For a large number C of copies of a state I$), 1 CE(I$)) - 5 10gZ(v$c) + O ( l ) , (11) Efi-SSR(I$)"c) where V$ is the variance in Alice's particle number for a single copy. Thus

(I$)" >

C+oO lim Efi-SSR

lE

(I$)"">

= 1.

(12)

6. Example: Ensemble QIP

Ensemble &I processing means (i) there are N >> 1 identical copies of a system ("a molecule") containing M qubits, and (ii) all operations are collective (i.e., affect each molecule identically). For example, in NMR QIP l1 each molecule contains M atoms having a spin-: nucleus. The collective operations use (in general) spatially uniform RF magnetic pulses for unitaries and fixed external antennae for measurements. In liquid-NMR the molecules can only be prepared in highly mixed states. We show that even if pure states could be produced, the above restrictions imply that the useful entanglement per molecule goes to zero as N -+ 00. The restriction on operations 0 can be formulated as the SSR

o[~(Pbp+(P)l = ~(P)[OPlpt(P)

(13)

Here p is a permutation of the N molecules and ?(p) is the unitary operator that implements that permutation. We call this the SN-SSR, as the N ! permutations p form a group called the Symmetric group SN. We define the Spinvariant (that is, randomly permuted) state 1 Pp = P(p)pP+(p).

N!

c

PESN

To understand the above, consider a simple example. Say M = 2 (two nuclei, A and B , per molecule) and N = 2 (there are two molecules, 1 and 2), and the state is

I$)

= ITf4TL)letli).

(15)

313

We consider that the As belong to Alice and the Bs to Bob, and the 5'2-SSR applies independently to Alice and to Bob. Now if Alice's local operations (acting only on nucleus A ) cannot distinguish molecules 1 and 2, then this state is equivalent to

* 2)1$J) = lL%3)lt;Ji). (16) Under the action of PA (or PB),I+) goes to an equal mixture of these two PA(1

states, and all correlations are lost. Now consider a more interesting example, where N = 2J and each molecule of the above sort is prepared in a pure Bell state

)@I

I

+ I JAJB).

(17) How much entanglement do Alice and Bob have at their disposal? The naive answer (no restrictions) is N ebits - 1 per molecule. By contrast, the constrained entanglement is = fAfB)

so the entanglement per molecule goes to zero as N -+

00.

7. Discussion In this work we have argued as follows. Bipartite entanglement is a resource that enables the two parties to do certain quantum tasks, independent of the medium that holds it. For many systems there are restrictions upon the physical operations, so naive calculations of entanglement may overestimate it. For such systems we operationally define the entanglement as the amount of distillable entanglement that can be produced between two conventional (i.e., unrestricted) registers. If the restrictions can be formulated as a generalized superselection rule (which we have defined) then we can derive an explicit expression for this entanglement. We have considered two SSRS in detail, those associated with the group generated by local particle-number, and the group of local permutations of particles. We have applied the first to a two-mode BEC (with Alice owning mode A and Bob mode B ) , and find that the entanglement shared by Alice and Bob is identically zero. We have applied the second to a pure NMR "ensemble Bell-state" (with Alice owning nucleus A and Bob nucleus B ) , and find that their entanglement per molecule is asymptotically zero. There are many other aspects of our work discussed in the papers Our work also opens up many avenues of future investigation. First, there 697.

314

are the relations with reference frames and quantum communication 12, and with quantum nonlocality 13. Second, there is the question of how t o treat physical restrictions not expressible as a SSR (as defined by us). In particular, QIP in NMR is more restricted than implied by our SSR because there are no controllable inter-molecular interactions. Thirdly, there is an apparent duality between our conclusion, that in the presence of a SSR, a nonseparable p does not imply that the system is entangled, with the conclusion of Verstraete and Cirac 14, that in the presence of a SSR, a separable p does not imply that the system is locally preparable. It is clear that SSRs place severe contraints on &I processing, and our operational definitions of SSRs and entanglement constrained by them provide a new understanding and a valuable tool t o &I science.

Acknowledgments This work was supported by the Australian Research Council.

References 1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, Cambridge, 2000). 2. A. Sorenson, L.-M. Duan, J. I. Cirac and P. Zoller, Nature 409,63 (2001). 3. B. Julsgaard et al., Nature 413,400 (2001). 4. A. P. Hines, R. H. McKenzie, and G. J. Milburn, Phys. Rev. A 67,013609 (2003). 5. K. Burnett, personal communication. 6. H. M. Wiseman and J. A. Vaccaro, quant-phJ0210002, to be published in Phys. Rev. Lett. (2003). 7. S . D. Bartlett and H. M. Wiseman, quant-ph/0303140, to be published in Phys. Rev. Lett. (2003). 8. G. C. Wick et al., Phys. Rev. 88, 101 (1952). 9. H. M. Wiseman, Proceedings of SPIE 5111 Fluctuations and Noise i n Photonics and Quantum Optics, Eds. D. Abbott, J. H. Shapiro, and Y . Yamamoto (SPIE, Bellingham, WA, 2003), pp 78-91. quant-ph/0303116. 10. M. Horodecki, P. Horodecki and R. Horodecki, Phys. Rev. Lett. 84 2014 (2000). 11. D.G. Cory et al., Proc. Natl. Acad. Sci. USA 94,1634 (1997); N.Gershenfeld and I. L. Chuang, Science 275,350 (1997). 12. S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Phys. Rev. Lett. 91,027901 (2003). 13. S. M. Tan, D. F. Walls and M. J. Collett, Phys. Rev. Lett. 66,252 (1991). 14. F. Verstraete and J. I. Cirac, Phys. Rev. Lett. 91,010404 (2003).

PHOTON NUMBER DIAGRAM FOR CHARACTERIZING CONTINUOUS VARIABLE ENTANGLEMENT

W. P. BOWEN, M. T. L. HSU, T. SYMUL, A. M. LANCE, B. C. BUCHLER, R. S . SCHNABEL, N. TREPS, H.-A. BACHOR AND P. K. LAM Quantum Optics Group, Department of Physics, Faculty of Science The Australian National University, Canberra, ACT 0200, Australia E-mail: [email protected]

T.C. RALPH Centre for Quantum Computer Technology, Department of Physics The University of Queensland, St Lucia, QLD 4072, Australia We present an experimental analysis of continuous variable entanglement produced from a pair of amplitude squeezed beams. The correlation matrix of the entanglement is characterized within a set of assumptions. The entanglement is represented on a photon number diagram that provides an intuitive and physically relevant description of the state. We show efficacy contours for three quantum information protocols on this diagram, and use them to predict the effectiveness of our entanglement in those protocols.

1. Introduction

Although optical entanglement has been widely regarded as a basic resource in many quantum information protocols, the characterization of continuous variable (CV) entanglement still remains a complex enterprise and the subject of much debate in the community. Unlike optical squeezing which is quantifiable using a single parameter (the amount of noise reduction achieved below the quantum limit), the quantum nature of entanglement is harder to quantify. The performance of different quantum information protocols may exploit different quantum characteristics of entanglement and as a result there is no consensus on a single most appropriate entanglement measure Experimental realizations of CV entanglement, however, have to date been limited to states with Gaussian statistics, and for these states well-defined characterization techniques do exist. In this article, we present a brief description of CV entanglement produced from the interference of two squeezed beams at a beam splitter. In Sect. 3, we begin our characterization of quadrature entanglement. By making a set of reay’.

315

316

sonable assumptions, we argue that the correlation matrix of the entangled state can be completely characterized by making four independent measurements on the pair of entangled beams. Using the familiar “ball-on-stick” picture of the quantum noise in phase space, we delve into the uncertainty ellipses of the entangled beams to seek out the physical relevance of these four parameters. By considering the average number of photons present in the sidebands of the entangled beams, we translate these parameters into a photon number diagram that characterizes an entangled state in terms of entanglement strength, entanglement purity and entanglement symmetry. We conclude by showing three quantum information protocols where efficacy contours can be plotted on the photon number diagram. 2. Production of Continuous Variable Entanglement A single mode of the electromagnetic field is defined by its field annihilation operator ii(t),with the commutation relation [si(t),ii+(t)] = 1. ii(t) = o(t) ( 6 X + ( t ) i 6 2 - ( t ) ) / 2 can be expanded in terms of Hermitian amplitude 6 2 + ( t )and phase 6 2 - ( t )quadrature noise operators, and its coherent amplitude o(t)= (ii(t)).In this article, we consider quadrature entanglement generated by combining two quadrature squeezed beams with orthogonal squeezing on a 50150 beam splitter. In general, the two beam splitter outputs &(t) and iiy(t) are given by

+

+

where iisqzl(t) and iisq22(t) are the annihilation operators of the constituent squeezed beams, 8 defines the relative phase between them, q5z and q5v are phase shifts that rotate the operators such that CY, ( t )and oy( t )are real, and throughout this article the sub-scripts 2 and y denote the beams being interrogated for entanglement. We transfer our analysis from this time domain to the frequency domain of relevance to our experiments by taking the Fourier transform. The resulting operators are distinguished by replacing the decoration - with a We take one input beam to be amplitude squeezed (A2XhZ1< 1) and the other to be phase squeezed (A2XGZ2< 1)with osqz2(t) = 0. We set 8 = 0 so that the squeezed quadratures are orthogonal at the beam splitter. The frequency domain amplitude and phase quadratures of the output beams x and y can then be expressed as

We see that as the squeezing of the input beams approaches perfect (A2XAZl,A2X&,, -+ 0), the amplitude and phase quadratures of the output

317

beams become perfectly anti-correlated and correlated, respectively. In this limit, the beams are therefore perfectly entangled.

3. Characterization of Continuous Variable Entanglement

3.1. Gaussian Entanglement and the Correlation M a t h Any Gaussian CV bi-partite state can be fully characterized by its amplitude and phase quadrature coherent amplitudes a$, a:, and the correlation (or covariance) matrix. Since a: and are easily characterized and do not contribute to the strength of entanglementexhibited by the state we neglect them here and focus on the correlation matrix. The correlation matrix C is given by

at

C=

‘CZ. c,+, C-f c-XX 22

(3)

c g c,; C-f c-‘ YX

YX

Each term in this matrix is the correlation coefficient between two of the variables X:, X;, X:, and X;, defined as CK, = (dX&dXA dXAdX&)/2, with { k , l } E {+,-}, and {rn,n} E {z,y}. The symmetry in the form of CZn dictates that in general CEn = Clk nm. The Correlation matrix is therefore fully specified by ten independent coefficients. The entangled beams considered in this article were generated symmetrically, and encountered identical loss before detection. Assuming that the squeezed fields used to generate the entangled state are uncorrelated, we find that A2Xh = A2X$ = A2XZ, so that C$$ = A2X* and that no cross-quadraturecorrelations exist between the beams C$z = 0. The correlation matrix is then given by

+

0

C=

c,

0

c,

c.+y+ 0

c,+,+

(4)

0

and requires only characterization of A2X+,AzX- and (dX$dX$tdX$dX$).

3.2. Noise Distribution Diagrams Using the recipe for generating quadrature entanglement given in Sec. 2, we can adopt the “ball-on-stick” picture commonly used to represent the uncertainty ellipses in phase space to analyse the quantum correlations in quadrature entanglement 3. Figure 1 illustrates the noise variances of the entangled beams z and y. We

318

X x+

x'

Figure 1. The 'ball-on-stick' picture for entanglement. (i) is the quantum noise limit; (ii) and (iii) are the uncertainties of the two constituent squeezed beams. The amount of squeezing on the beams are purposely chosen to be different for generality; (iv) and (v) are the uncertainty of the same two squeezed beams if they were pure states. (vi) is the uncertainty ellipse of one of the entangled beam and (vii) is the corresponding uncertainty ellipse if the constituent squeezed beams were pure.

note that the total noise ellipse (vi) of one beam is a phase quadrature axis reflection of the other, thus indicating that both beams are anti-correlated in amplitude but correlated in phase. When looking within the uncertainty ellipses of the entangled beams, we can attribute the noise to the constituent squeezed beams, (ii) and (iii). In the limit of perfect squeezing, the contribution of an individual squeezed beam is limited to strictly one dimension in the phase space (0" or go"), whilst the anti-squeezing quadrature then requires the total noise ellipse to be infinite. In a real experiment, however, squeezing is always finite. Furthermore, any inefficiency would ultimately lead to an increase in the mixedness of the squeezed beams; that is, the squeezed beams are no longer minimum uncertainty states with A2X&,,,A2X&,,, > 1and A2X,f,,,,A2X&,, > 1. We can represent the mixedness of the constituent squeezed states in the diagram by assuming that a lack of purity is equivalent to adding noise to the anti-squeezing quadrature, with pure squeezed states being represented by (iv) and (v). Based on a simple geometric argument, we can extract four quantities of interest summarized in Fig. 2.

3.3. The Photon Number Diagram Since generation of noise on an optical field requires energy in the form of photons, these noise distribution diagrams can be re-interpreted from a photonic perspective. Each of the four defining features of the noise distribution diagrams contributes some mean number of photons per bandwidth-time to the entangled state. The mean number of photons per bandwidth-time in the sideband w of an optical beam is given by

~ ( w=)(Lit(w)ii(w)) = 101'1~

+ A2X-) + 1 0 1 - 1 ~ + -(A2X+ 1 4

- -1

2'

(5)

319

Figure 2. Noise distribution of an entangled beam. Shaded area in (a) corresponds to the noise ellipse generated by 2 pure squeezed states, and (b) corresponds to the mixedness of the entangled beam. Ratio of the widths shown in (c) comesponds to the asymmetry or the bias present in the quantum correlations, and in (d) the bias present in the mixedness between the two quadratures.

As the coherent amplitudes have no relevance to the correlation matrix characterizing our entanglement and are easily accounted for, we can neglect contributions from them by setting = 0. The total number of photons in a pair of entangled beams, fitotal, is simply the sum of the number in beams z and y

Obviously, some fraction of fitotal (termed aimin here) is required to maintain the strength of the entanglement. This is equivalent to the requirement of a mean number of photons per bandwidth-time to generate the squeezed states used to produce entanglement, and therefore correponds to the size of the shaded uncertainty circle in Fig. 2 (a). The bias between the amplitude and phase quadrature correlations of the entangled state, which corresponds to the difference of the two widths shown in Fig. 2 (c) on the noise distribution diagrams, requires some mean number of photons per bandwidth-time fibias. Further contributions are made by the mixedness of the squeezed states used to generate the entanglement and the different degree of mixedness between the amplitude and phase quadratures of the entangled states. These contributions correspond to the quantities mentioned in Fig. 2 (b) and Fig. 2 (d) on the noise distribution diagrams, respectively. We sum them into one parameter fiexcesshere. Of course, the photons in a quadrature entangled state are indistinguishable from one another so that a definite separation of photons into distinct categories is not possible. This separation is possible,

320

however, when the average number of photons within a quadrature entangled state per bandwidth-time is considered. For entanglement symmetric between beams z and y, the average number of excess photons per bandwidth-time, fiexcess, can be found by considering the squeezed beams used to generate our entanglement. Any mixedness causes these beams to be non-minimum uncertainty (i.e., A2X&z1,2A2X&l,2 > 1). We can simply compare the mean number of photons in the entangled state fitotal to the number that would be in the state if it was perfectly pure, Tipure, to determine the average number of photons in the entangled state due to mixedness Tiexcess = Titotal - fipure.Here, Tipure is the average number of photons per bandwidth-time required to generate two pure squeezed beams with the same level of squeezing as the two input beams, and is therefore given by

Furthermore, fipurecan be separated into components required to maintain and to bias the entanglement, Tiipure = fimin fibias. Here Amin is directly dependent on the strength of the entanglement which can be characterised using the degree of inseparability introduced by Duan et al. which in this case is given by Z =

+

d

m

,

214

and is therefore independent of local reversible operations performed individually on beams z and y. The photons resulting from bias can be eliminated from finpureby performing equal local squeezing operations on beams z and y 5. After performing these operations fipure becomes

where g is the gain introduced by the pair of OPAs. fibure is minimized, and therefore fibias is eliminated, when g2 =

where fimin is the minimum mean number of photons per bandwidth-timerequired to generate entanglement of a given strength Z. fimin is completely determined by Z and is monotonically increasing as Z -+ 0 (i.e., as the strength of the entanglement increases). The average number of photons per bandwidth-time present as a result of bias can also be deterimined fibias = fipure - fimin. We can now separate the average total photon number per bandwidth-time in a quadrature entangled state into three categories: photons required to maintain the

321

entanglement Amin, photons produced by bias between the amplitude and phase quadratures Rbias, and excess photons resulting from mixedness Aexcess. An entangled state can then be conveniently and intuitively analyzed on a three dimensional diagram with Amin, f i b i s , and Aexcess forming each of the axes. Some of our recent experimental results are shown on such a plot in Fig. 3.

Figure 3. Representation of the entangled state on the photon number diagram. Points shown are experimental entanglement results obtained from a spectral analysis of the entangled beams from a frequency of 2 MHz to 10 MHz. The dashed lines indicate projection of points onto the iiexcessa m i n , fiexcess-fibias and fibias-fimin P h e S .

In a manner analogous to that performed for Aipure above, Tiexcess can be separated into two components: the average number of photons required to produce the mixedness of the entanglement, and the average number of photons as a result of mixedness bias. Including this extra parameter, the correlation matrix of Sec. 3.1, and the noise distribution diagram of Sec. 3.2 can both be fully characterized by these photon number parameters. An analogy can be drawn between the jimin-Aexcessplane of the photon number diagram and the tangle-linear entropy analysis often performed for discrete variable entanglement6. In both cases the entanglement is represented on a plane with one axis representing the strength of the entanglement (fi,in for CV, tangle for discrete variables), and the other axis representing the purity of the state (AeXCeSS for CV, linear entropy for discrete variables). Unlike the discrete variable case where the region of the tangle-linear entropy plane occupied by physical states is bounded, the set of CV entangled states spans the entire Amin-fiexcessplane. The difference occurs because the discrete quantum states analyzed on the tangle-linear entropy plane involve a finite

322

and fixed number of photons. This restriction limits both the strength of the entanglement (the tangle) and the purity (the linear entropy). CV entangled states on the other hand have no such limitation. (a)

4

.I 2

I

0

u

0.1

0.2.

03

04

o

01

02.

03

nmin

0.4

o

0.1

02-

03

nmin

04

Figure 4. Efficacy contour of quantum information protocols on the fiimin-fiexcess plane of the photon number diagram. The contour plots are (a) the degree of EPR paradox’ given by the product of conditional variances between beams I and y, (b) the fidelity of a coherent quantum teleportation protocola given by the overlap between the teleporter input and output and states, and (c) the ratio of the small signal channel capacity achievable using the entangled state, to that achieveable using a squeezed state when each state is used as an information channelg. The unshaded regions in the plots are the region in which the EPR paradox is demonstrated, the no cloning limit of teleportation is surpassed lo,and dense coding is demonstrated, respectively. For details on the derivation of these contours see Ref. 11

4. Conclusion We have discussed some physically relevant descriptions of quadrature entanglement. The photon number diagram (e.g. Fig. 3), in particular, may be used to analyse the efficacy of a given entanglement resource in quantum information protocols *. Figure 4 shows some examples of this analysis for our entanglement.

References 1. See,for example, V. Giovannetti et al., Phys. Rev. A 67, 022320 (2003); M. S . Kim et al., Phys. Rev. A 66,030301 R (2002); S . Parker et al., Phys. Rev. A 61,032305 (2000). 2. L-M. Duan, G. Giedke, J. I. Cirac and P. Zoller, Phys. Rev. Lett. 84,2722 (2000). 3. G. Leuchs, T. C. Ralph, C. Silberhorn and N. Korolkova, J. Mod. Opt. 46,1927 (1999). 4. W. P. Bowen et al., Phys. Rev. Lett. 90,043601 (2003). 5. W. P. Bowen, P. K. Lam and T.C. Ralph, J. Mod. Opt. 50, 801 (2003). 6. A. G. White et al., Phys. Rev. A 65,012301 (2001).

323 7. M. D. Reid and P. D. Drummond, Phys. Rev. Lett. 60, 2731 (1988); M. D. Reid, Phys. Rev. A 40,913 (1989). 8. A. Furusawaet al., Science 282,706 (1998);W. P. Bowen et al., Phys. Rev. A 67,032302 (2003). 9. S. L. Braunstein and H. J. Kimble, Phys. Rev. A 61,042302 (2000). 10. F. Grosshans and P. Grangier, Phys. Rev. A 64,010301 R (2001). 11. W. P. Bowen, R. Schnabel, P. K. Lam and T. C. Ralph, quant-ph10309013.

CONTINUOUS VARIABLE TELEPORTATION WITHIN STOCHASTIC ELECTRODYNAMICS

H. J. CARMICHAEL AND HYUNCHUL NHA Department of Physics, University of Auckland Private Bag 92019, Auckland, New Zealand E-mail: [email protected] Stochastic electrodynamics provides a local realistic interpretation of the continuous variable teleportation of coherent light. Timedomain simulations illustrate broadband features of the teleportation process.

1. Introduction

Semiclassical theories of the radiation field have a long history, going all the way back to the old quantum theory. The proposal of Bohr, Kramers, and Slater,' in which a classical electromagnetic field determines the rate of quantum jumps between discrete states of matter, is possibly the earliest of relevance to quantum optics. Its failure sets the boundary between the so-called classical and nonclassical states of light; although, today, the boundary is more commonly defined by the existence, or otherwise, of a positive-definite Glauber-Sudarshan probability density. The neoclassical theory of Jaynes2 is also widely known. Stochastic electrodynamic^^^^ is a less familiar example from the pantheon of semiclassical theories. Along with the others, it considers the electromagnetic field to be made up of classical Maxwell waves, but with the unique feature that it complements these waves with vacuum fluctuations, treating the electromagnetic vacuum as a real stochastic process. It, like neoclassical theory, was for some time considered as an alternative to quantum electrodynamics. Recent claims are more m ~ d e s t but , ~ still challenge traditional views on nonclassical phenomena in quantum optics, asserting that a local realistic interpretation can be given for them. No doubt, many regard this claim to be in error. To qualify this judgment, however, it must be recognized that the usual explanation of quadrature squeezing holds it to be due to a straightforward deamplification of the vacuum fluctuations. 324

325

Stochastic electrodynamics accepts the explanation at face value; according to it, the deamplification literally takes place. As a concrete model of the accepted explanation of quadrature squeezing, it is possibly more rooted in the “quantum optics psyche” than initial impressions suggest. In this paper we assess the claims of stochastic electrodynamics as they apply to continuous variable teleportation of coherent light.6~7~8~9 We show that stochastic electrodynamics does provide a local realistic interpretation, and, moreover, an insightful view of broadband features in the teleportation process. Implications for the role of entanglement and nonclassicality in continuous variable teleportation are discussed. 2. Quadrature Squeezing We begin with a review of the squeezing of broadband classical noise by a degenerate parametric oscillator (DPO), adopting the scheme illustrated in Fig. 1. Input and output fields are expanded in traveling-wave modes satisfying periodic boundary conditions on an interval of length 2L, with &(z,

t)=

cm c(m +

fw,-i(wO+w)(t-zlc),

(1)

w

Eout(z,t ) =

fw

&aw)

e-+o+w)(t-=’c),

(2)

w

where wo is the resonance frequency of the cavity, and the f w are random amplitudes, of zero mean, and covariances

- f wfw, = f :

f;,

= 0,

f: fwl

= fidwwI.

(3)

The input-field noise spectrum is flat with an average strength of ii photons per mode (fields are expressed in photon flux units). At the cavity output,

Figure 1. Schematic of the DPO squeezer.

326

Ein(z,t ) is superposed with the intracavity field a(t) = ‘&a,e-i(wo+w)t. Amplitude a, is driven by f,, damped at the rate K , and coupled to a?, through the x(’)nonlinearity of the intracavity crystal. Thus, a, and a?, satisfy

+ KAa:, - iw)a*_, + K A a ,

0 = -(K - iw)a, 0 = -(K

m & G f,, -m & G f:,.

-

(4)

(5)

+

Introducing quadrature amplitudes X , = (f, fZ,)/2, Y, = - i ( f , fZ,)/2 and x, = ( a , a?,)/2, y, = -i(a, - a?,)/2, we then solve Eqs. (4)and (5) for

+

and the spectra of squeezing are given by the intensities

Z I ~ X+ ~ ,x

, l = n- 1[ K ( l + 2

C

[K(1

A)I2 + w2

- A)]2

+ w2’

(7)

The Y-quadrature exhibits squeezing over a bandwidth 2 ~ Asymptotically, . the spectrum (8) is flat, while squeezing appears as a Lorentzian dip around w = 0. The degree of squeezing increases with pump parameter A, and the dip goes all the way to zero (perfect squeezing) for X = 1, w = 0. This perfect squeezing has a straightforward explanation: a complete cancelation of the input noise occurs due to the interference, at the cavity output, of &y,=o = 0). the input and intracavity fields ( m y , = , The extension to quantum fields may be made in various ways. No matter how it is made, the effect is to replace the A in Eqs. (7) and (8) by iz This change may be passed all the way back to the A in Eq. (3). The same analysis then holds for a vacuum-state input. In the standard view, a calculation in the Wigner representation has been made. Stochastic electrodynamics regards the added vacuum fluctuations to be real; its fundamental assumption is that the electromagnetic field is classical, but asymptotic inputs are nonzero and stochastic with variance per It is more instructive from this point to work in the time domain. To this end, we replace Eq. (1) by a complex Weiner increment dEi,, with

+

+ a.

a

-dEindEin = dEi*,dEk = 0,

dEi*,dEi, = t d t ,

(9)

327

and compute realization of the intracavity and output fields from the stochastic differential equations dEout = 6 a d t

+ d&in,

da = -V;(CY - Xa*)dt - 6 d & i n .

(10)

It is important to realize that the output field is inherently broadband (bandwidth dt-’ for a time step d t ) ; thus, only filtered fields are usefully plotted; here and elsewhere we denote filtered fields by q ( t ) i p ( t ) . An example of the filtered output field is presented in Fig. 2. Note that the squeezing ellipse is not a static object (quantum state), but built up from the field fluctuations as they explore phase space over time.

+

4

2-

-

o

P 2-

Figure 2. Fluctuations of the filtered quadrature amplitudes of the DPO squeezer outPut; for a pump parameter = 0.4 and filter band width 10-4 . It might appear that it is illegitimate to view q ( t ) and p ( t ) simultaneously, since they are represented in quantum mechanics by noncommuting operators. Certainly, we may not interpret them as simultaneously measured quantities. They may have simultaneous status, however, as hidden variables. They are “hidden” in the usual sense of providing a hidden yariable theory-i.e., p ( q ) is the hidden variable that resolves the quantum distribution over q ( p ) into dispersionless states. They are also “hidden” in the operational sense that no violation of the Heisenberg uncertainty principle can occur, since to measure q ( t ) and p ( t ) simultaneously, it is necessary to divide the field q ( t ) + i p ( t ) on a 50/50 beam splitter and measure the X quadrature in one output and the Y - in the other. The process unavoidably

328 2-

PO 2-

I 1 _

3Mm

4wo

Kl

4 - 2 0 ~

2

4

Figure 3. Fluctuations of the filtered quadrature amplitudes of the DPO squeezer output before (inner frames) and after (outer frames) its division at a 50/50 beam splitter (filter bandwidth lO-*n).

introduces an additional vacuum field through the second beam-splitter input, contaminating the “hidden” q ( t ) and p ( t ) as shown in Fig. 3.

3. Teleportation of Coherent Light Our model for the teleportation of coherent light is laid out in Fig. 4. For simplicity, and in contrast to the experirnent~,~l~ no modulation of the input light is employed; Alice’s detection is centered at d.c.; her detection bandwidth equals the bandwidth of the squeezing. There are three independent vacuum-field inputs, with covariances as in Eq. (9), one each for the two squeezers (pump parameters XX and X y ) , and a third (displaced) to model the coherent field €in. These inputs are mapped through the network using Eqs. (10) for the squeezers and the standard beam splitter transformation. Alice’s homodyne detection is modeled as a simple filtered “reading” of the chosen quadrature amplitude (no additional noise). Unit gain displacement of Bob’s field produces the output field Eout. Figure 5 presents results of typical simulations when (a) vacuum fields are distributed to Alice and Bob, and (b) the distributed fields are moderately squeezed. In each case, fields at six strategic points are plotted, filtered for visualization. In (a), although the mean amplitude p i q = 10(1+ i) is

+

329

reproduced at the output, the output field fluctuates with a variance three times that of the input. In (b) these fluctuations are significantly reduced.

Figure 4.

I

Schematic of the teleporter.

I

I

Figure 5 . Operation of the teleporter for (a) AX = Xy = 0.0 and (b) AX = Xy = 0.4. The input field (bottom left) is compared with the output field (bottom right). The squeezer outputs are shown before (top) and after (middle) the 50/50 beam splitter. All fields filtered with a bandwidth of

330

Figure 6 shows results for an even higher squeezing level. Here, not only is the fluctuation variance at the output equal to that at the input, but the output fluctuations track the input field over time, at least to a good approximation. As already stated in relation to Fig. 2, the variance is not a static object (quantum state), but a feature of a finite bandwidth noise process that explores phase space over time. Within stochastic electrodynamics, this tracking of the fluctuations is a trivial consequence of the correlation between the fields distributed to Alice and Bob. For unit gain, the mapping from &in to Eoutis summarized as

14 12

- out

14

in

6

qAlice

8

10

12

1.

PAlice

Figure 6. (a) Comparison of filtered input and output fields for A X = X y = 0.8 (filter (b) Correlation of the fields distributed to Alice and Bob. bandwidth

33 1

+

where qAlice 4-ipAlice and qBob ipBob are the said correlated fields. When the terms on the right-hand sides are independent vacuum fields (displaced for qin and pi,,), the fluctuation variance of qout (pout)is clearly three times that of qin (pin). On the other hand, for the conditions of Fig. 6(a) we have the correlations displayed in Fig. 6(b), with qBob x qAlice, pBob M - p ~ l i ~ ~ ; then from Eq. (11), qout M qin p o u t M p i n . 9

4. Entanglement and Nonclassicality

As demonstrated by the simulations, stochastic electrodynamics provides a local realistic model explaining the teleportation of coherent light. What does it say about nonclassicality and entanglement? This is likely the question of most interest to readers. Unfortunately, there is little space to explore it and we must limit ourselves to a few brief comments. First, it is known that stochastic electrodynamics implements the Wigner representation, presuming positive definite Wigner functions. It is known also that the squeezed fields distributed to Alice and Bob mimic the original EPR state,1° for which a positive Wigner function and hence a local realistic hidden variable theory exists.’l1l2 Our claim of such a theory is not therefore new, but we do present a more thorough treatment of it than before, emphasizing, in particular, its broadband character. Though proposals to demonstrate nonlocality for the original EPR state (in its optical form) have been made,l37l4 they address discrete (photon) features of the field and are not relevant for continuous variable measurements. Second, concerning entanglement , the separability requirement is Wab(qa

pj WF) ( q a 7 ?)a)Wb(j)(qbr p b ) 3

q b ;P a , P b ) =

(12)

j

for Wigner functions W a b , Wi’), wb(j) and some set of probabilities Pj . Since stochastic electrodynamics reproduces Wigner probabilities, any Gaussian state that is nonseparable in Wigner function terms-i.e. , quantum mechanically-is nonseparable within stochastic electrodynamics as well; this is so, even though the hidden variables permit separation in the form W a b ( q a ,q b ; p a , p b ) =

s

dpdvdqd<Wab(P, v ;77,