Laser Spectroscopy: Proceedings of the XVIII International Conference: ICOLS 2007: Telluride, Colorado, USA, 24-29 June 2007 9789812813190, 9812813195

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Proceedings of the XVlll International Conference

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2007

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Proceedings of the XVlll International Conference

ICOLS

2u07

Telluride, Colorado, USA

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24 - 29 June 2007

editors

Leo Hollberg, Jim Bergquist NIST-Boulder, USA

Mark Kasevich Stanford University, USA

KS World Scientific

N E W JERSEY

- L O N D O N - SINGAPORE

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- SHANGHAI

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H O N G KONG

TAIPEI

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CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA ofice: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601

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

LASER SPECTROSCOPY Proceedings of the Eighteenth International Conference Copyright 0 2008 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereoJ may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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PREFACE The eighteenth International Conference on Laser Spectroscopy was held 24-29 June, 2007 in Telluride, Colorado. Telluride, which is nestled in a box canyon near the south-west corner of the state, proved to be an exceptionally beautiful site for ICOLS-07. Although its remote location and high altitude (2909 m) did present some challenges for our participants, all in all, the setting and facilities were well worth the trip. We were also fortunate to experience warm, clear weather for the entire week of the conference. In keeping with its rich tradition, ICOLS-07 was truly an international gathering with 173 delegates and 34 accompanying guests from 21 countries (Australia, Austria, Canada, China, Denmark, France, Germany, Ireland, Israel, Italy, Japan, Netherlands, New Zealand, Poland, Russia, South Africa, Sweden, Switzerland, Taiwan, United Kingdom, and the United States). The technical program consisted of 34 invited talks arranged in the general topic areas of degenerate quantum gases, quantum information and control, precision measurements, fundamental physics and applications, ultra-fast control and spectroscopy, novel spectroscopic applications, spectroscopy on the small scale, cold atoms and molecules, single atoms and quantum optics, optical atomic clocks. We are indebted to the ICOLS-07 program committee for their time and effort in putting together an exceptional and broad technical program; but most importantly, we are indebted to all of the ICOLS-07 attendees whose participation and vibrant exchange of ideas provided the real strength and foundation of the Conference. In addition to the daily oral sessions, Monday and Tuesday evenings were spent in active discussions around 200 outstanding contributed posters. With only limited time available for the oral and poster sessions, ICOLS-07 necessarily focused on but a few areas of the ever expanding field of laser spectroscopy. Unfortunately, with the time demands imposed by the rich technical program, we could only block out a few hours Wednesday afternoon to sample the abundant outdoor activities and recreational opportunities around Telluride. Wednesday evening we joined forces with two local science education organizations in Telluride, the Pinhead Institute and Telluride Science Research Center, for a town talk on quantum computing presented by Rainer Blatt. After the technical sessions on Thursday, we gathered in the Telluride Town Park for an informal banquetlpicniclparty, complete with music, good food, hula-hoops and slack-lines. V

VI

Following the lead of the 2005 conference, we have continued the webbased presence and archive for ICOLS at www.lasersDectroscoDy.org. This year, in addition to this publication by World Scientific of the manuscripts submitted by our invited speakers, we have compiled a DVD that contains pdf images of most of the ICOLS posters and some of the oral presentations. Hopefidly, these records will provide a useful reference and at least a snap-shot of some major research activities in Laser Spectroscopy, circa 2007. The DVD and website also contain some memorable photos from ICOLS-07. Not surprisingly, there were also a few “issues” along the way. Serious delays were caused by the late loss of a major lodging provider which negatively impacted attendance, caused logistic problems, and resulted in the use of condominium-style lodging for most attendees. It was also most unfortunate and sobering that some delegates (at least 5 from China and one from India) were forced to cancel plans to attend ICOLS-07 because of excessive delays in obtaining entry visas to the U.S. The very slow and selective security procedures now in place continue to dampen international scientific exchanges; this is all too reminiscent of the cold-war days of the early ICOLS meetings. Many people and organizations contributed to the ultimate success of ICOLS-07. The conference was only possible because of the generous support of our industrial and organizational sponsors listed below. For local organization, we are especially indebted to Svenja Knappe, Ying-Ju Wang, Lisa Barnes, Lindsey Wilson, Viki Bergquist and Andrew Novick. A very special thanks goes to Bill Fairbank and Siu-Au Lee for assistance with logistics that went well beyond the call of duty. The staff of the Telluride Conference Center and the excellent audio-video services of Curt Rousse allowed us to concentrate on the science rather than on the facilities, meals and hardware. Wayne Itano deserves special recognition for his selfless contribution of time and energy in constructing and maintaining the ICOLS-07 website. Didi Leibfried did an outstanding job with all aspects of the poster sessions. Our sincere thanks also go Erling Riis, Allister Ferguson and Ed Hinds, the organizers of ICOLS-05, for invaluable advice and carryover support. Last but not least, we were extremely fortunate to have connected with the Telluride Science Research Center for organizational arrangements and conference operations. It was a real pleasure to work with Nana Naisbitt, Kari Koch and others of TSRC who did a splendid job in bring all the pieces together efficiently and in addressing numerous issues. Our sincere thanks to all involved with ICOLS-07. Leo Hollberg and Jim Bergquist, NZST-Boulder Mark Kasevich, Stanford University October 2007

In memoriam Herbert Walther (1935 - 2006)

An impassioned physicist Laser  spectroscopy  in  Germany  and  throughout  the  world  is  closely  connected with Professor Herbert Walther, who organized the 4th International Conference on  Laser  Spectroscopy  (ICOLS)  held  in  Rottach­Egern  in  1979.  A  giant  in the field,  he  was  a widely recognized  and  well  respected participant  of most  of the bi­annually  held  ICOLS  until  his  untimely  death  last  year.  On  the  occasion  of the  first  ICOLS  without  Professor  Walther,  it  is  appropriate  and  fitting  that  we look back at the  scientific oeuvre of a scientist who produced world­class results throughout  his  career.  Remarkably,  his  research  covered  a  surprisingly  wide spectrum of different topics with great scientific depth as evidenced by the  large number  of citations  his  publications  have  attracted.  The  story  of his  scientific career tells  in  large part also the history of high resolution spectroscopy. For  his  PhD  thesis  in  1962,  Professor  Walther  analyzed  the  Doppler­free transverse  fluorescence  from  an  atomic  beam  with  a plane  parallel  Fabry­Perot interferometer  to  learn  about  the  nuclear  quadrupole  moment  of the  manganese isotope  55Mn.  Lifetime measurements and level crossing experiments  followed. His  publication  in  1970  with  Dr.  John  L.  Hall  that  describes  the  development and  performance  of a  narrow­band  dye  laser  marked  the  beginnings  of tunable, high­resolution,  laser spectroscopy  and was  an  important milestone  in  Professor Walther's  scientific  life.  Dye  lasers became  a central  component in many of his experiments and in  1991,  on the occasion of the 25th anniversary of the first dye laser,  at a meeting held  at his  favorite retreat  in  Ringberg  castle,  he  is  pictured with  those  representing  the  Who's  Who  of  dye  laser  spectroscopists  of  that period.  The  year  he  passed  away  marked  the  40th  anniversary  of the  dye  laser,

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but by then it had largely been phased out, often replaced with semiconductor lasers or semiconductor-laser-pumped solid-state lasers. We have seen the spectacular development towards single-cycle pulses of ultrafast lasers, and Professor Walther again contributed with innovative experiments. He demonstrated the carrier-envelope phase dependence in the spatial distribution of photoelectrons comprising an attosecond double slit experiment. In addition to his many beautiful experiments in high-resolution spectroscopy and metrology, Professor Walther conducted a large number of experiments with notable results on resonance fluorescence, Rydberg-atoms, the one-atom maser, the spectroscopy of excimer molecules, radiation pressure, ion trapping, delayed choice experiments, above-threshold ionization, as well a whole series of early atmospheric LIDAR measurements, .. . the list is longer than the space on this page. Almost needless to say, Professor Walther was awarded a large number of prestigious prizes in recognition of his monumental contributions. Professor Walther believed great benefit could be derived from bringing together the world’s best scientists and he strove successfully to make the MaxPlanck Institute of Quantum Optics an international meeting place of laser scientists and quantum opticians. He himself collaborated with many internationally-renowned scientists, some over long periods of time. Professor Walther was also an outstanding teacher and mentor. Many of his former students now hold distinguished positions in academia and industry. With his passing, the fields of laser science and spectroscopy lost a great champion whom we will gratefully and fondly remember. We are decidedly pleased with the joint establishment of the Herbert Walther Award by the Optical Society of America and the Deutsche Physikalische Gesellschaft, which recognizes and will annually remind us of Professor Herbert Walther’s outstanding scientific contributions and his exceptional leadership. There is no better way to show our community’s appreciation. Gerd Leuchs September 2007

ICOLS-07 gratefully acknowledges the generous support of our corporate sponsors:

TQPTICA

fHOtOHICS

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Our thanks go to the following institutions and agencies that provided administrative, logistical and financial assistance to ICOLS-07 ||

ARO

OARPA

Conference Services Provided by;

The views and opinions, and/or findings contain in this report are those of the author(s) and should not be construed as the official policy or decision of any of the organizations listed above.

ICOLS-07 Program Committee E. Arimondo V. Bagnato K. Baldwin R. Ballagh V. Balykin J.C. Bergquist I. Cirac W. Ertmer A. Ferguson H. Fielding L. Hollberg

M. Inguscio W-H. Jhe H. Katori M. Leduc C. Salomon P. Schmidt F.T. Shimizu S. Svanberg W. Ubachs M-S. Zhan

Italy Brazil Australia New Zealand Russia U.S. Germany Germany Scotland England

us.

Italy Korea Japan France France Austria Japan Sweden Netherlands China

ICOLS Steering Committee members serving in recent years F.T. Arecchi E.A. Arimondo H.-A. Bachor R. Blatt N. Bloembergen C.J. Borde R.G. Brewer S. Chu W. Demtroder M. Ducloy M.S. Feld A. Ferguson J.L. Hall P. Hannaford T.W. Hansch S. Haroche

S.E. Harris E.A. Hinds L. Hollberg M. Inguscio V.S. Letokov A. Mooradian E. Riis Y.R. Shen F.T. Shimizu T. Shimizu K. Shimoda B.P. Stoicheff S. Svanberg H. Walther Y.Z. Wang Y.R. Shen

Italy Italy Australia Germany U.S. France U.S. U.S. Germany France U.S. Scotland U.S. Australia Germany France

U.S. England U.S. Italy Russia U.S. Scotland U.S. Japan Japan Japan Canada Sweden Germany China U.S.

Participants in the ICOLS-07 planning committee meeting J.C. Bergquist F.T. Shimizu J.L. Hall A. Ferguson R. Blatt D. Liebfried P. Hannaford S. Haroche

E.A. Hinds E. Riis T.W. Hansch K. Baldwin S-A. Lee H. Yoneda M. Ritsch-Marte D. Leibfried

U.S. Japan U.S. Scotland Germany U.S. Australia France xi

England Scotland Germany Australia U.S. Japan Austria U.S. zyxwvutsrqponm

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CONTENTS Degenerate Gases

1

Probing Vortex Pair Sizes in the Berezinskii-Kosterlitz-Thouless Regime on a Two-Dimensional Lattice of Bose-Einstein Condensates V. Schweikhard, S. Tung, G. Lumporesi and E.A. Cornell

3

Interacting Bose-Einstein Condensates in Random Potentials P. Bouyer, L. Sanchez-Palencia, D. Cle'ment, P. Lugan and A. Aspect

11

Towards Quantum Magnetism with Ultracold Atoms in Optical Lattices I. Bloch

23

Precision Measurement and Fundamental Physics

37

T-Violation and the Search for a Permanent Electric Dipole Moment of the Mercury Atom E.N. Fortson

39

Quantum Information and Control I

51

Quantum Information Processing and Ramsey Spectroscopy with Trapped Ions C.F. Roos, M Chwalla, T. Monz, P. Schindler, K. Kim, M. Riebe and R. Blatt

53

63 Quantum Non-Demolition Counting of Photons in a Cavity S. Haroche, C. Guerlin, J . Berm, S. Deleglise, C. Sayrin, S. Gleyzes, S. Kuhr, A zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 4 Brune and J.-M. Raimond Ultra-fast Control and Spectroscopy

73

Frequency-Comb- Assisted Mid-Infrared Spectroscopy P. de Natale, D. Mazzotti, G. Giusfredi, S. Bartulini, P. Cancio, P. Mudduloni, P. Malara, G. Gagliardi, I. Gulli and S. Borri

75

xiii

XIV

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Precision Measurement and Applications

87

Precision Gravity Tests by Atom Interferometry G.M. Tino, A. Alberti, A . Bertoldi, L. Cacciapuoti, M. De Angelis, G. Ferrari, A. Giorgini, V. Ivanov, G. Lamporesi, N. Poli, A4 Prevedelli and F. Sorrentino

89

Novel Spectroscopic Applications

101

On A Variation of the Proton-Electron Mass Ratio 103 W. Ubachs, R. Buning, E.J. Salumbides, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP S. Hannemann, H.L. Bethlem, D. Bailly, M. Vewloet, L. Kaper and M. T. Murphy Quantum Information and Control I1

111

Quantum Interface between Light and Atomic Ensembles H. Krauter, J.F. Sherson, K. Jensen, T. Fernholz, J.S. NeergaardNielsen, B.M. Nielsen, D. Oblak, P. Windpassinger, N. Kjaergaard, A.J. Hilliard, C. Olausson, J.H. Miiller and E.S. Polzik

113

Degenerate Fermi Gases

125

An Atomic Fermi Gas Near a P-Wave Feshbach Resonance D.S. Jin, J.P. Gaebler and J. T. Stewart

127

Bragg Scattering of Correlated Atoms from a Degenerate Fermi Gas R.J. Ballagh, K.J. Challis and C. W. Gardiner

138

Spectroscopy and Control of Atoms and Molecules

151

Stark and Zeeman Deceleration of Neutral Atoms and Molecules S.D. Hogan, E. Vliegen, D. Sprecher, N. Vanhaecke, A4 Andrist, H. Schmutz, U. Meier, B.H. Meier and F. Merkt

153

Generation of Coherent, Broadband and Tunable Soft X-Ray Continuum at the Leading Edge of the Driver Laser Pulse A. Jullien, T. Pfeijer, M.J. Abel, P.A4 Nagel, S.R. Leone and D.M Neumark Controlling Neural Atoms and Photons with Optical Conveyor Belts and Ultrathin Optical Fibers D. Meschede. W. Alt and A. Rauschenbeutel

167

175

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Spectroscopy on the Small Scale

185

Wide-Field Cars-Microscopy C. Heinrich, A. Hofer, S. Bernet, and M Ritsch-Marte

187

Atom Nano-Optics and Nano-Lithography KI. Balykin, P.N. Melentiev, A.E. Afanasiev, S.N. Rudnev, A.P. Cherkun, V.S. Letokhov, P. Yu Apel, V.A. Skuratov and V.V. Klimov

195

Pinhead Town Talk, Public Lecture and Mountainfilm

205

The Quantum Revolution - Towards a New Generation of Supercomputers R. Blatt

207

Cold Atoms and Molecules I

217

Ultracold & Ultrafast: Making and Manipulating Ultracold Molecules with Time-Dependent Laser Fields C.P. Koch, R. Koslofl E. Luc-Koenig, F. Masnou-Seeuws and R. Moszynski

219

Bose-Einstein Condensates on Magnetic Film Microstructures M Singh, S. Whitlock, R. Anderson, S. Ghanbari, B. V. Hall, M Volk, A. Akulshin, R. McLean, A. Sidorov and P. Hannaford

228

Cold Atoms and Molecules I1

241

Ultracold Metastable Helium-4 and Helium-3 Gases W. Vassen, T. Jeltes, J.M. McNamara, A.S. Tychkov, W. Hogeworst, K.A.H. Van Leeuwen, V. Krachmalnicofl M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect and C.I. Westbrook

243

Single Atoms and Quantum Optics I

257

Recent Progress on the Manipulation of Single Atoms in Optical Tweezers for Quantum Computing A. Browaeys, J. Beugnon, C. Tuchendler, H. Marion, A. Gaetan, Y. Miroshnychenko, B. Darquik, J. Dingjan, Y.R.P. Sortais, A.M. Lance, M.P.A. Jones, G. Messin and P. Grangier

259

xvi

Progress in Atom Chips and the Integration of Optical Microcavities E.A. Hinds, M. Trupke, B. Darquik, J. Goldwin and G. Dutier

27 1

Single Atoms and Quantum Optics I1

283

Quantum Optics with Single Atoms and Photons H.J. Kimble

285

Optical Atomic Clocks

295

Frequency Comparison of Al' and Hg' Optical Standards T. Rosenband, D.B. Hume, A. Brusch, L. Lorini, P.O. Schmidt, T.M. Fortier, J.E. Stalnaker, S.A. Diddams, N.R. Newbury, W.C. Swann, W.S. Oskay, K M Itano, D.J. Winelandand J. C. Bergquist

297

Sr Optical Clock with High Stability and Accuracy A. Ludlow, S. Blatt, M. Boyd, G. Campbell, S. Foreman, M. Martin, M H. G. De Miranda, T. Zelevinsky and J. Ye

303

Author Index

317

DEGENERATE GASES

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PROBING VORTEX PAIR SIZES IN THE BEREZINSKII-KOSTERLITZ-THOULESS REGIME ON A TWO-DIMENSIONAL LATTICE OF BOSE-EINSTEIN CONDENSATES V. SCHWEIKHARD, S. TUNG, G. LAMPORESI, and E. A. CORNELL

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J I L A , National Institute of Standards and Technology and University of Colorado, and Department of Physics, University of Colorado, Boulder, Colorado 80309-0440, USA jilawww. colorado. edu/bec/ We present results of a study of vortex proliferation in the BerezinskiiKosterlitz-Thouless (BKT) regime on a two-dimensional (2D) array of Josephson-coupled Bose-Einstein condensates. In our lattice system, tunneling between nearest-neighbor condensates provides a Josephson coupling J which acts t o keep the condensates' relative phases locked. A cloud of uncondensed atoms, on the other hand, interacts with the condensates and induces thermal phase fluctuations, which we observe as vortices. As long as the Josephson energy J exceeds the thermal energy T, the array is vortex-free, while with decreasing J / T , thermally activated vortices appear. We give an extended description of a time-to-length mapping technique that allows us t o obtain information on the size of vortex pairs as J/T is varied.

Keywords: Vortices; Bose-Einstein condensates; Josephson-junction array.

1. Introduction

Two dimensional (2D) superfluids undergo a thermal phase transition to a normal state, which proceeds through the unbinding of vortex-antivortex pairs, i.e. pairs of vortices of opposite circulation. Our theoretical understanding of this transition is due to work by Berezinskii' and Kosterlitz and Thouless2 (BKT). The BKT picture applies to a wide variety of 2D systems, among them Josephson junction arrays (JJA), i.e. arrays of superfluids in which phase coherence is mediated via a tunnel coupling J between adjacent sites. Placing an isolated (free) vortex into a J J A is thermodynamically favored if its free energy F = E - T S 5 0. In an array of period d the vortex energy diverges with array size R as E M J l ~ g ( R / d ) , ~ but may be offset by an entropy gain S M log(R/d) due to the available

3

~ R2/d?  sites.  This  leads  to  a  critical  condition (J/T)crit  sa  1  indepen­ dent  of system  size,  below  which  free  vortices  will  proliferate.  In  contrast, tightly bound vortex­antivortex pairs  are  less  energetically  costly  and  show up  even  above (J/T)crn.  The  overall  vortex  density  is  thus  expected  to grow  smoothly  with  decreasing J/T  in  the  BKT  crossover  regime. The  BKT  transition  in  ultracold  gases  has  been  the  subject  of much  ex­ perimental4"6  and theoretical7"9  work,  following the observation of concur­ rent  thermal  phase  decoherence  and  vortex  formation4  in  a  continuous  2D Bose  gas.  Our  work  is  focused  on  a more  detailed  understanding of vortex­ formation,  collected  in  a  2D  array  of  Bose­Einstein  condensates  (BECs) with  experimentally  controllable  Josephson  couplings.  Parts  of  our  results have  been  published  previously.5

2. Experimental System and Procedure

Fig.  1.  (a)  Experimental 2D  optical lattice system.  In the white­shaded area a lattice of Josephson­coupled BECs is created.  The central box marks the double­well potential shown  in  (b).  The  barrier  height VOL  and  the  number  of  condensed  atoms  per  well, Nweii,  control  the  Josephson  coupling J,  which  acts  to  lock  the  relative  phase  A  T,  whereas  in  (iii)  for  J  < T  vortices  (dark  spots)  appear  as remnants of the thermal fluctuations in the array.

5

W e create an array of Josephson-coupled BECs by adiabatically loading a partially Bose-condensed sample of 87Rb atoms into a 2D hexagonal optical lattice of period d = 4.7pm in the x-y plane, as shown in Fig.l(a). The resulting potential barriers between adjacent sites [Fig.l(b)] rise above the condensate's chemical potential, splitting it into an array of condensates which now communicate only through tunneling. Each of the central wells contains NzuellM zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC 7000 condensed particles. By varying the optical lattice depth VOLin a range between 500 H z and 2 k H z we tune J, the collective Josephson coupling," between 1.5p K and 5 n K . The temperature T of the array can be adjusted between 30 - 7 0 n K . The "charging" energy E, due to repulsive mean field interactions, defined in Ref. 11, is on the order of a few p K , much smaller than both J and T . These parameters place our array in the Josephson regime," where J >> E, but E, >> J/Niezl. In this regime the Josephson coupling energy J(l - cos(A4)) acts to lock the relative phases Ag5, and if dominant will ensure at least local phase coherence in the array. A cloud of uncondensed atoms at temperature T on the other hand induces thermal fluctuations of the relative phases of order Aq5~hM zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA The charging energy iE,(ANw,11)2 disfavors population imbalances between sites. In the Josephson regime however, with J >> E,, the resulting quantum fluctuations of the relative phase are quite negligible," of order A& M (E,/4J)1/4. After allowing time for thermalization we probe the array. Because we do not have direct experimental access to the condensate phases in the array, we turn down the optical lattice on a time-scale t,, which is fast enough to trap phase winding defects, but slow enough to allow neighboring condensates to merge, provided their phase difference is small. Phase fluctuations are thus converted to vortices in the reconnected condensate, We then expand as has been observed in the experiments of Scherer et the condensate and take a destructive image in the x-y plane.

m.

3. Earlier Results

Figure l ( c ) illustrates our observations: When J/T < 1 vortices occur in the BEC, as remnants of the thermal fluctuations in the array. In an earlier publication5 we proved thermal activation as the origin of these phase fluctuations. We studied vortex activation while varying J at distinct temperatures T , and showed that vortex proliferation is controlled almost exclusively by the ratio J/T, with a steep rise of vortex number around J/T 1, just as suggested by the free energy arguments presented above.

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6 4. Inferring Vortex-Antivortex Pair Sizes

Fig.  2.  Vortex­antivortex  pairs,  imaged  just  prior  to  their  annihilation.  Following  the optical lattice ramp­down, tightly bound pairs annihilate faster than loosely bound pairs, providing  a  time­to­length  mapping  that  allows  to  extract  information  on  vortex  pair sizes.

Here  we  describe  a  technique  that  allows  us  to  infer  vortex  antivortex pair sizes. As in our earlier work we use as a robust vortex­density surrogate the  "roughness" T>  of  the  condensate  images  (see  Fig.  1)  caused  by  the vortex cores.  This vortex density T> by itself provides no distinction between bound vortex­antivortex pairs and free vortices.  In the following we exploit our  time­dependent  control  of  the  optical  potential  to  distinguish  free  or loosely bound vortices from tightly bound vortex­antivortex pairs.  We make use  of the  fact  that,  once  the  optical  lattice  potential  has  been  turned  off, vortices and antivortices annihilate in the bulk condensate over a w  100 ms timescale.  Figure  2  shows  an  example  image  of pairs  of vortices just  prior to  their  annihilation.  It  is  intuitively  obvious  that  tightly  bound  vortex pairs  will  annihilate  on  a  much  faster  timescale  than  loosely  bound  pairs. A  "slow"  optical lattice ramp­down therefore allows time for tightly bound pairs to  annihilate  before they  can  be  imaged.  By slowing down the  ramp­ down  duration r  [inset  of Fig.  3  (a)],  we  can  thus  selectively  probe  vortex pairs of increasing size. Figure 3 shows vortex activation curves, probed with two different ramp­ down  times.13  A  slow  ramp  compared  to  a  fast  one  shows  a  reduction  of the  vortex  density  Z>> • 10 Fig. 3.  Vortex density £> probed at different optical lattice ramp­down timescales T. A slow ramp provides time for tightly bound vortex­antivortex pairs to annihilate, allowing selective  counting  of  loosely  bound  or  free  vortices  only,  whereas  a  fast  ramp  probes both  free  and  tightly  bound  vortices.  A  fit  to  the  vortex  activation  curve  determines its midpoint (J/T) 50% , its 27% ­ 73% width A(J/T) 2 7­73, and the limiting values T>< (£>>) well below (above) (J/T) 50% .

annihilated  on  the  long  ramp,  but  not  on  the  fast  one. To  map  the  experimental  ramp­down  time­scale  to  theoretically  more accessible vortex­antivortex pair sizes, we compare the observed number of vortices  in fully randomized arrays  at  low J/T to simulations of vortex dis­ tributions  in  a  hexagonal  array  with  random  phases.  In  these  simulations, following Ref.  12, we count a vortex if all three phase differences in an ele­ mental  triangle  of junctions  are  E  (0,  TT),  or  if all  are  G  (—7r,0).  A  snapshot of  a  simulated  vortex  distribution  is  shown  in  Fig.  4(a).  Within  the  cen­ tral  20  lattice  sites,  comparable  to the experimental  region  of interest5  we find,  on average,  a total of 10 vortices.  6 vortices occur in nearest­neighbor vortex­antivortex pairs  [configuration I in Fig.  4(b)],  1.7 (0.4)  occur in con­ figuration  II  (III)  respectively,  and  1.9  occur  in larger  pairs  or  as  free  vor­ tices.  To  relate  these  time­independent  simulations  to  the  experiment,  we show in Fig.  4(d)  the relevant  cumulative vortex distributions,  i.e.  all vor­ tices  occurring  in  pairs  larger  than  a  given  lower  cutoff size.  For  a  given experimental ramp­down duration,  we expect only those vortex configura­ tions  to  survive  which  are  above  a  lower  cutoff  pair  size  imposed  by  the ramp­down  rate. In  Fig.  4(e)  we  compare  the  simulated  cumulative  vortex  distributions to  experimentally  measured  vortex  numbers  as  a  function  of  ramp  down

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9

timescale, to obtain the desired time-to-length mapping. Downward triangles show the decrease of the experimentally measured saturated (low-J/T) vortex density V < with increasing ramp timescale r. The right axis shows the inferred number of vortices that survived the ramp. M 11 vortices are observed for the fastest ramps, in good agreement with the total number of vortices expected from the simulations (indicated as grey bars). For just somewhat slower ramps of T M 5 m s , only 3 vortices survive, consistent with only vortices in configuration I1 & zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM I11 or larger remaining (indicated in Fig. 4(e), top axis). For T 2 30ms ramps less than 2 vortices remain, according to our simulations spaced by more than 2 d / & Thus we infer that ramps of r M 30ms or longer allow time for bound pairs of spacing 5 2 d / a to decay before we observe them.

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With this time-to-length mapping we now return to the observations in Fig.3. For the slower ramp we observe vortex activation at lower ( J / T ) 5 0 % , confirming that free or very loosely bound vortices occur only at higher T (lower J ) . In Fig. 5 we plot the midpoint ( J / T ) 5 0 %of vortex activation curves versus the applied ramp-down time. The data quantitatively show a shift of (J/T)So%from 1.4 for fast ramp times when all vortices are expected to contribute to the signal, to 1.0 for slow ramp times when only loosely bound vortices survive. The data therefore reveal that loosely bound pairs of size larger than 2 d / a , or indeed free vortices, do not appear in quantity

10

until J / T 5 1.0, whereas more tightly bound vortex pairs appear in large number already for J / T 5 1.4. This result clearly illustrates the mechanism of vortex-antivortex unbinding with increasing temperature or decreasing superfluid coupling, which underlies BKT theory.

Acknowledgments We acknowledge illuminating conversations with Leo Radzihovsky and Victor Gurarie. This work was funded by NSF and NIST.

References

1. V. Berezinskii, Sov. Phys.-JETP zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP 32,493 (1971); 34,610 (1972). 2. J. Kosterlitz, D. Thouless, J. Phys. C 6, 1181 (1973). 3. M. Tinkham, Introduction t o Superconductivity, McGraw-Hill, Inc., New York (1996). 4. Z. Hadzibabic et al., Nature 441,1118 (2006). 5. V. Schweikhard et al., Phys. Rev. Lett. 99, 030401 (2007). 6. P. Kriiger et al., Phys. Rev. Lett. 99, 040402 (2007). 7. A. Polkovnikov et al., Proc. Natl. Acad. Sci. U. S. A. 103,6125 (2006). 8. T. Simula and P. Blakie, Phys. Rev. Lett. 96, 020404 (2006). 9. L. Giorgetti et al., Phys. Rev. A. 76, 013613 (2007). 10. J is obtained from 3D numerical simulations of the Gross-Pitaevskii equation for the central double-well system, self-consistently including meanfield interactions of both condensed and uncondensed atoms. A useful approximation for J in our experiments is: J ( V ~ L , N , , ~ ~ , T M ) zyxwvutsrqponm Nwell x 0.315nKexp[Nw,ll/3950 - V o ~ / 2 4 4 H z ] ( l +0.59T/100nK). 11. A. Leggett, Rev. Mod. Phys. 73,307 (2001). 12. D. Scherer et al., Phys. Rev. Lett. 98, 110402 (2007). 13. Within a dataset, the ramp-down rate is kept fixed, t , = r x V o ~ 1 1 . 3 kHz.

INTERACTING BOSE-EINSTEIN CONDENSATES IN RANDOM POTENTIALS P. Bouyer, L. Sanchez-Palencia, D. ClBment, P. Lugan, A. Aspect

Laboratoire Charles Fabry de l’lnstitut d’optique, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP CNRS and Univ. Paras X I , Campus Polytechnique, 2 av Fresnel, 91128 PALAISEAU cedex, France We investigate the transport properties of an interacting Bose-Einstein condensate in a speckle random potential. At equilibrium in a trapping potential and for the considered small disorder, the condensate shows a Thomas-Fermi shape modified by the disorder. When the condensate is released from the trap, a strong suppression of the expansion is obtained as observed in recent experiments. For the parameters of the experiment, it is shown to result from the competition between the disorder, the interactions and the kinetic energy. A scenario for disorder-induced trapping is proposed and analyzed. Numerical calculations performed in the mean-field approximation agree with the analytical results derived on the basis of this scenario. Keywords: Anderson Localisation, Random Potential, Bose-Einstein Condensate

1. Introduction : disorder and ultracold atoms Disorder can dramatically change the properties of quantum systems and result in a variety of non-intuitive phenomena, many of which are not yet fully understood. Striking examples are Anderson localization,’ percolat i o q 2 disorder-driven quantum phase transitions and the corresponding B o s e - g l a ~ sor ~ >spin ~ glass5 phases. It is known from the Bloch theory of solid state6 that all eigenstates of non-interacting particles in a periodic potential extend over the full system (as in free space). In contrast, it has been shown by Anderson‘ that the single-particle eigenstates in a random potential can be localized in regions significantly smaller than the size of the system. This effect is particularly dramatic in one-dimensional (1D) systems as it can be rigorously established that almost all eigenstates are locali~ed.~,~ Quantum disordered systems are of practical interest in modern condensed matter physics (CM). Indeed, since ‘Nature is never perfect’, the main periodic structure of real solids has to be completed by additional 11

12

quenched r a n d o m potentials. Understanding of quantum transport in amorphous solids is thus one of the main issues in this context, related to electric and thermal conductivities. The basic knowledge is that contrary to Bloch's theory which predicts a (frictionless) transport of non-interacting particles6 as a consequence of the extension of all eigenstates in a periodic crystal, localization effects in disordered potentials result in a strong suppression of the electronic transport in amorphous solids.' On the experimental front, the persistence and the stability of the superfluid phase have been studied g in systems such as 4He on Vycor glasses and dirty electronic materials." Ultracold atomic gases are now widely considered to revisit standard problems of CM with unique control possibilities. Dilute atomic BoseEinstein condensates (BEC) and degenerate Fermi gases (DFG) are currently produced taking advantage of the recent progress in cooling and trapping of neutral atoms. In particular, periodic potentials (optical lattices) with no defects can be designed in a wide variety of geometries." For example, in periodic optical lattices, transport has been widely investigated. l2 Controlled disordered potentials can also be produced by a variety of techniques, for instance speckle optical fields,'3p16 the use of magnetic traps designed on atomic chips with rough wires, localized impurity atoms," or radio-frequency fields.18 Optical speckle ptentials are of interest as both amplitudes and correlation functions with submicron correlation length14 can be controlled at will. For instance, the atom-atom interactions can be treated almost exactly in a tractable mean-field approximation or in other many-body theories. In addition of providing priviledge playgrounds for textbook models, ultracold gases in random potentials are also of fundamental interest as they introduce novel viewpoints related to finite size effects, inhomogeneities and, probably the most important, the possibility of investigating non-equilibrium phenomena and dynamical response funct ions. l9 Within the context of quantum gases, many recent theoretical efforts have considered disordered or quasi-disordered optical lattices. In these systems, one expects a large variety of phenomena, such as the Bose-glass phase transition, localization and the formation of Fermi-glass and quantum percolating and spin glass phases (for a recent review, see27).Localisation properties in interacting Bose gases a t equilibrium in speckle potentials (without an underlying lattice potential) have been discussed in." Further effects have also been addressed in connection to superfluid flows through disordered media. In particular, the reduction of the superfluid fraction

13

and a significant shift as well as the damping of sound waves has been calculated in ref^.^^)^' More recently, the coherent transport of a BEC along a disordered g ~ i d e ~ ' )and ~ ' the propagation of a soliton in a BEC in the presence of disorder have been in~estigated.~' The interplay between the kinetic energy, the atom-atom interactions and disorder is a challenging question that is relevant for interacting matterwaves in random potentials. We investigate here the transport properties of an interacting one-dimensional (1D) Bose-Einstein condensate in a speckle random potential. We focus on a regime where the interatomic interactions strongly dominate over the kinetic energy (hydrodynamic or Thomas-Fermi regime), a situation that significantly differs from the textbook Anderson localization problem and that is relevant for almost all current experiments with BECs (see for instance recent works on disordered BECs13-16). zyxwvutsrqponm 2. Suppression of expansion of a condensate in a speckle random potential

The question of the coherent dynamics of interacting matterwaves in random media is currently attracting significant experimental attentionl4)l5 mainly related to the search for a suppression of transport similar to that related to Anderson localization.' In this section, we consider the transport of an interacting BEC in a random potential in a tight binding 1D guide. We assume (i) that the chemical potential of the BEC is larger than the depth of the additional potential, p > VR,and (ii) that the correlation length oR of the potential V is much larger than the healing length of the BEC and much smaller than the initial size of the BEC, E ^' whose entanglement may be connected t o a multiparticle entangled state using e.g. superexchange interaction between the initially disconnected pairs. In the context of quantum information, such large entangled quantum states have been shown to be powerful resources for quantum computing.14 The control of superexchange interactions along different lattice directions also offers novel possibilities for the generation of topological many-body states for quantum information p r o ~ e s s i n g . ~ ~ ~ ~ ~

References

1. D. Jaksch and P. Zoller, Annals ofphysics 315,p. 52 (2005). 2. I. Bloch, Nature Physics 1, 23 (2005). 3. M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. S. De and U. Sen, Adv. Phys. 56,243 (2007). Physik A 31,p. 253 (1925). 4. E. Ising, Zeitschrift fiir zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJ 5. W. Heisenberg, Zeitschrzft fiir Physik A 38, 411 (1926). 6. A. Auerbach, Interactzng Electrons and Quantum Magnetism (Springer, 2006). 7. P. A. M. Dirac, Proc. Roy. SOC.,Ser. A 112,661 (1926). 8. P. A. M. Dirac, Proc. Roy. SOC.,Ser. A 123,714 (1929). 9. W. Heisenberg, Zeitschrift fiir Physik 49,619 (1928). 10. H. A. Kramers, Physica 1, 182 (1934). 11. P. Anderson, Phys. Rev. 79,350 (1950). 12. P. Lee, N. Nagaosa and X.-G. Wen, Rev. Mod. Phys. 78,17 (2006). 13. J. R. Petta, A. Johnson, J. Taylor, E. Laird, A. Yacoby, M. Lukin, C. Marcus, M. Hanson and A. Gossard, Science 309,2180 (2005). 14. H. J. Briegel and R. Raussendorf, Phys. Rev. Lett. 86,910(Jan 2001). 15. A. Kitaev, Ann. Phys. 321,2 (2006). 16. L.-M. Duan, E. Demler and M. D. Lukin, Phys. Rev. Lett. 91, p. 090402 (2003). 17. E. Altman, E. Demler and M. D. Lukin, Phys. Rev. A 70,p. 013603 (2004). 18. S. Folling, F. Gerbier, A. Widera, 0. Mandel, T. Gericke and I. Bloch, Nature 434,481 (2005). 19. M. Greiner, C. Regal, J . Stewart and D. Jin, Phys. Rev. Lett. 94,p. 110401 (2005). 20. R. H. Brown and R. Q. Twiss, Nature 177,27 (1956). 21. R. H. Brown and R. Q. Twiss, Nature 178,1447 (1956).

34 22. G. Baym, Act. Phys. Pol. B 29, 1839 (1998). 23. M. Yasuda and F. Shimizu, Phys. Rev. Lett. 77, 3090 (1996). 24. A. Ottl, S. Ritter, M. Kohl and T. Esslinger, Phys. Rev. Lett. 95, p. 090404 (2005). 25. M. Schellekens, R. Hoppeler, A. Perrin, J. V. Gomes, D. Boiron, A. Aspect and C. I. Westbrook, Science 310, 648 (2005). 26. W. D. Oliver, J. Kim, R. C. Liu and Y . Yamamoto, Science 284, 299 (1999). 27. M. Henny, S. Oberholzer, C. Strunk, T. Heinzel, K. Ensslin, M. Holland and C. Schonenberger, Science 284, 296 (1999). 28. H. Kiesel, A. Renz and F. Hasselbach, Nature 418, 392 (2002). 29. M. Iannuzzi, A. Orecchini, F. Sacchetti, P. Facchi and S. Pascazio, Phys. Rev. Lett. 96, p. 080402 (2006). 30. T. Jeltes, J. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect and C. Westbrook, Nature 445, 402 (2007). 31. F. Werner, 0. Parcollet, A. Georges and S. Hassan, Phys. Rev. Lett. 95, p. 056401 (2005). 32. P. Lee, N. Nagaosa and X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006). 33. W. Hofstetter, J. I. Cirac, P. Zoller, E. Demler and M. D. Lukin, Phys. Rev. Lett. 89, p. 220407 (2002). 34. L. Mathey, E. Altman and A. Vishwanath, cond-mat/0507108 (2005). 35. A. Polkovnikov, E. Altman and E. Demler, PNAS 103, 6125 (2006). 36. D.-W. Wang, M. D. Lukin and E. Demler, Phys. Rev. A 72, p. 051604 (2005). 37. V. Ahufinger, L. Sanchez-Palencia, A. Kantian, A. Sanpera and M. Lewenstein, Phys. Rev. A 72, p. 063616 (2005). 38. A. M. Rey, I. I. Satija and C. W. Clark, cond-mat/0601307 (2006). 39. V. W. Scarola, E. Demler and S. D. Sarma, Phys. Rev. A 73, p. 051601(R) (2006). 40. D. Jaksch, H. J. Briegel, J . I. Cirac, C. W. Gardiner and P. Zoller, Phys. Rev. Lett. 82, 1975 (1999). 41. A. Sgirensen and K. Mgilmer, Phys. Rev. Lett. 83, 2274(Sep 1999). 42. 0. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hansch and I. Bloch, Nature 425, p. 937 (2003). 43. M. Anderlini, P. J. Lee, B. L. Brown, J. Sebby-Strabley, W. D. Phillips and J. V. Porto, Nature 448, 452 (2007). 44. E. Jan6, G. Vidal, W. Dur, P. Zoller and J. Cirac, Quantum Inf. Comput. 3, 15 (2003). 45. J. Sebby-Strabley, M. Anderlini, P. Jessen and J. Porto, Phys. Rev. A 73, p. 033605 (2006). 46. J. Sebby-Strabley, B. Brown, M. Anderlini, P. Lee, P. Johnson, W. Phillips and J. Porto, Phys. Rev. Lett. 98, p. 200405 (2007). 47. S. Folling, S. Trotzky, P. Cheinet, M. Feld, R. Saers, A. Widera, T. Muller and I. Bloch, Nature 448, 1029 (2007). 48. A. B. Kuklov and B. V. Svistunov, Phys. Rev. Lett. 90, p. 100401 (2003). 49. A. Micheli, G. K. Brennen and P. Zoller, Nat. Phys. 2, 341 (2006). 50. C. F. Roos, G. P. T. Lancaster, M. Riebe, H. Haffner, W. Hansel, S. Gulde,

35 C. Becher, J. Eschner, F. Schmidt-Kaler and R. Blatt, Phys. Rev. Lett. 92, p. 220402 (2004).

51. C. Langer, R. Ozeri, J. D. Jost, J. Chiaverini, B. DeMarco, A. Ben-Kish, R. B. Blakestad, J. Britton, D. B. Hume, W. M. Itano, D. Leibfried, R. Reichle, T. Rosenband, T. Schaetz, P. 0. Schmidt and D. J. Wineland, Phys. Rev. Lett. 95, p. 060502 (2005).

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PRECISION MEASUREMENT AND FUNDAMENTAL PHYSICS

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T-VIOLATION AND THE SEARCH FOR A PERMANENT ELECTRIC DIPOLE MOMENT OF THE MERCURY ATOM

E. N. FORTSON* zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR Department of Physics, University of Washington, Seattle, W A 98195, USA *E-mail: [email protected]. edu

There has been exciting progress in recent years in the search for a permanent electric dipole moment (EDM) of an atom, a molecule, or the neutron. An EDM along the axis of spin can exist only if time reversal symmetry ( T ) is violated. Although such a dipole has not yet been detected, mainstream theories of possible new physics, such as Supersymmetry, predict the existence of EDMs within reach of modern experiments. After a brief survey of current and planned EDM searches worldwide and the implications of current results for the existence of new T-violating (and hence CP-violating) interactions, I review recent work on our own EDM experiment with mercury atoms, describing the newest version of this experiment and discussing current measurements. We have instituted a fixed blind offset that permits us t o test for systematic errors while insuring that any cuts in the data are made objectively. Compared with 2.1 x 1OP2*ecm, an improvement by a factor of our 2001 result l d ( l g 9 H g ) / 2They set what seemed at the time a remarkably small upper limit, ld(n)I < 5 x 10W2'ecm. Since then, there have been many further searches for an EDM of the neutron, with ever increasing precision. Likewise there have been continually improved searches for an EDM of an atom or a molecule - including experiments sensitive to an intrinsic EDM of the electron. Thus far, all EDM experiments have yielded a null result. Nevertheless, elementary particle theories that attempt to go beyond the Standard Model,3 most notably Supersymm e t r ~ predict ,~ that EDMs should exist and be large enough to detect by experiments now underway or soon to begin.5,6 The existence of an EDM of any non-degenerate quantum system would 39

40

imply a breakdown of time-reversal symmetry (T), and through the CPT theorem, a violation of CP-symmetry as ell.^>^ (C is charge conjugation, or particle/antiparticle symmetry, and P is parity, or space-inversion symmetry.) CP-violation was first discovered in the decays of KO mesons 40 years ago,7 and has recently been confirmed in B meson decay^.^)^ For many years after the initial discovery, the search for a neutron EDM provided an exacting test of theories put forward to account for the &, and ruled out most of them as the experimental upper limit on the neutron moment steadily decreased to its current value." Atomic and molecular EDM experiments made equally striking advances as well, starting in the 1960s" and leading up to recent work that includes measurements on thallium" and mercury,13 with a host of new experiments now planned or underway. At current accuracies, the atomic and neutron experiments set comparable and complementary bounds on Supersymmetry and other theories of new physics. The rest of this paper is divided into the following sections: 2. Underlying Theory; 3. Survey of EDM Experiments; and 4. The lg9Hg EDM Measurement in Seattle. The reader interested mainly in experimental details first can skip to section 4, and afterwards read about the significance and the relation to other work in sections 2 and 3 if desired. 2. Underlying Theory

It is now generally accepted that a satisfactory explanation of the observed CP-violation (and equivalently, T-violation) in the KO and Bo systems is given by the Standard Model, in which CP-violation occurs as a complex phase factor (the KM phase) in the interaction of quarks with W bosons. Particle EDMs can be calculated from this mechanism, but due to cancellations at the lowest orders, the Standard Model gives negligibly small prediction^.^>^ (In the Standard Model there is one other phase which can lead to physically observable effects, the OQCD term of the QCD Lagrangian. However, the limits on the neutron and atomic EDMs indicate that O Q c D < lo-', and the usual assumption now is to take OQCD = 0 in connection with the existence of a postulated new light particle, the axion.5) Thus the Standard Model by itself predicts EDMs far too small to be observed in current or contemplated experiments. If an EDM is found, it will be compelling evidence for the existence of some sort of physics beyond the Standard Model. There is no shortage of theories of such new physics, but by far the most cherished among particle theorists is Supersymmetry (SUSY).4>14

41

SUSY incorporates quantum gravity consistently, and also solves the gauge hierarchy problem, i.e. it protects the huge energy gap between grandunification/quantum-gravity at 10l6 - lo1' GeV and the electroweak scale at 100GeV. Another reason to expect that new sources of CP violation, as in SUSY, will eventually be found is that Standard Model CP violation is too small to explain the matter-antimatter asymmetry of the universe.15 A feature of SUSY and most other models of new physics that is of great importance for EDMs is the existence of many new particles with CP-odd phase angles that do create EDMs in lowest order and have no natural reason to be small, just as the KM phase angle is about 7r/4 in the Standard Model. SUSY requires that for every particle there exist a superpartner particle, with spin that differs by one half. The emission and reabsorption of virtual spin-0 superpartners tends to generate EDMs in lowest order,5 which will automatically be of observable size if the lowest superpartner mass scale is in the 100 - 1000 GeV range required for SUSY to protect the gauge hierarchy. A number of authors have pointed out that EDM searches therefore have a good chance of being the first experiments to discover SUSY or whatever new physics does lie beyond the Standard Model.

Fig. 1. Allowed values of C P violating phases for the MSSM, assuming a superpartner mass scale of M= 500 GeV. For SUSY to protect the guage hierarchy, M should be in the range 100 - 1000 GeV. The EDM sensitivity scales as M-', so if M= 1000 GeV the angle bounds would be four times larger. The figure is adapted from Ref. 16, updated by M. Pospelov (2003).

As shown in Fig. 1, EDM predictions from SUSY models are already

42

worrisomely large when compared to experiment. The Minimally Supersymmetric Standard Model (MSSM), with “natural” values (of order unity) for its two additional C P violating phases, gives EDMs that are between 10 and 100 times larger than current experimental limits. Fig. 1 shows the allowed phase values in the MSSM when the neutron,” electron (determined from the atomic thallium EDM limit12), and mercury13 EDM limits are considered. The combined limit constrains both phases to be very near zero, which indicates that the MSSM requires some degree of “fine tuning” to be a valid model. Further improvements in the precision of EDM experiments will continue to inform SUSY models, and in general can be considered a sensitive method of probing for C P violating new ~ h y s i c s . ~ zyxwvu

3. Survey of EDM Experiments The way T(or CP)-violation at the fundamental elementary particle level would generate an observable EDM depends upon the system under study. The neutron is sensitive almost exclusively to T-violation in the quark sector, while atoms and molecules have bound electrons and are therefore sensitive to T-violation in the lepton sector as well as the quark sector. In atoms and molecules there are actually a number of ways that Tviolating interactions at the particle level could give rise to an EDM, and all are enhanced considerably in heavy atoms.6 Calculations have been made of the atomic EDM due to an EDM distribution in the nucleus, to a T violating force between electrons and nucleons, and to an intrinsic EDM of the electron itself, corresponding respectively to hadronic (quark-quark), semi-leptonic (electron-quark), and purely leptonic interactions as the chief source of T-violation. Which of the possible effects will predominate in a given atom or molecule depends upon the net electronic angular momentum J . In systems with J = 0 (i.e. systems with only closed electronic shells, such as Hg, Xe, and Ra), the EDM vector points along the nuclear spin I, and the greatest sensitivity is to purely hadronic T-violation inside the nucleus. In this case, the important quantity is the nuclear Schiff M ~ m e n t ,which ~ > ~ measures the part of the nuclear EDM that is not completely shielded from the outside world by the atomic electrons. Although shielding does reduce the size of EDMs in closed shell atoms, it turns out that this loss can be more than compensated by the extra experimental EDM sensitivity attained in these atoms. Another source of an EDM along I could in principle be a tensor-pseudotensor form of electron-nucleon T - ~ i o l a t i o n . ~ > ~ In systems with non-zero J (i.e. paramagnetic systems such as Cs, T1

43

or open-shell molecules) the EDM has a component parallel to J , and the greatest sensitivity is to an intrinsic electron EDM, or to a scalarpseudoscalar form of electron-nucleon T - ~ i o l a t i o n .The ~ > ~great atomic theory discovery here, made in the 1960s by Sandars,17 is that the effect of an electron EDM is actually enhanced in a heavy atom, by over a factor of 100 in cesium and considerably more in thallium and other heavier atoms. An additional enhancement, also discovered by Sandars," takes place in polar molecules due to the large internal electric field in these molecules that can couple to an EDM. This field can be of order lo4 zyxwvutsrqponmlkjihgfedcbaZ lo5 times available laboratory fields, yielding a corresponding enhancement. The field axis of a polar molecule can generally be aligned in a relatively modest laboratory field. New experiments, some of which are shown in Table 1, are expected to improve over current EDM sensitivity by factors of 10 - 100. Table 1. Some EDM experiments underway or planned Spin

System

Method

Location

Nuclear

lggHg lZ9Xe Ra Neutron

4-cell vapor Liquid cell Optical trap Superfluid He bath Neutron cell

Seattle Princeton Argonne Los Alamos, SNS Grenoble, ILL, PSI

Electron

YbF PbO Other molecule

Beam Cell Optical and ion traps Optical lattice traps Macroscopic B or E

Imperial College Yale Oklahoma, Boulder Penn St, Austin Amherst, Yale, Indiana

133cs

Magnetic Crystal

All experiments are based on what should happen when a spinning elementary particle, atom or molecule having an EDM is placed in the electric field that exists between two oppositely charged parallel plates. In the manner of a spinning top, the spin will precess about the electric field axis due to the electric torque on the dipole. The longer the spin remains in the electric field without being otherwise disturbed, i.e. the longer the spin relaxation time Tz, the larger will be its angle of precession due to an EDM and the more sensitive will be the experiment. When the electric field direction is reversed by reversing the sign of the voltage between the plates, the sense of spin precession about the field axis also reverses. This behavior helps distinguish the precession due to an EDM from that due to

44

other torques. 4. The lg9Hg EDM Measurement in Seattle

lg9Hg has a 6 '5'0 ground state electronic configuration, and a nuclear spin I = Because the ground state carries no electronic angular momentum, an EDM search in mercury is primarily sensitive to T-violation associated with the quarks in the nucleus. The T-violating nature of an EDM is apparent from the Hamiltonian describing the interaction of the mercury spin with external magnetic and electric fields:

i.

H

=

-(dE

+ pB) . I/I,

(1)

where d is the electric dipole moment and p is the magnetic dipole moment. Under time reversal, H must change since I and B change sign while E does not. A search for an EDM of lg9Hg has been underway in our laboratory at the University of Washington for over 20 years. Our last experiment,13 which used a frequency-quadrupled laser diode on the 254 nm mercury absorption line to orient the lg9Hg nuclear spins, yielded the 2001 EDM result: d(lggHg)= -(1.06 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJ f 0.4gStatf 0.4OsYst)x 10-28ecm, which set an upper bound on the EDM of Id(199Hg)I< 2.1 x 10-28e cm (95% confidence level)

As shown in Fig. 1 above, the leading theoretical extension to the Standard Model, Supersymmetry, is expected to generate a lg9Hg EDM comparable to our experimental limit. By increasing the precision of our result, we could provide important information about the model parameter space of Supersymmetry and other theories or of course possibly observe a nonzero EDM. ~

4.1. 4-cell Experiment With such motivations in mind, upon completion of our 2001 measurement we undertook a major improvement in the lg9Hg EDM experiment. We began with a study of the spin relaxation in our vapor cells, which led us to construct new cells that on average have 1.5 times longer spin coherence times. However, the main improvement to the experiment was the construction of an apparatus that incorporates a stack of four vapor cells (See the cutaway view in Fig. 4 below). Previous versions of the experiment have all compared the spin precession frequency between two vapor cells, where the

45

cells are in  a common magnetic field and oppositely directed electric  fields. In  the  current  experiment  the two  additional  cells  are  at  zero  electric  field and  are  used  as  magnetometers  above  and  below  the  EDM  sensitive  cells. They  help  to  improve  our  statistical  sensitivity  by  allowing  magnetic  field gradient  noise  cancellation,  and  they  are  also  used  to  cancel  out  possible magnetic systematic effects  (See Fig.  5 below).

Fig.  2.  Simplified diagram of the 199Hg EDM apparatus.

As before, to search for an EDM, we measure the Larmor spin precession frequency of 199Hg.  A common magnetic field produces Larmor precession in a vapor of spin polarized mercury in each cell,  and a strong electric  field applied in opposite directions in the middle two cells modifies the precession frequency  by  an  amount  proportional  to  the  electric  dipole  moment.  From Eq.  (1), an EDM would cause a frequency shift of 2Ed/h, with opposite sign in  the  two  cells;  so  the  magnitude  of the  EDM  is  given  by d = h8v/(^E), where 5v is the difference in precession frequency between the two cells. The  current  version  of the  experiment  is  shown  in  Fig.  2.  We  spin  po­ larize the  199Hg nuclei by optical pumping on the 253.7 nm absorption line in mercury.  Strong laser beams line up the nuclear spins,  and weaker probe beams monitor the free­precession frequency in each cell.  The pump­probe pattern  is  shown  in  Fig.  3.  Since  the  light  beam  is  transverse  to  the  pre­ cession  axis,  the  circularly  polarized  pumping  light  is  modulated  at  the

46

0

»

>

16

_a

'S "2 4 1

0

20

40

60

SO

100

120

Time (sec) Fig.  3.  Pump­probe  sequence  showing  the  Larmor  precession  frequency  expanded  in the  inset.

Larmor  frequency to  synchronously pump the precessing spins.  The  probe beam  is  made  linearly  polarized  and  the  back  and  forth  optical  rotation at  the  Larmor  frequency is  used  to measure the  frequency.  The ultraviolet light  for  this  transition  is  obtained  by  quadrupling  the  output  of  an  in­ frared  diode  laser.  Our  laser  system  produces  several  milliwatts  of stable, tunable  UV  radiation  with  good  spatial  characteristics.  This  system  has operated continuously and problem­free for several years, and requires only occasional maintenance.  We lock the  laser frequency to  the  absorption line in  a separate  Hg vapor  cell. The  cells  are  held  as  shown  in  Fig.  4  inside  a  sealed  vessel  filled  with about I  bar of SFg  or Nj  gas to reduce the  leakage currents.  The vessel and electrodes are constructed of conductive polyethylene, which we found had exceptionally low magnetic impurity content.  The vapor cells have been al­ tered  slightly since  our  last  publication,  containing  a  100%  CO  buffer gas, instead  of the  95%  N2  /  5%  CO  mixture  used  for  the  2001  measurement. Our  studies  of spin  relaxation  in  mercury  vapor  cells19  indicated  that  the wax coating on the interior of the cells could be damaged by collisions with excited  metastable mercury atoms.  The  CO  buffer gas efficiently quenches these  metastable  states  and  thus  helps  prevent  damage  to  the  coating.  The end  result  is  that  we  can  achieve  polarization  lifetimes  that  are  a factor  of 1.5  longer  than was  possible with  the  old  vapor  cells.  With  these  improve­ ments, we are now sensitive to spin precession frequency shifts on the 10"10 Hz scale. We have made an extensive effort to assess the noise performance,

47

with  the  goal  of improving  the  sensitivity  still  further.  The  shot  noise  con­ tribution is modeled using computer simulations, which show we are within a  factor  of  three  of  the  shot  noise  limit.  While  the  modeling  also  shows that  further  improvements  to  reduce  the  shot  noise  itself  are  possible,  we must  first  eliminate  the  current  extraneous  noise  limiting  the  experiment. We  are  pursuing  these  goals  while  at  the  same  time  we  are  accumulating EDM  data  with  the  present  sensitivity.

Fig. 4.  Cutaway view of the EDM cell­holding vessel. High voltage (± 10 kV) is applied to  the  middle  two  cells  with  the  ground  plane  in  the  center,  so  that  the  electric field is opposite  in  the  two  cells.  The  outer  two  cells  are  enclosed  in  the  HV  electrodes  (with light  access holes as shown here for the bottommost cell),  and are at zero electric field. A uniform magnetic field is applied in the vertical direction.

In order to reach the 1x10 2g e cm.  level we must place tight bounds on any systematic effects  in the measurement.  The most  dangerous  effects  are those  which  generate  magnetic fields that  are  correlated  with  the  direction of  the  applied  electric  field.  Leakage  currents  across  the  cell  when  high voltage is applied are one prime example. We continually monitor all leakage currents,  and  with  careful  cleaning  and  preparation we  limit  such  currents to  the  pA  level.  Our  measurements  continue  to  suggest  that  this  is  below the  level  that  could  cause  a  problem  at  the  present  level  of  sensitivity. An  important  new  safeguard  is  possible  now  that  we  have  4  cells.  As  one example,  the  "leak­test"  combination  of  individual  cell  measurements,  as shown  in  Fig.  5,  is  sensitive  to  leakage  current  fields  while  canceling  any

48

EDM effect. Another possible problem could be high voltage sparks which might change the field of a trace magnetic impurity located near the cells or electrodes. We have constructed the apparatus from materials that are as free of such impurities as possible. Again, some combinations of cell frequencies will be sensitive to such local fields, and can reveal the presence of impurities. 4.2. Blind Analysis Because of the need to cut some data (for example, when magnetic impurities do appear), we initiated a blind analysis procedure for all data taken after March 2006. The analysis program adds a fixed, blind HV correlated offset to the middle cell fitted frequencies, +6/2 to the middle top cell and -6/2 to the middle bottom cell, which gives an artificial EDM-like signal of size 6, randomly generated between f 2 x e cm (our previous upper bound). This range is large enough to insure the analysis is blind, but small enough to reveal any large spurious signals that might appear due to the changes made when the blind analysis began. Once selected, the blind offset remains fixed throughout all data, and therefore does not interfere with tests for systematic effects (e.g. correlations with leakage currents, etc). And of course it guards against human bias in decisions about making data cuts, etc. We are now taking data for a new measurement of the lg9HgEDM. Thus far the accumulated statistical error is f 1 . 5 x 10-29ecm, over a factor of 10 below the upper limit of our 2001 measurement. It remains to be seen how small a systematic error will emerge from this measurement. 4.3. The Ig9Hg Stark Interference Eflect

A static electric field applied to an atom with an El (electric dipole) optical transition induces M1 (magnetic dipole) and E 2 (electric quadrupole) transitions. The presence of these additional transitions leads to an interference effect of a particular vector character. For a F = 4 F = El transition, such as the one we use in the lggHg EDM search, the fractional change in the absorptivity a is of the form,

;

;

where a is a factor denoting the strength of the effect, E is the direction of the electric field vector of the light driving the transition, k is the propagation direction of the light, Es is the static electric field, and B is the atomic spin

49

Frequency combinations Middle cell difference:

(

0- ~

~

cancels common made noisc cqnivaknt to zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF 2001 mcasurrmcnt

Anti-symmetric combination 1 (EDMcombo): zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG (%-qd$I%­%)

.

cancels np to 2nd order grndicnt noisc same EDM rcsponsc ils middic =n d i & r ~ o c ~

Symmetric combination (LeakTest combo): ((DMT + %)-

( ( ~ a+ r

cancels linear giadicnt noise

givcszcfoforahucEDM scmitivc to lcakagc currents and other -tic systematics

Other combinations can also help reveal the presence of spurious magnetic effects

Fig. 5.

Frequency combinations with 4 cells.

polarization direction of the ground state. The factor a has been calculated t o be -6.6 x lo-' (kV/cm)-120 for the 254 nm El transition in lg9Hg. This Stark interference effect is of interest for the EDM search because it can lead to a light shzft (also called an ac-Stark shift), an apparent Larmor frequency shift that is linear in the strength of the applied electric field; in other words, it can mimic an EDM. The effect can be measured with the present EDM apparatus with only minor modifications, and a preliminary result agrees in order of magnitude with the calculated effect. A satisfying feature of the result is the confirmation that we see an effect that, like an EDM, is linear in Es while using the same apparatus with almost the identical procedure and analysis as used in the EDM experiment itself. A more precise measurement is currently underway. It is crucial to guard against the Stark interference appearing as a systematic effect. One way we have exploited to control the problem is t o use the probe laser at two different wavelengths where the Stark interference light shift has opposite sign, and average the results to cancel out the Stark interference. A way to completely eliminate the Stark interference problem is to evaluate the Larmor frequency 'in the dark' between two probe laser pulses (which establish the Larmor phase at the beginning and end of the dark period). We are currently implementing such a scheme.

50

Acknowledgments

I wish to thank Clark Griffith, Blayne Heckel, Tom Loftus, Mike Romalis, Matthew Swallows, a n d m y other colleagues on the mercury EDM experiment over the years. This work was supported by NSF Grant PHY 0457320. References 1. 2. 3. 4. 5. 6. 7. 8.

E. M. Purcell and N. F. Ramsay, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML Physical Review 78,p. 807 (1950). J. H. Smith, E. M. Purcell and N. F. Ramsey, Phys. Rev. 108,120 (1957). S. M. Barr, International Journal of Modern Physics A (1993). G. L. Kane, Perspectives on Supersymmetry (World Scientific, Singapore, 1998), p. xv. N. Fortson, P. Sandars and S. Barr, Physics Today 56,33(June 2003). I. P. Khriplovich and S. K. Lamoreaux, C P Violation Without Strangeness (Springer, Berlin, 1997). J. H. Christenson, J. W. Cronin, V. L. Fitch and R. Turlay, Physical Review Letters 13, 138 (1964). Babar Collaboration, B. Aubert, et al., Physical Review Letters 87, 091801/1

(2001). 9. Belle Collaboration, K. Abe, et al., Physical Review Letters 87, 091802/1 (2001). 10. P. G. Harris, C. A. Baker, K. Green, P. Iaydjiev, S. Ivanov, D. J. R. May, J. M. Pendlebury, D. Shiers, K. F. Smith and M. van der Grinten, Phys. Rev. Lett. 85,904 (1999). 11. P. G. H. Sandars and E. Lipworth, Physical Review Letters 13, 718 (1964). 12. B. C. Regan, E. D. Commins, C. J. Schmidt and D. DeMille, Phys. Rev. Lett. 88 (2002). 13. M. V. Romalis, W. C. Griffith, J. P. Jacobs and E. N. Fortson, Phys. Rev. Lett. 86,2505 (2001). 14. J. H. Schwarz and N. Seiberg, Reviews of Modern Physics 71,S112 (1999). 15. M. Trodden, Reviews of Modern Physics 71, 1463 (1999). 16. T. Falk, K. A. Olive, M. Pospelov and R. Roiban, Nuclear Physics B 560,3 (1999). 17. P. G. H. Sandars, Physical Review Letters 14, 194 (1964). 18. P. G. H. Sandars, Physical Review Letters 19, 1396 (1967). 19. M. V. Romalis and L. Lin, Journal of Chemical Physics 120, 1511 (2004). 20. S. K. Lamoreaux and E. N. Fortson, Physical Review A 46,7053 (1992).

QUANTUM INFORMATION AND CONTROL I

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QUANTUM INFORMATION PROCESSING AND RAMSEY SPECTROSCOPY WITH TRAPPED IONS C. F. ROOS, M. CHWALLA, T. MONZ, P. SCHINDLER, K. KIM, M. RIEBE, and R. BLATT Institut fur Experimentalphysik, Universitat Innsbruck, Technikerstr. 25, A-6020 Innsbruck, Austria and Institut fur Quantenoptik und Quanteninformation, Osterreichische Akademie der Wissenschaften Otto-Hittmair-Platz 1, A-6020 Innsbruck, Austria High-resolution laser spectroscopy and quantum information processing have a great deal in common. For both applications, ions held in electromagnetic traps can be employed, the ions’ quantum state being manipulated by lasers. Quantum superposition states play a key role, and information about the experiment is inferred from a quantum state measurement that projects the ions’ superposition state onto one of the basis states. In this paper, we discuss applications of Ramsey spectroscopy for quantum information processing and show that techniques developed in the context of quantum information processing find useful applications in atomic precision spectroscopy. Keywords: Trapped ions, quantum information processing, precision spectroscopy, Ramsey spectroscopy, entanglement

1. Introduction

Single trapped and laser-cooled ions held in radio-frequency traps constitute a quantum system offering an outstanding degree of quantum control. The ions’ internal as well as external quantum degrees of freedom can be controlled by coherent laser-atom interactions with high accuracy. At the same time, the ions are well isolated against detrimental influences of a decohering environment. The combination of these two properties have enabled spectacular ion trap experiments aiming at building better atomic c l o c k ~ , l - ~ creating entangled state^^)^ and processing quantum i n f ~ r r n a t i o n . ~ > ~ At a first glance, the construction of a quantum computer and of an atomic clock might not seem to have much in common. However, there are close ties linking the two fields of research. The implementation of en- zyxwvuts

53

tangling quantum gatesg-I3 with ultra-high fidelity necessitates a precise knowledge of the Hamiltonian governing the dynamics of the atomic system and its interaction with the laser beams applied for steering it. Equally important are the characterization of decohering or dephasing mechanisms arising from the interaction of the atoms with fluctuating electromagnetic fields. For this task, Ramsey spectroscopy turns out to be an extremely important tool. In an atomic clock, the transition frequency between two atomic levels is measured by exciting the atom with laser pulses. If the excitation is done in a Ramsey experiment, probing of the clock transition can be described in the language of quantum information processing as a phase estimation algorithm. For this purpose, the use of multi-particle entangled states has been shown to be of interest.14>15In addition, entangling interactions have found applications in atomic clock measurements for quantum state detection of a system that is otherwise difficult to measure" and for the detection of small energy level shifts by preparing a system of two ions in a manifold of entangled states that are part of a decoherence-free subspace. l7 In the first part of this paper, generalized Ramsey experiments investigating ion-ion couplings which are important in the context of high-fidelity quantum gates will be presented. In its second part, experiments aiming at making quantum information processing more robust against environmental noise will be discussed. We will show how to apply (quantum mechanically) correlated states of two ions for precision measurements of atomic constants. These ion-trap experiments demonstrate high-precision spectroscopy in a decoherence-free subspace using a pair of calcium ions for a determination of energy level shifts and transition frequencies in the presence of phase noise. For the measurement, maximally entangled ions are advantageous for achieving a good signal to noise ratio. As the preparation of these states is more involved than single-ion superposition states, we explore the possibility of using classically correlated ions for achieving long coherence times. 2. Experimental setup

In our experiments, two 40Ca+ ions are confined in a linear Paul trap with radial trap frequencies of about w1/27r zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ = 4 MHz. By varying the trap's tip voltages from 500 to 2000 V, the axial center-of-mass frequency w, is changed from 860 kHz to 1720 kHz. The ions are Doppler-cooled on the Sl12 H P112 transition. Sideband cooling on the Sl/2 zyxwvutsrqponmlkjihg H D5/2 quadrupole transition18 prepares the stretch mode in the motional ground state Simultaneous cooling of stretch and rocking modes is accomplished

55

by alternating the frequency of the cooling laser exciting the quadrupole transition between the different red motional sidebands. Motional quantum states are coherently coupled by a laser pulse sequence exciting a single ion on the IS)= zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Sl12(rn= -1/2) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM c-) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJ ID) = D 5 p ( r n = -1/2) transition with a focused laser beam on the carrier and the blue sideband. Internal state superpositions (IS) ei@ID)) 10)can be mapped to motional superpositions lD)(lO)+ei@Il))by a 7r pulse on the blue motional sideband and vice versa. We discriminate between the quantum states Sl/2and D512 by scattering light on the Sl/2 H Pllz dipole transition and detecting the presence or absence of resonance fluorescence of the individual ions with a CCD-camera. A more detailed account of the experimental setup is given in

+

3. Ramsey spectroscopy techniques for quantum information processing

In trapped ion quantum computing, continuous quantum variables occur in the description of the joint vibrational modes of the ion string. The normal mode picture naturally appears when the ion trap potential is modelled as a harmonic (pseudo-)potential and the mutual Coulomb interaction between the ions is linearized around the ions’ equilibrium positions.20 In this way, the collective ion motion is described by a set of independent harmonic oscillators with characteristic normal mode frequencies. The normal modes are of vital importance for all entangling quantum gates as they can give rise to effective spin-spin couplings in laser-ion interaction^.^ All entangling ion trap quantum gates demonstrated so far use laser beams that intermittently entangle the internal states of the ion with a vibrational mode of the ion string. At the end of the interaction, the vibrational mode returns t o its initial state and the propagator describing the entangling gate operations is an operator acting only on the ions’ internal degrees of freedom. In most gate operations, the fidelity of the gate suffers if the vibrational state of the ion string couples to an environment that heats or dephases the ion motion. In previous experiments investigating the coherence of the center-ofmass mode of a two-ion crystal, we had observed heating rates of about 100 ms/phonon and coherence times for superpositions (0) 11) of vibrational states of about the same order of magnitude. For the coherence measurement, a Ramsey experiment was performed where first a carrier 7 ~ / 2pulse was applied to the ions in state IS)(S)lO)followed , by a 7r-pulse on the blue sideband of the center-of mass mode in order to create the state IS)lD)(lO) 11)).After a variable delay T , the inverse pulse sequence mapped the state IS)lD)(lO) ei@ll))onto a superposi-

+

+

+

56

tion IS)(cos(+ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA +o)lS) +sin(+ - +o)lD))IO).A Ramsey fringe pattern was recorded by scanning the phase $0 which was achieved by switching either the phase of the second blue sideband pulse or the phase of the second carrier pulse with respect t o the phase of the corresponding first pulse. Surprisingly, when this kind of Ramsey experiment was applied to investigate the coherence of the stretch mode of the two-ion crystal, the measured coherence time was found to be nearly two orders of magnitude shorter than for the center-of-mass mode. Fig. 1 (a) shows the contrast C(T)of the Ramsey fringe pattern as a function of the delay time. In this experiment, a coherence time of less than 2 ms was measured at a trap frequency wl(27r) = 1486 kHz. We found that the loss of contrast could be attributed to the nonlinear terms in the Coulomb interaction between the ions giving rise to a cross-coupling between the normal modes.21 For a two-ion crystal, this leads to a dispersive coupling between the stretch mode and the rocking mode where the ions oscillate out of phase in the transverse direction. As a result, the bare stretch mode frequency v$j is lowered slightly by an amount that is proportional t o the number of rocking mode phonons so that (0) v,t, = vst, - xn,,,k. After cooling the rocking modes to the ground state

02-

Fig. 1. Experiments probing the coherence of the stretch mode. (a) Ramsey experiment. (b) Spin echo experiment.

before repeating the experiment shown in Fig. 1 (a), we observed coherence times similar to the ones found for the center-of-mass mode. The fact that the stretch mode frequency is varying from experiment to experiment for ( n r O c k ) # 0 but constant within a single experiment is also revealed by a spin echo experiment probing the stretch mode coherence. For the experiment shown in Fig. 1 (b), a pulse sequence similar to the sequence for (a)

57

is used, but with additional pulses in the middle of the sequence that swap the population of the two lowest quantum states of the stretch mode. This makes the experiment insensitive against small changes of the stretch mode frequency so that the contrast decays to half of its initial value only.after 100 ms. To confirm that the observed spread in vibrational frequencies is indeed due to the postulated mechanism, we could even measure the shift induced by a single rocking mode phonon by performing a spin echo experiment and increasing the rocking phonon number by exactly one at the beginning of the second spin echo period by a blue sideband pulse on the rocking mode. The extra phonon shifts the Ramsey fringe pattern by an amount that can be related t o the strength x of the cross mode coupling. In our experiments, we find frequency shifts of up to 20 Hz per phonon.21 These shifts dramatically reduce the fidelity of Cirac-Zoller gate operations making use of the stretch mode as long as the rocking modes are cooled only to the Doppler limit.22 For the realization of high-fidelity quantum gate operations, this observation points to the necessity of either cooling the rocking modes to the ground state or using the center-of-mass mode for mediating the ion-ion coupling. 4. Quantum information processing techniques for precision spectroscopy

In atomic high-resolution spectroscopy, dephasing is often the most important factor limiting the attainable spectral resolution. Possible sources of dephasing are fluctuating electromagnetic fields giving rise to random energy level shifts but also the finite spectral linewidth of probe lasers. Under these conditions, two atoms located in close proximity to each other are likely to experience the same kind of noise, i.e. they are subject to collective decoherence. The collective character of the decoherence has the important consequence that it does not affect the entangled two-atom state

as both parts of the superposition are shifted by the same amount of energy by fluctuating fields. Here, for the sake of simplicity, 19) and le) denote the ground and excited state of a two-level atom. Because of its immunity against collective decoherence, the entangled state Q+ is much more robust than a single-atom superposition state L(1g) le)). This properties makes Jz states like Q+ interesting candidates for high-precision spectroscopy. In the following, we will first discuss how t o use Bell states for the measurement

+

58

of energy level shifts. Then, it will be shown that certain unentangled twoatom states can have similar advantages over single atom superposition states albeit at lower signal-to-noise ratio.

4.1. Spectroscopy with entangled states

In a Ramsey experiment, spectroscopic information is inferred from a measurement of the relative phase q5 of the superposition state Ig) +eZ41e)). zyxwvutsrqponm

&(

The phase is measured by mapping the states &(lg) zyxwvutsrqponmlkjihgfedc f) .1 to the measurement basis { lg),) .1 by means of a 7 r / 2 pulse. In close analogy, spectroscopy with entangled states is based on a measurement of the relative phase 4 of the Bell state Q4 = &(lg)le) e'4le)Ig)). Here, the phase is determined by applying 7r/2 pulses t o both atoms followed by state detection. 7r/2 pulses with the same laser phase on both atoms map the singlet state 1 (1g)le) - 1e)Ig)) to itself whereas the triplet state (1g)le) 1e)lg)) is fi mapped to a state (1g)Ig) ez"le)le)) with different parity. Therefore,

+

-&

&

+

+

oL1)op)

yields information about the measurement of the parity operator relative phase since (oL1)oL2)) = cos 4. If the atomic transition frequencies are not exactly equal but differ by an amount 6, the phase will evolve as a function of time 7 according to q5(7)= q50 ST. Then, measurement of the phase evolution rate provides information about the difference frequency 6. To keep the notation simple, it was assumed that in both atoms the same energy levels participated in the superposition state of eq. (1). In general, this does not need to be the case and the phase evolution is given , ~ the atomic by q5(-r) = ( ( w A ~- w h l ) f ( W A ~- w h 2 ) ) 7 . Here, W A ~ denote transition frequencies of atom 1 and atom 2, and w ~ are~the, laser ~ frequencies used for exciting the corresponding transitions. The minus sign applies if in the Bell state the ground state of atom 1 is associated with an excited state of atom 2 and vice versa. If the Bell state is a superposition of both atoms being in the ground state or both in the excited state, the plus sign is appropriate.

+

4.2. Spectroscopy with unentangled states of two atoms

One may wonder whether entanglement is absolutely necessary for observing long coherence times in experiments with two atoms. In fact it turns out that the kind of measurement outlined above is applicable even to completely unentangled atoms.23 If the atoms are initially prepared in the

59 product state

this state will quickly dephase under the influence of collective phase noise. The resulting mixed state

appears to be composed of the entangled state 9+ with a probability of 50% and the two states Igg) and lee) with 25% probability each. If the state @+ is replaced by the density operator p p in the measurement procedure described in subsection 4.1, the resulting signal will be the same apart from a 50% loss of contrast. The states 1g)Ig) and 1e)Ie) do not contribute to the signal since they become equally distributed over all four basis basis states by the 7r/2 pulses preceding the state detection. Their only effect is t o reduce the signal-to-noise ratio by adding quantum projection noise since only half of the experiments effectively contribute to the signal.

'

­1 0

I 50

100

150

200

Time (ms) Fig. 2. Parity oscillation caused by the interaction of a static electric field gradient with the quadrupole moment of the D512 state of 40Ca+. The first data point significantly deviates from the fit since the quantum state has not yet decayed to a mixed quantum state.

1 h

2 2

(b)

0.5- zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA T

v)

0 v)

of

0

* T 0 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK L - zyxwvutsrqponm t zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED -

-0.5I

10 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF 20 30 40 50

Field gradient (V/rnrn2) Fig. 3. Electric quadrupole shift measured with a pair of atoms in a product stat,e. (a) The shift varies linearly with the applied electric field gradient. (b) Residuals of the electric quadrupole shift measurements. The plot shows deviations of the data points measured with unentangled ion (open circles) and entangled ions (filled circles) with respect to the fit obtained from the entangled state data.

4.3. Measurement of a n electric quadrupole m o m e n t We applied the method outlined in subsection 4.2 to a measurement of the quadrupole moment of the D5l2 state. For this, we prepared the state 1 9, = + 5 / 2 )

+ I-W)

8 (1+3/2) zyxwvutsrqponmlkjihgfedcbaZYXWVUTS +I-1~)) (4)

61

and let it decohere for a few milliseconds. Here, Im) = m) denotes the Zeeman sub-level of D512 with magnetic quantum number m. After a waiting time ranging from 0.1 to 200 ms, 7r/2 pulses were applied and a parity measurement performed. Fig. 2 shows the resulting parity oscillation whose contrast decays over a time interval orders of magnitude longer than any single atom coherence time in 40Ca+.A sinusoidal fit to the data reveals an initial contrast of 48(6)% and an oscillation frequency v = 38.6(3) Hz. For the fit, the first data point at t=O.lps is not taken into account. At this time, the quantum state cannot yet be described by a mixture similar to the one of eq. (3) as some of the coherences persist for a few milliseconds and thus affect the parity signal. The parity signal decays exponentially with a time constant Td = 730(530) ms that is consistent with the assumption of spontaneous decay being the only source of decoherence (in this case, one would have T d = T D , / , / ~ FZzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON 580 ms where T D ~ is/ ~the lifetime of the metastable state). The quadrupole moment is determined by measuring the quadrupole shift as a function of the electric field gradient El. The latter is conveniently varied by changing the voltage applied to the axial trap electrodes. For a calibration of the gradient, the axial oscillation frequency of the ions is measured. Further details regarding the measurement procedure are provided in ref.17 Fig. 3(a) shows the quadrupole shift AvQs as a function of the field gradient (the small offset at El = 0 is caused by the second-order Zeeman effect). By fitting a straight line to the data, the quadrupole moment can be calculated provided that the angle between the orientation of the electric field gradient and the quantization axis is known. Setting AvQs = aE‘, the fit yields the proportionality constant ~1 = 2.977(11) Hz/(V/mm2). The quadrupole shift had been previously measured using a pair of ions in an entangled state. Both measurement give consistent results and thus confirm the validity of the approach based on correlated, unentangled atoms. 5 . Conclusion

Techniques developed for atomic clock measurements turn out to be very useful for precisely characterizing quantum interactions in a system of trapped ions dedicated to quantum information processing. For the realization of ultra-high fidelity quantum gates even small effect like the crosscoupling between vibrational modes that is not predicted by the simple normal mode picture become important. Quantum gates based on nonresonant excitations of vibrational sidebands are less affected than those relying on a resonant excitation. Still, to approach the precision required

62 for fault-tolerant quantum operations might require cooling all modes to the ground state. On the other hand, precision spectroscopy itself can profit from concepts developed for processing of quantum information by making use of more advanced detection schemes.

Acknowledgments We acknowledge support by the Austrian Science Fund (FWF), the European Commission (SCALA, CONQUEST networks), and by the Institut fur Quanteninformation GmbH. K. K acknowledges funding by the LiseMeitner program of the FWF.

References H. S. Margolis et al., Science 306, 1355 (2004). T. Schneider, E. Peik, and Chr. Tamm, Phys. Rev. Lett. 94, 230801 (2005). W. H. Oskay et al., Phys. Rev. Lett. 97, 020801 (2006). T. Rosenband et a]., Phys. Rev. Lett. 98, 220801 (2007). 5. D. Leibfried et al., Nature 438, 639 (2005). 6. H. Haffner et al., Nature 438, 643 (2005). 7. M. Riebe et al., Nature 429, 734 (2004). 8. M. D. Barrett et al., Nature 429, 737 (2004). 9. D. Leibfried et al., Nature 422, 412 (2003). 10. F. Schmidt-Kaler et al., Appl. Phys. B 77,789 (2003). 11. M. Riebe et a]., Phys. Rev. Lett. 97, 220407 (2006). 12. P. C. Haljan et al., Phys. Rev. A 72, 062316 (2005). 13. J. P. Home et al., New J. Phys. 8, 188 (2006). 14. J. J. Bollinger, W. M. Itano, D. J . Wineland, and D. J. Heinzen, Phys. Rev. A 54, R4649 (1996). 15. D. Leibfried et al., Science 304, 1476 (2004). 16. P. 0. Schmidt, T. Rosenband, C. Langer, W. M. Itano, J. C. Bergquist, and D. J. Wineland, Science 309, 749 (2005). 17. C. F. Roos, M. Chwalla, K. Kim, M. Riebe, and R. Blatt, Nature 443, 316 (2006). 18. C. F. Roos et al., Phys. Rev. Lett. 83, 4713 (1999). 19. F. Schmidt-Kaler et al., J. Phys. B: At. Mol. Opt. Phys. 36, 623 (2003). 20. D. F. V. James, Appl. Phys. B 66, 181 (1998). 21. C. F. Roos, T. Monz, K. Kim, M. Riebe, H. Haeffner, D. F. V. James, and R. Blatt, preprint, arXiv:0705.0788. 22. F. Schmidt-Kaler et al., Nature 422, 408 (2003). 23. M. Chwalla, K. Kim, T. Monz, P. Schindler, M. Riebe, C. F. Roos, and R. Blatt, preprint, arXiv:0706.3186.

1. 2. 3. 4.

QUANTUM NON-DEMOLITION COUNTING OF PHOTONS IN A CAVITY

S.HAROCHE*, C. GUERLIN, J. BERNU, S. DELEGLISE, C. SAYRIN, S.GLEYZES, S.KUHRt ,zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM M. BRUNE, and J.-M.RAIMOND Laboratoire Kastler Brossel, ENS, CNRS, UPMC 24 rue Lhomond, 75005 Paris, France Collbge de France 11 place Marcelin Berthelot, 75005 Paris, France *E-mail: haroche0lkb. ens.fr t Permanent adress: Johannes Gutenberg Universitat, Institut fur Physik, Staudingerweg 7, 55128 Maint, Germany The photons of a microwave field stored in a high-Q cavity are detected nondestructively by a beam of circular Rydberg atoms crossing the cavity one by one. The field collapses into a Fock state as information is progressively extracted by the atoms. The photon number subsequently decays through a succession of quantum jumps under the effect of cavity damping. The QND detection of photons could be used for the preparation and study of various kinds of non-classical fields localized in one or two cavities. Keywords: Cavity Quantum Electrodynamics, Quantum Non Demolition Measurement

1. Introduction Counting photons is generally a destructive process, since light is usually absorbed by photo-sensitive materials. It does not have to be so, however, and it has been known for a long time that light intensity can in principle be measured by photon non-absorbing quantum non-demolition (QND) methods.' Such QND procedures have been successfully used to analyze the fluctuations of relatively intense light beams containing many light quanta,2 but have so far been unable to pin-down discrete photons numbers. Taking advantage of the very strong light-matter coupling provided by Cavity Quantum electrodynamic^,^ we have recently been able to count photons in a high-Q cavity, in a way which fulfills all the conditions of an ideal QND measurement. The breakthrough for realizing these experiments has been the development of a very high-Q superconducting Fabry-Prot resonator 63

64

made out  of precisely machined  copper mirrors sputtered with  a thin  layer of Niobium.4 A microwave field is trapped between these mirrors for 0.13 s on average, a time long enough to let thousands of circular Rydberg atoms of Rubidium  cross  the  cavity  and  extract  progressively  information  from the field. The  measurement  induces  the field to  collapse  into  a  Fock  state with  a well­defined  number  of photons.  The field remains  in this  state  for a  while,  until  relaxation  makes  the  photon  number  cascade  down  to  zero by undergoing successive quantum jumps at random times. This QND pro­ cedure  has  allowed  us  to  detect  ­  for  the  first  time  in  a  real  experiment  ­ the staircase­like field­intensity signals that were previously exhibited  only by Monte Carlo simulations of quantum field evolution.5 We briefly present here  the  results  of these  experiments  which  have  been  recently  published in  two  papers,6'7  and  discuss  the  perspectives  they  open  for  the  study  of non­classical states of light. 2. Principle of experiment: atoms as clocks to read out the number of photons stored in a box

S

Fig. 1.  Schematic view of the circular Rydberg atom­superconducting cavity set­up for photon  QND  counting  (from ref.  6)

The  measurement  is  based  on  the  detection  of the  phase­shift  induced by  the field on  the  atomic  coherence  between  the  circular  Rydberg  states

65

Ie) and 19) of rubidium atoms crossing one by one the cavity (1e)and zyxwvutsrqpo 19) have principal quantum numbers 51 and 50 respectively). Fig. 1 presents an artist's view of our experimental set-up. A stream of Rydberg atoms, prepared in state Ie) in box B , are sent one at a time through the superconducting cavity C. This cavity sustains a Gaussian transverse profile mode at 51 GHz, nearly resonant with the transition between the states le) and 19). The atoms are subjected to classical pulses of microwave emitted by the pulsed source S and applied in the auxiliary cavities R1 and Rz sandwiching C. The combination of these two pulses constitute a Ramsey interferometer. After leaving Rz, the atoms are detected by a state selective field-ionization detector D . The velocity of each atom and its preparation time are controlled through a pulsed optical pumping process involving properly tuned laser beams. The central part of the set-up, from B to Rz, is cooled to a temperature of 0.8 K by a He cryostat. This ensures the good operation of the superconducting cavity and suppresses most of the thermal radiation background. More experimental details about this set-up can be found in refs. 3,6 and 8. The atoms and the field in C are slightly off-resonant, the frequency offset 6 being at least of the order of the atom-cavity vacuum Rabi frequency R (R/27r = 50 kHz). In spite of this relatively small detuning, no real atomic transitions can occur, because the atom-field coupling varies adiabatically as the atoms travel across the Gaussian profile of the cavity mode. The method is thus truly quantum non-destructive for the field. Precise tuning of the atomic transition is achieved by applying across the cavity mirrors a small electric field which Stark-shifts the circular states. This tuning field is added to a constant directing field applied across the mirrors to protect the circular states from unwanted transitions towards non-circular levels. The necessity to apply these fields to the atoms while they cross C precludes for this experiment the use of closed cavity structures. Due to the very strong coupling of the Rydberg atoms to microwaves, the phase-shift per photon accumulated by the atomic coherence during cavity crossing reaches values of the order of 7 r , for an atom-field detuning 6/27r = 70 kHz and an atomic velocity 'u = 250 m/s. Smaller phase shifts are easily obtained by merely increasing 6. In order to analyse our QND procedure, which is a variant of a method we had proposed in the early 1 9 9 0 ~ , it~ 1is~convenient ~ to describe each atom crossing C as a spin 1/2, the circular levels Ie) and Ig) corresponding to the spin states I+)z and I-)z respectively, along the direction Oz. The atomic states can then be represented as Bloch vectors whose tips are

66 on a Bloch sphere. Just before entering C, the atomic Bloch vector, initially prepared along Oz (state 1.) ), is rotated by the pulse R1 along the transverse direction Ox [the corresponding state is the linear superposition = ).1( Ig))/fi]. The Bloch vector then starts to rotate in the equatorial plane of the Bloch sphere, in full analogy with the ticking of a clock’s hand. Due to the light-shifts, this clock is delayed by the presence of the field in the cavity, so that the spin ends up in different directions depending upon the photon number. If the atomic phase shift per photon is adjusted to r/q (q integer), the spin’s hand when the atom leaves the cavity points in 2q directions spanning 360 degrees for photon numbers ranging from n = 0 to n = 2q - 1. The QND method consists in reading out these directions. For n 2 2q, the spin’s positions repeat themselves, so that the method is measuring n modulo 2q. Detecting a single atom provides in general only partial information, since the 2q final spin states are non-orthogonal (a notable exception is q = 1, see section 3 ) . Suppose that we decide to detect the spin component along the direction Ou(p) making the angle 4 ( p ) = p r / q ( p integer) with Ox.If the cavity contains p photons, the spin ends up in the state I+) u ( p ) and the probability for finding the result - l / 2 for the spin component along Ou(p) is zero. Conversely, if one finds the spin in this state, the probability that C contains p photons must obviously vanish. If, on the other hand, the spin is found in the +1/2 state along O u ( p ) , it is the probability for finding p+q photons which cancels. In other words, the atom detected along the direction Ou(p) provides information enabling us to suppress either the value p or the value p q from the photon number distribution. This logical argument, which allows us to infer the change in the photon probability distribution due to an acquisition of knowledge on the final spin’s state, is an expression of Bayes law in probability theory. By choosing for the next atom another detection direction Ou(p’), different photon numbers are decimated. With q different detection directions adjusted for successive atoms crossing C, we find out which photon number survives out of 2q initially possible values. In practice, the transverse spin is detected along a given direction making an angle 4with Ox by mapping out this direction onto Oz with the pulse Ra, applied to the atom after cavity exit, before performing a measurement of the atom in the energy basis. The angle 4 is fixed by properly choosing the phase of the R2 pulse. A measurement is thus ideally constituted by a sequence of q atoms crossing C, providing each a +1/2 or -1/2 reading, associated to one out of q different detection angles, i.e. different phases of the R2 pulse. In a real

+

+

67

situation,  some  redundancy  is  necessary  and  more  atoms  are  required  to pin down n without  ambiguity,  because of ­Ri  and R-2  pulses imperfections and of unread atoms due to limited detection efficiency. 3. A simple situation: counting single photons and detecting field quantum jumps

0.0

0.5

1.0 1.5 Tim e (s)

2.0

2.5

Fig. 2.  QND detection of a single photon. Upper and lower bars show the signal, a se­ quence of atoms detected in e) and \g) respectively. The photon observed here is excep­ tionally long­lived (about three cavity damping times). Erroneous counts (\e) detections in vacuum and \g) detection when 1 photon is present) are due to the imperfections of the  Ramsey  interferometer  (adapted  from  ref.  6).

We  have  first  applied  this  method  to  the  measurement  of the  residual field  produced  in C  by  the  thermal  excitation  of the  mirrors.6  According to  Planck's  law,  the  cavity  contains  on  average  0.05  photons  at T =  0.8 K,  this  mean  value  resulting  from  random  fluctuations  of the  number  of light  quanta  between  zero  and  one.  The  probability  that C  contains  more than one photon is negligible at this low temperature. We thus have merely to distinguish between two  photon number values  (0  and  1).  We choose in this  case q =  1,  which corresponds to a  TT  phase­shift  per photon.  There  is then  only  one  detection  direction  and  the  jf?2  pulse  has  an  unique  phase, mapping the transverse Bloch vectors corresponding to 0 and 1 photon onto the +  and  —  directions along Oz respectively. An  atom  detected  in \g}  thus  signals  0  photon  and  an  atom  detected  in e)  one photon. Fig.  2 shows a sequence of 2200 atomic detections recorded over  a  2.5  second  interval.  The  upper  and  lower  vertical  bars  correspond to  atoms  found  in \e)  and \g)  respectively.  A  long  sequence  of atoms  de­ tected  mostly  in \g)  indicates  that  the  field  is  in  vacuum.  Then,  around t =  1.05  s,  the  telegraphic  signal  suddenly  changes.  The  atoms  are  then detected  mostly  in e),  signalling  the  appearance  of one  photon.  The  pho­ ton number has undergone a quantum jump from n = 0 to n = 1, followed

68 about half of a second later by a jump in the opposite direction, marking the annihilation of the photon. Thousand of similar signals have been recorded, whose statistical analysis is in complete agreement with the predictions of Planck’s law and quantum electrodynamics theory. In another test, we have first prepared a photon in C by having a first resonant atom emit it, then detected this photon by sending across C a long sequence of QND-detector atoms. Repeating the experiment many times, we have analysed the statistical distribution of these single-photon survival times and obtained an exponential distribution, with a mean life time equal to the cavity damping time T, (with a small well-understood correction due to the effect of residual radiative thermal processes). 4. Progressive field state collapse and stochastic evolution of the photon number

In order to count larger photon number^,^ we inject in C a coherent field produced by a microwave source. This field is coupled in the cavity via diffraction on the mirrors edges. Its photon number has a Poisson distribution, with an average nozyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH = 3.84. The probability for finding more than 7 photons is 3.5%. The task of the QND procedure is thus to distinguish between 8 consecutive values of n comprised between 0 and 7. For this, we . detection choose q = 4 and adjust the phase shift per photon to ~ / 4 The phase is, from one atom to the next, adjusted to four different angles corresponding to the directions of the Bloch vectors associated to n = 6 , 7 , 0 and 1. The atomic data are processed by exploiting Baysian logic, extracting information from a long sequence of detection events. The atomic detection rate is about 5 atoms per milliseconds. At time t = 0, the photon probability distribution is flat, since no a priori knowledge is assumed, except that the photon number is bounded by seven. Then, as atoms are successively detected, photon numbers are decimated, until the distribution has converged to a single integer value. This corresponds to the collapse of the field state, induced in a step-by-step process by the progressive acquisition of information provided by the atomic readings. The measuring sequence corresponds to 110 atoms, detected within 26 milliseconds. This number is a compromise. It is large enough to let the photon number converge on most sequences, and small enough for the measurement time to remain short compared to T,. At the end of this collapse stage, the procedure is resumed. We drop information provided by the first atom and add information extracted from the lllth one and so on.. . In this way, the data are decoded continuously, N

69

using at each time information provided by the last  110  atoms.  An example of  signal  is  shown  in  Figure  3,  as  a  3­D  histogram.  The  photon  number is  plotted  along  one  horizontal  direction  and  the  atom  number  along  the other. Photon numbers from 0  (foreground) to seven (background) are rep­ resented  by  channels  of  different  shades  whose  heights  (representing  the corresponding  probabilities)  evolve  from  left  to  right.  Out  of the  initially uniform  distribution,  a  single  channel (n =  5)  surges,  as  the  others  decay to  zero.  This  is  the  state­collapse process.  The  selected  channel  remains  at a plateau­level for a while, illustrating the repeatability of a QND measure­ ment.  Cavity damping then takes over.  The n =  5  channel suddenly drops to zero while the n = 4, 3, 2,1 ones successively and transiently surge.  This describes  a photon­number  cascade towards  zero  occuring  through  sudden quantum  jumps.  The  evolution  ends  with  a  steady n = 0  channel  (field in  vacuum).  For  clarity,  the  time  scale  ­  and  hence  the  calibration  of the atomic  axis  in  Figure  3  ­  are  non­linear.

Fig. 3.  Three dimensional histogram showing the evolution of the photon number distri­ bution under repeated QND measurement.  Note the non­linear calibration of the atomic axis which makes visible the fast field collapse stage.

We  have  observed  thousands  of such field trajectories.  The  histograms of the  n­values  obtained  at  the  end  of the  collapse  stage  reproduce,  to  an excellent approximation the Poisson distribution of the initial coherent  field.

70

This illustrates the quantum postulate about the statistics of measurement outcomes. This experiment has generated for the first time Fock states of radiation with photon numbers larger than 2.

5. Perspectives for the study of non-classical field states in one or two cavities

This QND measurement opens novel perspectives for the generation of nonclassical states of light. If the initial photon number distribution spans a range of ns larger than 2q, the decimations induced by successive atoms do not distinguish between n and n zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM 2q. The field then collapses in a coherent superposition of the form Encn+zqln 2q). For instance, ~ 0 1 0 ) c2,12q) represents a field coherently suspended between vacuum and 2q photons. This superposition of states with energies differing by many quanta is a new kind of Schrodinger cat state of light. Other kinds of Schrodinger cat states are produced during the QND sequence. As the photon number is pinned-down, its conjugate variable, the field’s phase, gets blurred. After the first atom’s detection, the initial state collapses into a superposition of two coherent states with different phases.”>” Each of its components is again split into two coherent states by the next atom and so on, leading to complete phase uncertainty when the photon number has converged.” The evolution of the Schrodinger cat states generated in the first steps of this process could be studied by measuring of superpositions of coherent the field Wigner function.12 De~oherence’~ states containing many photons could be monitored in this way. Finally, we intend to extend these experiments t o the generation and study of field states belonging to two high-Q cavities, successively crossed by a beam of circular Rydberg atoms.14>15We could for instance prepare the field in a superposition of the form ( a ,0) 10, a ) representing a coherent field of complex amplitude a which is “at the same time” in the first cavity and in the second.16 If a beam of circular Rydberg atoms is used to measure the global photon number of the two cavities in a QND way, this field will collapse into a two-cavity Fock state of the form In, 0)+ 10, n ) ,corresponding to n photons being in a superposition of the state in which they all belong to the first cavity with the state in which they all belong to the second. These strange non-classical states, which have been recently generated in different c o n t e ~ t s , ~ will ’,~~ be very interesting to investigate in this Cavity QED situation.

+

+

+

+

71

Acknowledgements

We acknowledge funding by Agence Nationale pour la Recherche (ANR), by the Japan Science and Technology Agency (JST), by the EU under the I P projects “SCALA and ‘CONQUEST. C.G and S.D are funded by a grant from Dklkgation Gknkrale zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJI B 1’Armement (DGA). JMR is a member of Institut Universitaire de France (IUF) References 1. V. B. Braginsky and Y. I. Vorontsov, Usp. Fiz. Nauk, 114,41 (1974) [Sov. Phys. Usp. 17,644 (1975)]; K. S. Thorne, R. W. P. Drever, C. M. Caves, M. Zimmerman and V. D. Sandberg, Phys. Rev. Lett. 40,667 (1978). 2. P. Grangier, J. A. Levenson and J.-P. Poizat, Nature 396,537 (1998). 3. S. Haroche and J. M. Raimond, Exploring the Quantum: Atoms, Cavities and Photons (Oxford Univ. Press, Oxford, UK, 2006). 4. S. Kuhr et al., Appl. Phys. Lett. 90,164101 (2007). 5. H. Carmichael, An open system approach to quantum optics (Springer, Berlin, 1993). 6. S. Gleyzes et al., Nature 446,297-300 (2007). 7. C. Guerlin et al., Nature in press (2007). 8. J. M. Raimond, M. Brune and S. Haroche, Rev. Mod. Phys. 73,565 (2001). 9. M. Brune, S. Haroche, V. Lefkvre, J. M. Raimond and N. Zagury, Phys. Rev. Lett. 65,976-979 (1990). 10. M. Brune, S. Haroche, J. M. Raimond, L. Davidovich and N. Zagury Phys. Rev. A. 45,5193 (1992). 11. M. Brune et al, Phys. Rev. Lett. 77,4887 (1996). 12. P. Bertet et al, Phys. Rev. Lett. 89,200402 (2002). 13. W. H. Zurek, Rev. Mod. Phys. 75,715 (2003). 14. L. Davidovich, M. Brune, J. M. Raimond and S. Haroche, Phys.Rev.A, 53, 1295 (1996). 15. P. Milman, A. Auffeves, F. Yamagushi, M. Brune, J. M. Raimond and S. Haroche, Eur. Phys. J . D., 32,233 (2005). 16. L. Davidovich, M. Brune, J.-M. Raimond and S. Haroche, Phys. Rev. Lett. 71,2360 (1993). 17. M. W. Mitchell, J. S. Lundeen and A. M. Steinberg, Nature, 429,161 (2004). 18. P. Walther, J. W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni and A. Zeilinger, Nature, 429,158 (2004).

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ULTRA-FAST CONTROL AND SPECTROSCOPY

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FREQUENCY-COMB-ASSISTED MID-INFRARED SPECTROSCOPY P. DE NATALE*, D. MAZZOTTI, G. GIUSFREDI, S. BARTALINI and P. CANCIO Istituto Nazionale di Ottica Applicata (INOA) - CNR and European Laboratory for Nonlinear Spectroscopy (LENS) Via Carrara 1 , 50019 Sesto Fiorentino FI, Italy * e-mail: paolo. [email protected], web: http://www.inoa.it

P. MADDALONI, P. MALARA and G. GAGLIARDI Istituto Nazionale di Ottica Applicata (INOA) - CNR and European Laboratory for Nonlinear Spectroscopy (LENS) Via Campi Flegrei 34, 80078 Pozzuoli NA, Italy I. GALL1 and S. BORRI Dipartimento di Fisica, Universitb di Firenze and European Laboratory for Nonlinear Spectroscopy (LENS) Via Sansone 1 , 50019 Sesto Fiorentino FI, Italy A new class of IR coherent sources and IR frequency combs, that combine optical frequency-comb synthesizers (OFCSs) and optical parametric up/downconversions, is already available and still progressing at a very fast pace. Peculiar features for IR radiation produced by difference-frequency-generation (DFG) set-ups or quantum-cascade lasers (QCLs) can he achieved when they are phase and frequency controlled by the OFCS. Indeed, their frequency is accurately known against the primary frequency standard and their linewidth is highly narrowed thanks to the transferred OFCS coherence even for laser sources whose frequencies are several THz apart. These features, together with their wide tunability and their small intensity fluctuations (down to the shotnoise limit), make these IR sources well suited for a wide range of applications, in particular for spectroscopic ones. Very high sensitivity for trace-gas detection has been achieved when combined with enhancement absorption techniques as high-finesse Fabry-Perot cavities or multipass cells. Moreover, the large number of fundamental ro-vibrational transitions of many stable and transient molecular species accessible with this spectrometers, make them particularly attractive for environmental applications, especially considering their compactness and ruggedness when a fiber-based set-up is chosen. Their unique capabilities in terms of achievable precision for absolute frequency measurements can he used to create a “natural” grid of secondary frequency standards of IR molecular absorptions, frequency measured with these high-resolution spectrometers. More important, we have directly generated an IR frequency comb around

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76 3 pm by DFG conversion of an OFCS. The generated comb can be employed both as a frequency ruler and as a direct source for molecular spectroscopy.

Keywords: mid IR, optical frequency comb; difference-frequency generation; quantum-cascade laser; molecular spectroscopy.

1. Introduction The molecular “fingerprint” region, roughly located in the 2.5-10 ,urn interval of the IR spectrum, can be considered the natural “gateway” for any molecular-based spectroscopic study or sensing. Indeed, the strongest rovibrational transitions generally lie in this range, for most simple molecules, thus guaranteeing a high detection sensitivity. In addition, Doppler-limited linewidths are narrower than in the visible/near-IR range, thus providing a better selectivity. However, very few tunable sources have been available until recently, thus favoring overtone-transitions-based investigations, relying on relatively cheap and compact near-IR telecom sources. Nonetheless, high-quality frequency standards have been developed, as He-Ne/CH4 laser, relying on fortuitous coincidences between laser lines and molecular transitions. The introduction, only a few years ago, of the OFCS and the parallel development of high-Q frequency standards more and more aiming at UV frequencies, with a more favorable A u / u value, has suddenly revolutionized frequency metrology. Molecular standards have been quickly replaced by trapped-ions-based ones and nowadays neutral atoms trapped in optical lattices are also under study and very promising. All this progress is quickly pushing frequency standards towards values of 10-16-10-17 for A u / u and even better values are already foreseen. If new ideas and novel optical technologies have suddenly changed the world and the perspectives of frequency metrology, also the community of people using spectroscopy has literally boomed. In particular, trace molecular sensing is becoming the primary tool for any quantitative assessment in environmental sciences, like greenhouse effect studies, atmospheric studies, anthropogenic as well as natural release of gases in the atmosphere, or also homeland security problems, just to mention some. The common requirements for these and other spectroscopic applications are always high resolution (to get high selectivity) and high sensitivity (to get highly accurate concentration values). For all such applications, as explained above, the spectral window of choice is the IR, and the primary concern is not the absolute frequency determination. On the other hand, a steep development has also been undergone by IR technologies. More specifically, at least two classes of new coherent sources, emitting in this spectral window, have emerged: sources

77 based on nonlinear generation in periodically-poled crystals and QCLs. In the first class are included optical parametric oscillators (OPOs) and DFG radiation sources. They are both very widely tunable sources, OPOs being mainly limited by the nonlinear crystal transparency range and DFG sources by the crystal or by the overall tunability range of pump/signal lasers. Instead, QCLS’ are not widely tunable but proper design of the quantum well structure may have them emitting in the range that roughly goes from 3.5 to several hundreds microns. Continuous-wave (CW) midIR QCLs need generally to be operated around liquid-N2 temperatures, but also room-temperature operation has recently been demonstrated.2 All such sources are very species-selective because linewidths are generally several orders of magnitude narrower than mid-IR Doppler profiles and often sufficient to detect saturated-absorption line shape^.^^^ Moreover, very high ~ > ~the very molecular detection sensitivities have been d e m ~ n s t r a t e dand wide tunability range allows to freely move throughout this rich portion of the spectrum containing fundamental ro-vibrational bands. It is also worth noticing that the mid-IR spectral window is “naturally” endowed with an ultra-wide comb of lines, represented by the manifolds of ro-vibrational transitions that can be easily saturated and that often have natural widths of a few tens of Hertz. Therefore, whatever the OFCS IR extension is realized, once the lines of interest are measured, they can be directly used as secondary frequency standards, similarly to what has been done until now with I2 lines. In this completely new situation, concerning the midIR spectral coverage, moderate-quality standards, easy to realize wherever is required in the IR, are probably the right choice for the ever increasing community of spectroscopy end-users. Partly to this purpose, several groups have been working, in the last few years, to an IR extension of OFCSs. So far, direct broadening of the spectrum of fs mode-locked lasers through highly-nonlinear optical fibers has succeeded in extending combs up to a 2.3 pm ~ a v e l e n g t hFor . ~ longer wavelengths, a few alternative schemes have been devised, essentially based on parametric generation processes in nonlinear crystals. A 270-nm-span frequency comb a t 3.4 pm has been realized by DFG between two spectral peaks emitted by a single uniquely-designed Ti:sapphire fs laser.8 In our group we have focused the attention onto the development of DFG- and QCL-based spectrometers and we will review, in the next paragraphs, the main results achieved. We report three different schemes which exploit a nonlinear optical process to transfer the metrological performance of a visible/near-IR OFCS to the mid IR. In the first scheme (Sec. 2), the

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metrological performance of a Ti:sapphire OFCS is extended to the mid IR by phase-locking the pump and signal lasers of a DFG source to two nearIR teeth of an optical comb. Then, the generated IR radiation is used for high-resolution spectroscopy providing absolute frequency measurements of molecular lines at 4 pm. However, a drawback of this approach is the impossibility of comb-referencing for laser sources directly emitting in the mid-IR, such as QCLs. In Secs. 3 and 4 we demonstrate two novel schemes that overcome this limitation, based respectively on optical parametric upand down-conversion. In Sec. 3 a QCL at 4.43 pm has been used for producing near-IR radiation at 858 nm by means of sum-frequency generation with a Nd:YAG source in a periodically-poled LiNbO3 (PPLN) nonlinear crystal. The absolute frequency of the QCL source has been measured by detecting the beat note between the sum frequency and a diode laser at the same wavelength, while both the Nd:YAG and the diode laser were referenced to the OFCS. Vice versa, in Sec. 4 a frequency comb is directly created at 3 pm by nonlinear mixing of a near-IR fiber-based OFCS with a CW laser.g Possible applications for the generated comb are as a clockwork to transfer IR-frequency standards to other spectral regions, as a frequency ruler for high-precision molecular spectroscopy or telecommunications, and as a direct source for molecular spectroscopy. lo 2. DFGat 4 p m

Our OFCS-referenced DFG source at 4 pm is described About 200 p W of idler radiation at 4.2 pm is generated by nonlinear frequency mixing in a PPLN crystal of about 130 mW from a diode laser (pump laser) operating between 830 and 870 nm, and about 5 W from a fiber-amplified Nd:YAG laser (signal laser) at 1064 nm. We follow the scheme13 shown in detail in Fig. 1, to control the frequency and phase of the generated IR radiation against our mode-locked Ti:sapphire-laser-based OFCS, which covers an octave in the visible/near-IR region (500t1100 nm).

Both pump and signal lasers are beaten with the closest tooth of the OFCS ( N p and N,), with residual R F beat notes Aupc and Ausc respectively. The contribution of the OFCS carrier-envelope-offset (CEO) frequency u, is canceled out of these beat notes by standard RF mixing, yielding Aupc-zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA u, and Au,, - u, respectively. A low bandwidth (- 10 Hz) phase-locked-loop (PLL1) is used to control the long-term frequency fluctuations and drifts of the Nd:YAG laser. As a result, the signal-laser frequency is u, = N,u, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA + y o ,where y o is the R F frequency of the local oscillator used

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v,=Nsvr+vLO To DFG/SFG

To

Apo3

Fig.  1.  Schematic  of the  OFCS­DFG/SFG  phase  lock

in  the  PLL1  loop,  and vr =  1  GHz.  In order  to  control the  frequency  and phase  of the  diode  laser  against  the  Nd:YAG  one,  a direct­digital  synthesis (DDS)  multiplying  the  A^sc  — v0  frequency  by  a factor Np/Ns  is  used  as a  local  oscillator  in  a  second  PLL  circuit  (PLL2)  with  a  wide  bandwidth (~  2  MHz).  Then  the  pump  frequency  is vp = (Np/Ns)vB,  without  any contribution  from  the  OFCS  parameters (v0  and vr).  As  a  consequence, the  absolute  frequency of the  generated  idler  radiation  is  given  by Vi-vp-vs

= (Np -

N,

(1)

with a precision and accuracy limited only by the reference oscillator of our OFCS.  The  latter  consists  of a  Rb/GPS­disciplined  10­MHz  quartz  with  a stability of 6 • 10~13 at 1 s and a minimum accuracy of 2 • 10~12. Moreover, continuous scans of  i/j  can be performed by properly sweeping vr. Because in the above DDS­PLL scheme, the pump laser linewidth Az/p is a  factor Np/Ns  higher  than  the  narrow  Nd:YAG  linewidth  Az^ s ,  a  residual (Np/Ns —  l)A^ 5  idler  linewidth  is  expected.  We  have  measured  a  idler linewidth  AI/J  ~  11  kHz  by  coupling  the  OFCS­locked  4­/zm  beam  to  a high­finesse  optical  cavity  (FSR=150  MHz,  finesse  >  17000),  as  shown  in Fig.  2.  It is more than 30 times narrower than the DFG without  any OFCS

80 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

zyxwvuts zyxwvutsrqp Voigt fit with fixed Lorentz

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Frequency (20 kHz/div)

Fig. 2. High-finesse Fabry-Perot transmission of the OFCS-DFG source at 4 pm and Voigt function fit. Total acquisition time 5 ms. The fit that takes into account a Lorentz contribution from the Fabry-Perot with a fixed linewidth (9 kHz measured with CRDS) and a Doppler contribution from the idler radiation. The linewidth of the OFCS-DFG source extrapolated by the fit is 10.8(1) kHz.

control. Such a narrow linewidth can satisfy most of the spectroscopic needs and can still be improved with a proper choice of signal/pump lasers. This idler radiation has been used both for high-precision“ and highsensitivity” spectroscopy of CO2 molecular transitions around 4 pm. In the former case, we have performed saturated-absorption spectroscopy with a medium-finesse optical cavity (FSR=1.3 GHz, finesse > 500) of even verylow-populated rotational levels ( J > 80). Absolute frequency measurements of these transitions with an accuracy of about are proposed as a “natural” grid of secondary frequency standards in this spectral region. For trace-gas detection, we perform cavity-ring-down spectroscopy (CRDS) by coupling the idler beam to a high-finesse optical cavity (FSR=150 MHz, finesse > 17000), filled with the COz gas. In this case, the OFCS control of the DFG spectrometer helps not only t o get a narrow-linewidth idler radiation (thus increasing the cavity-coupling), but also to interrogate the molecular absorption at the same resonant frequency for long times, due to the high reproducibility of the OFCS-referenced IR frequency. In this way, minimum absorbances a L of the order of few parts in lo8 (i.e. linestrengths of about cm) can be achieved with few hours of integration. The power of this technique can be used, e.g., to detect C02 isotopologues with very low natural abundance, or to search for highly-forbidden C02 transitions, as those due t o the wave-function symmetry under the bosonic l60exchange.

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3. QCL-based spectrometer QCLs  operating  in  the  mid­IR  region  can  represent  a  valid  alternative  to DFG systems, especially when the application requires high emission power and very compact designs. It makes QCLs very appealing not only for high­ sensitivity  spectroscopy  experiments,  but  also  for  a huge  variety  of indus­ trial  and  commercial  applications.  On  the  other  hand,  their  capabilities cannot  be  fully  exploited  at  present,  due  to  the  lack  of precise  references in most of the IR region used to control their absolute frequency.  Here, we illustrate an experiment14 which overcomes this problem: the frequency of a  4.43­/Ltm  QCL  was  measured  against  our  Ti:sapphire  OFCS  by  means  of a  parametric  up­conversion  process. The  set­up  is  shown  in  Fig.  3.  The  QCL  is  a  CW,  liquid­N2­cooled,

1||P Oetedtor Nd­.YAO @ 1C84 nm

Dsfectal Grating $ • = OFB4SG Sgssr

Sum­Frequency Frequency­Comb

I  ­'"" t• o  Filter JSSSBrar

Fig. 3.  Schematic of the experimental apparatus, focused on the SFG generation process providing the  optical  link  between  the  QCL  and  the  OFCS.

distributed­feedback (DFB) device at 4.43 /mi. The collimated QCL beam is split into two parts: 1 mW is used for CC­2 Doppler­absorption spectroscopy and  2  mW  are  used  for  the  nonlinear  up­conversion  process.  The  latter  is achieved by mixing the QCL and a fiber­amplified Nd:YAG laser in a PPLN crystal  for  a sum­frequency  generation  (SFG)  process  to  produce  858­nm radiation.  With  about  1.2  W  of Nd:YAG  power  and  only  2  mW  of  QCL

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radiation incident on the nonlinear crystal, about 10 p W of SFG radiation has been obtained. This radiation is beaten with an external-cavity diode laser (ECDL) working at the same wavelength, yielding a beat note A U + ~ that can be easily counted (40 dB S/N at 500 kHz resolution bandwidth). Both the ECDL and the Nd:YAG lasers are the same of the DFG source at 4 pm, and are phase-locked to the OFCS following the DDS-PLL scheme described in Sec. 2 . Then, the QCL absolute frequency can be expressed as:

where the symbols have the same meaning as in Sec. 2 . Moreover, since is the beat note between the sum frequency v+ = vi v, and up, it can be used t o measure the phase/frequency noise of the QCL similarly to what has been described for the DFG source at 4 pm. We used this OFCS-referenced QCL for several absolute frequency measurements of two 13C02 Doppler-broadened ro-vibrational transitions, the (OOol-00'0) P(30) and the (Ol11-0l1O) P(17). A recording of the latter, weaker line is shown in Fig. 4. Each point of the trace results from the

+

0.50

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Absolute frequency [GHz]

Fig. 4. Absolute frequency measurement of the 1 3 C 0 2 (Ol11-0l1O) P(17) line. The gas pressure in the cell was 3 mbar. A Voigt fit of the data is also shown.

simultaneous measurements of the amplitude of the absorption signal, and the QCL frequency measured by counting with a spectrum analyzer. The acquisition time for each point is 500 ms, during which an average on both the amplitude and frequency measurements is performed. The uncertainty associated to the absolute frequency measurement of each point

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is due to the frequency fluctuations of the free-running QCL during the 500 ms single-point acquisition time. For our measurements we obtain a frequency uncertainty of about 2 MHz. Several spectra have been acquired for the same transition, even at slightly different gas pressures. Each set of data has been fitted t o a Voigt profile (Fig. 4) to determine the corresponding line-center frequency. The final precision of these absolute frequency measurements is 3 . mainly limited by the above mentioned QCL jitter. This number can be heavily improved (at least 3 orders of magnitude) with proper frequency stabilization of the QCL. This upgrade will match the implementation of high-precision spectroscopic techniques such as Doppler-free detection. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK

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4. 3-pm comb generation

The apparatus devised to create the 3-pm frequency comb,g is shown in Fig. 5. The nonlinear down-conversion process occurs in a PPLN crystal ECDL

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4 Fig. 5. Layout of the optical table. A 3-pm frequency comb is created by DFG in a PPLN crystal between a near-IR OFCS and a CW laser. A fast, 100-pm-diameter HgCdTe detector is used to characterize the generated mid-IR comb.

(with a period around 30 pm) between a near-IR OFCS and a CW tunable laser. The generated mid-IR frequency comb covers the region from 2.9 to 3.5 pm in 180-nm-wide spans with a 100-MHz mode spacing and

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keeps the same metrological performance as the original comb source. Such a scheme can be easily implemented in other spectral regions by use of suitable pumping sources and nonlinear crystals. The near-IR OFCS is an octave-spanning (1050-2100 nm) fs modelocked fiber-laser-based system, but for the DFG process only the OFCS fraction (25 mW) covering the 1500-1625 nm interval is used as combsignal laser. The power of this comb-signal beam is enhanced by amplifying it with an external Er-doped fiber amplifier (EDFA). The amplified combsignal beam has an overall power of 0.7 W and spans from 1540 to 1580 nm with a 100 MHz spacing, corresponding to nearly 50000 teeth (i.e. about 14 p W per tooth). The pump beam is an external-cavity diode laser (ECDL) at 1030-1070 nm, amplified up to 0.7 W by an Yb-doped fiber amplifier (YDFA). Both pump and comb-signal lasers are mixed in a PPLN crystal, whose period and temperature are chosen depending of the pump wavelength satisfying the quasi-phase-matching (QPM) condition for the center wavelength (1560 nm) of the comb-signal laser. The comb-idler beam is detected by filtering out the unconverted near-IR light and focusing it on to a liquid-Nz-cooled, 150-MHz-bandwidth HgCdTe detector. In this way, a RF beat note at v, = 100 MHz is recorded by a spectrum analyzer, which is the sum of the beat signals between all pairs of consecutive teeth in the generated DFG comb. The latter has a span of 180 nm (5 THz), limited by the comb-signal coverage, and is centered in the 2.9 to 3.5 pm interval, depending on the pump wavelength used. The teeth on both sides of the near-IR comb are involved in many DFG processes, with a conversion efficiency decreasing according to the well-known sinc2 law.15 The overall measured power of the 5-THz-spanning radiation is about 5 pW. This value corresponds to a power of nearly 100 pW per mode of the mid-IR comb. We phase-lock the pump laser to the closest tooth of the near-IR OFCS, in order to cancel out the CEO vo frequency in the generated mid-IR comb and to fix the beat-note frequency Avpcbetween the pump and the closest near-IR comb frequencies. In this way the frequency of the generated IR modes is

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and, hence, with the same metrological performance of v,. In the following we discuss the application of the mid-IR comb as an absolute frequency ruler at 3 pm. For this purpose, we have beaten this comb with a CW laser at 3 pm. The CW radiation is generated by a second DFG process which uses most of the same set-up used to produce the

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mid-IR comb. Indeed, a CW extended-cavity diode laser at 1520-1570 nm, is amplified by the EDFA simultaneously with the fraction of the near-IR OFCS used to generate the 3-pm comb. As described above, this amplified CW laser is DFG mixed with the 1-pm pump laser to generate mW-powerlevel CW idler radiation around 3 pm, co-propagating with the DFG comb. A beat note Aucwc between the CW DFG radiation and the mid-IR comb can be detected by sending directly the generated light to the HgCdTe fast detector. As the frequency of each tooth of the mid-IR comb is well known (Eq. 3), the mid-IR CW laser frequency can be measured by counting Aucwc. Furthermore, because the 1-pm pump laser is comb-locked, Aucwc can be used to phase-lock the mid-IR CW radiation to the DFG comb by feeding back proper phase corrections to the 1.5-pm laser. The S/N ratio of Aucwc was measured when the ECDL wavelength was tuned from 1540 to 1570 nm ( ( X ~ ) C W from 3.22 to 3.35 pm), in order to characterize the effective DFG-comb span suitable for use in phase-locked systems and frequency counting. Such value reaches a maximum of 40 dB at the center wavelength, while decreases almost symmetrically down to less than 20 dB at the upper and lower edges, limiting in principle to about 130 nm the interval in which a mid-IR source can be locked. Actually, the 180-nm span can be fully exploited, as stronger beat notes are expected when two different DFG apparata are used for the mid-IR comb and CW lasers. Moreover the beat-note S/N can be improved if filtering of the comb modes not contributing to the beat is done before and after the nonlinear conversion. The described mid-IR DFG comb has demonstrated to be a suitable absolute frequency ruler in this spectral window and may be strategic for future metrological applications with direct mid-IR lasers as QCLS." On the other hand, the generation of a frequency comb in the mid IR leads straightforwardly to consider its use as a direct spectroscopic source. In this sense, several schemes involving coherent coupling to high-finesse cavities, as well as Fourier-transform molecular spectroscopy schemes have already been demonstrated in the near IR, and may be now extended, taking advantage of the detection sensitivities achievable in the fingerprint region. zyxwvutsr

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References

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son, D. L. Sivco, J. N. Baillargeon and A. Y. Cho, Opt. Express 7,p. 243 (2000). A. Castrillo, E. De Tommasi, L. Gianfrani, L. Sirigu and J. Faist, Opt. Lett. 31, p. 3040 (2006). Y. A. Bakhirkin, A. A. Kosterev, R. F. Curl, F. K. Tittel, D. A. Yarekha, L. Hvozdara, M. Giovannini and J. Faist, Appl. Phys. B 82, p. 146 (2006). S. Borri, S. Bartalini, P. De Natale, M. Inguscio, C. Gmachl, F. Capasso, D. L. Sivco and A. Y. Cho, Appl. Phys. B 85, p. 223 (2006). I. Thomann, A. Bartels, K. L. Corwin, N. R. Newbury, L. Hollberg, S. A. Diddams, J. W. Nicholson and M. F. Yan, Opt. Lett. 2 8 , p. 1368 (2003). S. M. Foreman, A. Marian, J. Ye, E. A. Petrukhin, M. A. Gubin, 0. D. Mucke, F. N. C. Wong, E. P. Ippen and F. X. Kartner, Opt. Lett. 30, p. 570 (2005). P. Maddaloni, P. Malara, G. Gagliardi and P. De Natale, New J . Phys. 8, p. 262 (2006). M. J. Thorpe, K. D. Moll, R. Jason Jones, B. Safdi and J . Ye, Science 311, p. 1595 (2006). D. Mazzotti, P. Cancio, G. Giusfredi, P. De Natale and M. Prevedelli, Opt. Lett. 30, p. 997 (2005). D. Mazzotti, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi and P. De Natale, J . Opt. A 8, p. S490 (2006). H. R. Telle, B. Lipphardt and J. Stenger, Appl. Phys. B 74, p. 1 (2002). S. Bartalini, P. Cancio, G. Giusfredi, D. Mazzotti, P. De Natale, S. Borri, I. Galli, T. Leveque and L. Gianfrani, Opt. Lett. 32, p. 988 (2007). P. Maddaloni, G. Gagliardi, P. Malara and P. De Natale, Appl. Phys. B 80, p. 141 (2005). F. Capasso, C. Gmachl, R. Paiella, A. Tredicucci, A. L. Hutchinson, D. L. Sivco, J. N. Baillargeon, A. Y. Cho and H. C. Liu, IEEE J . Sel. Top. Quantum Electron. 6, p. 931 (2000).

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PRECISION MEASUREMENT AND APPLICATIONS

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PRECISION GRAVITY TESTS BY ATOM INTERFEROMETRY

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G. M. TINO*, A. ALBERTI, A. BERTOLDI, L. CACCIAPUOTI~,

M. DE ANGELISt, G. FERRARI, A. GIORGINI, V. IVANOV,

G. LAMPORESI, N. POLI, M. PREVEDELLIf, F. SORRENTINO

Dipartimento di Fisica and L E N S Laboratory - Universitb d i Firenze Istituto Nazionale d i Fisica Nucleare, Sezione d i Firenze via Sansone 1, Polo Scientifico, I-5001 9 Sesto Fiorentino (Firenze), Italy

We report on experiments based on atom interferometry to determine the gravitational constant G and test the Newtonian gravitational law at micrometric distances. Ongoing projects to develop transportable atom interferometers for applications in geophysics and in space are also presented.

1. Introduction

Advances in atom interferometry led to the development of new methods for fundamental physics experiments and for applications. In particular, atom interferometers are new tools for experimental gravitation as, for example, for precision measurements of gravity acceleration [l]and gravity gradients [2], for the determination of the Newtonian constant G [3,4], for testing general relativity [5,6] and l / r 2 law [7-lo], and for possible applications in geophysics. Ongoing studies show that future experiments in space will allow to take full advantage of the potential sensitivity of atom interferometers [ll].The possibility of using atom interferometry for gravitational waves detection was also investigated (see [12] and references therein). In this paper, we report on experiments we are performing using atom interferometry t o determine G and test the Newtonian gravitational law a t

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*E-mail: [email protected] - Web: www.lens.unifi.it/tino tpermanent address: ESA Research and Scientific Support Department, ESTEC, Keplerlaan 1- P.O. Box 299, 2200 AG Nordwijk ZH, The Netherlands $On leave from: Istituto Cibernetica CNR, 80078 Pozzuoli (Napoli), Italy f Permanent address: Dipartimento di Chimica Fisica, Universita di Bologna, Via del Risorgimento 4, 40136 Bologna, Italy zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON

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micrometric distances. We also present ongoing projects to develop transportable atom interferometers for applications in geophysics and in space. zyxwvutsr 2. Determination of G by atom interferometry

The Newtonian constant of gravity G is one of the most measured fundamental physical constants and at the same time the least precisely known. The extreme weakness of the gravitational interaction and the impossibility of shielding the effects of gravity make it difficult t o measure G, while keeping systematic effects well under control. Despite the numerous experiments, the uncertainty on G has improved only by one order of magnitude in the last century [13]. Many of the experiments performed to date are based on the traditional torsion pendulum method, direct derivation of the historical experiment performed by Cavendish in 1798. Recently, many groups have set up new experiments based on different concepts and with completely different systematics. However, the most precise measurements available today still show substantial discrepancies, limiting the accuracy of the 2006 CODATA recommended value for G t o 1 part in lo4. From this point of view, the realization of conceptually different experiments can help to identify still hidden systematic effects and therefore improve the confidence in the final result. We use atom interferometry to perform precision measurements of the differential acceleration experienced by two samples of laser-cooled rubidium atoms under the influence of nearby tungsten masses. In our experiment, specific efforts have been devoted to the control of systematic effects related to atomic trajectories, positioning of source masses, and stray fields. In particular, the high density of tungsten and the distribution of the source masses are crucial in our experiment to compensate for the Earth gravity gradient and reduce the sensitivity of the measurement to the initial position and velocity of the atoms. The measurement, repeated for two different configurations of the source masses, is modeled by a numerical simulation which takes into account the mass distribution and the evolution of atomic trajectories. The comparison of measured and simulated data provides the value of the Newtonian gravitational constant G. Proof-of-principle experiments with similar schemes using lead masses were already presented in [3,4]. In our interferometer, laser pulses are used to stimulate s7Rb atoms on the two-photon Raman transition between the hyperfine levels F = 1 and F = 2 of the ground state [14]. The light field is generated by two counter-propagating laser beams with wave vectors kl and kz E -kl aligned along the vertical direction. The laser frequencies v1

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91

and  1/2  match the resonance condition v\—vi = v§, where hi/o  is the energy associated  to  the F =  1  —)• F =  2  transition.  The  atom  interferometer,  ob­ tained  with  a  7T/2 — TT—7T/2  sequence  of Raman  pulses,  drives  the  atoms  on two  independent  paths  along  which  the  quantum  mechanical  phases  of the atomic wavepackets independently evolve. In the presence of a gravity  field, atoms experience a phase shift $  = k­g  T2, where k = kl—k2, 2T is the du­ ration of the  pulse  sequence,  and  g  is  gravity  acceleration.  A  measurement of the  phase $  is  equivalent  to  an  acceleration  measurement.  The  gravity gradiometer  consists  of  two  absolute  accelerometers  operated  in  differen­ tial  mode.  Two  spatially  separated  atomic  clouds  aligned  along the  vertical direction  are  simultaneously  interrogated  by  the  same  ?r/2 — ir — ?r/2  pulse sequence. The difference of the phase shifts detected on each interferometer provides  a  direct  measurement  of  the  differential  acceleration  induced  by gravity  on  the  two  atomic  samples.  This  method  has  the  major  advantage of being highly insensitive to noise sources  appearing in  common mode on both  interferometers.  In  particular,  any  spurious  acceleration  induced  by vibrations or seismic noise on the common reference frame identified by the vertical Raman beams is efficiently rejected by the differential measurement technique. Figure  1  shows  a  schematic  of the  experiment.  The  gravity  gradiome­ ter  set­up  and  the  configurations  of  the  source  masses (Ci  and  C^)  used for the G measurement  are visible.  The  atom  interferometer apparatus  and

upper gravimeter

/

1 ^

lower gravimeter

1 1

detection beams

Fig.  1.  Schematic  of the  experiment  showing the  gravity gradiometer set­up with the Raman beams  propagating  along  the  vertical  direction.  For the  measurement  of G,  the position of the source masses is alternated between configuration C\ (left) and C-2 (right).

92

the source masses assembly are described in detail elsewhere [3,15]. In the vacuum chamber at the bottom of the apparatus, a magneto-optical trap (MOT) collects rubidium atoms from the vapor produced by getters. After turning the MOT magnetic field off, the atomic sample is launched vertically along the symmetry axis of the vacuum tube by using the moving molasses technique. The gravity gradient is probed by two atomic clouds moving in free flight along the vertical axis of the apparatus and simultaneously reaching the apogees of their ballistic trajectories a t 60 cm and 90 cm above the MOT. Such a geometry, requiring the preparation and the launch of two samples with high atom numbers in a time interval of about 100 ms, is achieved by juggling the atoms loaded in the MOT. The interferometers are realized at the center of the magnetically shielded vertical tube shown in Fig. 1. The three-pulse interferometer has a duration of 2T = 320ms. The 7r pulse lasts 48 ps and occurs 5 ms after the atomic clouds reach their apogees. In this configuration, only one pair of counterpropagating laser beams with frequencies ul and v2 and crossed linear polarizations is able to stimulate the atoms on the two-photon transition. At the end of their ballistic flight, the population of the ground state is measured by selectively exciting the atoms in both hyperfine levels of the ground state and detecting the light-induced fluorescence emission. We typically detect lo5 atoms on each rubidium sample at the end of the interferometer sequence. Even if the phase noise induced by vibrations washes out the atom interference fringes, the signals simultaneously detected on the upper and lower accelerometer remain coupled and preserve a fixed phase relation. Therefore, when the trace of the upper accelerometer is plotted as a function of the lower one, experimental points distribute along an ellipse. The differential phase shift is then obtained from the eccentricity and the rotation angle of the ellipse fitting the data [16]. The Allan deviation shows the typical behavior expected for white noise. The instrument has a sensitivity of 140 mrad at 1s of integration time, corresponding to a sensitivity to differential accelerations of 3.5 . lop8 g in 1s. The source masses [15] are composed of 24 tungsten alloy (INERMET IT180) cylinders, for a total mass of about 516 kg. They are positioned on two titanium platforms and distributed in hexagonal symmetry around the vertical axis of the tube (see Fig. 1). The value of G was determined from a series of gravity gradient measurements performed by periodically changing the vertical position of the source masses between configuration C1 and C2 while keeping the atomic trajectories fixed. Because of the high density of tungsten, the gravitational

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93 field produced by the source masses is able to compensate for the Earth gravity gradient. As a consequence, the acceleration becomes less sensitive to the positions of the atomic clouds around extremal points, allowing for a better control of systematic effects and a relaxation of the requirements on the knowledge of atomic trajectories. Figure 2 shows a data sequence used for the measurement of G. Each phase measurement is obtained by fitting a 24-point scan of the atom interference fringes t o an ellipse. After an analysis of the error sources affecting our measurement, we obtain a value of G = 6.667.10-11 m3 kg-l s - ~ with , a statistical uncertainty of f O . O 1 l . l O - l l m3 kg-' s-' and a systematic uncertainty of 3~0.003. m3 kg-' s - ~ .Our measurement is consistent with the 2006 CODATA value within one standard deviation. The main contribution to the systematic error on the G measurement derives from the positioning accuracy of the source masses. This error will be reduced by about one order of magnitude by measuring the position of the tungsten cylinders with a laser tracker. Eventually, uncertainties below the 10 ppm level could be reached with this scheme using for the source mass a material with a higher density homogeneity.

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Fig. 2. Typical data set showing the modulation of the differential phase shift measured by the atomic gravity gradiometer when the distribution of the source masses is alternated between configuration C1 and C Z .Each phase measurement is obtained by fitting a 24-point scan of the atom interference fringes to an ellipse.

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3. Precision gravity measurements at pm scale with

laser-cooled Sr atoms in an optical lattice

The extremely small size of atomic sensors can lead to applications for precision measurements of forces at micrometer scale. The investigation of forces at small length scales is indeed a challenge for present research in physics for the study of surfaces, of the Casimir effect, and in the search for deviations from Newtonian gravity predicted by recent theories beyond the standard model. We showed [lo] that using laser-cooled **Sr atoms in an optical lattice, persistent Bloch oscillations can be observed for a time 10 s, because of remarkable properties of immunity of this atom from perturbations due to stray fields and interatomic collisions. This system can reach an unprecedented sensitivity as sensor to measure gravity acceleration on micrometer scale with ppm precision opening the way to the investigation of small-scale gravitational forces in regions so far unexplored. The experiment starts with trapping and cooling 5 x lo7 *'Sr atoms at 3 mK in a magneto-optical trap (MOT) operating on the lSo-lP1 blue resonance line at 461 nm. The temperature is then further reduced by a second cooling stage in a red MOT operating on the 1S0-3P1 narrow transition at 689 nm and finally we obtain 5 x lo5 atoms at 400 nK. After this preparation phase, the red MOT is switched off and a one-dimensional optical lattice is switched on adiabatically in 50 ps. The lattice potential N

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Fig. 3. Bloch oscillation of "Sr, atoms in the vertical 1-dimensional optical lattice under the effect of gravity. Two quantities are extracted from the analysis of the data: The vertical momentum of the oscillating atoms (a) and the width of the atomic momentum distribution (b).

95

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is originated by a single-mode frequency-doubled Nd:YV04 laser (XL = 532 nm). The beam is vertically aligned and retro-reflected by a mirror producing a standing wave with a period X L / ~= 266 nm. We obtain a diskshaped sample of lo5 atoms at T 400 nK with a vertical rms width of 12 pm and a horizontal radius of 150 pm. We observe Bloch oscillations in the vertical atomic momentum by releasing the optical lattice at a variable delay, and by imaging the atomic distribution after a fixed time of free fall. Figure 3 shows the signal recorded for 7 s, corresponding t o 8000 oscillations. The coherence time for the Bloch oscillation is 12 s. These values are the highest ever observed for Bloch oscillations in atomic systems. Measuring the oscillation frequency we determine the vertical force on the atoms, that is, Earth gravity with a resolution of 5 x lop6. In the effort to increase the sensitivity, recently we investigated strontium atoms trapped in phase-modulated optical lattices. We found that we can induce a broadening of the atomic distribution in the lattice potential with a phase modulation of the lattice at frequencies multiple of Bloch frequency. We observed a resonant broadening up to the 5th harmonic which corresponds to a hop through 5 lattice sites (Fig. 4). All the resonance spectra exhibit a Fourier-limited width for excitation times as long as 2 s. The resulting high-resolution measurement of Wannier-Stark levels of the atomic wavefunction in the gravity potential allows to determine the local gravity acceleration with a relative precision lop6. When studying atom-surface interactions, one key point is the precision of sample positioning close to the surface. In our experiment, the optical lattice is also used for an accurate positioning of the sample close to the zyxwvutsr

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96

surface.  We  translate  the  atomic  sample  along the  lattice  axis  by  applying a  relative  frequency  offset  to  the  counterpropagating  laser  beams  produc­ ing  the  lattice.  In  this  way,  we  place  the  atoms  close  to  a  transparent  test surface  placed  ~  45 mm  far  from  the  MOT.  We  measure  the  atoms  num­ ber  and  the  phase  of  the  Bloch  oscillation  with  absorption  imaging  after bringing the  atoms  back to the  original position.  In  Fig.  5.a),  we  show  the number of atoms recorded after an elevator round­trip,  as a function of the travelled  distance.  A  sudden  drop,  corresponding  to  the  loss  of atoms  hit­ ting the  test  surface,  is  clearly visible.  The  plot  provides a measurement of the vertical size of the atomic sample. This scheme directly applies to trans­ parent  materials.  In  order  to  study  metallic  surfaces,  the  atomic  sample  in the optical lattice is  displaced by means of optical components mounted on micrometric  translation  stages.  In  that  case,  the  optical  lattice  is  produced by  retroreflecting  the  laser  beam  on  the  test  surface  itself.  The  minimum attainable atom­surface distance is limited by the vertical size of the atomic distribution.  For  experiments  at  distances  below  10  /zm,  we  compress  our sample using an optical tweezer. This is obtained with a strongly astigmatic laser beam with the vertical focus centered on the atoms. Figure 5.b)  shows an  image  of the  atoms  trapped  in  the  optical  tweezer. Deviations  from  the  Newtonian  gravity  law  are  usually  described  as­ suming a Yukawa­type potential with two parameters, a giving the relative strength of any new effect  compared to Newtonian gravity and  A its range. Experiments  searching for  possible  deviations  have  set  bounds  for  the  pa­ rameters a  and  A.  Recent  results  using  microcantilever  detectors  lead  to extrapolated limits a ~  104  for A  ~  10  /zm  [17,18].  The small size  and high sensitivity  of the  atomic probe  allow  a direct,  model­independent  measure­

Fig. 5.  a) Number of atoms in the lattice versus vertical displacement. The inset shows the region close to the test surface. The vertical displacement is varied by changing the duration  of the  motion  at  uniform  velocity,  b)  In­situ  absorption  image  of the  atomic sample trapped in the optical tweezer.  Untrapped falling atoms are also visible.

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ment at distances of a few pm from the source mass with no need for modeling and extrapolation as in the case of macroscopic probes. This allows to directly access unexplored regions in the Q - X plane. Also, in this case quantum objects are used to investigate gravitational interaction. If we consider a thin layer of a material of density p and thickness d , the Newtonian gravitational acceleration due to the source mass is a = 2nGpd; for d 10 pm and p 2: 20 g/cm3, as for gold or tungsten, the resulting acceleration is a 10-l' msP2. Measuring v~ at a distance of 10 pm from the surface will then provide a direct test of present constraints on a [18]. For smaller distances, around 5 pm, it is possible to improve present limits on (Y by more than two orders of magnitude in the corresponding X range. Even shorter distances could probably be accessed, also considering a related scheme based on a Sr lattice clock [19]. Non-gravitational effects (Van der Waals, Casimir forces), also present in other experiments, can be reduced by using a conductive screen and performing differential measurements with different source masses placed behind it. For this experiment we developed a source mass made of a sandwich of A1 and Au layers covered by an Au layer. This layer acts as the mirror t o produce the optical standing wave and as the conductive screen. Placing the atoms in proximity of the different materials with different mass densities, mass-dependent effects can be investigated. Also, by performing the experiment with different isotopes of Sr, having different masses but the same electronic structure, gravitational forces can be distinguished from other surface interactions. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM

-

-

4. Transportable atom gravimeters for geophysics applications and future experiments in space Inertial and rotational sensors using atom interferometry display a potential for replacing other state-of-the-art sensors for e.g. geophysics and space applications. The intrinsic benefits making direct use of fundamental quantum processes promise significant advances in performance, usability, and efficiency, from the deployment of highly optimized devices on satellites in space or from the use of ground based transportable devices.

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4.1. Geophysics applications

The knwoledge of the value of g and its time and space variations is of interest to a wide field of physical sciences connected to geophysics and geodesy. Surface gravity measurements on local scale can help in under-

98

standing active tectonics areas dynamics, faulting during the interseismic phase, fluid migration/diffusion due to stress changes. A variety of maninduced physical and chemical processes are known to produce substantial vertical displacements of the Earth’s surface and are related to mass or fluid extraction or to subsurface pore fluid flows. Gravity measurements are used in acquifier and reservoirs monitoring, monitoring of fluid infiltration and water table rise near nuclear repositories and monitoring of subsidence and mining effects. Recently, micro-gravimetric observations have found an important field of application in volcanoes monitoring. Variations in the local gravity field were observed prior and during eruptive episodes of variable size at a number of volcanoes worldwide. It appears that crucial mass redistribution in geodynamics occur over time scales spanning the 1 - l o6 s interval and have amplitudes ranging from 10 microgals up to hundreds of microgals (I Gal = 1 cm/s2). Some of these phenomena are observed very early and occur before other phenomena as strain deformations or seismic signals. These considerations indicate that the continuous observation of the local gravity field using sensitive instrumentation with comparable accuracy on long periods is a major goal to be attained towards a better understanding of active volcanic systems and prediction of eruptive activity. Prototypes of transportable gravimeters based on atom interferometry have been realized by different labs. In Stanford, a transportable system was developed and used to measure the components of the gravity tensor. At JPL, a compact gravity gradiometer is being developed. A transportable gravimeter based on atom interferometry was developed for metrological applications at SYRTE in Paris. Our group is involved in these activities in the frame of a European STREP/NEST project (FINAQS) and with support from the Istituto Nazionale di Geofisica e Vulcanologia (INGV). We are developing a transportable atomic gravimeter that will be used for geophysics applications. Our main interest is in vulcanoes monitoring t o investigate the possibility of predicting eruptions. The high sensitivity and long term stability achievable with an atom gravimeter are important characteristics for this application.

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4.2. Space applications

Future experiments in space will allow to take full advantage of the potential sensitivity of cold atom interferometers as acceleration or rotation sensors. Indeed, atomic quantum sensors can reach their ultimate performances if operated in space because of the extremely long achievable interaction time and the vibration-free environment.

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In Europe, the interest on application of atomic quantum sensors in space is demonstrated by the activities initiated by ESA and by national space agencies, CNES, ASI, DLR. The study conducted by ESA on HYPER mission, proposed in 2000 in response to the call for the second and third Flexi-missions (F2/F3), showed the feasibility of cold atom interferometry in space both for inertial sensing and fundamental physics studies. SAI, a new project funded by ESA, started in 2007 [20]. The project intends t o exploit the potential of matter-wave sensors in microgravity for the measurement of acceleration, rotations, and faint forces. SAI aims to push present performances to the limits and t o demonstrate this technology with a transportable sensor which will serve as a prototype for the space qualification of the final instrument. The atom interferometer will be used to perform fundamental physics tests and t o develop applications in different areas of research (navigation, geodesy). These activities are financed by the HME Directorate in the frame of the ELIPS2 program. Several pieces of technology for this activity are common to those developed for other ESA projects based on cold atom clocks, namely, ACES (Atomic Clocks Ensemble in Space) and SOC (Space Optical Clocks). Different proposals based on the utilization of matter-wave interferometers and atomic clocks for fundamental physics studies were submitted to ESA in the context of the ”Cosmic Vision 2015-2025” program. The applications of atomic quantum sensors in space are interdisciplinary, covering diverse and important topics. In fundamental physics, space-based cold atom sensors may be the key for new experiments, e.g., accurate tests of general relativity, search for new forces, test of l / r 2 law for gravitational force at micrometric distances, neutrality of atoms. Possible applications can be envisaged in astronomy and space navigation (inertial and angular references), realization of SI-units (definition of kg, measurements of Newtonian gravitational constant G, h/m measurement), GALILEO and LISA technology, prospecting for resources and major Earth-science themes. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH

Acknowledgments For the experiment on G, G.M.T. acknowledges seminal discussions with M.A. Kasevich and J. Faller and useful suggestions by A. Peters. M. Fattori, T. Petelski, and J. Stuhler contributed to setting up the apparatus. This work was supported by INFN (MAGIA experiment), EU (contract RII3CT-2003-506350 and FINAQS STREP/NEST project), INGV, ESA, ASI.

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References

1. A. Peters, K. Y. Chung and S. Chu, Nature 400,p. 849 (1999). 2. J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden and M. A. Kasevich, Phys. Rev. A 65,p. 033608 (2002). 3. A. Bertoldi, G. Lamporesi, L. Cacciapuoti, M. D. Angelis, M. Fattori, T. Petelski, A. Peters, M. Prevedelli, J. Stuhler and G. M. Tino, Eur. Phys. J . D 40,p. 271 (2006). 4. J. B. Fixler, G. T. Foster, J. M. McGuirk and M. Kasevich, Science 315, p. 74 (2007). 5. S. Fray, C. A. Diez, T. W. Haensch and M. Weitz, Phys. Rev. Lett. 93,p. 240404 (2004). 6. S. Dimopoulos, P. Graham, J. Hogan and M. Kasevich, Phys. Rev. Lett. 98, p. 111102 (2007). 7. G.M. Tino, in 2001: A Relativistic Spacetime Odyssey - Proceedings of J H Workshop, Firenze, 2001 (I. Ciufolini, D. Dominici, L. Lusanna eds., World Scientific, 2003). Also, Tino G. M., Nucl. Phys. B 113, 289 (2003). 8. S. Dimopoulos and A. A. Geraci, Phys. Rev. D 68,p. 124021 (2003). 9. D. M. Harber, J. M. Obrecht, J. M. McGuirk and E. A. Cornell, Phys. Rev. A 72,p. 033610 (2005). 10. G. Ferrari, N. Poli, F. Sorrentino and G. M. Tino, Phys. Rev. Lett. 97,p. 060402 (2006). 11. G. M. Tino, L. Cacciapuoti, K. Bongs, C. J. Bordk, P. Bouyer, H. Dittus, W. Ertmer, A. Gorlitz, M. Inguscio, A. Landragin, P. Lemonde, C. Laemmerzahl, A. Peters, E. Rasel, J. Reichel, C. Salomon, S. Schiller, W. Schleich, K. Sengstock, U. Sterr and M. Wilkens, Nucl. Phys. B (Proc. Suppl.) 166, p. 159 (2007). 12. G. Tino and F. Vetrano, Class. Quantum Grav. 24,2167 (2007). 13. P. J. Mohr and B. N. Taylor, Rev. Mod. Phys. 77-1,42 (2005). 14. M. Kasevich and S. Chu, Appl. Phys. B 54,p. 321 (1992). 15. G. Lamporesi, A. Bertoldi, A. Cecchetti, B. Dulach, M. Fattori, A. Malengo, S. Pettorruso, M. Prevedelli and G. Tino, Rev. Sci. Instrum. 78,p. 075109 (2007). 16. G. T. Foster, J. B. Fixler, J. M. McGuirk and M. A. Kasevich, Opt. Lett. 27, p. 951 (2002). 17. J. C. Long, H. W. Chan, A. B. Churnside, E. A. Gulbis, M. C. M. Varney and J. C. Price, Nature 421,p. 922 (2005). 18. S. J. Smullin, A. A. Geraci, D. M. Weld, J. Chiaverini, S. Holmes and A. Kapitulnik, Phys. Rev D 72,p. 122001 (2005). 19. P. Wolf, P. Lemonde, A. Lambrecht, S. Bize, A. Landragin and A. Clairon, Phys. Rev. A 75,p. 063608 (2007). 20. G. M. Tino et al., Space Atom Interferometers (SAI) (AO-2004-64), Proposal in response to ESA Announcement of Opportunity in Life and Physical Sciences and Applied Research Projects, ESA-AO-2004.

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NOVEL SPECTROSCOPIC

APPLICATIONS

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ON A VARIATION OF THE PROTON-ELECTRON MASS RATIO

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W. UBACHS', R. BUNING', E. J. SALUMBIDES', S. HANNEMANN', H. L. BETHLEM', D. BAILLY2, M. VERVLOET3, L. KAPERIs4,M. T. MURPHY' 'Laser Centre Vrije Universiteit Amsterdam, The Netherlands 2Laboratoire de Photophysique Mol&culaire, Universitg de Paris-Sud, Orsay, France 3Synchrotron Soleil, Gif-sur-Yvette, France 41nstituut Anton Pannekoek, Universiteit van Amsterdam, The Netherlands 'Institute of Astronomy, Cambridge University, UK Recently indication for a possible variation of the proton-to-electron mass ratio p=mp/me was found from a comparison between laboratory H2 spectroscopic data and the same lines in quasar spectra. This result will be put in perspective of other spectroscopic activities aiming at detection of variation of fundamental constants, on a cosmological as well as on a laboratory time scale. Furthermore the opportunities for obtaining improved laboratory wavelength positions of the relevant H2 absorption lines, as well as the prospects for obtaining a larger data set of HZabsorptions at high redshift will be presented. Also an experiment to detect Ap on a laboratory time scale will be discussed.

1. Introduction

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Recently the finding of an indication for a decrease of the proton-to-electron mass ratio p=mp/m by 0.002% in the past 12 billion years was reported [l]. Laser spectroscopy on molecular hydrogen, using a narrow-band and tunable extreme ultraviolet laser system resulted in transition wavelengths of spectral lines in the B-X Lyman and C-X Werner band systems at an accuracy of 5 x for the best lines. This corresponds to an absolute accuracy of 0.000005 nm. A database of 233 accurately calibrated H2 lines is produced for future reference and comparison with astronomical observations. Recent observations of the same spectroscopic features in cold hydrogen clouds at redshifts z=2.5947325 and z=3.0248970 in the line of sight of two quasar light sources (Q 0405-443 and Q 0347-383) resulted in 76 reliably determined transition wavelengths of H2 lines at accuracies in the range 2 x to zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Those observations were performed with the Ultraviolet and Visible Echelle Spectrograph at the Very Large Telescope of the European Southern Observatory at Paranal, Chile [2]. A third ingredient in the analysis is the calculation of an improved set of sensitivity coefficients Ki, a parameter 103

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associated with each spectral line, representing the dependence of the transition wavelength on a possible variation of the proton-to-electron mass ratio. Details of the methods are reported in Ref. [3]. zyxwvutsrqponmlkjihgfedcbaZYX A statistical analysis of the data yields an indication for a variation of the proton-to-electron mass ratio of Adp = (2.45 f 0.59) x for a weighted fit and A d p = (1.98 5 0.58) x for an unweighted fit. This result has a statistical significance of 3.50. The redshifts of the hydrogen absorbing clouds can be converted into look-back times of 11.7 and 12 billion years with the age of the universe set to 13.7 billion years. Mass-variations as discussed relate to inertial or kinematic masses, rather than gravitational masses. The observed decrease in p corresponds to a rate of change of per year, if a linear variation with time is assumed. This dlngdt = -2 x remarkable result should be considered as no more than an indication for a possible variation of p. Only a very limited data set is available: two quasar systems with a total of 76 spectral lines.

In the following we put these results in perspective of other spectroscopic activities concerning variation of fundamental constants, and present possibilities to obtain confirmation of the findings in the near future by producing improved laboratory data for H2 and extend the data set of H2 astronomical observations.

2. Variation of dimensionlessfundamental constants: a and p Renewed interest in the possibility of temporal variation of fundamental constants arose through the findings of Webb et al. [4]. Based on highly accurate spectroscopic observations of atomic and ionic resonance lines at high redshift (from the HIRES-Keck telescope at Hawaii) a variation of the fine structure constant a was deduced. This breakthrough could be made through implementation of the so-called Many-Multiplet method, which allows for using many transition wavelengths in the analysis [5], rather than just separations between fine structure lines, as in the alkali-doublet method. By this means the sensitivity to detect A a is improved. These findings on a lower value of a in the past were disputed by competing teams who found essentially a null result on A a from data obtained with the UVES-VLT on the southern hemisphere [6,7]. Meanwhile the Webb-Murphy-Flambaum team extended their data set to some 150 quasar systems, obtaining a more than 5 0 effect with Aala = (-0.574 f 0.102) x [8]. The discrepancy in the findings by different teams on Aala were resolved by the recent reanalysis of the UVES-VLT data set by Murphy et al. [9]; flaws in the fitting procedures were uncovered and a reanalysis yields a revised value of A d a = (-0.44 f in agreement with the values of [8]. 0.16) x

105

The invention of frequency comb lasers has immensely increased the accuracies in atomic spectroscopy, to the extent that absolute precision at the can be obtained, in fact limited by the Cs time and frequency level of standard. From atomic precision experiments on various systems boundaries to the rate of change in the fine structure constant dln aldt were set to the per year by performing laboratory laser spectroscopic level of 2 x studies with time intervals of one or a few years. The NET-Boulder group set a limit of 1.2 x lo-'' per year from measurements on a singly trapped '99Hg+ ion [lo]. The Munich group deduced a similarly small rate from calibrating the H-atom (1s-2s) transition against the Paris portable Cs fountain clock [ l l ] , as did the PTB-Braunschweig team from 171Yb+ions [12]. Very recently the NIST-Boulder group pushed the boundary on dln p/dt to 1.3 x 10-16per year by comparing Hg' against Cs [ 131; at the ICOLS 07 even a tighter limit was presented at the 2 x level from a comparison of Hg' and Al' clock transitions (see this book).

zyxw zyxw zyx

It has been hypothesized that the changes in the proton-electron mass ratio p and the fine structure constant a are linked and that p would change faster by an order of magnitude or more; this hypothesis [14] is based on Grand Unification Theories. From a recent analysis of microwave spectra from the astrophysical object B0218+357 at redshift z = 0.68 Flambaum and Kozlov [15] put a limit to the variation of the mass-ratio at ANp = (0.6 5 1.9) x Data on the inversion motion of ammonia (23 GHz) were compared to microwave transitions in other molecules.

Hence there is evidence for a variation of a , and some indication for a variation of p at high redshifts (z > l), while the laboratory studies seem to put strict boundaries on Aa. At the same time the recent findings of high redshift ammonia put a strict boundary to Ap at a redshift of z = 0.68. In this context the hypothesis of a phase transition occurring in the history of the universe, going from a matter-dominated (dust era) to a dark energy dominated (curvature era) universe may play a role [16]. Barrow hypothesized that only before this transition, which may have occurred near z=0.5, the fundamental constants may have changed.

3. Extension of the database of molecular hydrogen at high redshift There exists only a limited data set of H2 absorptions at high redshift. Of the tens of thousands identified quasar sources some 600 are known to be associated with a damped Lyman-a system; such systems are characterized by a fully saturated and broad L-a absorption feature from a relatively dense cloud of atomic hydrogen with a column density of N(H) > 2 x 10'' ern-'. Such systems display metal absorptions and in some cases also H2

106

zyxw zyxwvu

absorptions. For the investigations probing A a some 200 systems have been spectroscopically analyzed (from metal lines Mg, Si, Zn, etc), but in only 14 of them H2 has been detected [3]. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO Lyman­a of quasar emission

I

3795

3500

'I

3800

I 4000

I

I

4500

zyxwv zyxw I

i

5ooo

I

I

5500

6ooo

1

I zyxwvutsrqponm 6500

Fig. 1. Typical spectrum of a damped Lyman-a quasar system, in this case Q2348-011, as recorded by Noterdaeme et al. with the Ultraviolet-Visible Echelle Spectrometer at the Very Large Telescope at Paranal, Chile [18]. The large emission peaks can be assigned to H-La and to C IV, and give the redshift of the quasar emission (z=3.0236). Spectra are recorded with a certain setting of a dichroic, detecting blue light on a single CCD, and the red part on two distinct CCDs. A damped L a absorption line is found at z = 2.426 and also H2 absorption is found at this redshift. Hence the Lyman and Werner absorption lines (in the laboratory at 90-112 nm) are shifted to below 380 nm. Part of the H2 window is enlarged in the left upper corner, displaying the complicated velocity structure of the H2 cloud: at least 7 velocity sub-components are visible for each absorption line (LOR0 is shown).

Obtained spectra in existing databases have been surveyed and besides the two systems used in our previous analysis (Q0347 and Q0405) three others have a potential to play a role in detecting Ap if spectra at sufficient SNR and resolution, with optimum wavelength calibrations were to be obtained. The system Q2348-011 at z= 2.426 (an archived spectrum shown in Fig. 1) will be observed under such conditions at UVES-VLT in August 2007. Other appropriate systems would be Q0528-250 at z=2.81 and Q1443+272 at z=4.22; the latter is the system with H2 detected at the highest redshift [17]. Of course there should be many damped L-a systems with H2 at high redshift

zyxw 107

in the universe that can be implemented in Ap analyses. They 'just' need to be found and subjected to high resolution observation; as a figure of merit, at current dish sizes of 8 m typical observation times in access of 20 h on target (depending on brightness of the quasar background source) are needed to obtain spectra at resolutions of R = 60000 and SNR of 50. In view of the importance of the subject the data set will be extended in coming years; currently a number of observation stations, HIRES-Keck (Hawaii), UVESVLT (Chile), and HDS-Subaru (Japan), are suitable for the purpose. In 2009 the PEPSI-LBT system in Arizona, equipped with two 8 m dishes, will become available for detection of H2 at high redshift.

4. Improving the laboratory accuracy of the Lyman and Werner lines

zyx zyxw

The prominent electronic absorption systems, also detected in quasars, are the B'C,+- X'C; or Lyman and the C'l3,- X'C; Werner band systems. At zero redshift these lie in the difficult to access wavelength range of 90-112 nm. With the use of a narrowband and tunable extreme ultraviolet (XUV) laser source the lines could be calibrated to an accuracy of 5 x [ 191.

zyxwvutsr zyxwvut zyxwv

0 7840

0.7860

0.7880

0.7900

Wavenumber - 991 64.0 ( 6 ' )

Fig. 2. Recording of the Qo two-photon line in the EF-X (0,O) band of H2 at 99164.78691(11) cm-I. Note that the resonance width is 36 MHz, determined by the linewidth of the laser at its 8'h harmonic.

We have devised an alternative spectroscopic scheme to derive the wavelengths in the B-X and C-X systems via combination differences. This method is based on two independent spectroscopic measurements. First the lowest energy levels in the EF'C;, v=O state are determined via a Dopplerfree two-photon-excitation scheme in the deep-UV at h=202 nm, that was previously described [20]. Using various advanced techniques, such as calibration against a frequency comb laser, a Sagnac configuration to avoid Doppler shifts, and on-line frequency chirp evaluation for each of the laser

108

pulses  an  absolute  accuracy  of 3.5  MHz  on the resonances  (see Fig.  2  for a spectrum of the Q0 line) was obtained, which translates to a relative accuracy of A A A = l x  10"9.

6778 8780

8782

8784 67S8 8788 S7W W92 cm

its

dull 6450

6500

..JaUk^jujylluJj

lltjiL ijLJli.lLw­uJi

6550

6600

6650

6700

6750

6800

6850

cm'

Fig.  3.  A  portion  of the  FT  infrared  emission  spectrum  of H2  in  the  range  6400­ 6900 cm"1  displaying lines in the EF­B (0,2) and (1,4) bands as indicated.

A second experiment entails Fourier­Transform infrared and visible emission spectrocopy performed on a low pressure electrodeless discharge in H2. In the near  infrared  domain  ranging  between  0.5­4  jam  many  lines  in  the  EF­B (v',v") are observed (a portion of the spectrum is shown in Fig.  3). Although the  spectral lines are Doppler­broadened (0.02  ­ 0.2  cm"1),  the high SNR and the  fact  that  each  energy  level  is  connected  to  10  or  more  other  quantum levels  produces  a  consistent framework of energy  levels  at  accuracies  in  the 10"3 ­  10"4  cm"1  range. Level energies in the C state are determined, somewhat less accurate, through transitions in the l'ng­C, j'Ag­C, H1Sg+­C and GK'lg*­ C  systems.  Systematic  effects  are  addressed  by  absolute  wavelength calibration in a wide range using Ar and CO lines. This work in progress will yield  relative  level  energies  that  can  be  combined  with  the  level  energies  of the  lowest EF,  v=0  levels;  from the  combined  set transition  wavelengths  of most  of the relevant Lyman and Werner lines  can be  calculated  at accuracies in the range  AAA =  1­5  x  10"9.  Bearing  in  mind  that the  uncertainties  in the current  quasar  absorption  data  are  at  the  level  of  AX/X = 2  x  10"7,  these

109

studies provide a laboratory or zero­redshift data set for H2 which is exact for the purpose of comparison. 5. A molecular fountain for precision studies and detection of A|i Variation  of  the  proton­electron  mass  ratio  may  be  detected  from comparisons  on  a  cosmological  time  scale,  but  also  on  a  laboratory  time scale.  Since  the  intervals  are  reduced  to  years  the  required  spectroscopic precision  has  to  be  much  higher  in  the  latter  case.  In  view  of  the  fact  that quantum tunneling phenomena  scale  exponentially  with  mass,  the  inversion splitting  in  ammonia  is  extremely  sensitive  for  Au..  This  also  underlies  the tight constraint to A|i from ammonia spectra at z=0.68 discussed above [15].

mijrowave cavity

ion detector

skimmer ffi

nozzle

Fig. 4. Design of the molecular fountain under construction in Amsterdam.

In order to obtain long effective measurement times in a Ramsey­type scheme a  molecular  fountain is under construction,  in  which  NH3  molecules  will be launched, after deceleration by the Stark technique [21].

110

Acknowledgement We thank Prof. G. Meijer (Fritz Haber Institut, Berlin, Germany) for the collaboration on the molecular fountain project.

References

zyx zyxwvuts zyx

E. Reinhold, R. Buning, U. Hollenstein, A. Ivanchik, P. Petitjean, W. Ubachs, Phys. Rev. Lett. 96, 151101 (2006). 2. A. Ivanchik, P. Petitjean, D. Varshalovich, B. Aracil, R. Srianand, H. Chand, C. Ledoux, P. BoisseC, Astron. Astroph. 440,45 (2005). 3. W. Ubachs, R. Buning, K. S. E. Eikema, E. Reinhold, J. Mol. Spectr. 241, 155 (2007). 4. J. K. Webb, V. V. Flambaum, C. W. Churchill, M. J. Drinkwater, J. D. Barrow, Phys. Rev. Lett. 82, 884 (1999). 5. V. A. Dzuba, V. V. Flambaum, J. K. Webb, Phys. Rev. Lett. 82, 888 (1999). 6. R. Srianand, H. Chand, P. Petitjean, B. Aracil, Phys. Rev. Lett. 92, 121302 (2004). 7. R. Quast, D. Reimers, S. Levshakov, Astron. Astroph. 415, L7 (2004). 8. M. T. Murphy, J. K. Webb, V. V. Flambaum, Mon. Not. Roy. Astr. SOC. 345,609 (2003). 9. M.T. Murphy, J. K. Webb, V. V. Flambaum, arXiv:astro-ph/0612407vl. 10. S. Bize, et al. Phys. Rev. Lett. 90, 150802 (2003). 11. M. Fischer et al. Phys. Rev. Lett. 92,230802 (2004). 12. E. Peik, B. Lipphardt, H. Schnatz, T. Schneider, C. Tamm, S. G. Karshenboim, Phys. Rev. Lett. 93,170801 (2004). 13. T. M. Fortier et al. Phys. Rev. Lett. 98,070801 (2007). 14. X. Calmet, H. Fritsch, Eur. J. Phys. C 24, 639 (2002). 15. V. V. Flambaum, M. G. Kozlov, Phys. Rev. Lett. 98,240801 (2007). 16. J. D. Barrow, H. B. Sandvik, J. Magueijo, Phys. Rev. D. 65, 063504 (2002). 17. C. Ledoux, P. Petitjean, R. Srianand, Astroph. J. 640, L25 (2006). 18 P. Noterdaeme, P. Petitjean, R. Srianand, C. Ledoux, F. Le Petit, Astron. Astroph. 469,425 (2007). 19. J. Philip, J. P. Sprengers, P. Cacciani, C. A. De Lange, W. Ubachs, E. Reinhold, Can. J. Chem. 82, 713 (2004). 20. S. Hannemann, E.J. Salumbides, S. Witte, R. T. Zinkstok, E.-J. Van Duijn, K. S. E. Eikema, W. Ubachs, Phys. Rev. A 74,062514 (2006). 21. H.L. Bethlem, G. Berden, G. Meijer, Phys. Rev. Lett. 83, (1999). 1.

zyx

QUANTUM INFORMATION AND CONTROL 11

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QUANTUM INTERFACE BETWEEN LIGHT AND ATOMIC ENSEMBLES

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Hanna Krauter, Jacob F. Sherson, Kasper Jensen, Thomas Fernholz, Jonas S.

Neergaard-Nielsen, Bo Melholt Nielsen, Daniel Oblak, Patrick Windpassinger, Niels Kjzergaard, Andrew J . Hilliard, Christina Olausson, Jorg Helge Miiller, Eugene S. Polzik

Quantop -Danish Research Center f o r Quantum Optics, Niels Bohr Institute, Copenhagen University, Denmark

1. Introduction

Recent years have witnessed astounding progress in the ability to control quantum systems making the vision to create working quantum networks more realistic than ever. A key component in any quantum network is certainly the interface between stationary and flying carriers of information. One avenue towards a reliable interface makes use of macroscopic atomic ensembles to distribute fragile quantum states over many particles. We review here our experiments and recent progress with atomic samples in different temperature regimes and with non-classical light sources of suitable spectral characteristics for efficient coupling to atomic ensembles. 2. Quantum interface between Cesium atoms at room

temperature and light 2.1. Canonical variables

In the first group of experiments we study a quantum interface between an ensemble of lo1' Caesium atoms at room temperature and a coherent light beam. We use dispersive atom light interaction as a versatile tool for quantum communication protocols and as a method to read out the atomic state via light. We describe the quantum interface in the language of canonical variables for light and atoms.' The atomic ensemble is characterized by the collective spin of the Cesium atoms J = j i , where j i represents the spin zyxwvuts

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113

114

zyxwv zyx zy zyxwvu zyxw zyxwvut

of a single atom. In the experiment, the atoms are oriented along the xdirection achieved by optically pumping the atoms into the F=4, m=4 state of the 6S1/2 ground state of Cesium. Following the commutation relation for the spin components and with J, being a large classical number, the Heisenberg uncertainty principle reads V a r ( j , ) . V a r ( j , ) 2 For light we consider the polarization state, characterized by the Stokes-operators Sz, Sy and S,,with [Sy,SZ]= is,, where S, is treated classically for a strong beam with a large polarization in y-direction. In order to have a common language for the light and the atoms, we introduce canonical operators: 2 = A, a Ij = A and y = 6'ij = L, 6 where each set of operators follows the commutation relation of canonical operators. Initially atoms and light will be in a minimum uncertainty state where the variables have a Gaussian probability distribution with variance $. A magnetic field is added in direction of the macroscopic spin leading to a Larmor precession of the transverse spin components around the x axis. The relevant atomic variables will then be the spins in the rotating frame. For light the cosine and sine modes at the Larmor frequency R will be of interest: GC = :J SZ(t)c o s ( ~ t ) d t ,ijs = :J S z ( t )sin(Rt)dt, = ...

zy $.

a

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cc

For the interaction of light and atoms, we consider a beam of light coupled off-resonantly to the 6 S 1 / 2 , ~ = 4+ 6P3/2 transition. Via the Faraday interaction the polarization of light is rotated proportional to the spin component in the propagation direction. At the same time the atomic spin is rotated due to the angular momentum of light. The atomic quadratures after an interaction of duration T become:2

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with the coupling-parameter IF. d m . The two spin components rotate in and out of the interaction direction and are thus affected by the cosine and sine modes of S z . For the light, the equations look a little more complicated:

Here and ts,l are higher order temporal modes, which enter the equations because of the back-action of light on itself mediated by the precessing atoms. From these equations it is evident, that light and atoms leave an

115

imprint  on each other — they become entangled. In  a  modified  experimental  setting  two  cells  with  oppositely  oriented macroscopic  spins  are  used; J% =  —J.2. = Jx.  By  introducing  non­local J 1 —j 2  * jl+j2 canonical  operators  such  as: X = v,n , v  and P —  *,„ , *  the  back­action V2Jz 

V2Jx

terms  cancel  out  and  the  input  output  relations  simplify  to: \rout • v\ 

yin

—  .A.

put =  .in

pout __ pi q

out

in

= q . 

(4) (5)

Here one of the input quadratures of light is directly mapped on the atoms (eq. 4), while one of the atomic quadratures is read out by the light (eq. 5). 2.2. Teleportation of a quantum state of light onto atoms

Fig.  1.  In  (a)  the teleportation  setup of the experiment  is shown.  The first stage of the teleportation is the entanglement. For this a strong entangling pulse, seen on the left side, is sent  through the  atomic  ensemble  (eqs.  1,  2  and  3),  which  is  kept  by  Bob,  while  the light pulse is sent to Alice. Alice also has an unknown input state created with an electro­ optical  modulator  (EOM)  and  characterized  by Y  and Q.  Then  the joined  measurement (also known as Bell measurement) is performed, where her entangled beam and the input beam are mixed at a beamsplitter and the two quadratures are measured via polarization homodyning  at  the  two  output  ports.  The  outcomes  of those  measurements  are  sent  to Bob,  who uses RF­coils to perform feedback on his atoms,  thus  recreating Alice's input state. In  (b)  the  gain was  extracted  by comparing the  first  pulse  measurements  to  the  second pulse measurements.

The  concept  of teleportation of continuous variables was  introduced by Vaidman,3  where  the  canonical  variables Y  and  Q  of  a  quantum  system (held  by Alice)  are transferred onto another  (Bob's)  system with the  help of  a  shared  Einstein­Podolsky­Rosen  (EPR)  entangled  pair.  To  complete

116

the state transfer, a joint measurement on the input state and Alice’s part of the entangled pair is followed by classical communication and a local transformation on Bob’s remaining part of the EPR pair. This general procedure can be applied to teleport a quantum state of light onto atoms4 The principle of teleportation in our setup2 is illustrated in Fig.1 (a). Since the states we consider here are Gaussian, we only have to verify the mean value transfer and find the variances of the outcomes of the atomic states to characterize the quality of the teleportation. For the read-out a second pulse is sent through the atoms after the completion of all teleportation steps (for the timing see inset in Fig.l(a)). Now one can compare the mean value of the first pulse measurements, which bear the mean value of the input state, with the mean value of the second pulse, with which the atomic state was read out via light (eq. 2, 3 ) . In Fig.l(b) one can see the calibration for one of the two quadratures, for which the input was scanned. The gain = < z t e l e p o r t e d > can thus be extracted and set to one.

The remaining task is to detect the variances of the final state. Again the second pulse is used as the read out of the atomic state, but this time the input is not varied but held constant. The variance of the atomic state can be retrieved from the light measurement with help of eqs. 2 and 3 . In Fig.2(a) one can see the outcomes of a light measurement. From this the atomic state can be reconstructed as shown in 2 (b). In the case of the displayed measurement, where the displacement corresponds to a photon number of fi = 5, the fidelity is F = 0.58 f .02, which clearly lies above the classical limit of 0.5.5 For a limited range of input states the fidelity is maximized with a gain different from one. For input distributions with widths of < n >= 2,5,10,20,200 fidelities of F 2 = 0.64 f 0.02, F 5 = 0.60 f 0.02, F ~= o 0.59 f 0.02, F ~ = o 0.58 f 0.02 and I7200 = 0.56 f 0.03 can be extrapolated from our measurement^,^ which should be compared with the achievable classical fidelities5 F t l a s s i c a l - 0.60, ~ g c l a s s i c a l= 0.545, F $ a s s i c a l - 0.52, F ; m s i c a l -

zyxw zyxwv zyxwvu zyx z zyxw

0.51 and F;&jssical= 0 .50 . There are different possibilities to improve the performance of this experiment. The protocol as it is suffers from residual noise introduced by the initial entangling interaction. By including higher order temporal modes and utilizing squeezed light in the entangling arm those extra noise contributions can be lessened and the fidelity increased.

117 (a)

(b) probability density .! • .

I

"'... ..' 'I ' ' J. - i.'-'t- i- '•- .

verifying pulse: ^

Fig.  2.  From  the  light  measurement  (a)  the  atomic  state  can  be  reconstructed  (b). There  are  two  probability  distributions  indicated.  The  one  without  coloring  is  the  best possible  classical  teleportation.  It  has  been  shown,5  that  the  best  possible  achievable classical fidelity F  is  0.5.  Compared to  the best  classical teleportation  (uncolored  graph) the teleported  state  (colored graph)  is narrower.

2.3. Single atom squeezing In  section  2.1  the  two­cell  setup  was  discussed  briefly.  From  eqs.  4  and 5  the  possibility  of a  mapping  protocol  of  light  onto  atoms  arises,  where after  the  interaction  one  of the  light  quadratures  is  automatically  written on  the  atoms.  Then y  of the  light  is  measured  and  fed  back  to  the  atoms, so that: Pout = Pout + g • yout = -yin, if the gain is adjusted properly. This experiment  has been  conducted successfully.6  However,  the fidelity of the  mapping protocol is  limited  by the  residual  input  noise  from  the  other atomic  quadrature  (see  eq.  4).  This  can  be  partly  overcome  by  squeezing Xin. Here  the  multilevel  structure  of the  Cesium  atoms  is  utilized  to  reach this  goal.  By  creating  a  suitable  superposition  of  the  even  magnetic  sub­ levels m = 4, 2,0,...  a spin squeezed state7  can be  achieved,  at the expense of a decrease in the macroscopic spin Jx.  As a result,  one of the normalized transversal  spin  components x  or p  has  a variance  smaller  than  ^. Experimentally  such  a superposition  state  can  be  obtained  by  inducing Raman  transitions  with  two  light  beams.  The  important  features  of  the experiment  are sketched in Fig.3(a).  Figure 3(b)  shows preliminary results for  the  experiment.  The  crosses  are  measured  variances  of  the  squeezed quadrature,  obtained  for  10000  measurements  with  equal  pulse  duration, power  and  detuning.  For  first  tests  a  single  cell  setup  was  used  and  the Faraday interaction utilized for the read out of the two  atomic quadratures. A noise reduction of (30 ± 10)%  compared to the standard quantum limit is achieved.  The dotted line shows the maximum achievable squeezing with

118

Fig.  3.  In  (a)  the  squeezing  experiment  is  shown,  where  two  beams,  which  are  off res­ onantly  coupled  to  the  Dl  line  (A  =  —550  MHz),  are  sent  through  the  atoms  along the  macroscopic  atomic  orientation  to  create  a  superposition  of m  =  4, 2,0,....  The  two beams  are  detuned  by  twice  the  magnetic  sublevels  splitting,  given  by  the  Larmor  fre­ quency Ci =  322  kHz.  The  Faraday  interaction  is  used  to  determine  the  noise  reduction of the atomic quadratures. Figure  (b)  shows  measurement  results.  The  solid  line  indicates  the  noise  level  of a coher­ ent  spin  state  (CSS)  with  the  equivalent  macroscopic  spin.  The  crosses  and  the  circles show  the  measured  squeezed  and  anti­squeezed  variance,  whereas  the  dotted  line  gives the  calculated  optimum  squeezing.

the used interaction in the absence of decoherence predicting approximately 70% noise reduction. Different  effects  limit  the  performance  of the  experiment.  The  decay  of the  created  state  during  the  preparation  and  the  read­out  introduces  con­ straints. Furthermore, the second order Zeeman and Stark shifts lead to the fact that the atomic spin can not be described by a Larmor precession with a single frequency.  Experimentally,  one needs to strike a balance between the  limiting  effects  by  choosing  an  optimal  power  and  timing,  as  well  as detuning,  of the  Raman beams to  achieve optimal  noise  reduction.

3. Dispersive measurements on dipole trapped cold Cs atoms In a different experimental setup we investigate non­destructive probing of laser  cooled  Cs  atoms  confined  by  a  focussed  laser  beam  and  restricted  to populate  the  two  ground  level  clock  states (F — 3,mp =  0)  and (F = 4, mp = 0)  and superpositions thereof.  As  is well known such a two level system is formally equivalent to spin 1/2 system and we can form a collective pseudo­spin  for  the  atomic ensemble  in  much the  same way as  introduced in  section  2.1.  The  system  is  conveniently  illustrated  in  the  Bloch  sphere picture Fig. 4.

119

fz ]F=4,mF=0>

Spin Squeezed State

1

!F=3,m,=0> Uncertainty disk prior to (2ND measurent

Fig.  4.  Bloch sphere representation of collective pseudo­ spin.  Beginning from situation with all the atoms in one of the clock states  (a pole on the sphere) and applying a Tf/2­ pulse produces an equal superposition of both clock states  (the equatorial plane ­ here assumed to point  in the a;­direction)  with an uncertainty  disk associated  to  it.  Such  a so­called coherent spin state will give rise to projection noise when measuring the z­axis spin projection. Performing a QND measurement produces a spin squeezed state.

3.1. Atom-light interaction A  probe  laser  beam  propagating through  the  trapped  atoms  will  expe­ rience  an  index  of refraction  according  the  population  of  atoms  in  either clock  state.  This  leads  to  a  state  dependent  phase  shift  of light  which  we can measure by comparing to a reference beam in free space using a Mach Zehnder interferometer.8 Again, light in the two­path interferometer can be described  using  an  angular  momentum  algebra in  a  similar  way  as  for  the polarization  states  in  section  2.1.  It  turns  out  that  the  interaction  between the collective angular momenta for light and atom leads to a quantum non­ demolition  (QND)  measurement of an atomic pseudo spin projection when the  phase  shift  of light  is  detected.  The  outcome  of this  measurement  can in principle be used to infer spin squeezing,  i.e.  a reduction of uncertainty in  one  spin  component (Sz)  on  the  expense  of  an  increased  uncertainty  in another  component (Sy)9'w  (See  Fig.  4). 3.2. Rabi Oscillations As  a  first  step  towards the  observation of projection  noise  and  production of  spin  squeezed  states  on  the  clock  transition,  we  have  studied  the  co­ herent  evolution  of a  spin  polarized  sample.11  The  atoms  are  prepared  in the (F =  3, mp =  0)  state  and  Rabi  oscillations  on  the  clock  transition are driven with a resonant 9.1  GHz microwave field. During Rabi flopping

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the  atoms  are  probed with  short  (200  ns)  off resonant  (150  MHz  blue  de­ tuned  from  6Sri/2(­F  =  4)  —> 6P3/2(F' =  5)  transition.)  laser  light  pulses. The  probe  laser  beam will  be  shifted  in  phase  proportionally to  the  num­ ber  of  atoms  in  the (F =  4, mF =  0)  state.  Figure  5  shows  an  example of non­destructive  probing of Rabi  oscillation using our  Mach  Zehnder  in­ terferometer.  Using  this  method  makes  it  possible  to  follow  the  coherent evolution of the ensemble quantum state in "real time". Having  established  that  the  atomic  ensemble  can  be  controlled  coher­ ently,  we  have  the  tools  at  hand  to  produce the  coherent  spin  state  Fig.  4. The  challenge  is  now  to  detect  the  atomic  quantum  fluctuations  in  the recorded  phase  shift  for  independently  prepared  ensembles  and  that  this variance  can  be  reduced  by  using  information  gained  in  a  previous  QND measurement.

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4. Non-classical states of light 4.1. Gaussian states With the purpose of being a resource for the experiments presented in sec­ tions  2  and  3,  we  have  a  setup  for  generating  various  non­classical  states of  light.  The  heart  of  the  experiment  is  an  optical  parametric  oscillator (OPO)  pumped  below  threshold.  Employing  a  nonlinear  PPKTP  crystal (periodically  poled  potassium  titanyl  phosphate),  the  blue  pump  beam  at frequency  2o>o is down­converted into several longitudinal cavity modes cen­ tered  around the  frequencies ujk = w0 +

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121

Fig.  6.  Results  of homodyne  measurement  and  reconstruction  of the  two  different  non­ Gaussian  states;  Wigner  function and  density matrix  in number  state representation. Left: Photon subtracted squeezed vacuum (kitten state). Note the predominance of odd photon numbers in the density matrix diagonal. Right: Single photon state.

where  WA  is  the  free  spectral  range  of the  cavity.  The field emitted  from the  OPO  in  the  degenerate  mode U>Q  is  in  a  squeezed  vacuum  state  with one of the field quadratures being less noisy than the vacuum level.  We  are currently  able  to  produce  a  very  pure  state  with  ­6.5  dB  squeezing  versus 10  dB  anti­squeezing.  As  mentioned  in  section  2.2,  the  squeezed  vacuum can be used to improve the fidelity of the memory protocol.  Apart from the u>o  mode,  the  non­degenerate  longitudinal  modes  are  pairwise  correlated such  that  e.g.  cj_i  and  w+i  are  in  a  two­mode  squeezed  state.  In  the  OPO the  two  modes  are  produced  in  the  same  spatial  and  polarization  modes, but  since  they  have  different  frequency they  can  be  separated via  a cavity resonant  on  w_ 12 4.2. Non-Gaussian states The  single  mode  and  two­mode  squeezed  states  are  indeed  non­classical states, but they are still  Gaussian.  As demonstrated by Wenger et al,13  it is possible  to  de­gaussify the  states  by  conditioning  on detection  of a photon in  a  part  of the  field.  In  the  case  of the  squeezed  vacuum,  we  reflect  on  a beam  splitter  a  small  fraction  towards  a  single  photon  counter.  When  this photon  counter  clicks,  we  have  effectively  subtracted  a  photon  from  the remaining  transmitted field, turning  it  into  a  superposition  of odd­photon number  states  (the  initial  squeezed vacuum  is  ideally  an  even­photon num­ ber superposition),  sometimes referred to  as a  'Schrodinger kitten'.14 If we instead focus  on the two­mode squeezed state,  detection of a pho­ ton in the w-\  mode will de­gaussify the correlated u}+\  mode.  If the pump intensity is sufficiently low, the generated state in w+i  will be a single pho­

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detector Fig.  7.  Experimental  configuration  for  the  detection  of  backscattered  light  from  a trapped  BEC.  The  inset  illustrates  momentum  detection  after  time  of  flight.

ton state.15 This scheme for a single photon source is similar to the standard down­conversion scheme of triggering on one photon from a down­converted pair, except that they are usually performed by single pass pulsed pumping. In  that  case  the  bandwidth  of the  state  will  be  several  GHz  or  nm  wide. Due to the cavity enhancement  by the  OPO, we can reduce the  bandwidth to the order of 10 Mhz while still keeping a high production rate of ~10,000 s"1.  This  property ­  which  is  shared  by  the  kitten  states  ­  is  very  impor­ tant  for  future  perspectives,  where  storage  of such  non­Gaussian  states  in atomic memories is a possibility.  The wavelength of the source is frequency tunable  around  the  852  nm  Cs  line. In order to characterize the  non­Gaussian states,  we measure them in a broadband  homodyne  detection  setup.  The  conditional  states  appear  in  a temporal mode  centered  around the trigger time  and with  a shape  roughly determined by a double­sided exponential decay with the decay constant of the  OPO.  Hence  it  is  necessary to  temporally  filter  the  continuous  homo­ dyne  detection  signal  to  properly  measure  the  non­Gaussian  state  and  not the  background  squeezed  state.  After  acquiring  several  thousand  quadra­ ture points at different  phase angles we can reconstruct the density matrix and  Wigner  function  of the  state.  The  results,  presented  in  Fig.  6,  shows clearly non­Gaussian Wigner  functions which even  have deep  negative re­ gions  around  the  origin.  The  purity  of  the  states  are  between  60%  and 70%. 5. Atom-Light interface with quantum degenerate atoms

The dispersive coupling between Cs atoms and light discussed in the previ­ ous sections can,  of course,  also be applied to different  atomic species.  The coupling strength K  introduced in  Sec.2  between the collective variables of atoms  and  light  can  be  conveniently  expressed  as K2  =  ao?7,  the  product of  on­resonance  optical  depth  ao  and  time  integrated  spontaneous  emis­ sion rate r\.  For a cold atomic sample optical depth is monotonous in phase

123

Fig.  8.  Left:  atomic  momentum  distribution  measured  after  45  ms  of  free  expansion with  the  depleted  original  condensate  to  the  right  and  the  recoiling  atoms  to  the  left surrounded  by  an  isotropic halo,  populated  by  light  scattering  and  by  s­wave  collisions during expansion;  Right:  Background  corrected  photodetector  signal  showing  the time dependent  reflectivity of the  atomic sample;

space  density,  so  that  quantum  degenerate  bosonic  samples  (BEG)  offer the ultimate  coupling strength for a given amount of dissipation by sponta­ neous emission.  Furthermore,  the  absence of Doppler  broadening allows to resolve the excited state hyperfine structure in the optical excitation,  which gives rise to  a different  effective interaction between atomic internal states and  light  polarization  states  with  more  complex  input/output  relations.16 For  atomic  samples  far  below  the  recoil  temperature  the  momentum  de­ gree  of freedom  becomes  accessible  as  an  additional  quantum  system with a  well  denned  initial  state,  where  the  coupling  between  light  and  atomic momentum  states  can  be  formulated  in  a language  analogous to  the  polar­ ization/angular  momentum  case.  In  first  experiments  we  try  to  assess  the achievable  coupling  strength  by  studying  super­radiant  Rayleigh  scattering off EEC's.17 Experimentally we prepare EEC's of 87Rb atoms by standard rf evapo­ ration  techniques  inside  a  magnetic  trap  of the  loffe­Pritchard type.18  The cigar  shaped  clouds,  containing  ~  5  •  105  atoms  with  no  discernible  ther­ mal  fraction,  are  probed  along  the  long  axis  with IfjW  pulses  of circularly polarized  light,  focused  to  a  waist  of 20fj,m  and  detuned  by —2.6GHz  from the 5si/2(F =  1)  —>  5pi/2(F'  =  2)  transition  (see  Fig.7).  The  elongated geometry  of  the  sample  favors  repeated  scattering  and  super­radiant  gain in the  direction  of the  long  axis.  We  detect  light  scattered  in the  backward direction and measure the  atomic momentum distribution by shadow imag­ ing after  time­of­flight  expansion  (see  Fig.8)  clearly demonstrating that  the

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super-radiant regime is reached. Ongoing experiments aim to verify the strict correlations between scattered photons a n d recoiling atoms implied by momentum conservation.

6. Acknowledgements

This research is funded by DG and through the EU projects COVAQUIAL,

QAP, a n d EMALI. References

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1. J. F. Sherson, B. Julsgaard and E. S. Polzik, Deterministic atom-light quantum interface, in Adv. At. Mol. Opt. Phys. 54, eds. P. Berman, C. Lin and E. Arimondo (Elsevier, 2006) pp. 82-131. 2. K. Hammerer, E. S. Polzik and J . I. Cirac, Phys. Rev. A 72, 064301 (2005). 3. L. Vaidman, Phys. Rev. A 49, 1473 (1994). 4. J. F. Sherson, H. Krauter, R. K. Olsson, B. Julsgaard, K. Hammerer, I. Cirac and E. S. Polzik, Nature 443, 557 (2006). 5. K. Hammerer, M. M. Wolf, E. S. Polzik and J. I. Cirac, Phys. Rev. Lett. 94, 150503 (2005). 6. B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek and E. S. Polzik, Nature 432, 482 (2004). 7. M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993). 8. P. G. Petrov, D. Oblak, C. L. Garrido Alzar, N. Kjaergaard and E. S. Polzik, Phys. Rev. A 75,033803 (2007). 9. A. Kuzmich, N. P. Bigelow and L. Mandel, Europhys. Lett. 42, 481 (1998). 10. D. Oblak, P. G. Petrov, C. L. Garrido Alzar, W. Tittel, A. K. Vershovski, J. K. Mikkelsen, J. L. Sorensen and E. S. Polzik, Phys. Rev. A 71,043807 (2005). 11. P. Windpassinger et al., in prep. (2007). 12. C. Schori, J. L. S~rensenand E. S. Polzik, Phys. Rev. A 66, 033802 (2002). 13. J. Wenger, R. Tualle-Brouri and P. Grangier, Phys. Rev. Lett. 92, 153601 (2004). 14. J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mdmer and E. S. Polzik, Phys. Rev. Lett. 97, 083604 (2006). 15. J. S. Neergaard-Nielsen, B. M. Nielsen, H. Takahashi, A. I. Vistnes and E. S. Polzik, Opt. Express 15,7940 (2007). 16. 0. S. Mishina, D. V. Kupriyanov, J. H. Miiller and E. S. Polzik, Phys. Rev. A 75,042326 (2007). 17. S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, J. Stenger, D. E. Pritchard and W. Ketterle, Science 285,571 (1999). 18. T. Esslinger, I. Bloch and T. W. Hansch, Phys. Rev. A 58,R2664 (1998).

DEGENERATE FERMIGASES

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AN ATOMIC FERMI GAS NEAR A P-WAVE FESHBACH RESONANCE D . S. JIN, J. P. GAEBLER, AND J. T. STEWART JILA, Quantum Physics Division, NIST and Department of Physics, University of Colorado, Boulder, 440 UCB Boulder, CO 80309-0440, USA

Abstract: Atomic scattering resonances, called Feshbach resonances, have been used to create molecular Bose-Einstein condensates and Fermi superfluids. Past work has focused on s-wave, or non-rotating, pairs created from two fermionic atoms. Here we report on investigations of pair creation in an ultracold Fermi gas of 40Katoms near a p-wave Feshbach resonance.

1. A p-wave Feshbach Resonance 1.1. Introduction and Motivation

Ultracold gases of atoms are powerful model systems for exploring many-body quantum phenomena. A unique feature of these systems is that the experimenter can actually control the interactions between the particles through the magic of a Feshbach resonance. By going to the strongly interacting regime in a Fermi gas of atoms, it is now possible to create a Fermi superfluid state [l]. This state results from the pairing of atoms; a pair of correlated fermions is itself a composite Bose particle, which can form a Bose condensate and thus give rise to superfluidity. This basic phenomenon can be seen in many other Fermi systems including superconductors, superfluid liquid 3He, and nuclear matter. The simplest type of pairing is s-wave pairing. In this case, the pairing is isotropic in space and does not involve orbital angular momentum. The Fermi superfluid state realized in ultracold gases, using either 40K atoms or 6Li atoms, is an s-wave superfluid. For ultracold gases, unlike for dense Fermi systems, we understand extremely well the microscopic origin of the interactions. Indeed, we control those interactions in order to form the Fermi superfluid state. If, as in the case of current experiments, the interactions are controlled using an swave Feshbach resonance: then the resulting pairs are obviously s-wave. Now, let us consider the possibility of using a non-s-wave Feshbach resonance to create non-s-wave pairs. We know that non-s-wave pairing occurs in high T, superconductors (d-wave) and in superfluid liquid 3He (p-wave). These condensed matter systems have some unique properties because of the non-s-wave pairing. Non-s-wave pairing is anisotropic and can give rise to an zyxwvutsrqp a

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An s-wave Feshbach resonance, as discussed here, couples an s-wave scattering state of two atoms to an s-wave (non-rotating) molecule. The p-wave Feshbach resonance discussed in this paper couples a p-wave scattering state of two atoms to a p-wave (rotating) molecule. zyxwvutsrqponmlkjihgfedcb

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anisotropic pairing gap. In addition, there are now different possible quantum numbers describing the pairs; these correspond to the different projections of the orbital angular momentum of the pair. For example, in the case of p-wave pairing, with one quanta of orbital angular momentum (L=l), the projection, mL, can be - 1 , 0, or 1. This allows for multiple superfluid states, and the opens the possibility of quantum phase transitions between distinct superfluid states. In addition to providing access to these intriguing features in a uniquely controllable model system, p-wave superfluidity in an ultracold Fermi gas has also been discussed as an interesting system for topological quantum computing.

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1.2. Not One, But Two Resonances

In ultracold Fermi gases, magnetic-field tunable p-wave Feshbach resonances have been observed for both 40Katoms and 6Li atoms. In 2003 our group reported the observation of a p-wave resonance between spin-polarized 40K atoms [3]. As a first step toward pursuing the possibility of using this resonance to create correlated fermion pairs and p-wave superfluidity, we discuss here experiments studying weakly bound p-wave molecules created using this Feshbach resonance. More details about these experiments can be found in Ref. [3].

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A Feshbach resonance arises from the coupling of two-particle scattering states to a bound state. And, therefore, it turns out that one can use a variety of

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experimental techniques near a magnetic-field tunable Feshbach resonance to very efficiently convert atoms painvise into weakly bound molecules. Perhaps the simplest technique is simply to set the magnetic-field strength to a value near the resonance position and wait. Figure 1 shows the loss in the number of atoms that we observe for such an experiment near the 40Kp-wave Feshbach resonance. A striking feature in this data is the splitting of the p-wave resonance into two distinct resonances. This was first observed in Ref. [2] and explained in Ref. [4]. The two resonances correspond to p-wave scattering with different projections (onto the direction of the magnetic field) of the pair orbital angular momentum mL. The lower field resonance corresponds to mL = +1 or mL= -1, while the higher field resonance corresponds to mL=O. These two resonances are separated by about 0.5 Gauss; this splitting is due to a small energy difference that comes from the magnetic dipole interaction between atoms. The loss in the observed number of atoms seen in Figure 1 is consistent with the conversion of atoms into weakly bound Feshbach molecules. Since we measure the atom number using resonant absorption imaging of the gas, and since molecules, in general, do not absorb the same color light as free atoms, we expect that the creation Feshbach molecules would appear as a loss of atoms in our measurment. More direct evidence of p-wave Feshbach molecule creation is presented later in section 2.2.

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2. p-wave Molecules

2.1. Molecule Energy

Another technique that has been used to create s-wave Feshbach molecules is to set the magnetic-field strength to a value near the resonance position and then add a small-amplitude sinusoidal modulation to the magnetic field. This oscillating magnetic field can resonantly couple free atoms pairs to the Feshbach molecule state[5][6]. By varying the frequency of the magnetic-field modulation and looking for the resonant loss of atoms, we can determine the resonant frequency for a particular magnetic-field detuning from the Feshbach resonance. Then, repeating the measurement for a variety of magnetic-field values, we mapped out the energy difference between free atom pairs and Feshbach molecules. This is shown in Figure 2. Compared with similar plots of molecule energies for an s-wave resonance, Figure 2 illustrates several new features of the p-wave resonance.

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The p-wave resonance is split into two distinct resonances, as discussed in the previous section. Here, we directly measure the energy splitting of the two p-wave Feshbach molecule states. 2. For magnetic fields above the Feshbach resonance, there exists a metastable state with a well-defined energy. Such a state is not seen for the s-wave resonances that have been used for Fermi superfluidity. The Feshbach molecule has a binding energy that depends linearly on the 3. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA magnetic-field. In contrast, for the s-wave resonances that have been used for Fermi superfluidity, the s-wave Feshbach molecule energy depends quadratically on the magnetic-field detuning from resonance. zyxwvutsrqponmlkjihgfed

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Figure 2. Energy of the Feshbach molecule relative to free atoms vs. magnetic-field detuning from the (lower) Feshbach resonance. This plot is taken from Ref. [3]. A negative energy corresponds to a true bound-state while a positive energy corresponds to a metastable “quasi-hound” resonance state.

2.2. “Seeing”p-wave Molecules

The existence of the quasi-bound state for fields above the Feshbach resonance provides a way to more directly probe the p-wave Feshbach molecules. After creating molecules (by holding the magnetic-field for a few ms near the resonance) we can increase the magnetic-field to a value above the resonance. The molecule state then adiabatically becomes a quasi-bound state with an energy greater than that of free atoms. This quasi-bound state is metastable

131

because  of  the  p­wave  centrifugal  barrier.  The  height  of  this  centrifugal barrier,  in  units  of  frequency,  is  about  5.8  MHz.  This  is  much  larger  than typical kinetic  energies  in the gas  (the  Fermi  energy is typically about  10 kHz) and  also  much  larger  than  the  quasi­bound  state  energies  explored  here  (see Figure  2).  However,  the  quasi­bound  state  will  eventually  dissociate  into  free atoms  by  quantum  mechanical  tunneling  through  the  centrifugal  barrier.  The resulting  free  atoms  will  have  a  relative  kinetic  energy  that  is  defined  by  the quasi­bound state energy.

Figure  3.  Images  of  dissociated  p­wave  Feshbach  molecules.  A  linear  grayscale indicates  the  optical  depth,  with  white corresponding to more absorption of the resonant probe  light.  The  images  are  taken  after ballistic  expansion  from  the  trap  and  therefore show  the  velocity  distribution  of the  atoms  resulting  from  dissociation  of the  Feshbach molecules.  The  left  image  corresponds  to  the  mL  =  ±1  resonance  and  the  right  image corresponds  to  the  mL  =  0 resonance.  The  image  plane  is  transverse  to  the  quantization axis,  which  is  defined  by  the  external  magnetic  field.  The  images  show  the  expected angular distributions for p­wave pairs.

Figure  3  shows  images  of p­wave  molecules  taken  using  this  technique. After the  Feshbach molecules  are  created,  we  remove  all  remaining  free  atoms using  resonant  laser  light.  Since  the  molecules  do  not  absorb  this  light,  they remain  unperturbed.  We  then  increase  the  magnetic­field  strength  to  a  value above  the  resonance  and  allow  the  now­quasi­bound  molecules  to  dissociate. The resulting free atom clouds are shown in Figure  3.  The left and right images are  taken using the mL  = ±1  resonance  and the  mL =  0 resonance,  respectively. Both  images  are  taken  after  ramping  the  magnetic­field  to  the  same  positive detuning  from  the  relevant  resonance  and  for  the  same  expansion  time  after

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turning the trap off. The difference in the clouds’ sizes and shapes then reflects the different angular distribution of atoms in the p-wave pairs. zyxwvutsrqponmlkjihgfedcba

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As mentioned in section 1.2. it is possible to produce p-wave molecules simply by setting the magnetic field to a value near the resonance and waiting. With our method to see the molecules we could dynamically probe this process. It should be noted that we do not currently understand this molecule creation process. Moreover, the data presented in this section represent our first investigations, for the p-wave resonance, of the region where many-body effects can be expected to play an important role in determining the behavior of the gas. zyxwvutsrq

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Figure 4. (a) Measured atom number as a function of time that the magnetic field is held at the mL = i 1 resonance. (b) Measured molecule number for the same hold time on resonance. The plot is taken from Ref. [3]. The inset shows the timing sequence for this experiment. The number of molecules is measured using the dissociation technique described in the text (solid line.) The number of atoms not in molecules is measured by ramping the field below the resonance (dashed line).

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Figure 4 shows the atom and molecule populations as a function of time. In this experiment, we quickly changed the magnetic field from a value far from the resonance to a value near the resonance and held for a variable amount of time. The data was taken for the magnetic-field value where we observe the highest conversion efficiency to molecules. It can be seen that the molecule population quickly reaches its maximum value in about 1 ms, and then slowly decays on a timescale of order 10 ms. The atomic population monotonically decays, indicating the presence of an inelastic decay process. zyxwvutsrqponmlkjihgfedcbaZ

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If we set the hold time at the near resonant magnetic field to be 1 ms and then vary the value of the field, we observe that molecule creation occurs only over a small range of magnetic-field values near the resonance; this can be seen in Figure 5 . The molecule creation feature is observed to be asymmetric with a width that scales with the Fermi energy of the gas. This suggests that the measured width of the resonance feature reflects the atomic kinetic energy distribution rather than an intrinsic energy width of the resonance. In other words, the energy width of the resonance is narrower than the distribution of kinetic energies in the gas. The s-wave resonances that have been used to

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realize super fluidity in Fermi gases have been in the opposite, broad resonance limit where the resonance simultaneously affects all collision energies in the gas. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

2.4. Molecule Lifetimes

Using the technique to see the p-wave molecules we could measure their lifetimes. Note that our imaging technique is state selective and we only detect the p-wave Feshbach molecules (and not other molecule states or free atoms). We could also measure the quasi-bound molecule lifetimes. The result of these measurements is shown in Figure 6. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP

-200

-100

zyxw zyxw 0

I0

Energy (kHz)

Figure 6 Lifetimes of the p-wave molecules as a function of their energy This plot is taken from Ref [3] Negative energies (left) correspond to bound molecules, while positive energies (nght) correspond to quasi-bound pairs Data for the mL = 0 resonance are shown in open symbols, while mL= *1 are closed symbols The two dotted lines on the left indicate the averages of the measured bound state lifetimes The solid line on the nght is a theory curve for the quasi-bound lifetimes

For the bound molecules (negative energy relative to the free atoms), we find that the lifetime does not depend strongly on the binding energy (and therefore does not depend strongly on the magnetic-field detuning from

135 resonance). This is very different from the case of the s-wave resonances that have been used for Fermi superfluidity. For those s-wave molecules, the lifetime has been seen to change by orders of magnitude over a similar range of binding energies. The dotted lines in Figure 6 indicate the average measured lifetimes for bound molecules created at each resonance. On the right side of Figure 6 we show the measured lifetimes for the quasibound p-wave molecules. Here, we see a very strong dependence on the pair energy. The solid line in Figure 6 shows a zero-free-parameter theory prediction by John Bohn [ 3 ] for the lifetime due to quantum mechanical tunneling through the p-wave centrifugal barrier. The expected (and measured) dependence of the lifetime on energy is a power law with a power of -312. The good agreement between the experiment and the theory show that the dominant decay mechanism for the quasi-bound p-wave pairs is dissociation into free atoms by tunneling through the centrifugal barrier. It should be noted that this is not an inelastic process and so does not cause any heating of the gas.

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2.5. Future Prospects

One motivation for exploring a p-wave Feshbach resonance in an ultracold Fermi gas is the possibility of creating a p-wave superfluid state and exploring the many-body behavior of this new state. The measured lifetimes of the pwave Feshbach molecules tell us something about how difficult this will be to achieve. For bound p-wave molecules, our measured lifetimes of 1 or 2 ms are short compared to thermalization times for the trapped gas. This short lifetime is a serious problem for future prospects for creating Bose-Einstein condensates of these p-wave molecules. There are two different decay mechanisms for our pwave molecules. One is collisional vibrational quenching, where a molecule collides with a free atom or with another molecule. This inelastic collision produces a more tightly bound, and therefore lower energy, molecule state and releases a relatively large amount of energy. The second decay mechanism is not due to collisions, but rather can occur for single molecules in isolation. This decay is due to dipolar spin relaxation, where the molecule decays into a pair of free atoms whose hyperfine spin states have lower energy than the original internal states of the two atoms paired in the molecule. This process only exists when the original atom states are not the lowest energy states in a magnetic field. This is the case for our 40K p-wave resonance. It is possible to consider creating p-wave molecules using a different pwave resonance where dipolar spin relaxation of the molecules would not be

136

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zyxwvuts time (ms)

Figure 7. P-wave Feshbach molecule lifetime after removal of free atoms. The molecules were created at the mL=Oresonance. The solid line is a fit to an exponential decay, which gives a lifetime of 7 1 ms.

*

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possible. While such a resonance does not exist for 40K atoms, it does for fermionic 6Li atoms. However, collisional vibrational quenching of the molecules would remain as a possible inelastic decay channel. To see if collisional decay of the p-wave molecules was important we measured the molecule lifetime after removing all remaining free atoms. (The conversion efficiency from atoms to molecules was about 20%, so the remaining free atoms would be the most likely collision partner for the molecules.) Free atoms were removed using a pulse of resonant light to heat them out of the trap. The molecule lifetime was then measured as described previously. The results of this measurement are shown in Figure 7. The measured molecule lifetime without atoms present was 7 + 1 ms. This is significantly longer than the previously measured molecule lifetime, but still relatively short. This measured lifetime without atoms present is consistent with a prediction by John Bohn for the molecule lifetime due to dipolar relaxation [3]. This result suggests that atom-molecule collisions do play an important role in limiting the lifetime of the p-wave Feshbach molecules. For future work it will be important to understand if it is possible to create non-s-wave molecules with better stability against collisional decay. Arguably, the most interesting many-body physics will occur on the quasibound molecule side of the resonance. Here, there lifetimes shown on the right side of Figure 6 are even shorter, but this is not necessarily a bad thing. The

137

measured lifetime of the quasi-bound molecules comes from dissociation into free atoms by tunneling through the centrifugal barrier. This is an elastic process and, in fact, this tunneling is the process that can establish the pairing required for Fermi superfluidity. Very close to the resonance, where one would like to explore the many-body physics of the Fermi gas, inelastic collisional decay may become important. Exploring this behavior very close to resonance will be important in assessing the possibility for achieving p-wave Fermi superfluidity. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

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2.6. Conclusion

Using a Fermi gas of 40K atoms, we have been able to create and detect pwave Feshbach molecules. We have explored several novel aspects of this resonance, such as the existence of a quasi-bound state that dissociates by tunneling through the p-wave centrifugal barrier. Measurements of the molecules lifetime suggest that inelastic collisional decay presents a serious challenge for future experiments attempting to investigate equilibrium manybody physics with this system.

Acknowledgments

This work was supported by the National Science Foundation and by NASA. References

1. C. A. Regal and D. S. Jin, Adv. Atom. Mol. Opt. Phys. 54, 1 (2007). 2. C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Phys. Rev. Lett. 90, 053201 (2003). 3. J. P. Gaebler, J. T. Stewart, J. L. Bohn, and D. S. Jin, Phys. Rev. Lett. 98, 200403 (2007). 4. C. Ticknor, C. A. Regal, D. S. Jin, and J. L. Bohn, Phys. Rev. A 69,042712 (2004). 5. M. Greiner, C. A. Regal, and D. S. Jin, Phys. Rev. Lett. 94,070403 (2005). 6. S. T. Thompson, E. Hodby, and C. E. Wieman, Phys. Rev. Lett. 95, 190404 (2005).

BRAGG SCATTERING OF CORRELATED ATOMS FROM A DEGENERATE FERMI GAS

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R. J. BALLAGH, K. J. CHALLIS, and C. W. GARDINER Jack Dodd centre for Photonics and Ultra-Cold atoms Physics Department, University of Otago, Dunedin, New Zealand *E-mail: [email protected] www.physics. otago. ac. nz/research/jackdodd

We formulate a treatment of Bragg scattering of a Fermi gas in a BCS state, based on the time-dependent Bogoliubov de Gennes equations. We solve these equations in three dimensions to present a quantitative analysis of the scattering. We find that, in addition to the expected scattering of atoms by the Bragg momentum transfer tzq, a comparable fraction of atoms is scattered into a spherical shell in momentum space, centered at &/2. The atoms in the scattered shell are pair correlated, and we present an analytic model that provides an interpretation of the correlated scattering mechanism, and explains the key parameter dependencies.

1. Introduction

Superfluidity in fermion systems arises due to momentum correlations between pairs of particles (the phenomenon of Cooper pairing). Techniques for probing these pair correlations in ultra-cold atomic gases have concentrated primarily on observations of the energy gap associated with the pairing (e.g., Ref. 1).Bragg scattering has been suggested in a number of theoretical studies (e.g see Refs. 2-8) as a means for characterising Fermi gases. In this paper we show that Bragg scattering can be used t o coherently probe and manipulate a degenerate Fermi gas, and that it reveals a unique signature of the pairing correlation. Bragg scattering, with a light potential of the form Acos(q . r - w t ) , has proven to be a versatile tool for manipulating and characterising BoseEinstein condensates. A long Bragg pulse is used in the so-called momentum spectroscopyg , where a narrow momentum range of the condensate is selected and incremented in momentum by an amount hq. Short Bragg pulses zyxwvut

z

138

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on the other hand can coherently divide the condensate spatial wavefunction, giving one of the packets a momentum kick of fiq. This has enabled the implementation of atomic beam-splitters and mirrors, and the exploitation of the large scale spatial coherence of condensates in atom interferometry" . The correlations in a Fermi gas are of a different nature: they are due to attractive collisional interactions, and reside primarily on the Fermi surface. Observations of the associated energy gap using single photon processes destroy the correlation. However, Bragg scattering, which is a coherent two-photon process, has the potential to manipulate the correlation into a different form which, as we show in this paper, may be directly observed. The regime we will consider is for weak collisional interactions, for which the initial equilibrium state and its pair correlations have been successfully described by the BCS model" . This provides an appropriate starting point for our treatment, but we find the Bragg scattering is sensitive to the value of the collisional interaction at the Fermi surface, and a better treatment of the collisional interaction is required than for the conventional BCS theory, as we outline in section 2. The dynamical behaviour of the atom field under the influence the Bragg potential is calculated using the formalism of time-dependent Bogoliubov de Gennes equations. We sketch the derivation of these equations in section 2.2, and present numerical solutions for the case of a homogeneous three dimensional gas. As expected, the Bragg potential induces scattering of some of the atoms by momentum fiq. However, the key result of our calculations is that a new phenomenon of Bragg scattering atoms occurs, giving rise to a scattered spherical shell of atoms centered at momentum fiq/2. There is a threshold Bragg frequency for this shell t o form, and its radius increases with w . The atoms in the shell are correlated spin-up spin-down pairs, scattered from Cooper pairs on the Fermi surface. In the final section of the paper, we develop an analytic model that describes the underlying mechanisms for this new signature for Cooper pairing in a degenerate Fermi gas. We show in detail how it results from laser mediated transfer of the initial correlation to a different region of momentum space.

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2. Time-dependent Bogoliubov de Gennes equations 2.1. Treatment of the BCS state

Our treatment is based on a BCS-like approach for the description of a degenerate Fermi gas, but with an improved treatment of the interatomic

140

collisions. In the conventional BCS approach collisions are described by a contact interaction potential, which has an infinite momentum-space range, and leads to the divergence of the pairing mean-field potential (a central object of the theory). The pair potential is typically renormalised to a finite value by introducing a momentum-space cutoff. The BCS ground state is cutoff independent, but in the limit that the cutoff tends to infinity, the renormalisation process requires that the collisional interaction strength becomes arbitrarily small. The renormalisation process described above is not sufficient for the case of Bragg scattering. We find that the correlated-pair scattering depends quantitatively on the initial pair correlations at the Fermi surface (see section 4.2), and therefore on the value of the collisional interaction potential at that momentum. We use a more sophisticated treatment of the interatomic collisions, in which the momentum-space range of the collisional interaction potential appears explicitly and is determined from experimental measurements. We assume a homogeneous Fermi gas with equal numbers of spin ‘up’ and spin ‘down’ atoms (denoted here as and -, respectively). In the absence of the Bragg field, the many-body Hamiltonian has the form

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4cy

zyxw

where is the field operator for spin state a , ‘Ho = ( - f i 2 / 2 M ) V2, M is the atomic mass, and p is the chemical potential. The interaction Hamiltonian f i c o l l n is the most general translationally invariant two-body collisional Hamiltonian for s-wave interactions between fermions of opposite spins12 ;

x

4-(Y (R

-

Here R is the centre-of-mass coordinate, r and r‘ are the relative coordinates of the two particles, and V ( r , r’) is a non-local collisional interaction potential. Our approach is to approximate V ( r , r’) by the separable potentia1l3 ,

V(r, r’) = g F ( r ) F * ( r ’ ) , (3) where F(r) has a finite range 0,even parity, and is normalised to 1. For a particular form of F(r),the interaction V(r,r’) is characterised by the interaction strength g and the range 0.

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For a given range CT,the interaction strength g can be determined by solving the Lippman Schwinger equation for the separable potential. At low energy this gives

where T ~ B = 47rh2a/M is the two-body T-matrix, with a the s-wave scattering length. The parameter y is given by -

if(k)i2d3k

M

( 2 4 3 ~

k2

'

(5)

and depends on the range c7 through the function f ( k )= F(r)e-"k"d3r. Szymariska et al. assume a Gaussian form for the function F(r),and then determine g and c7 for a particular pair of colliding atoms, far from a Feshbach resonance. For computational convenience, we approximate the collisional interaction potential using the step function f ( k ) = Q ( k c - lkl), and then choose the wave-vector cutoff k , to match the parameters g and y with those used by Szymariska et al. By this method, we find that for 40K atoms prepared in the ( F = 9 / 2 , m ~= -9/2) and ( F = 9 / 2 , m ~= -7/2) Zeeman states, the wave-vector cutoff is k , = 0.0154/a~,h,. Therefore, for a typical Fermi momentum of k~ 0.5 x 10-3/ag0h,14 , we find that k, 30k~. In the spirit of BCS, we introduce mean-field potentials and approximate kcolln by the sum of the dominant single-particle interaction terms, which are of the form (GiGa)G',&a and ( ~ ~ ~ ~ a Assuming ) ~ a ~ that p a the BCS correlation length greatly exceeds the range of the collisional interaction potential, we obtain the Heisenberg equations of motion

-

-

The quantities W and X contain the usual Hartree and pair potentials, but with 'smearing' functions that sample the field operators over the collsional range, i.e.,

X ( r , r ' , t ) = A ( r , t ) F * ( r- r'),

(7)

.

142

with the Hartree and pair potentials given by

The prefactors in these mean-field definitions (i.e., T ~ B and g, respectively) are chosen here in order to agree with the standard renormalised BCS theory. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

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2.2. The Bragg formalism

We assume that the Bragg field does not change the particle spin, and describe the effect of the Bragg field on the Fermi gas by adding V B = ~ ~ ~ Acos(q.r-wt) to 7-10. The new single particle Hamiltonian 7-1 = 7 - 1 o + V ~ ~ ~ ~ ~ is used in the dynamic field equations (6). To solve those equations we introduce a time-dependent form of the well-known Bogoliubov transformation,l1>l5i.e.,

In equation (8), the 9 k e are t i m e - i n d e p e n d e n t quasi-particle annihilation operators defined to obey the standard fermion commutation relations. The dynamic evolution of the gas is described by the evolution of the amplitudes U k ( r , t ) and Z)k(r,t ) . The quasi-particle modes are populated according to the equilibrium mean-value rules (?ka?k,p) t = bkk’bapfik and (?ka?k,p) = 0. The Fermi function is f i k = l/[exp(Ek/kBT) 11, with T the temperature and the quasi-particle energies Ek of the gas in the ground state being measured relative to the chemical potential p. The equations of motion for the amplitudes U k ( r , t ) and ?&(I-, t ) are derived from equation (6) giving

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+ and

-

1

W(r, r’, t ) U k ( r ’ ,t)d3r’

J W(r, r’,

t)’Uk(r’,t)d3r’

+

+

J

J

X(r,r’, t ) U k ( r ’ , t)d3r’,

X*(r,r’, t ) U k ( r ‘ ,t)d3r’,

in which W and X act as projectors in momentum space, with ranges 2k, and k , respectively. We solve the coupled sets of equations (9) and (10) in three dimensions, for a large set of modes k,within the regime k ~ l u l< 1, for which the BCS theory is valid.

143 The  initial  state  is  a  standard  BCS  ground  state,  found  from  the time­independent  form  of  equations  (9)  and  (10)  with  the  Bragg  field off.  Figure  l(a)  shows  the  momentum  space  column  density  /n(k, t)dkz, where  the  number  density  of  the  gas  at  momentum Kk  is  n(k, t) = C(^(k, £) a (k, t ) } .  The  momentum  space  field  operator  is  defined as  (/>a(k, t)  =  /^ Q (r,i)e~ l k ' r d 3 r/L 3 / 2 ,  where L3  is  the  computational  vol­ ume,  and the normalisation constant C is chosen such that J n(k, t)d?k  =  1. The  unit  of  energy  is  chosen  to  be  the  Fermi  energy EF = hujp,  and  the unit  of  wavevector  is  the  Fermi  wavevector  defined  in f f k p / Z M  = Ep.  In this paper we consider only zero temperature,  in which case the Fermi en­ ergy  is  related  to  the  chemical  potential  via EF  =  /z +  (1  — gjT^B^U .  The momentum  radius  of  the  cloud  is  approximately hk'F,  where  the  effective Fermi wave vector k'F is defined  in terms of the effective chemical potential by fj,' = EF - U = H2k'F2/2M  (e.g.,  Ref.  16). Column  Density 

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