Laser Spectroscopy: Proceedings of the XVII International Conference Aviemore, Scotland, UK 19 - 24 June 2005 981-256-659-7

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LASER SPECTROSCOPYPY

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Proceedings of the XVII International Conference

LASER SPECTROSCOPYY Aviemore, Scotland UK 19-24 June 2005

editors

E. A. Hinds Imperial College, London

Allister Ferguson Erling Riis University of Strathcldye, Glasgow

vp World Scientific N E W JERSEY

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LONDON

SINGAPORE

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BElJlNG

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SHANGHAI

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HONG KONG

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TAIPEI

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CHENNAI

Published by

World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore596224 USA once: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK once: 57 Shelton Street, Covent Garden, London WCZH 9HE

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

LASER SPECTROSCOPY Proceedings of the XVU International Conference Copyright Q 2005 by World ScientificPublishing Co. Pte.Ltd. AU rights reserved. This book, or parts there% 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 permissionJrom the Publisher.

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Preface

Following the tradition set by previous meetings, the 17th International Conference on Laser Spectroscopy (ICOLS 05) provided a tranquil and remote setting for researchers to discuss the latest developments and applications of laser spectroscopy. It was held during 19-24 June 2005 in the Highlands of Scotland at the Hilton Coylumbridge Hotel near Aviemore in the Heart of the Cairngorms National Park. This was the first ICOLS to be held in the United Kingdom, previous meetings having been in Vail, Megkve, Jackson Lake, Rottach-Egern, Jasper Park, Interlaken, Maui, Are, Bretton Woods, Font-Romeu, Hot Springs, Capri, Hangzhou, Innsbruck, Snowbird and Palm Cove. The conference was attended by 250 delegates from 26 countries including, Austria, Australia, Brazil, Canada, China, Denmark, France, Germany, Greece, Ireland, Israel, Italy, Japan, Latvia, Netherlands, New Zealand, Pakistan, Poland, Russian Federation, South Korea, Spain, Sweden, Switzerland, Taiwan, United Kingdom and United States. The scientific programme comprised 13 topical sessions with 34 invited talks, 8 “hot topics” presentations and two poster sessions with a total of 194 posters. In these proceedings we gather a collection of the invited talks and a selection of the contributed presentations from ICOLS 05. Subjects covered at the conference included 0 0

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Atomic and molecular laser spectroscopy Precision spectroscopy and metrology Laser cooling and trapping Bose-Einstein condensation and degenerate Fermi gases Cold collisions and cold molecules Atom optics and interferometry Quantum optics and quantum information Nonlinear optics Laser sources Ultrafast spectroscopy and atoms in high fields Novel applications of laser spectroscopy

The invited talks were grouped under the headings of High Precision Measurements, Symposium on Cold Atoms & Molecules, Atomic Clocks & InV

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terferometers, and Quantum Control & Quantum Information. The Symposium on Cold Atoms & Molecules provided a wonderful opportunity for the UK Cold Atom Network (UKCAN) to hold a satellite meeting. The conference also incorporated a very full social programme, including the traditional Wednesday afternoon excursion (a cruise on Loch Ness, a climb up Cairngorm, and golfing at Boat of Garten were among the activities on offer), a mid-Summer barbeque where many of the delegates were introduced to the finer points of Ceilidh dancing, and the Highland Banquet where several delegates took the opportunity to appear in full Highland dress. We thank all the participants for making ICOLS 05 both memorable and exciting. We also thank the Programme Committee for their expert help and advice in putting together an outstanding scientific programme. We are indebted to our financial sponsors listed on the next page. The principal corporate sponsor was Coherent Inc., whose generous sponsorship enabled a total of 18 students from outside the UK to attend the conference. The Research Council EPSRC supported 24 UK students. Finally, we would like to thank those people in the background, whose efforts were crucial in ensuring that ICOLS 05 was a successful and memorable event. We thank Julie Kite for her devoted support in bringing the programme together and in editing these Proceedings, Vicki Grant and Mairi-Claire Fitzpatrick for the smooth running of the conference and the staff of the Hilton Coylumbridge for their warm hospitality and dedicated support of the conference and the social programme. Ed Hinds Imperial College, London Allister Ferguson and Erling Riis University of Strathclyde, Glasgow September 2005

Programme Committee

E.A. Hinds (Chair), A. Aspect, H. Bachor, R.J. Ballagh, V. Balykin, R. Blatt, E.A. Cornell, W. Ertmer, A.I. Ferguson, H.H. Fielding, B. Girard, P. Hannaford, T.W. Hansch, M. Inguscio, W. Jhe, M. Leduc, M. Raizen, A. Sanpera, F. Shimizu, G.V. Shlyapnikov, M. Zhan

List of Sponsors Coherent (UK) Ltd Coherent Inc Agilent Technologies Elliot Scientific Engineering and Physical Sciences Research Council (EPSRC) Highlands and Islands Enterprise Institute of Physics QEP Group IoP Journal of Physics B: Atomic, Molecular and Optical Physics Imperial College London National Physical Laboratory (NPL) Newport Sacher Lasertechnik Scottish Enterprise Spectra-Physics Toptica Photonics University of Durham University of Oxford University of Strathclyde

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The 2005 Nobel Prize for Physics

While this volume was being prepared, it was announced that two very active participants, indeed founding fathers, of this conference, John L. Hall and Theodor W. Hansch, have been awarded the Nobel prize “for their contributions to the development of laser-based precision spectroscopy, including the optical frequency comb technique”, together with Roy Glauber “for his contribution to the quantum theory of optical coherence”. This is wonderful news and a very proper recognition of their immense contributions to the subject of this conference. On behalf of our whole community we offer our warmest congratulations to all of them.

Jan Hall and Ted Hansch at the ICOLS 05 barbecue.

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Contents

High Precision Measurements 3

Improving Laser Coherence J.L. Hall, M. Notcutt and J. Ye Precision Measurement Meets Ultrafast Control J. Ye, S. Blatt, M.M. Boyd, S.M. Foreman, T. Ido, R.J. Jones, A.D. Ludlow, A . Marian, K . Moll, M. Notcutt, M. Stowe, M. Thorpe and T. Zelevinsky Long Baseline Gravitational Wave Detectors-Status and Developments

14

25

J. Hough Precision Spectroscopy of Hydrogen and Femtosecond Laser Frequency Combs Th. Udem, P. Fendel, M. Fisher, N . Kolachevsky, J. Alnis, M. Zimmermann, Ch. Gohle, M. Hemmann, R. Holzwarth and T. W. Hansch Precision Spectroscopy of Helium Atom P. Cancio Pastor, G. Giusfredi, D. Mazzotti, P. De Natale, V. Krachmalnicoff and M. Inguscio Laser Spectroscopy of Antiprotonic Helium Atoms - Weighing the Antiproton R.S. Hayano Bloch Oscillations of Ultracold Atoms: A Tool for Metrological Measurements S. Guellati-Khe'lifa, P. Clade', E. De Mirandes, M. Cadoret, C. Schwob, F. Nez, L. Julien and F. Biraben Astrophysical Laser Spectroscopy S. Johansson and V. Letokhov

38

52

62

72

79

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xii

Quantum Interference Metrology at Deep-UV Wavelengths Using Phase-Controlled Ultrashort Laser Pulses R.Th. Zinhtok, S. Witte, W. Ubachs, W. Hogervorst and K.S. E. Eikema Spectroscopic Determination of the Boltzmann Constant: First Results C. Daussy, S. Briaudeau, M. Guinet, A . Amy-Klein, Y. Hermier, Ch.J. Borde' and C. Chardonnet

94

104

Helium 23P Fine Structure Measurement in a Discharge Cell T. Zelevinsky, D. Farkas and G. Gabrielse

112

Fundamentals and Applications of Slow and Fast Light R. W. Boyd, N . Lepeshkin, A . Schweinsberg, P. Zerom, G. Piredda, Z. Shi and H. Shin

123

Towards a New Measurement of the Electron's Electric Dipole Moment J.J. Hudson, P. C. Condylis, H. T. Ashworth, M.R. Tarbutt, B.E. Sauer and E. A. Hinds

129

Symposium on Cold Atoms and Molecules BEC - The First 10 Years C.E. Wieman Observation of Bose-Einstein Condensation in a Gas of Chromium Atoms A. Griesmaier, J. Stuhler and T. Pfau Disordered Complex Systems Using Cold Gases and Trapped Ions A . Sen, U.Sen, M. Lewenstein, V. Ahufinger, M. Pons and A. Sanpem Correlations and Collective Modes in Fermions on Lattices P.D. Drummond, J. F. Corney and X.- J . Liu and H. Hu

139

149

158

167

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Cooling and Trapping in Cavity Quantum Electrodynamics G. Rempe, M. Hijlkema, A. Kuhn, P. Maunz, K. Muw, S. Nussmann, P. W.H. Pinkse, T. Puppe, I. Schuster, N. Syassen and B. Weber

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Controlling Strings of Single napped Atoms A. Rauschenbeutel, W. Alt, I. Dotsenko, L. Forster, M. Khudaverdyan, Y. Miroshnychenko, D. Schrader and D. Meschede

189

Bose-Einstein Condensates Studied with a Linear Accelerator Ch. Buggle, J. Leonard, W. von Klitzing and J. T.M. Walraven

199

Electric Field Spectroscopy of Ultracold Polar Molecular Dimers J.L. Bohn and C. Ticknor Optical Production of Ultracold Polar Molecules D. DeMille, J.M. Sage, S. Sainis and T. Bergeman Thermodynamics and Mechanical Properties of a Strongly-Interacting Fermi Gas J.E. Thomas, J. Kinast and A. Turlapov Coherent Spin Oscillations Driven by Collisions in an Optical Lattice A . Widera, F. Gerbier, S. Folling, 0. Mandel, T. Gericke and I. Bloch Inhibition of Transport of a Bose-Einstein Condensate in a Random Potential J.A. Retter, A.F. Vdron, D. Climent, M. Hugbart, P. Bouyer, L. Sanchez-Palencia, D. Gangardt, G. V. Shlyapnikov and A. Aspect A Continuous Raman Output-Coupler for an Atom Laser N.P. Robins, A.K. Morrison, C. Figl, M. Jeppesen and J.D.Close

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213

223

239

248

256

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Interactions in an Ultracold Gas of Rydberg Atoms M. Weidemuller, M. Reetz-Lamour, T. Amthor, J. Deiglmayr, K. Singer and L.G. Marcassa Bose-Einstein Condensates on a Permanent Magnetic Film Atom Chip B.V. Hall, S. Whitlock, F. Scharnberg, P . Hannaford and A. Sidorov Fermionic Atoms with Tunable Interactions in a 3D Optical Lattice T. Stoferle, H. Moritz, C. Schori, K.J. Gunter, M. Kohl and T. Esslinger Sympathetic Cooling of Fermionic Lithium via a Bosonic Rubidium Gas S. Gunther, C. Silber, C. Marzolc, B. Deh, Ph. W. Gourteille and C. Zimmemnann The Hanbury Brown Twiss Effect for Atoms C.I. Westbrook, M. Schellelcens, A. Perrin, R. Hoppeler, J. Viana Gomes, D. Boiron and A . Aspect Loading of Selected Sites in an Optical Lattice Using Light-Shift Engineering P.F. Grifin, K.J. Weatherill, S.G. Macleod, R.M. Potvliege and C.S. Adams

264

275

283

291

299

307

Atomic Clocks A Strontium Ion Optical Frequency Standard with Hz-Level Uncertainty H.S. Margolis, G.P. Barwood, G. Huang, H.A. Klein, S.N. Lea, K. Hosalca, K . Szymaniec and P . Gill Simulate Ion Traps with Neutral Atoms: Stark Atom Chip and Optical Lattice Clock H. Katori, M. Takamoto, H. Hachisu, T. Kishimoto, R. Higashi and F.-L. Hong

317

327

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337

Microfabricated Atomic Clocks and Magnetometers S. Knappe, P.D.D. Schwindt, V. Gerginov, V. Shah, H. G. Robinson, L. Hollberg and J. Kitching

Quantum Control and Quantum Information Attosecond Physics: Controlling and Tracking Electron Dynamics on an Atomic Time Scale R. Kienberger and F. Krausz

349

Managing Continuous Variables for Single Photons L. Zhang, LA. Walmsley, C. Silberhorn, A.B. U'ren, and K. Banaszek

360

Giant Atoms for Explorations of the Mesoscopic World J.M. Raimond, T. Meunier, P. Bertet, S. Gleyzes, P. Maioli, A. Auffeves, G. Nogues, M. Brune and S. Haroche

371

Entanglement of Trapped Ions C. Becher, J. Benhelm, D. Chek-Al-Kar, M. Chwalla, W. Dur, 0. Guhne, H. Haffner, W. Hansel, T. Korber, A . Kreuter, G.P.T. Lancaster, T.Monz, E.S. Phillips, U.D Rapol, M. Riebe, C.F. ROOS,C. RUSSO, F. Schmidt-Kaler and R. Blatt

381

'

Quantum Control, Quantum Information Processing, and Quantum-Limited Metrology with Trapped Ions D.J. Wineland, D. Leibfned, M.D. Barrett, A. Ben-Kish, J. C. Bergquist, R.B. Blakestad, J. J. Bollinger, J. Bm'tton, J. Chiaverini, B. Demarco, D. Hume, W.M. Itano, M. Jensen, J.D. Jost, E. Knill, J. Koelemeij, C. Langer, W. Oskay, R. Ozeri, R. Reichle, T. Rosenband, T. Schaetz, P.O. Schmidt and S. Seidelin The Atomic Spirograph: Atomic Wave F'unction and Laser Pulse Shape Measurements from Coherent Transients B. Chatel, A. Monmayrant and B. Girard

393

403

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Controlled Photon Emission and Raman Transition Experiments with a Single Trapped Atom M.P.A. Jones, B. Darquig, J. Beugnon, J. Dingjan, S. Bergamini, Y. Sortais, G. Messin, A. Browaeys and P. Grangier

413

Ion Trap Networking: Cold, Fast and Small D.L. Moehring, M. Acton, B.B. Blinov, K.-A. Brickman, L. Deslauriers, P.C. Haljan, W.K. Hensinger, D. Hucul, R.N. Kohn Jr., P. J. Lee, M. J. Madsen, P. Maunz, S. Olmschenk, D. Stick, M. Yeo, C. Monroe and J. A. Rabchuk

421

Author Index

429

High Precision Measurements

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IMPROVING LASER COHERENCE JOHN L HALL*, MARK NOTCUTT, AND JUN YE+ JILA, University of Colorado and National Institute of Standards and Technology Boulder, CO 80309-0440 USA

The convenient approximation of a real laser field by a Coherent State is again a relevant topic of interest, as laser spectroscopy scenarios are being developed in which remarkably long atomic lifetimes and extended interaction times (-100 s) can be enjoyed. Years ago, appropriate locking techniques were shown to allow precise locking of a laser field to a cavity, even in the milliHz domain, but lab vibrations modulated the cavity length and so the obtained optical frequency. Methods such as mechanical isolation (on a heroic scale) or active anti-vibration approaches are sufficiently productive such that, by now several groups have developed visible optical sources with -Hz linewidths. Still, linewidths in the 100 milliHz domain have seemed very challenging - all the margins have been used up. We discuss mounting systems for an optical reference cavity, particularly an improved one based on implementing vertical symmetry, which provides dramatic reduction in the vibration sensitivity and can yield sub-Hz linewidths on an ordinary optical table in an ordinary lab. Interesting and commanding new issues - such as temporally-dependent spatial structure of the EO-modulated probe beam, and thermallygenerated mechanical position noise - are found to dominate the laser phase errors in the sub-Hz linewidth domain. The theoretical scaling - and the spectral character - of this thermal noise motion of the cavity mirror surfaces have been studied and confirmed experimentally, showing an -1 ~ 1 0 ' m/d(Hz) '~ thermal noise amplitude at 1 Hz, with a I/$ amplitude spectral density, with f being the Fourier frequency of this noise process. For effective temperature stabilization, multi-point thermal control and dual thermal shells provide stable operation near the ULE thermally-stationary point. Spectral filtering in the optical and vacuum paths is critically important to prevent ambient thermal radiation from entering the inner shell. The observed frequency drift-rate of -0.05 H d s is not yet ideally stable, but it appears possible to compensate drift accurately enough to allow 1 radian coherence times to approach -100 s if other problems such as the thermal noise can be adequately suppressed. Recent JILA spectra of lattice-trapped cold Sr atoms show an excellent prospect for ultrahigh resolution spectroscopy and highly stable optical atomic clocks and make us anxious to perfect improved phase-stable laser sources for the 'So - 3P0doubly-forbidden transition at 698 nm. These laser developments are aided by optical comb techniques, allowing useful phase comparison of several prototype stable laser sources, despite their various different wavelengths.

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*

Formerly also with Quantum Physics Division, National Institute of Standards and Technology. Also with Quantum Physics Division, National Institute of Standards and Technology.

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1.1. Optical Frequency Coherence - In the Beginning Initially, laser coherence was quantified in “Gillette” units. It was not the razor blade’s sharp edges that could help define a position, but rather their darkened color and convenient thinness such that a diffraction-limited Ruby laser of about % Joule was already able to punch a hole through a blade’s thickness. Spatially less-coherent lasers would require about 1 Joule/Gillette. Regrettably, every Qswitched laser pulse was different: “Doing science” with early laser tools was indeed exciting! Nonlinear physics was clearly visible, but hard to study in a fully controlled way. A JILA study of optically-induced level shifts in potassium vapor showed rates scaling like Angstroms per MW/cm2 , but with frustrating irreproducibility due to uncontrolled spectral and geometrical structure of the ruby laser’s output. We chose to go fishing in calmer waters, where kHz shifts per milliWatt/cm* would be the scale. It would be - and still is - done with cw lasers and can be highly quantitative. And now frequency comb technology makes it almost easy! The early cw gas lasers had relatively narrow emission linewidths, since their gain media were stable and their low gain required use of low loss cavities. The milliWatt output power with such a cavity gave Schawlow-Townes phasediffusion linewidths calculated to be in the milliHz domain. Of course it was quickly understood that vibrations were influencing the length of the laser cavity in a profound way, so the precision level was more like kHz or even MHz, rather than the expected milliHz. So early in the laser’s history anti-vibration activities began in earnest, using remote facilities such as the old wine cellar at MIT and inactive gold mines, such as the Poorman’s Relief Gold Mine, in the mountains about 20 minutes west of Boulder [ 11. But the approach of “hiding away” was destined to be limiting: By the mid 1970’s interesting new tunable dye lasers could achieve almost any wavelength, but the electric power and cooling water required for their Argon-ion pump lasers would be enough to support the good life in a whole neighborhood. Already in 1973, the first year of cw dye lasers, opto-electronic servo-control methods were used to make even a dye laser achieve a narrow linewidth [ 2 ] . The concept was to use a reference Fabry-Perot cavity of adequate finesse, such that the shot-noise limit in finding its resonance center could be adequately small. We were thrilled to achieve fast linewidths in the order of 1 kHz, unfortunately contaminated by vibration-induced frequency noise perhaps one or two orders larger. A wonderful collaboration between spectroscopists and Gravitational Wave Detector communities led to the now-standard scheme [3] of using RF modulation sidebands as the phase reference for measuring the

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phase-shift of the leakage field emerging again from the input mirror of the reference cavity. This RF-based discriminator, sometimes referred to as the Pound-Drever-Hall technique, has many wonderful properties as an optical phase/frequency discriminator, as did its conceptual predecessor that had been used by Robert Pound more than a quarter-century earlier for cavity stabilization of microwave oscillators in Radar systems. Locking to optical cavity modes at the Hz level was soon demonstrated [4] .

1.2. Stable Lasers are Needed! But the attained laser frequency stability soon grew to be inadequate as ion trapping and cold atom magneto-optics traps (MOT’S) became widespread. As a community, we arrived at the funny place that we could manage vacuums in the 10.’’ torr range, trap ions for hours, cool them to nearly the ground state of vibration in their trapping potential - serious high-tech achievements. But almost no group had the sub-100 Hz laser needed to probe such delicious spectroscopic samples. Of course, major labs could and did attack the laser vibrational incoherence problem with great energy - and money - by building elaborate and expensive vibration isolated rooms. Pioneering work in this regard was done by Jim Bergquist and his colleagues in NIST Boulder, and their published 0.22 Hz linewidth [5] of a laser beatnote remains the world record. Another anti-vibration approach would be active anti-seismic servo systems: eventually our JILA labs were able to record 60 dB reductions of the vertical accelerations of an optical table around 1 Hz. But mode coupling is an important issue when the MultiInput-MultiOutput (MIMO) control system has been carefully tuned for orthogonalization of the responses. A screwdriver for Allen cap screws, carelessly left on the table, could cost a factor 2 in the noise level at the cavity, due to changed coupling with a less-well-cooled tilt mode. A better active anti-vibration campaign has been pursued by MPQ [6] and by NPL [7], using recently-available 6 Degrees-of-Freedom active vibrationreducing modules 181, which offer more than 20 dB improvement over the band 3-100 Hz. Starting from a quiet environment, or if “stacking” such attenuators, the ultimate limit is set by the acceleration sensing noise, -100 ng/d(Hz) . As an alternative, passive isolating platforms are available [9] with -% Hz resonance and >30 dB isolation above 3 Hz. The Q-5 resonance leads to a l/e decay time "

Date (2005)

Figure 1. The measured 'So - 3P0 transition frequency versus (a) lattice intensity ( l o = 35 kW/cm2), (b) atomic density, and (c) magnetic field. (d) JILA measurements over a 3 month period, with each data point representing an averaged daily frequency measurement. The results reported in this work (lower bars) and in Ref. 15 (upper bars) are both shown with the total (outer box) and statistical (inner shaded area) errors.

dependent effects by either varying the total 698 nm light intensity incident on the atoms or changing the relative carrier-sideband amplitude through the EO modulation index. We determine each of these effects within 2 Hz uncertainty. The frequency reference used for the optical measurement is a hydrogen maser directly calibrated by the NIST F1 Cs fountain clock, available to us by an optical fiber link from NIST to J1LA.24 The approximate 14 m lower elevation of our JILA Sr experiment relative to the NIST fountain clock introduced a < 1 Hz gravitational shift. Including all systematic effects discussed here and others with much smaller magnitudes, the total uncertainty (added in quadrature) is 20 Hz.The final number we report for the s7Sr '5'0 - 3P0 transition frequency is 429, 228, 004, 229, 867 f 20(sys) & 2.8(stat) Hz.We note this result disagrees by three standard deviations with that of Ref. 15 measured with a GPS-calibrated referen~e.~

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3. United Time Frequency Spectroscopy Recent work has demonstrated that an optical frequency comb is a highly efficient tool for precise studies of atomic s t r ~ c t u r e . ~A~ 'phase-stabilized femtosecond comb has been used as an effective tool t o perform direct spectroscopy of one- and two-photon transitions in ultracold Rb atoms, permitting high-resolution spectroscopy of all atomic transitions anywhere within the comb bandwidth. By measuring the previously unmeasured absolute frequency of the 5S112 -+ 7S1/2 two-photon transitions in 87Rb,we show that prior knowledge of atomic transition frequencies is not essential for this technique to work, and indicate that it can be applied in a broad context. Additionally, this approach enables precise studies of time domain dynamics, coherent accumulation and interference, and quantum contr01.~ In this contribution we discuss only one particular aspect of direct frequency comb spectroscopy (DFCS) , namely coherent accumulation and population transfer effects in multilevel systems probed by multiple comb components. The importance of understanding these dynamics for precision spectroscopy is highlighted by the measurement of the 5P states. The 5s1/2 + 5P1/2,3p transitions are probed both directly via one-photon DFCS (Fig. 2(a) left panel) and indirectly via two-photon DFCS of the 5S5D two-photon transition that enjoys resonant enhancement when comb components are scanned through the intermediate 5P states (Fig. 2(a) right panel). Comparison between these two approaches clearly demonstrate the importance of including the dynamic population changes arising from pulse-accumulated population transfer in this indirect measurement. A density-matrix based theoretical model describing the interaction of the femtosecond comb with atoms accounts for detailed dynamics of population transfer among the atomic states involved in transitions within the comb bandwidth. Impulsive optical excitation followed by free evolution and decay is used to model the interaction with each pulse in the train. The density matrix equations are solved to a fourth order perturbative expansion in the electric field and an iterative numerical scheme is employed to obtain the state of the atomic system after an arbitrary number of p ~ l s e s .This ~ ~ >model ~ is applied t o accurately predict the coherent population accumulation in the relatively long-lived 5D or 7s states, followed by incoherent optical pumping. Especially important for the indirect 5P measurements is the incoherent optical pumping to the ground state hyperfine levels, which depends critically on the 5P state detunings. The one-photon DFCS employs a single optical comb component and

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5s F=t one-photon DFCS

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5P State Detuning ( M e )

0 -20 -10 0 10 20 5P State Detunlng (MHt)

Figure 2. (a) Schematic of one- and two-photon DFCS, used for measuring singlephoton transition frequencies. (b) Lineshape of the 551,~F=2 + 5P112 F’=2 transition obtained from a scan of fo for a fixed value of f r , by one-photon DFCS. (c) Raw counts for the same lineshapes.a in (b) by two-photon DFCS, along with a visual guide for the data. (d) Normalized lineshape corresponding to the raw data in (c), obtained by using results from theory simulations accounting for optical pumping effects.

makes radiative detections directly from the 5P states (Fig. 2(a) left panel). Frequency scans are carried out by stepping continuously the carrier-envelope offset frequency f, while keeping the repetition frequency fr fixed at a convenient value. This one-photon transition lineshape is shown in Fig. 2(b). The absolute optical frequency of the 5s1/2 F=2 4 5P1p F’=2 transition in the D1 manifold is determined.* For the corresponding two-photon DFCS experiment we map the 5S1/2 F=2 4 5P1/2 F’=2 -+ 5&/2 F”=3 two-photon transition. we use a set of different pairs of fr and fo specifically chosen to have varying detunings from the 5P state for each data point shown in Fig. 2(c), while at the same time satisfying the 5S-5D two-photon resonance. The lineshape in Fig. 2(c) is retrieved by detecting the 420 nm fluorescence signal originating from 5D as a function

21

of 5P state detuning produced by these (f,.,fo) pair selections, along with a visual guide for the data. The apparent linewidth is significantly broader than that associated with the 5P state. The pairs of fv and fo used to obtain each point in Fig. 2(c) lead to substantially different detunings of the other 5P states and subsequently, varying optical pumping to the F=l ground state. Indeed, the theory model applied to the actual experiment conditions predicts significantly different ground state population transfer dynamics. The asymptotic values of the F=2 ground-state population are not the same for symmetric detunings from the intermediate state. Figure 2(d) presents the Lorentzian lineshape resulting from the normalization of the raw data shown in Fig. 2(c) with respect to theoretical values of the ground state population. After implementing this normalization, the transition lineshape resumes the same linewidth as shown in Fig. 2(b) and the transition frequency measured by the two-photon DFCS agrees with the corresponding one-photon DFCS result within the error bar.8

4.

XUV frequency comb

Using a femtosecond laser coupled to a passive optical cavity, we can now address some of the issues associated with traditional high harmonic generation, namely average power, system size and cost, and spectral resolution. We demonstrate coherent frequency combs in the XUV spectral region from the generation of high-harmonics of the laser without any active amplification or decimation of the repetition frequency. The output from the laser is stabilized to a femtosecond enhancement cavity with a gas jet at the intracavity focus. The high-peak power of the intracavity pulse enables efficient HHG (Fig. 3). Since little of the fundamental pulse energy is converted, a fs enhancement cavity is ideally suited for HHG as the driving pulse is continually "recycled" after each pass through the gas target. HHG at high repetition rates opens the door for dramatic improvements in average power conversion efficiency. In addition, system cost and size are greatly simplified. Optical-heterodyne-based measurements reveal that the coherent frequency comb structure of the original laser is fully preserved in the high-harmonic generation process. These results lead the way for precision frequency metrology at extreme wavelengths and permit efficient HHG using only a standard laser oscillator. To sufficiently build up the intracavity pulse energy, the passive optical cavity needs to incorporate a number of important characteristics: i) a high finesse,26ii) low round-trip group-delay dispersion to allow ultrashort pulses

22

to be coupled into and stored inside the cavity,27 and iii) a robust servo to stabilize the two degrees of freedom of the incident pulse train to the corresponding cavity resonance modes.2s A standard modelocked femtosecond Ti:Sapphire laser with 100 MHz fr, 60-fs pulse duration, and 8 nJ pulse energy is used. The pulse train from the laser passes through a prism-based compressor before incident on the passive optical cavity. To investigate the peak intensity that can be obtained with this method, an empty fs enhancement cavity is initially characterized. With the laser locked to the cavity, the transmitted spectrum shows the effect of the residual cavity dispersion limiting the intracavity pulse bandwidth. Measurement of the transmitted pulse verifies that the pulse is nearly Fourier-transform-limited with a duration of 60 fs. The duration of the pulse is minimized inside the cavity by adjusting the compressor while measuring the current produced from a plasma in Xe at the focus. The intracavity pulse energy is enhanced up to 4.8 pJ for these short pulses, approximately a 600-fold increase from the incident pulse energy of 8 nJ. Based on these measurements we estimate a peak intracavity intensity of > 3 x 1013 W/cm2 is obtained at the intracavity focus. To couple the HHG light out of the cavity, a 0.7 mm-thick sapphire plate is placed at Brewster's angle (for the IR) inside the cavity. The Fresnel reflection coefficient of the intracavity plate rises at shorter wavelengths, with a reflectivity of 510% between 40 and 100 nm. The gas target is confined within a thin, hollow brass cylinder, with a 150 pm hole to allow the intracavity pulse to pass through (Fig. 3). The diffracted pattern of the HHG light, obtained with a MgF2-coated aluminum grating, demonstrates that at least the 9th order harmonic has so far been generated. The average power of the 3rd harmonic light generated inside the cavity reaches nearly 10 pW. The corresponding intracavity single-shot efficiency (,lo-') is comparable to traditional femtosecond-amplifier-based systems at similar intensity levels. This demonstrates the dramatic increase in high-harmonic power that can be accessed using a high repetition rate (100 MHz). Clearly there is significant potential to further improve this efficiency and produce harmonics far into the XUV simply by increasing the incident pulse energy, easily allowing access to intensities > 1014 W/cm2 at high repetition rates. In order to utilize the precision of the fs comb in the XUV spectral region, it must be verified that no detrimental phase/frequency fluctuations in the high-harmonic generation process exist. To test this we allow the original comb to drive two independent nonlinear processes, one through well characterized bound electronic nonlinearities based on second harmonic

23

Fundamental comb

.....

Does HHG preserve the comb structure Conventional nonlinear optics frequency

Figure 3. (a) Schematic setup for coherent heterodyne detection between HHG in Xe gas and bound optical nonlinearities in a BBO crystal. (b) Measurement of the optical beat signal between the two separately generated 3rd harmonics, indicating a coherent linewidth of 1 He, limited by the analyzer's resolution bandwidth. (c) Schematic setup of intracavity high-harmonic generation. The incident pulse train is stabilized to a high finesse cavity, enhancing pulse energy nearly three orders of magnitude while maintaining a high repetition frequency. A gas target at the cavity focus enables phase-coherent highharmonic generation, resulting in a phase-stable frequency comb in the XUV spectral region. The photo inset shows the actual spatial mode profile of the 3rd harmonic coupled out of the cavity.

and sum frequency generation via two BBO optical crystals and the other through the HHG process (Fig. 3). Two sets of the frequency combs at 266 nm that represent the third harmonic of the fundamental IR comb are then brought together in a Mach-Zehnder interferometer geometry for beat detection, after these two separate pulse trains are temporally overlapped. A 90 MHz acousto-optic modulator is inserted in one of the interferometer arms so that the beat detection is shifted to a convenient non-zero frequency. Pairs of corresponding comb components from each spectrum produce a coherent optical beat detected by a photomultiplier. The radio frequency spectrum of the beat note shows the clear presence of the comb structure in the UV (Fig. 3). The resolution bandwidth-limited 1 Hz beat signal demonstrates that the full spectral resolution and temporal coherence of the original near-IR comb has been faithfully transferred to higher

24

harmonics in this process.

Acknowledgments

We gratefully acknowledge t h e contributions of T. Loftus, R. Santra, E. Arimondo, and C. Greene o n Sr experiments, D. Felinto, F. Cruz, and J. Lawall on united time-frequency spectroscopy, S. Diddams, T. Parker, and L. Hollberg for help in t h e delivery of NIST Cs fountain-calibrated maser signals t o JILA, and J. Hall for the encouragement and support of our work. References 1. T. H. Loftus et al., Phys. Rev. Lett. 93, 073003 (2004); ibid. Phys. Rev. A 70, 063413 (2004). 2. X. Xu et al., Phys. Rev. Lett. 90, 193002 (2003). 3. T. Ido et al., Phys. Rev. Lett. 94, 153001 (2005). 4. A.D. Ludlow et al., arXiv:physics/0508041 (2005). 5. R. Santra et al., Phys. Rev. Lett. 94, 173002 (2005). 6. M. Notcutt et al., Opt. Lett. 30,1815 (2005). 7. A. Marian et al., Science 306, 2063 (2004). 8. A. Marian et al., Phys. Rev. Lett. 95, 023001 (2005). 9. R.J. Jones et al., Phys. Rev. Lett. 94, 193201 (2005). 10. C. Gohle et al., Nature 436, 234 (2005). 11. S. Cavalieri et al., Phys. Rev. Lett., 89, 133002 (2002). 12. S. Witte et al., Science, 307, 400 (2005). 13. C. W. Oates, E. A. Curtis, and L. Hollberg, Opt. Lett. 25 1603 (2000). 14. U. Sterr et al., Comptes Rendus Physique 5 , 845 (2004). 15. M. Takamoto et al., Nature 435, 321 (2005). 16. S. G. Porsev and A. Derevianko, Phys. Rev. A 69, 042506 (2004). 17. R. Santra et al., Phys. Rev. A 69, 042510 (2004). 18. T. Ido and H. Katori, Phys. Rev. Lett. 91, 053001 (2003). 19. H. Katori, T. Ido, and M. Kuwata-Gonokami, J . Phys. SOC.Japan 68, 2479 (1999). 20. H. J. Kimble et al., Proceeings of ICOLS’99 (XIV Inter. Conf. Laser Spectroscopy) Eds. R. Blatt, J. Eschner, D. Leibfried, F. Schmidt-Kaler, World Scientific, Singapore 1999, pp. 80. 21. I. Courtillot et al., Phys. Rev. A 68, 030501(R) (2003). 22. T. Hong et al., physics/O504216 (2005). 23. C. W. Hoyt et al., physics/0503240 (2005). 24. J. Ye et al., J. Opt. SOC. Am. B 20, 1459 (2003). 25. D. Felinto, L. H. Acioli, and S. S. Vianna, Phys. Rev. A 70,043403 (2004). 26. R.J. Jones and J. Ye, Opt. Lett. 27, 1848 (2002). 27. M.J. Thorpe et al., Opt. Express 13, 882 (2005). 28. R.J. Jones, I. Thomann, and J. Ye, Phys. Rev. A 69, 051803(R) (2004).

LONG BASE

- LINE GRAVITATIONAL WAVE DETECTORS STATUS AND DEVELOPMENTS JAMES HOUGH

Institute for Gravitational Research, University of Glasgow, Glasgow GI2 SQQ, UK Experiments aimed at searching for gravitational waves from astrophysical sources have been under development for the last 40 years, but only now are sensitivities reaching the level where there is a real possibility of detections being made within the next few years by the long baseline detectors LIGO, VIRGO, GEO 600 and TAMA 300. There are a number of limitations to performance of these detectors and to their proposed upgrades, one of the most significant being thermal noise associated with mechanical losses in the mirror coatings.

1. Introduction For many years physicists have engaged in the exacting experimental challenge of searching for gravitational waves. Predicted by Einstein's General Theory of Relativity to be produced by the acceleration of mass [l], they have remained elusive. However observation of gravitational wave signals from the heart of violent astronomical events will open up a new area of astronomy/astrophysics with the field holding considerable promise for increasing our understanding of the relativistic nature of the Universe. Gravitational waves are quadrupole in nature, having transverse orthogonal polarizations at 45 degrees to each other, and induce differential strains along perpendicular paths in space. In principle these lead to small apparent displacements of test mass mirrors hung as pendulums a distance apart. Detection relies on sensing the change in distance between these mirrors, and thus long baseline detectors on earth comprise multiple mirrors, each of mass of the order of 10 kg, hung to form two perpendicular arms as in figure 1, with optical interferometry used to sense fluctuations in the relative length of the arms [2]. Signals are predicted to be extremely small. In the frequency range 10 Hz to lo4 Hz accessible to gravitational wave detectors on earth, strain signals typically less than lo-*' over bandwidths of a few hundred Hz [3] have to be searched for against a background of noise from a number of sources [4],

25

26

making gravitational wave detection a challenge in ultra-sensitive displacement measurement.

2.

Long Baseline Interferometric Detectors on Earth

The idea of using laser interferometry for gravitational wave detection was first proposed in 1962 [ 5 ] but for implementation had been awaiting the availability of relevant laser and optical technology. Indeed the first laser interferometric prototype was built in the early 1970s at the Hughes Aircraft Laboratories in Malibu, although the sensitivity was limited by the short distance between the masses, mechanical disturbances, and the low power of the helium neon laser used [6]. The concept is very attractive in that it offers the possibility of very high sensitivities over a wide range of frequency, and an excellent analysis of the issues that affect sensitivity is given in [7].

Y

+

Suspension0

time

X

Figure 1. Schematic diagram of how gravitational waves of both polarisations interact with a ring of matter. The ‘quadrupole’ nature of the interaction can be clearly seen, and if the mirrors of the Michelson Interferometer on the right lie on the ring with the beamsplitter in the middle, the relative lengths of the two arms will change and thus there will be a change in the output light intensity.

This technique is based on the Michelson interferometer and is particularly suited to the detection of gravitational waves as they have a quadrupole nature (figure 1). Waves propagating perpendicular to the plane of the interferometer will result in one arm of the interferometer being increased in length while the other arm is decreased and vice versa. The induced change in the length of the interferometer arms results in a small change in the intensity of the light observed at the interferometer output.

27

With the increasing availability of argon-ion lasers and then neodymium YAG lasers with the capability of producing watts of single frequency light, a number of prototype detectors at MPQ, Glasgow, Caltech, MIT and Tokyo 18-16] were constructed leading to the funding and building of the current generation of long baseline instruments - LIGO, VIRGO, GEO 600 and TAMA 300 [ 17-20] - which will be described in a later section. In order to observe a full range of sources and initiate gravitational wave astronomy a sensitivity or noise performance in strain of below 10-23/dHzhas to be achieved over most of the proposed operating range from 10 Hz to a few kHz. For an Earth-based detector the distance between the test masses is limited to a few km by geographical and cost factors. If we assume an arm length of 3 to 4 km, detecting a strain in space of the above level implies measuring a residual motion of each of the test masses of around mldHz. This sets a formidable goal for the optical detection system at the output of the interferometer.

2.1. Main noise sources There are several main noise sources that limit the sensitivity of ground-based interferometric gravitational wave detectors [7]. Fundamentally it should be possible to build interferometric systems to monitor strains in space which reach or even bypass the Standard Quantum Limit (SQL), i.e. the limit set by the Heisenberg Uncertainty Principle [2 1,221. Indeed the proposed performance for the next generation of detectors is close to this limit in the frequency range around a few hundred Hz. However there are other practical issues that must be considered. Fluctuating gravitational gradients pose one limitation to the interferometer sensitivity achievable at low frequencies. While schemes to monitor such gradients and cancel out their effects on the interferometers have been proposed [23] these are still far away from implementation. It is the level of gravity gradient noise which dictates that experiments to look for gravitational wave signals below 10 Hz or so have to be carried out in space [24, 251. In general [26, 271, for the practical building of ground based detectors the most important limitations to sensitivity result from the effects of (1) seismic and other ground-borne mechanical noise, (2) thermal noise associated with the test masses and their suspensions 128, 291, (3) shot noise in the photocurrent from the photodiode which detects the interference changes, and (4) radiation pressure recoil effects on the interferometer mirrors, these last two being

28

intimately related with quantum limits to performance and leading to the SQL [21, 221.

2.2. Current situation with interferometric detectors The American LIGO project which sprang from the prototypes at MIT and Caltech comprises two detector systems with arms of 4 km length, one in Hanford, Washington State, and one in Livingston, Louisiana. One half length, 2 km, interferometer has also been built inside the same evacuated enclosure at Hanford. A birds-eye view of the Hanford site showing the central building and the directions of the two arms is shown in figure 2. Construction of LIGO began in 1996 and progress has been outstanding with one of the LIGO detectors - the Hanford 4 km instrument - currently being almost at its design sensitivity over much of its frequency range. Many research groups from the USA and other parts of the world are actively involved in the analysis of data from LIGO and in research towards future upgrades as part of the LIGO Scientific Collaboration (LSC).

Figure 2. Birds eye view of LIGO (Hanford). Image courtesy of the LIGO Scientific Collaboration

The FrenchLtalian VIRGO detector of 3 km arm length at Cascina near Pisa is designed to have particularly good low frequency performance, down to 10 Hz and is close to full operation. The Japanese TAMA 300 detector, which has arms of length 300 m, is operating at the National Astronomical Observatory near Tokyo. All the systems mentioned above are designed to use resonant cavities in the arms of the detectors to enhance sensitivity and use standard wire sling techniques for suspending the test masses. However the GermanBritish detector, GEO 600, is somewhat different. It makes use of a four-pass-delay-

29

line system with a signal enhancement system known as signal recycling [30, 311, and utilises fused silica suspensions of very low mechanical loss for the test masses to help reduce thermal noise [32]. GEO is expected to reach a sensitivity at frequencies above a few hundred Hz close to those of VIRGO and LIGO when they are in initial operation. GEO is now fully built and its sensitivity is being continuously improved. Currently it is within a factor of ten of design sensitivity over much of its frequency range. Four science runs, ranging from 17 to 70 days in length, have so far been carried out with these new interferometric detectors. All have involved the LIGO detectors, three have involved the GEO detector and two the TAMA instrument. The bar detector Allegro in Louisiana has also taken part in certain of these runs. From the first science run, upper limit results have been set on the signals from a number of potential sources such as pulsars, coalescing compact binary stars, as well as on burst events and the level of a stochastic background [33 - 361. Results from the second run are currently being published [37], and those from the third and fourth runs are being analysed. During the next few years we can expect to see a series of increasingly sensitive searches for gravitational wave signals at a sensitivity level of approximately for the steady signals for millisecond pulses or close to from pulsars, to take two examples. This latter level is equivalent to a neutron star having an ellipticity of which could quite possibly occur. Thus the detection of gravitational waves from pulsars in the near term is a real possibility. Further, the recent discovery of another compact binary system in the galaxy - the double pulsar 50737-3039 - has improved the statistics for the expected rate of binary coalescences by a significant factor, implying that the most probable rate of binary neutron star coalescences detectable by the LIGO system now lies between one per 10 years and 1 per six hundred years [38]. Many people expect the rate of binary black hole coalescences to be even higher. However, detection at the level of sensitivity of the initial detectors is in no way guaranteed; thus improvement of the order of a factor of 10 in sensitivity of the current interferometric detectors is essential to allow compact binary coalescences to be detected at a useful rate. Indeed, plans for Advanced LIGO, an upgrade of the present LIGO interferometers, are already mature and the project has recently been approved by the National Science Board in the USA. Plans are also well advanced for an underground detector (LCGT) with cooled test masses to be built in Japan [39]. The proposed design for Advanced LIGO has 40 kg silica test masses, suspended by fused silica fibers or ribbons, along with an improved seismic isolation system, increased laser power, 200 W, and

-

-

30

signal recycling [40]. The upgrade is now expected to commence in 2009 and it is exciting to note that the most probable rate of detectable binary neutron star coalescences is now expected to be in the range of 10 to 500 per year [38]. For GEO a different upgrade strategy is being adopted. A proposed upgrade will be targeted at the observation of the oscillations of neutron stars resulting from quakes in pulsars or magnetars, situations where there is an external trigger from other branches of astronomy, and detector improvement will be in the area of enhancing narrow-band sensitivity around a few kHz. Potential upgrades to the VIRGO detector are currently being discussed.

3. Thermal noise Thermal noise associated with the mirror masses and the last stage of their suspensions is, and is likely to continue to be, one of the most significant noise sources at the low frequency end of the operating range of long baseline gravitational wave detectors [28]. Current suspension designs have been based predominantly on modeling the behaviour of the resonant modes of test masses and suspensions as damped harmonic oscillators. The power spectral density of thermal displacement noise, S&), associated with a mode of resonant frequency& can then simply be written as [26]:

where &f) is the mechanical dissipation or loss factor of the oscillator of mass m, at temperature T . Inspection of the equation above indicates that at the resonant frequency fo the thermal displacement is very large for &f) > 1 where Q = l/&f). However at frequencies far from resonance the thermal displacement is proportional to d&f). The test masses and their suspensions in interferometric detections have thus been fabricated from materials of low mechanical dissipation and designed to have, where possible, resonant modes outwith the frequency band of interest for gravitational wave detection. In particular the longitudinal pendulum mode of a suspension is typically 1 Hz and the internal resonant modes of the test mass mirrors are tens of kHz. However the transverse ‘violin’ modes of the fibres suspending the test masses are typically hundreds of Hz and thus appear in the detection band. The off-resonance thermal noise of the pendulum and violin modes results from dissipation associated with the flexing of the suspension wires or fibres. High strength carbon steel wires are used in the LIGO, VIRGO and TAMA

-

31

suspensions. In contrast, as mentioned earlier, the GEO 600 test mass suspensions use cylindrical fused silica suspension fibres in the final stage since the intrinsic dissipation of silica is substantially lower than that of steel, 141 441 whilst having comparable tensile strength [451. It is important to note that for both the pendulum and violin modes, the resulting thermal noise is reduced over that expected simply from consideration of the internal dissipation of the fibre material through the fact that part of the potential energy associated with the flexing of each fibre is stored as gravitational potential energy in the Earth’s loss-less gravitational field, thus ‘diluting’ the suspension fibre dissipation [28]. Through careful suspension design the dissipation factors of the in-band violin modes can thus be made low enough that the associated thermal displacement occupies very little of the detector frequency band. Over much of the detector frequency band the off-resonance thermal noise from the test mass mirrors has a more significant impact on detector sensitivities. Levin [46] and others [47 - 501 using calculations based on a direct application of the Fluctuation Dissipation Theorem have developed techniques for calculating the thermal noise even in situations where the losses are not homogeneous. Results agree with the standard modal approach of [51] in the case of homogeneous loss, and give different results for localized loss showing in particular the significance of losses in the ion beam sputtered multilayer dielectric coating on the front of the test masses. However, before considering mirror coatings, attention must first be turned to the choice of a suitable substrate material. In the suspended test masses thermal noise appears in two forms, Brownian thermal noise resulting from the internal friction of the materials forming the test masses, coatings and suspension elements, and thermoelastic thermal noise. The latter results from thermodynamic fluctuations of temperature in the suspension systems, which then effectively couple to displacements predominantly through the expansion coefficient of the materials [52, 291. Both Brownian and thermoelastic dissipation can be of a level significant for interferometric detectors. All current interferometric detectors use fused silica as a mirror substrate material. This choice is a result of fused silica having suitable optical properties in addition to relatively low Brownian dissipation. Work from a number of researchers suggests that a subset of Suprasil fused silica (grades 31 1 and 312) has consistently lower mechanical loss than other available silicas [53, 541, with loss factors lower than having been measured and interpreted as arising from internal frictional losses in both the bulk and surface layers of the samples

1551.

32

An alternative material of current interest for transmissive substrates is sapphire. Sapphire has been demonstrated to have extremely low Brownian dissipation at frequencies above the gravitational wave detection band with dissipation factors as low as 2 x having been measured at room temperature [56]. However as recently pointed out by Braginsky and colleagues [29], the thermo-mechanical properties of sapphire are such that the thermoelastic thermal noise from sapphire test mass substrates can be significantly higher in the gravitational wave detection band than thermal noise from Brownian dissipation. As a result of this and the recent encouraging progress on understanding losses in silica, silica currently forms the baseline choice for the substrate material for the planned Advanced LIGO detector upgrade. As mentioned above dissipation associated with the dielectric mirror coatings added to the test masses is of particular significance for Advanced LIGO and future generations of interferometric detectors. From experiment and calculation, significant sources of coating mechanical dissipation exist in the form of Brownian dissipation intrinsic to the coating materials [57 - 611 and from a form of thermoelastic dissipation in the case where the thermomechanical properties of the coating multi-layers are different from those of the substrate. [62, 631. In both cases the magnitude of the resulting thermal noise depends on the relative properties of the coating and the substrate. Hence for example, the optimal coating for one substrate material may not be optimal for another. Studies of the most commonly used type of coating, formed from alternating multi-layers of ion-beam-sputtered SiOz and TazO5, suggest strongly that the coating dissipation is dominated by the Ta205component of the coating [60, 611. Current research is thus targeted at identifying ways to reduce the dissipation of Ta2O5 or finding an alternative high index material. It is interesting to note that recent work by Numata [64] has shown that coating thermal noise is also limiting the frequency stability of lasers locked to Fabry-Perot cavities and his predicted limitations are in broad agreement with experimental results presented by Hall at this meeting.

-

3.1. The application offlat-topped beams Analysis shows that Brownian noise, thermo-elastic noise and coating noise are all reduced to varying degrees if the radius of the laser beam sensing the test mass mirror motion can be increased (see [65] for a review of this). Simply scaling up the beam size of the Gaussian mode leads to increased optical loss as

33

the tails of the beams are truncated at the edge of the mirrors. However it is possible to use cavities in the arms of the interferometers supporting superGaussian optical modes where the beam intensity profile is significantly more flat-topped but where coupling from a standard input Gaussian mode is relatively efficient [66, 671. As an example, for a given mirror diameter anticipated reductions in thermoelastic thermal noise power spectral density of approximately 3 times are anticipated [66]. Research is underway to determine and study the practicability of the most appropriate mirror curvatures required to support the flat-topped modes [68].

4. Fundamental Issues It should be noted that the signal-recycling concept as currently used in GEO 600 and planned for Advanced LIGO has the potential of allowing measurements below the SQL [69]. The asymmetry introduced by narrowbanding the sensitivity offset on one side of the optical carrier introduces a correlation between the photoelectron shot noise and the effect of the backreaction. In this case quantum noise curves of the type included in figure 3 have been calculated by Buonanno and Chen. At its lowest point the quantum noise is better than would be predicted by the SQL. In principle the quantum noise limited sensitivity at different frequencies may be further improved by using

100

1000

f (Hzl Figure 3. Sensitivity curve, showing optical noise and the SQL, for an Advanced LIGO type detector system. (Buonanno and Chen)

34

squeezed light for illumination of the system and/or by using a long filtering cavity before the detection of the signal out of the system [70, 711. While these techniques for sensitivity enhancement beyond the SQL require losses in all parts of the main optical system to be very low and the quantum efficiencies of the photo-detection systems to be very high, they have real potential for the future and there is a growing experimental community dedicated to applying them to the detection of gravitational waves.

5. TheFuture The next stage forward in interferometric detectors is well defined with the design for Advanced LIGO incorporating silica fibre suspensions, signal recycling and higher power lasers being well advanced. On approximately the same timescale we can expect to see a similar upgrade to VIRGO, the rebuilding of GEO as a detector aiming at high sensitivity in the kHz frequency region and the building of a long-baseline underground detector, LCGT, in Japan. To go beyond this point, however, a number of challenges involving mechanical losses in coatings and thermal loading effects will have to be overcome, the latter possibly requiring the use of non-transmissive optics [72, 731 with materials of high conductivity such as silicon [74]. Research groups in the field are already looking towards the next generation of detectors that will herald the start of gravitational wave astronomy: ground based instruments making full use of squeezed light and techniques to bypass the standard quantum limit, and space-borne detectors such as LISA [25]and its extensions.

Acknowledgments I wish to thank colleagues in the LIGO Scientific Collaboration and the NSF funded LIGO Project for useful discussions, and the University of Glasgow and PPARC in the UK for support.

References 1. A. Einstein, Koeniglich Preussische Akademie der Wissenshaften Berlin, Sitzungsberichte: Erster Halbband, 688, (1916). 2. P.R. Saulson, “Fundamentals of Interferometric Gravitational Wave Detectors”, (World Scientific, Singapore), (1994). 3. L Grishchuk, Astrophysics Update in press, (Springer-Verlag) arXiv:gr-qc/030505 1. 4. J. Hough, S. Rowan and B. Sathyaprakash, J. Phys. B : At. Mol. Opt. Phys. 38 S497, (2005). 5. M.E. Gertsenshtein and V.I. Pustovoit , Sov. Phys. - JETP 16 433, (1962).

35

6. G.E. Moss, L.R. Miller andR.L. Forward, Appl. Opt. 102495b,(1971). 7. R. Weiss, Q. Prog. Rep. Res. Lab. Electron. MIT 105 54,(1972). 8. H. Billing, K. Maischberger, A. Riidiger, R. Schilling, L. Schnupp and W. Winkler, J. Phys. E: Sci. Instrum. 12 1043,(1979). 9. D. Shoemaker, R. Schilling, L. Schnupp, W. Winkler, K. Maischberger and A. Riidiger, Phys. Rev. D 38 423,(1988). 10. R.W.P. Drever, J. Hough, J.R. Pugh, W.A. Edelstein, H. Ward, G.M. Ford, N.A. Robertson, Proc. Roy. SOC.Lond. A368 11, (1979). 11. J. Hough, S. Hoggan, G.A. Kerr, J.B. Mangan, B.J. Meers, G.P. Newton, N.A.

12.

13. 14. 15.

16.

17. 18. 19. 20. 21. 22. 23. 24.

Robertson, H. Ward and R.W.P. Drever, Lecture Notes in Physics 212 (SpringerVerlag) 204,(1984). D. Robertson, E. Momson, J. Hough, S. Killbourn, B.J. Meers, G.P. Newton, K.A. Strain and H. Ward, Rev. Sci. Instrum. 66 4447,(1995). R.W.P. Drever et al., Laser Spectroscopy VEds A.R.W. McKellar et al., (SpringerVerlag) 33, (1981). A. Abramovici et al., Phys. Lett. A 218 157,(1996). E. Mizuno, N. Kawashima, S. Miyoke, E.G. Hein, K. Wada, W. Naito, S. Nagano and K. Arakawa, in Gravitational Waves, Sources and Detectors Eds. I. Ciufolini and F. Fidecaro, Volume 2 of Edoardo Amaldi Foundation Series, (World Scientific, Singapore) 108,(1997). K. Kawabe, S. Nagataki, M. Ando, K. Tochikubo, N. Mio, K. Tsubono, App. Phys. B 62 135,(1996). D. Sigg et al, Class. Quantum Grav. 21 S409,(2004);A. Abramovici et al, Science 256 325,(1992). F. Acernese et al., Class. Quantum Grav. 21 S385,(2004). B. Willke et al., Class. Quantum Grav. 21 S417,(2004). R. Takahashi et al., Class. Quantum Grav. 21 S403,(2004). V.B. Braginsky, Sov. Phys. JETP 26 831,(1968);V.B. Braginsky and F.Ya Khalili, Rev. Mod. Phys. 68 1, (1996). V.B. Braginsky et al., Phys. Rev. D 67 082001,(2003). F. Fidecaro, LIGO doc. G010317-00,(2001). P. Bender et al., “LISA Pre-Phase A Report” 2nd Edition MPQ Report No. 233,

(1998).

25. K. Danzmann and A. Riidiger, Class. Quantum Grav. 20 S1,(2003). 26. S. Rowan and J. Hough, Living Rev. Relativity 2 3

27. 28. 29. 30. 3 1. 32.

htt~://www.livingreviews.org/lrr-2000-3, (2000). N.A. Robertson, Class. Quantum Grav. 17 19,(2000). P.R. Saulson, Phys. Rev. D 42 2437,(1990). V.B. Braginsky, M.L. Gorodetsky and S.P. Vyatchanin, Phys. Lett. A 264 1, (1999). B.J. Meers, Phys. Rev. D 38 2317,(1988). H. Grote et al., Class. Quantum Grav. 21 S473,(2004). K. A. Strain et al., in Gravitational wave andparticle astrophysics detectors, Eds J. Hough and G.H. Sanders, Proceedings of SPIE Vol5500 (SPIE, Bellingham, WA)

25,(2004).

33. B. Abbott et al., Phys. Rev. D 69 082004,(2004). 34. B. Abbott et al., Phys. Rev. D 69 122001,(2004).

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35. B. Abbott et al., Phys. Rev. D 69 102001, (2004). 36. B. Abbott et al., Phys. Rev. D 69 122004, (2004). 37. B. Abbott et al., Phys. Rev. Lett. 94 181103, (2005). 38. V. Kalogera et al., Astrophys. J . 601 L179 or arXiv:astro-ph/0312101v3,(2004). 39. T. Uchiyama et al., Class. Quantum Grav. 21 S1161, (2004). 40. P. Fritschel, in Gravitational Wave Detection, Eds M. Cruise and P. Saulson, Proceedings of SPIE Vol4856 (SPIE. Bellingham, WA) 282, (2003). 41. V.B. Braginsky, V.P. Mitrofanov, K.V. Tokmakov, Phys. Lett. A 218 164, (1996). 42. S . Rowan, S.M. Twyford, R. Hutchins, J. Kovalik, J.E. Logan, A.C. McLaren, N.A. Robertson and J. Hough, Phys. Lett. A 233 303, (1997); G. Cagnoli, L. Gammaitoni, J. Hough, J. Kovalik, S. Mcintosh, M. Punturo and S. Rowan, Phys. Rev. Lett. 85 2443, (2000). 43. V. Tokmakov, V. Mitrofanov, V. Braginsky, S. Rowan and J. Hough, Gravitational Waves Proceedings of the Third Edoardo Amaldi Conference, Ed S. Meshkov, AIP Conference proceeding no. 523 (Melville, New York) 445, (2000). 44. P. Willems, V. Sannibale, J. Weel, V. Mitrofanov ,Phys. Lett. A 297 37, (2002). 45. N.A. Robertson et al, in Gravitational Waves Proceedings of the Third Edoardo Amaldi Conference, Ed S. Meshkov, AIP Conference proceeding no. 523 (Melville, New York) 313, (2000). 46. Y. Levin, Phys. Rev. D 57 659, (1 998). 47. Y.T. Liu and K.S. Thorne, Phys. Rev. D 62 1220022, (2000). 48. F. Bondu ,P. Hello and J-Y Vinet, Phys. Lett. A 246 227, (1998). 49. K. Yamamoto, M. Ando, K. Kawabe and K. Tsubono, Phys. Lett. A 305 18, (2002). 50. K. Yamamoto, S. Otsuka, Y. Nanjo, M. Ando and K. Tsubono, Phys. Lett. A 321 29, (2004). 5 1. A. Gillespie and F. Raab, Phys. Rev. D 52 577, (1995). 52. A S . Nowick and B.S. Berry, “Anelastic Relaxation in Crystalline Solids” Materials Science Series, (Academic Press, New York), (1972). 53. K. Numata, S. Otsuka, M. Ando and K. Tsubono, Class. Quantum Grav. 19 1697, (2002). 54. A. Ageev, B.C. Palmer, A. De Felice, S.D. Penn and P.R. Saulson, Class. Quantum Grav. 21 3887, (2004). 55. S. Penn, Private communication (2004). 56. V.B. Braginsky, V.P. Mitrofanov and V.I. Panov, “Systems with small dissipation” (University of Chicago Press, Chicago), (1986). 57. N. Nakagawa, A.M. Gretarsson, E.K. Gustafson and M.M. Fejer, Phys. Rev. D 65, 102001, (2002). 58. G.M. Harry et al., Class. Quantum Grav. 19 897, (2002). 59. D.R.M. Crooks et al., Class. Quantum Grav. 19 883, (2002). 60. S.D. Penn et al., Class. Quantum Grav. 20 2917, (2003). 61. D.R.M. Crooks etal., Class. Quantum Grav. 21 S1059, (2004). 62. V.B. Braginsky and S.P. Vyatchanin, Phys. Lett. A 312 244, (2003). 63. M.M. Fejer, S. Rowan, G. Cagnoli, D.R.M. Crooks, A. Gretarsson , G. H q , J. Hough , S. Penn, P. Sneddon and S.P. Vyatchanin, Phys. Rev D 70 082003, (2004).

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64. K. Numata, A. Kemery and J. Camp, Phys. Rev. Lett. 93 250602, (2004). 65. S. Rowan, J. Hough and D.R. M Crooks, Physics LettersA in press. 66. R. O’Shaughnessy, S. Strigin and S. Vyatchanin, submitted to Phys. Rev. D, arXive:gr-qc/0409050.

67. E. D’Ambrosio, R. O’Shaughnessy, S. Strigin, K.S. Thorne and S. Vyatchanin, submitted to Phys. Rev. D, arXive:gr-qc/0409075.

68. M. Bondarescu and K. S. Thorne, submitted to Phys. Rev. D , arXive:gr-qc/0409083. 69. A. Buonanno and Y. Chen, Phys. Rev. D 64 042006, (2001); Class. Quantum Grav. 19 1569-1574, (2002).

70. J. Harms, Y. Chen, S. Chelkowski, A. Franzen, H. Vahlbruch, K. Danzmann and R. Schnabel, Phys. Rev. D 68 042001, (2003). 71. A. Buonanno and Y. Chen, Phys. Rev. D 69 102004, (2004). 72. R.L. Byer, in Gravitational Astronomy: instrument Design and Astrophysical Prospects Eds. D.E. McClelland and H.A. Bachor (World Scientific, Singapore), (1990). 73. R. Drever, in Proceedings of the Seventh Marcel Grossman Meeting on General Relativity Eds M. Keiser and R.T. Jantzen (World Scientific, Singapore) 1401, ( 1995). 74. S. Rowan, R.L. Byer, M.M. Fejer, R.K. Route, G. Cagnoli, D.R.M. Crooks, J. Hough, P. Sneddon and W. Winker, in Proceedings of SPlE ~ 0 1 4 8 5 6GravitationalWave Detection Eds M. Cruise and P. Saulson (SPIE, Bellingham, WA) 4856, (2003).

PRECISION SPECTROSCOPY OF HYDROGEN AND FEMTOSECOND LASER FREQUENCY COMBS

TH. UDEM, P. FENDEL, M. FISCHER, N. KOLACHEVSKY, J. ALNIS, M. ZIMMERMANN, CH. GOHLE, M. HERRMANN, R. HOLZWARTH, T. w. HANSCH Max-Planck Institut f u r Quantenoptik, Hans-Kopfermann Strasse 1, 85748 Garching, Germany E-mail: [email protected]

The simple hydrogen atom has inspired many advances in quantum electrodynamics and experimental techniques. As the latest spin off, femtosecond optical frequency comb synthesizers have revolutionized the way optical frequencies are measured, and provide a reliable clock mechanism for optical atomic clocks. Pr+ cision spectroscopy of the hydrogen 1s-2s two-photon resonance has reached an accuracy of 1.4parts in and considerable future improvements are envisioned. Such laboratory experiments are setting new limits for possible slow variations of the electromagnetic and strong interaction. The frequency comb technique has recently been extended into the extreme ultraviolet and may allow similar measurements on hydrogen like ions that can be held in a trap.

1. Introduction Precision spectroscopy of the simple hydrogen atom has played a central role in the history of atomic physics, and it has inspired many advances in high resolution laser spectroscopy and optical frequency metro1ogy.l A recent new measurement of the absolute frequency of the 1s-2stwo-photon transition2i3 has reached an accuracy of 1.4 parts in Future experiments will permit ever more stringent tests of basic physics laws and they may ultimately reveal “new physics”, such as possible slow changes of fundamental constants or conceivable differences between matter and antimatter. Since 1990 it became possible to measure the frequency of light rather than the wavelength allowing to pass beyond the limits of interferometric wavelength metrology. About 6 years ago, optical frequency metrology has been revolutionized with the advent of femtosecond (fs) laser optical frequency comb synthesizer^.^ Very recently fs laser pulses have been converted 38

39

to much shorter wavelength (below 60 nm) by intracavity high harmonic generation516opening up the possibility for high resolution spectroscopy in this previously nearly unexplored spectral region, for example on hydrogen like ions or simple molecules like H i . 2. Frequency Combs

Fenitosecond frequency combs are most conveniently generated by the regular pulse train of a mode-locked laser. In the frequency domain such a train is the result of the phase coherent superposition of many continuous wave (CW)longitudinal cavity modes. These modes form a uniform series of frequency spikes that is a called frequency comb. In 1998 our lab demonstrated that individual modes can be isolated by phase locking other cw lasers to them, that the mode spacing is uniform across the comb with very high accuracy and that the mode spacing is equal to the pulse repetition rate.7 This made it clear that the modes of a mode-locked femtosecond (fs) laser can be used as a precise ruler in frequency space t o measure large frequency difference8 for example relative an optical frequency standard.gt’Otll We also demonstrated7 how to stabilize the whole frequency comb to a radio frequency standard by comparing the 7th with the 8th harmonic of an auxiliary laser , a technique that was later dubbed “self-referencing”. The frequencies w, of any frequency comb with a fixed mode spacing w, can be expressed by: Wn = nwr

+

WCE

(1)

where the integer n numbers the modes (usually chosen such that WCE < w,). For frequency combs generated by mode locked lasers the mode spacing is given by the pulse repetition frequency and the comb offset WCE is caused by unequal group and phase velocity of the pulse stored in the laser resonator (see figure 1). The spectral width of the fs frequency comb A w is roughly given by the inverse of the pulse duration so that for a typical fs laser (10 fs pulse duration, i.e. 100 THz spectral width and 100 MHz repetition frequency) lo6 optical modes w, are oscillating simultaneously. It is possible to stabilize all of these modes at the same time by locking only two radio frequencies w, and W C E . This provides a phase coherent link from the radio frequency domain, where the cesium atomic clock operates that defines the unit Hz, to the optical domain. Because on the order of lo6 optical modes may be referenced simultaneously this way to a precise radio frequency reference, such a device is equivalent to an optical synthesizer.4,7,14,17,ls,l9,2O,2l,2z,z3,24

40

Figure 1. Top: The pulse train emitted by a mode locked laser is made up of replicas of a single pulse that is stored and maintained inside the laser cavity. Inside this cavity the carrier wave at wc moves with the phase velocity while the pulse envelope moves with the group velocity so that the carrier wave shifts by Acp after each cavity round trip with respect to the envelope. Bottom: Spectrum of the pulse train. The continuous phase shift translates into a constant frequency offset WCE = Acp/T from the exact harmonics of wT = 2n/T which is called the carrier-envelope offset frequency. 12313,7,14,15,16

Whereas the pulse repetition rate is readily detected anywhere in the fs pulse train, the measurement of WCE requires more effort. The established technique (self-referencing) is to compare dzflerent harmonics of the same laser with the help of the frequency comb. Due t o the limited bandwidth of our frequency comb, we used in a first crude e ~ p e r i m e n t the ~ ~ 7th ~> and the 8th harmonic of a 3.39 pm laser t o determine WCE according to 8w, - 7w,r = (8n - 7n')w, WCE = WCE by choosing the mode numbers n and n' such that (8n - 7n') = 0. This requires a relative bandwidth of (8 - 7)/8 = 1/8 of the frequency comb. For our 70 fs Ti:Sapphire mode locked laser, operating around 905 nm this required a comb bandwidth of 44 THz that we could generate with the help of self phase modulation in a single mode fiber. To employ different combinations of harmonics for self referencing has also been discussed by H. R. Telle and coworkers16 and used by T . M. Ramond and coworkers.25 Since octave spanning frequency combs became available, the simplest approach t o fix the absolute position of the frequency comb is t o measure the gap between w, and wan of frequency com-

+

41

ponents taken from the frequency comb itself.17*18~19~20~22*21 In this case the carrier-envelope offset frequency W C E is directly produced by beating the frequency doubled red wing of the comb with the blue side of the comb such that 2w, - wnj = ( 2 n - n‘)w, W C E = WCE. This requires the modes with numbers n and n’ to be active in the laser cavity such that ( 2 n - n’) = 0. The necessary relative bandwidth of ( 2 - 1 ) / 2 = 1 / 2 and a full optical octave. The method of self-referencing effectively multiplies a radio frequency reference in a single step by six orders of magnitude. It has been predicted that such a large multiplication factor would lead to a so-called carrier collapse, i.e. the complete loss of coherence.26However, it turned out that the mode locked laser itself provides a stable enough flywheel in the optical region to greatly reduce this problem. In fact the comb properties, expressed by eqn.(l) have been verified within a few parts in 10l6 limited by statistics 0nly8921127128>29 and neighboring comb lines from several different frequency combs have been compared with uncertainties at the level.30 Using a frequency comb the accuracy of second harmonic generation and difference frequency generation has been tested to within 7 parts in lo1’ and 6 parts in 1021 31,32

+

Meanwhile frequency combs have been used for the measurement of a large variety of narrow optical resonances provided by neutral atoms2~33~34~35 and trapped ions.36~37~38~39 In our 1abora.tory we have measured the frequencies of the hydrogen 1 s - 2 s transition3, a methane stabilized HeNe laser at 3.39 pm40, the In+ 5s2 ‘SO- 5s5p3P0 clock transition41, several transitions in lodine and the cesium D1 lineg which is used to derive the fine structure constant from a recoil e ~ p e r i m e n t The . ~ ~ optical synthesizer can also be operated in reverse to produce a countable radio frequency output (w,) that is phase coherently linked to a sharp optical transition. The whole set-up, the optical “pendulum” and the frequency comb optical clockwork that counts its oscillations, make up an optical timepiece that is predicted to outperform even the current state of the art cesium fountain clocks. Optical clocks of this kind are now being constructed at several national labs.45>46

3. The Hydrogen 1s-2sTransition In our laboratory at Garching, we have long concentrated on precision spectroscopy of the particularly sharp 1s-2s two-photon resonance of hydrogen and deuterium. In the first set of measurements of an absolute optical

42

frequency with a femtosecond laser frequency comb in 1999, we have compared the ultraviolet 1S-2s frequency with the microwave frequency of a commercial cesium atomic clocklg and a cesium fountain clock developed at BNM-SYRTE in Paris2. Initially this measurement was limited by the commercial clock but with the cesium fountain in place the largest contribution to the total uncertainty of 1.8 x was eventually due to the hydrogen spectrometer. The contribution of the frequency comb was negligible.

Figure 2. Experimental setup for the 2003 comparison of the hydrogen 1S2S transition frequency with the primary Cs frequency standard from BNM-SYRTE. The 486 nm laser is stabilized to a reference resonator and steered with an acousto-optical modulator (AOM) before it is coupled into a linear enhancement cavity inside the vacuum chamber. The excitation of the hydrogen atoms takes place in a collinearly traveling atomic beam in the ultrahigh vacuum (UHV) zone. The frequency of the dye laser is measured with a self-referenced fs frequency-comb generator, which is controlled by a Cs clock.

In 2003, we have performed a second set of measurements3 as shown in fig. 2. At that time we where using a more compact 2w, - wzn selfreferencing rather than a 8w, - 7ws,/7 self-referencing as in 1999. In addition we implemented numerous improvements in the hydrogen spectrometer. As before, ultraviolet radiation at 243 nm was produced by frequency doubling of a blue dye laser. The dye laser frequency was locked to an

43

improved passive reference cavity by the Pound-Drever-Hall method. A spacer of ultra low expansion glass (Corning, ULE) reduces the cavity drift rate to about 0.5 Hz/s compared to 25 Hz/s obtained with the Zerodur spacer in 1999. The cavity housing with its temperature stabilization and acoustic shielding is mounted onto an active vibration isolation stage, reducing the laser line width to about 60 Hz a t 486 nm. Up t o 20 mW of UV light at 243 nm was coupled into a linear build-up cavity inside the vacuum chamber of the hydrogen atomic beam apparatus. Hydrogen atoms are produced in a microwave discharge and cooled to 5-6 K by a nozzle mounted to a helium cryostat. Afterwards, they travel collinearly in the standing wave field of the cavity, restricted by two narrow apertures. After a distance of about 10 cm they reach the 2s detection region where a quenching electric field forces the emission of Lyman-a photons, which are observed with a photo multiplier. The exciting UV light is periodically blocked with a chopper a t 160 Hz, and Lyman-a photons are detected only during the dark period, in order t o avoid background counts due to stray light. The signal counts are sorted into 12 bins corresponding to different delay times after the chopper blocked the light. This time-of-flight method makes it possible t o separate the contributions of different velocity classes whose signals are then fitted to a line shape model. The vacuum system has been upgraded t o allow differential pumping of the interaction region to a pressure of lo-' - l o p 7 mbar to reduce line shifts and the loss of slow atoms due to collisions with background gas atoms. About 10 mW of the 486 nm light is used to create a beat signal wb with the self referenced frequency comb. In this way the frequency of the dye laser W L is compared with a microwave frequency of a cesium atomic clock via W L = nw, W C E W b WLO where W L O is the sum of all local oscillators used in the phase locking process. As in 1999, the FOM atomic fountain clock built in Paris has been transported to Garching for this measurement. The relative accuracy of a t BNM-SYRTE47 but for the this clock has been verified to 8 x Garching experiments, we attribute a conservative accuracy of 2 x for both measurements. Fig.3 shows a comparison of measurement results from the two experiments during 10 days in 1999 and 1 2 days in 2003. Each data point represents the measurements of one day with statistical errors, after correcting for the AC Stark effect by extrapolating t o zero intensity. Both sets of data have been analyzed with the same theoretical line shape model and are therefore comparable. The quality of the data obtained during a given day is obviously much improved in 2003. However, the fluctuations from

+

+ +

44

w

5200

2 0

5100

8 0

5000

E.

I-

d

N

o

4900

7

r

0”

4800

W (0

4700

3 5

4600 4500

Figure 3. Results for the hydrogen 1S(F = 1, M F = H)---t 2S(F’ = 1,Mf;. = &i) transition frequency from the data sets in 1999 and 2003. Each data point represents one day worth of data. The mean values for both data sets where calculated without using the error bars as a weight.

day to day are similar to those of 1999, pointing at some uncontrolled systematic frequency shifts on the order of 40-80 Hz at 121 nm. After the completion of these measurements, we have performed a series of careful experimental tests to rule out many conceivable sources of such systematic line shifts, such as intra-beam pressure shifts, background gas pressure shifts, or line shifts due to stray microwaves from the hydrogen discharge. The most plausible candidate is the inadvertent excitation of (lossy) higher order transverse modes of the build-up cavity, or the distortion of wavefronts by a not ideally mode-matched traveling wave entering through the coupling mirror with its transmission of 2%. After quadratically adding a systematic error of 23 Hz, the frequency of the F = 1 -+F‘ = 1 hyperfine component of the 1S-2s transition obtained in 2003 is f(1S-2s)

= 2 466 061 102 474 851(34) Hz.

The uncertainty of 1.4 parts in l O I 4 is only slightly improved over that of the 1999 result f(1S-2s) = 2 466 061 102 474 880(46) Hz. Here, we have added a pressure shift of 10 Hz that had not been included in the 1999 analysis and a systematic error of 28 Hz. However, we now have two independent measurements of this important transition which agree

45

within their respective uncertainties. The difference of (-29 f57) Hz in 44 months corresponds to a drift of the 1s-2s frequency relative to the cesium per year, consistent with a zero frequency standard of (-3.2 f6.3) x drift.

4. Drifting Fundamental Constants

A nonzero-drift can arise if fundamental constants changed slowly with time. Any changes of non-gravitational constants are ruled out by Einstein’s equivalence principle, but may be permitted within string theories aiming for the unification of all interactions. The dramatic advances in the art of precision spectroscopy and optical frequency metrology have kindled much interest in laboratory searches for time variations of fundamental constant^.^^?^^ Recent studies of absorption lines in the light of distant quasars at the Keck observatory have been interpreted as an indication that the electromagnetic fine structure constant a has been somewhat smaller in the early universe.48 With the simplest assumption of a linear drift, the relative rate of change of a would be (6.4 f 1.4) x per year, too small to be directly observed in the present hydrogen experiment. However, Xavier Calmet and Harald Fritzsch, theorists at the University of Munich49 and others have argued, that a cannot simply change by itself, if one believes in grand unification. If all known forces are to remain unified at very high energies, other coupling constants have to change as well. Using the framework of quantum chromodynamics, Fritzsch points at an interesting amplification effect. He argues that the masses and magnetic moments of hadrons should change much faster than a: “We would expect that light emitted in hyperfine transitions should vary in time about 17 times more strongly than light emitted in normal atomic transitions, but in the opposite direction, i.e. the atomic wavelength becomes smaller with time, but the hyperfine wavelength increases.” This amplification should have produced an observable effect in the comparison of the two hydrogen measurements separated by almost 4 years, if the interpretation of the Keck observations were correct. However, new observations of quasar spectra at the Very Large Telescope in Chile do not support that evidence, but are consistent with a zero change of Regardless of such speculations, our experiments together with other recent precise measurements of the frequencies of optical clock t r a n s i t i o n ~ are already establishing new laboratory limits for possible time variations of fundamental constants. Our hydrogen results are consistent with zero

46

-6

-4

-2

0

2

4

6

&a [10-15yr-l] Figure 4. Limits for possible time variations of the fine structure constant and the cesium nuclear magnetic moment p (measured in units of the Bohr magneton p ~ es) tablished by comparing optical frequencies in 199Hg+36,hydrogen3 and 171Ybf37 with the microwave frequency of a Cs clock over a period of several years. The grey stripes are the limits set by each of the comparisons separately and the black ellipse marks the region consistent with all the data.

changes. However, nature could conspire against us and change both the electromagnetic fine structure constant cy and the magnetic moment of the cesium nucleus pcs in a correlated way such that we do not see any relative drift when comparing the hydrogen 1s-2sfrequency with the hyperfine frequency of a cesium clock. In fact, since pcS is determined mostly by the strong interaction Grand Unification Theories even suggest some form of correlation as explained above. To rule out this possibility and to derive drift rates of the strong or the electromagnetic interaction without assuming the other being constant one has to combine the results of different optical frequency measurements. Spectroscopy of the ultraviolet S-D electric quadrupole clock transition in lg9Hg+ has been perfected over decades by Jim Bergquist a t NIST in Boulder, and two absolute frequency measurements have been made in 2000 and 2002, using the laser frequency comb approach.36 In addition, an S-D quadrupole clock transition in the 171Yb+ ion has been measured at the PTB in Braunschweig in 2000 and in 2003.37 The combined data is shown in fig. 4. The shaded stripes indicate how these three measurements establish limits for possible changes of cy and

47

the cesium magnetic moment p = p c S / p (measured ~ in units of the Bohr magneton p ~ ) The . intersection of the three stripes defines a 1-sigma area which gives for the first time separate laboratory limits for possible drifts for the fine structure constant (Y and the magnetic moment of the cesium nucleus p51 of h

- = (-0.3

f 2.0) x

y~-l

and

(Y

4CL = (2.4 f 6.8) x

yr-'

respectively. We are at a rather early stage of such studies and continued measurements for example on the Sr+ clock transition3' will add further limits to the drift rates. In most industrialized countries, there are now strong teams working towards the development of more precise atomic clocks, based on sharp optical transitions. Because of their higher transition frequencies these clocks are expected to outperform the radio frequency clocks based on the Cs ground state splitting for example. As such clocks are perfected, with envisioned ultimate accuracies of lo-'' or beyond, a comparison of different types of clocks over extended periods might eventually reveal changes of fundamental constants down to levels of per year. Such changes would matter little for everyday life, but much for our understanding of the universe. But even if we do not find any time variations of fundamental constants, there are many good reasons for pushing the art of measuring time to the feasible limits.

5. Extending the Frequency Comb Very likely, there will be new unexpected discoveries, as the art of precision spectroscopy and optical frequency metrology continues to advance into the extreme ultraviolet and soft X-ray regime. So far there has not been a laser source suitable for high resolution laser spectroscopy in this spectral region. One possible route might be to upconvert a frequency comb to much shorter wavelengths. By exploiting the high intensity of fs pulses it is possible to create harmonics up to 1000th order of the f ~ n d a m e n t a l ~ ~ . Because of the comb structure it is also possible to perform high resolution spectroscopy even with ultra-short pulses. When driving a two photon transition the modes can add pair wise to the correct frequency. It has been predicted by Baklanov e t al. as early as 1974 that in such a case the transition is driven by the total power of all the modes while the line width is given by the spectral width of a single mode53 provided the atoms are sufficiently cold. Snadden et al. have demonstrated this type of spectroscopy

48

Figure 5 . The XUV comb set-up. The enhancement cavity is matched in length such that consecutive pulses from the laser interfere constructively at the input coupling mirror. The Brewster-angled sapphire window of the vacuum chamber that contains a Xe gas jet is used as an output coupler for the XUV laser beam via total external reflection. The inset shows the reflectance of sapphire for ppolarized XUV light at an incidence angle of 60.48’ (Brewster’s angle for 800 nm radiation).

in laser cooled Rubidium54 whereas Witte e t al. have used an upconverted frequency comb for that purpose for the first time.55 For an ion or atom to single out an individual mode of such a frequency comb its resonance linewidth must be much smaller than the mode spacing w,. Since we are primarily interested in narrow resonances this does not seem to impose a strong restriction. However, up to very recently higher laser harmonics could only be produced at kHz repetition rates. The process of high harmonic generation (HHG), requires the focusing of intense amplified pulses with intensities in excess of 1013 W/cm2 into a gas jet. The standard approach is to use Ti:Sapphire systems for post-amplification that effectively concentrates the available power of, say, 1 W into fewer pulses per second. To overcome the limited repetition rate we and our colleagues at JILA have now produced high harmonics in an enhancement cavity6i5, pretty much the way resonantly enhanced second harmonics have been done for many years. Inside the resonator, the circulating power is enhanced by a factor P (the resonator finesse divided by n)so that it can drive the HHG’s. The resonance condition in the resonator has to be fulfilled for each of the frequencies of the frequency comb that makes up the pulse train, simultaneously. This can be accomplished with a resonator of appropriate length and zero group velocity dispersion (GVD), whose CE frequency is matched to the laser. For increasing finesse, the stored pulse undergoes an increasing number of round trips and therefore dispersion compensation becomes more critical. Nevertheless, such cavities have been demonstrated recently to provide an enhancement factor of up to 70 for sub 50 fs pulses resulting in pulse energies of more than 200 nJ.56

49

Our setup (fig. 5) consists of a femtosecond oscillator with a repetition rate of 112 MHz, delivering 20 fs pulses with an average power of 850 mW. These pulses are shaped in a prism compressor arrangement (not shown) to yield near transform limited pulses in the gas jet. After mode matching the beam it is coupled to the enhancement resonator using a 1%input coupler. The resonator mode has a tight focus of about 5 pm 1/e diameter. The cavity is dispersion compensated by chirped mirrors and its length is electronically controlled using the polarization locking technique. The vacuum chamber that contains the gas jet has two Brewster angled sapphire windows that provide low transmission loss for the resonator in order to not compromise the cavity finesse. Simultaneously they act as a beam splitter for the collinearly generated XUV radiation. As the refractive index of that material is much smaller than one in the range from 30 to 120 nm total external reflection occurs in that wavelength range (see fig. 5). A Xenon gas jet is injected into the focus of the cavity mode using a small glass capillary with about 50 pm inner diameter, matched to the length of the focus. I

I

1

I I

50

60

70

80

90

100

110

120

wavelength [nm] Figure 6 . Harmonic spectrum measured directly (grey), and with O.1km A1 foil in the beam path to reduce stray light from the stronger harmonics (black). The close by 5s-7p resonances around 113 nm in Xenon modifies the shape of the 7th harmonic. Coherent population of long lived Rydberg states leads to a directed emission close to ionization limit of Xenon (dotted line).

In contrast to the usual HHG schemes, the power that is not converted into the XUV after a single pass through the medium is “recycled” and can contribute in subsequent passes. In other words, if we assume single

50

pass conversion efficiencies comparable to those observed in chirped pulse amplifier schemes we would expect orders of magnitude higher average XUV power since the repetition rate in this approach is also orders of magnitude higher. When the resonator is locked, we have about 12 MW peak power in pulses of about 30 fs duration available for harmonic generation. This leads t o a peak intensity in the focus exceeding 5 x 1013 W/cm2 sufficient t o generate harmonics up t o 15th order. The outcoupled radiation is subsequently analyzed in a grazing incidence monochromator. A spectrum obtained in this manner is shown in fig. 6. The total power generated in the shown range is approximately 10 nW which corresponds t o a conversion efficiency of about We hope that this new source of coherent XUV radiation will enable high resolution spectroscopy in that spectral region for the first time. For example the 1s-2stransition of singly charged helium may be probed with the 13th harmonic of a TiSapphire laser. This would provide an even more sensitive test of QED than the hydrogen 1s-2sexperiment 3. If these measurements are combined, a new precise value for the proton charge radius and the Rydberg constant may be obtained. Additionally one can envision many other applications of such a quasi-continuous compact and coherent XUV source, including XUV holography, microscopy, nano lithography and X-ray atomic clocks.

References 1. 2. 3. 4.

5. 6. 7.

8. 9. 10. 11. 12. 13. 14. 15.

B. de Beauvoir et al., Eur. Phys. J . D 12,61 (2000). M. Niering et al., Phys. Rev. Lett. 84,5496 (2000). M. Fischer et al., Phys. Rev. Lett. 92,230802 (2004). Th. Udem, R. Holzwarth, and T.W. Hansch, Nature 416,233 (2002). R. J. Jones et al. Phys. Rev. Lett. 94,193201 (2005). Ch. Gohle et al., Nature 436,234 (2005). Th. Udem et al., Proceedings of the 1999 Joint Meeting of the EFTF and the IEEE International Frequency Control Symposium, at Besanqon, France 1999, (IEEE-press) pp. 620-625. Th. Udem et al., Opt. Lett. 24,881 (1999). Th. Udem et al., Phys. Rev. Lett. 82,3568 (1999). S. A. Diddams et al., Opt. Lett. 25, 186 (2000). K. R. Vogel et al., Opt. Lett. 26,102 (2001). L. Xu et al., Opt. Lett. 21,2008 (1996). Th. Udem, Ph.D. Thesis, Ludwig-Maximilians-Universitat, Munich (1997). J. Reichert et al., Opt. Commun. 172,59 (1999). A.I. Ferguson, J.N. Eckstein, T.W. Hiinsch, Appl. Phys. 18,257 (1979).

51

16. H. R. Telle et al., Appl. Phys. B69, 327 (1999). 17. S.A. Diddams et al., Quantum Electronics and Laser Science Conference QELS 2000, OSA Technical Digest, p. 109. 18. R. Holzwarth et al., Quantum Electronics and Laser Science Conference QELS 2000, OSA Technical Digest, p. 197. 19. J. Reichert et al., Phys. Rev. Lett. 84,3232 (2000). 20. S.A. Diddams et al., Phys. Rev. Lett. 84,5102 (2000). 21. R. Holzwarth et al., Phys. Rev. Lett. 85,2264 (2000). 22. D. J. Jones et al., Science 288,635 (2000). 23. S. T. Cundiff, J. Ye, and J. L. Hall, Rev. Sci. Instr. 72,3749 (2001). 24. S. T. Cundiff, and 3. Ye, Rev. Mod. Phys. 75,325 (2003). 25. T. M. Ramond et al., Opt. Lett. 27,1842 (2002). 26. H. R. Telle in Frequency Control of Semiconductor Lasers (ed. M. Ohtsu) pp. 137167 (Wiley, New York, 1996). 27. R.Holzwarth, Ph.D. Thesis, Ludwig-Maximilians-Universitat,Munich (2001) 28. S. A. Diddams et al., Opt. Lett. 27 (2002). 29. Ph. Kubina et al., Opt. Expr. 13,904 (2005). 30. L. S. Ma et al., Science 303,1843 (2004). 31. J. Stenger et al., Phys. Rev. Lett. 8 8 , 73601 (2002). 32. M. Zimmermann et al., Opt. Lett. 29,310 (2004). 33. U. Sterr et al., Comptes Rendus Physique 5,845 (2004). 34. V. Gerginov et al., Phys. Rev. A 70, 42505 (2004). 35. M. Takamoto et al., Science 435,321 (2005). 36. S. Bize et al., Phys. Rev. Lett. 90,150802 (2003). 37. E. Peik et al., Phys. Rev. Lett. 93,170801 (2004). 38. H. S. Margolis et al., Science 306,1355 (2004). 39. P. Dube et al., Phys. Rev. Lett. 95,33001 (2005). 40. P. V. Pokasov et al., Proceedings of the Sixth Symposium on Frequency Standards and Metrology, P. Gill ed., World Scientific, Singapore p. 510 (2002). 41. J. von Zanthier et al., Opt. Lett. 25, 1729 (2000). 42. R. Holzwarth et al., Appl. Phys. B 73,269 (2001). 43. A. Yu. Nevsky et al., Opt. Commun. 192,263 (2001). 44. A. Wicht et al., Phys. Scr. T 102,82 (2002). 45. L. Hollberg et al., J . Phys. B 38,,3469 (2005). 46. P. Gill, and H. Margolis, Physics World p. 35, May 2005. 47. M. Abgrall, These de doctorat de l’universite Paris VI (2003). 48. M. T. Murphy, J. K . Webb, and V. V. Flambaum, Mon. Not. R . Astron. Soc. 345,609 (2003). 49. X. Calmet, and H. Fritzsch, Phys. Lett. B 540,173 (2002). 50. R. Srianand et al., Phys. Rev. Lett. 92,121302 (2004). 51. M. Zimmermann et al., Laser Physics 15,997 (2005). 52. J. Seres et al., Nature 433,596 (2005). 53. Y. V. Baklanov, V. P. Chebotayev, Appl. Phys. 12,97 (1977). 54. M. J. Snadden et al., Opt. Commun. 125,70 (1996). 55. S. Witte et al., Science 307,400 (2005). 56. R. J. Jones, J. Ye, Opt. Lett. 29, 2812 (2004).

PRECISION SPECTROSCOPY OF HELIUM ATOM

P. CANCIO PASTOR, G. GIUSFREDI, D. MAZZOTTI AND P. D E NATALE Istituto Nazionale di Ottica Applicata (INOA-CNR) and European Laboratory for Non-Linear Spectroscopy (LENS) V. KRACHMALNICOFF AND M. INGUSCIO LENS and Dipartimento di Fisica, Universitci Firenze, Via N. Carrara 1, I-50019 Sesto Fiorentino, ITALY E-mail: pcpoinoa. it Quantum electrodynamic theory (QED) of the simplest bound-three-body system is now stringently tested by precise spectroscopic frequency measurements in Helium. Alternatively, comparison between He measurements and theory could be used for accurate determination of some fundamental quantities, as for example, the fine structure constant a , and the differences of nuclear charge radii between two 3He and 4He. In the following, we review our precise spectroscopic measurements on the 1083 nm Helium transition, connecting the triplet 2 s and 2 P states. Frequency differences among the 1083 nm measured frequencies give precise values of the fine structure (FS) and hypefine structure (HFS) splittings of the 23P level in *He and 3He respectively, and of the isotope shift (IS) between these two isotopes. Implications of these results in the a determination as well as for nuclear charge radii and structure are discussed.

1. Introduction

The QED-theory knowledge for bound-three-body-systems is of great importance because, when compared with spectroscopic measurements, fundamental physical quantities can be determined . Comparisons between theory and measurements in He-like atoms or molecular-ions, being more complicated than He, have been reported in this conference'. In this way, antiproton-proton mass and charge ratios have been determined in antiprotonic He, but their final accuracy was limited, in part, by the present status of the QED-theory for this exotic atom'. In this frame, the accuracy achieved for frequency measurements of transitions in stable He isotopes" aThanks to the optical frequency combs, accuracies a t the have now been achieved2. 52

level or even better

53

can be used as a stringent test of the QED theory for the simplest boundthree-body system, and therefore contributes by analogy t o improve this theory for more complicated systems. Alternatively, this theory-experiment comparison applied to 4 H e has been used for more than three decades to give an accurate determination of the fine structure constant (Y by precise measurements and theory of the 2 3 P FS splittings3. Also, the nuclearcharge-radius determination of He and He-like isotopes, by comparison of precise IS measurements and theory4, is at present recognized as a powerful nuclear-model independent method, alternative to the usual one, i.e. e-nuclear scattering, almost for short-lived isotopes. The results have been used as a critical test of the effective low-energy nucleon-nucleon potential. The knowledge of this potential is fundamental for the theoretical calculus of the nuclear-charge-radius and other nuclear properties. Information about other nuclear properties can be achieved from precise measurements of hyperfine atomic transitions in He isotopes with non-zero nuclear spin. Among all He transitions, the one at 1083 nm is perhaps the most studied and measured using different experimental approaches. It is particularly convenient because it starts from a metastable state (about 7900 s lifetime) that can be populated by an electrical discharge and because it gives access to the 23P FS and HFS manifolds, by using microwaves as well as optical frequencie~~l~ Although - ~ ~ . the 3He nuclear-charge-radius has been determined by measurements of the 23S + 33P transition at 389 nm", most determinations are given by measurements at 1083 nrn1O>l2.In addition, the Lamb-shift contribution to the 23S and 23P levels is larger than for other simple atoms and can thus be measured with higher accuracy even with less precise measurements of the atomic transitions2. Here, we review the more than ten years long activity in Firenze on precise spectroscopy at 1083 nm27315. Preliminary results for absolute frequency measurements of the hyperfine 3He transitions at 1083 nm, and 3He-4HeIS at this wavelength are reported.

2. Experiment In our experiment, fluorescence saturation spectroscopy in a metastable He beam, in absence of external magnetic fields, has been implemented for precise line center measurement of the 1.6 MHz HWHM 23S1+23P~ transitions at 1083 nm. This experimental approach has been described in detail e l s e ~ h e r eand ~ ~it~is~ schematically ~ depicted in figure 1. The metastable beam of He* atoms is produced by a DC-discharge and both

54

Ophca,

DBR Diode laser

isolator

Master Laser

..................................

:

OFS-mastera c k

'He

'He

Figure 1.

Ophcal

Experimental set-up for precise He spectroscopy at 1083 nm.

3 H e and * H e isotopes can be used. In the former case, indeed, a gasrecirculating system was implemented in order to minimize the quantity of gas used, due to the high cost of 3He. Moreover, this system allows now to perform 3He-4He measurements with a controlled gas mixture. After production, these atoms interact with the saturating laser standing-wave, in a volume screened from external magnetic fields with a p-metal cylinder and three pairs of Heltzmolt coils, one for each spatial direction. The fluorescence from the excited z3Plevel as a function of the laser frequency scan is observed, as shown in Figure 2-a. A frequency modulation technique was used to enhance the saturation dip signal-to-noise (S/N) ratio and to eliminate the contribution of the residual Doppler fluorescence (see Figure 2-b/c). In our experiment, the line center of He transitions is determined by a fitting procedure. For that, it is important to get a precise frequency scale on the x-axis. To this purpose, we phase-lock the scanning laser (slave) t o a second (master) laser that, in turn, is frequency locked t o a stable reference. The master-slave frequency difference is covered by a GPS-disciplined RF-

55

. .. 550 200

(MHz,

. .-..

..

15000 15010 15020 15030 15040 276698590 276898610 276698830 Heterodynefrequency (MHr) Absolute frequency (HHz)

Figure 2. a- Saturated fluorescence signal of the 23S1,3/2--'23P,,1/, transition in 3He. b-Third derivative signal in the pwave frequency scale. c-Third derivative signal in the absolute frequency scale.

pwave synthesized frequency, that is changed by a computer, as shown in Figure 2-b. Differences of transitions at 1083 nm are thus measured with the same accuracy of the microwave synthesizer and the precision only limited by the master frequency stability. Moreover, an absolute scale is available for the He transitions if the absolute frequency of the master laser is known. Therefore, a quest for a good reference frequency for the master has characterized our experimental activity in these years. We first used a He-filled RF-discharge cel15>13to choose later the more stable molecular iodine hyperfine transitions at the doubled frequency (541 nm ~ a v e l e n g t h ) We ~~ then measured the absolute frequencies of He transitions with unprecedent accuracy by measuring the frequency of the iodine-locked-master against a GPS-disciplined-optical frequency comb synthesizer (OFS)2. In the present experimental Set-up, we directly phase-lock the master to the OFS, as shown in Figure 1. Our OFS is a self-reference comb in the visible (5001100 nm) with a 1 GHz repetition rate frequency, f,. The offset frequency fo, and f,. are locked to a stable 10 MHz GPS-disciplined quartz. With this OFS-locked-master system, an absolute frequency scale is given for the He spectra (see Figure 2-c), where the slave-frequency of each point is:

Here, N, is the order number of the OFS tooth nearest to the master and f,, is the beat-note frequency between master and OFS. The line center in this scale is measured with a precision of about 5-15 kHz, limited only by the S/N and not by the master reference. In fact, the OFS-locked-master stability at 1s is 160 Hz (Allan deviation of 6 x more than two orders

56

of magnitude better than the iodine-locked master-stabilityb. Moreover, a minimum accuracy of 5 ~ 1 0 - land ~ a better reproducibility of the He measurements are guaranteed by the new reference, used for the first time in the preliminary IS measurements presented here. It should be noticed that with this set-up we can perform "multiresonant precision spectroscopy", i.e. optical frequency measurements can be simultaneously measured together with their differences, by locking two different slave lasers resonant with different He transitions to two different teeth of the OFS15. Statistical values for the He frequencies with an uncertainty of about 1 kHz are obtained by averaging several tens of single line centers measured as indicated above. Systematic corrections and uncertainties are then estimated to get the final measured value. We refer to our previous publication^^)^?^ for a detailed description of systematic effects. Here, we simply note that the bigger effect in our measurements is due to mechanical effects of the light in the atom dynamics during the interaction time. These effects were well known to produce frequency shifts in accurate saturation spectroscopy measurements in atoms16, and they were observed and studied in He by us since the first FS measurement^'^.

3. Absolute frequency measurements of the 23S transit ions

---f

z3P

Lamb shift contributions to the He levels energy include QED and high order relativistic corrections to the non-relativistic ionization energy, which is calculated with high accuracy for this atom (about for the lower S states). Precise frequency measurements of transitions between these levels can be used to determine the Lamb shift contributions, and therefore to test the QED theory of ionization energies. We have performed this determination for what is, at present, the best measured Lamb shift in a simple atomic system, including Hydrogen and Deuterium: the 4He 23S23P Lamb shift2. The experiment measured the three 23S 4 Z 3 P (J=0,1,2) ~ transitions with unprecedent accuracy for optical He transitions, by using the iodine-locked-master-slave spectrometer and the OFS. The resulting "spin-averaged" frequency of the 23S + 23P transition has an accuracy 30 times better than the previous interferometric measurementls, as shown in Table 1. Lamb shift of this energy interval with an accuracy of 6 . 6 ~ 1 bDue to the negligible contribution of the residual frequency noise of all other frequencies of Equation 1, the OFS-locked-master stability is only limited by the f, stability (i.e. by the quartz stability).

57 Table 1. Comparison with previous measurements and theory. fc is the spin-averaged frequency of 23S+23P0,1,2 transitions. AEQEDand EQEDaxe, respectively, the 23S+23P and z3P Lamb-shift energies.

z3P FS3-7 Previous

276 736495 624.6

(2.4)

5 311.2354

(35)

-1 253.949

(58)

measurement^^^^^^

276 736 495 580 276736496 100

(70)

5 311.27 5312.3

(7) (1.1)

-1 253.9 -1253.75

(1) (40)

Theoryz0

(1100)

Note: Uncertainties are given in parentheses.

was calculated by substraction of the well known non-relativistic 23S and 23P ionization energies. Also, the Lamb shift of the 23P level was calculated removing the best published value of the largest contribution of the 23S Lamb shift, which limited the final accuracy. It is of particular interest to compare the accuracy of the experimental and theoretical determinations of these Lamb shifts. As shown in the Table 1, the theoretical values2' are between 7 and 300 times less accurate than the experimental ones, meaning a challenge for the He theory. Finally, we want to note that OFSassisted measurements, as the ones described above, can revolutionize the He frequency metrology, because He one/two photon transitions fall in the visible/near IR region, where the OFS operates. 4. Z3P fine structure splittings in 4He: towards a determination of a Perhaps the best measured energy intervals in He are the FS splittings of the 23P level in 4He due to the variety of experimental approaches used The interest of these measurements lies on the possibility to determine Q with an accuracy of 10 ppb by a measurement and a theoretical determination of the 23P0-23P1 ( ~ 0 1 )splitting with a 0.8 kHz accuracy. We have performed in the years two measurements of both uol and 1/12 splittings by using first the He-locked-master-slave spectrometer5 and then the 12-locked-master-slave spectrometer3. The results, shown in Figure 3, agree within one standard deviation. More important, an accuracy increment of a factor of two for the uol interval in the case of the 12-locked-master measurement was obtained due to the better master stability, that is the limiting factor for this interval. Also in this figure, the agreement with other measurements at the kHz accuracy level is shown. This agreement allows us to calculate a weightedmean value for each interval considering only published measurements with an uncertainty better than 3 kHz:

58 I

DGke"

Drake"

Mean*(

M "n , -8arvard8

Harvard

Firenze'

Fired-

-

Toronto6

Torodto6-

Texas.'

Texas'Fireme'

Firenze.'

Figure 3. in 4He.

cc_

Measurements (circle) and calculated (square) values of the Z3P FS intervals

~ 1 2= yo1

2291 175.3(8) kHz

= 29616951.8(6)

kHz

(350 ppb) (20 ppb)

a-1 = 137.035 987 2( 18) is calculated by using the above vol value and

by considering the last updated theory21. The result achieves the expected accuracy (13 ppb), but it is in disagreement with the present CODATA value22 by more than 90 standard deviations. Discrepancies between measured and calculated values (about 9 (T for vol and 14 (T for 1 4 2 ) must be clarified before an a value from He can be considered in a new adjustment for this constant. 5 . 3He-4He isotope shift in the 23S-23P transition:

nuclear-charge-radius determination The energy difference for a given transition for two different isotopes (i.e. isotope shift) is determined by the difference of nuclear mass and by the different finite nuclear volume and charge distribution.

+

A ~ I S= A ~ M SAvvs

(2)

The volume-shift term Avvs is dominated, in a first approximation, by an energy proportional to the difference between the square nuclear-chargeradius of both isotopes' - &, so that:

~-1"~

AvIS = AvMS + CnlLl-nZLZ(z)(&

- '&)

(3)

where the constant C depends on the nuclear charge 2, and on quantumnumbers nL of the levels involved in the transition. Therefore, an accurate CVolume-shift corrections due to nuclear polarization or other relativistic recoil effects are negligible in He for IS measurements at the kHz level accuracy4

59

0

4200

4550 4900

5250

6650

7000

11200 11550 11900

325503290039200 39550

Hyperfine structure (MHz) Figure 4. Hyperfine saturated-fluorescence spectra of the 3He transitions at 1083 nm.

A U Imeasurement ~ and A U Mcalculation ~ could determine an equally accurate value of r;l - r:2. Moreover, if the radius of one of the two isotopes is well known, the value for the other can then be calculated. This is the case of He, where the nuclear-charge-radius of the a particle has been well measured ( 6 ~ 1 0 - ~ ) ’The ~ . required accuracy must be, a t least, higher than the volume-shift contribution, which for He is a small quantity (about 1.7 MHz for the 33 GHz 3He-4He IS at 1083 nm). A U Mfor ~ He is calculated with sub-kHz accuracy, due to the cancellation of most of the Lamb-shift contribution, which is nuclear-mass independent4. With this A V Muncer~ tainty, our preliminary 5 kHz accurate measurement of A V I allowed ~ us - r2Hewith an uncertainty of 6x that is the same to determine uncertainty of the 4He nuclear-charge-radius presently known. To perform the A V I ~ measurement, we have measured the absolute frequencies of the 3He hyperfine transitions at 1083nm by using the OFSlocked-master-slave spectrometer. In Figure 4 we show a record of the nine saturated-fluorescence transitions allowed between hyperfine levels of the triplet 2s and 2P states. Although we measured the frequency of five of these nine components, only in 23S1,1/2-+23P~,1/2(VS) and 23s1,3p-+23P~,1/2 ( v g ) transitions, consistent statistics was considered to extract some preliminary results. The final uncertainty was 1 . 5 ~ for

60

CEI

Hatv8rd" (*,-2%J,] ,

I

=

Har+ard"@S$Pj I

.

, I

1.93

1.94

I .

I

1.95

'

I

1.96

'

I

.

1.97

I

.

1.98

1 39

3He nuclear charge radius (fm) Figure 5 . 3He nuclear-charge-radius measurements.

both transitions. More important, the v9-vs frequency differs by less than one standard deviation from the well known 23S HFS splitting24, which is used as an accuracy test. The difference between one of these two 3 H e frequencies, and our previous measurement of 23S1423P0 in 4He2 does not give the value of the AVIS, because the HF-shift must be removed from 243 or vg. Perturbative determination of HF-shifts is not possible due to the strong HF-interaction in 3He. HF-shifts have been measured by using a phenomenological Hamiltonian to describe the HF-interactiong, where HFS measurements or calculations enter as parameters. We have used the last published HF-shift values of the 23S1,1/2, 23S1,3/2and 23P0,1,21eve1s12,to obtain a preliminary value of AUIS= 33668074 (5) kHz. More accurate and theory-independent results will be obtained when the measurements of all components (i.e. the 23P HFS measurements) will be completed. The very good agreement of our result with the last published measurementlo, highlights one of the characteristics of our method to measure pwave frequency intervals as a difference of absolute optical frequencies: the primarystandard traceability. In fact, our 3He and 4He measurements, performed in very different time periods with respect to the same frequency standard, give the same IS value of a previous 3He-4He heterodyne measurement". Taking into account the last calculated values for A v ~ and s c 2 3 S - 2 3 p 4 , we have determined the 3He nuclear-charge-radius. The result, compared with

61

other determinations in Figure 5, has an uncertainty of ~ x I O - fm, ~ which is the second even more accurate value. Finally, we want to note t h a t t h e uncertainty of t h e r s H e - r 4 difference ~ ~ can be even smaller t h a n t h e r 4 uncertainty. Therefore, we can conclude t h a t accurate IS measurements in 3He-4He and H-Tritium could, in principle, improve t h e knowledge of t h e proton radius.

References 1. In this book of proceedings see "Precise laser spectroscopy of antiprotonic Helium-weighing the antiproton", R.S. Hayano and earlier references therein; and "Ultracold molecular ions: towards precision spectroscopy of HD' , P. Blythe et al.. 2. P. Cancio Pastor et al. Phys. Rev. Lett. 9 2 , 023001 (2004). 3. G. Giusfredi et al. Can. J. Phy. 8 3 , 301 (2005). This paper reviews the present status of a determination from He. 4. G.W.F. Drake et al. Can. J. Phy. 83, 311 (2005). 5. F. Minardi et al. Phys. Rev. Lett. 82, 1112 (1999). P. Cancio et al. Procc. 16 Atomic Physics Int. Conf., Edited by W.E. Baylis and G.W.F. Drake, AIP, New York., 42 (1998). 6. C.H. Storry et al. Phys. Rev. Lett. 84, 3274 (2000). M. C. George et al. Phys. Rev. Lett. 8 7 , 173002 (2001). 7. D. Shiner et al. Phys. Rev. Lett. 7 2 , 1802 (1994). J. Castillega et al. Phys. Rev. Lett. 84, 4321 (2000). 8. G. Gabrielse, Personal communication. To be published. 9. J.D. Prestage et al. Phys. Rev. A 3 2 , 2712 (1985) and earlier references therein. 10. D. Shiner et al. Phys. Rev. Lett. 74, 3553 (1995). 11. F. Marin et al. 2. Phys. D 3 2 , 285 (1995). 12. P. Zhao et al. Phys. Rev. Lett. 6 6 , 592 (1991). 13. M. Prevedelli et al. Opt. Commvn. 125, 231 (1996). 14. P. Cancio Pastor et al. Opt. Commun. 176, 453 (2000). N. Picqu6 et al. J . Opt. SOC.A m . B 18, 692 (2001). 15. D. Mazzotti et al. Optics Letters 30, 997 (2005) 16. M. G. Prentiss and S. Ezekiel Phys. Rev. Lett. 5 6 , 46 (1986). R. Grimm and J. Mlynek Phys. Rev. Lett. 6 3 , 232 (1989); 17. F. Minardi et al. Phys. Rev. A 6 0 , 4164 (1999). 18. D. Shiner, R. Dixson and P. Zhao Phys. Rev. Lett.72, 1802 (1994). 19. C. Dorrer et al. Phys. Rev. Lett.78, 3658 (1997). 20. K . Pachucki J . Phys. B 35, 3087 (2002). K. Pachucki Phys. Rev. Lett.84, 4561(2000). 21. G.W.F. Drake Can. J . Phys. 80, 1195 (2002) and earlier references therein. 22. P.J. Mohr and B.N. Taylor Rev. Mod. Phys. 77, 1 (2005). 23. E. Borie and G.A. Rinker Phys. Rev. A 18, 324 (1978). 24. S.D. Rosner and F.M. Pipkin Phys. Rev. A 1, 571 (1970). 25. A. Amroun et al. Nucl. Phys. A 579, 596 (1994).

LASER SPECTROSCOPY OF ANTIPROTONIC HELIUM ATOMS - WEIGHING THE ANTIPROTON -*

R. S. HAYANO Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan E-mail: hayanoC3phys.s.u-tokyo.ac.jp

Antiprotonic helium atom @He+) is a metastable (T N 3ps) neutral three-body Coulomb system consisting of an antiproton, a helium nucleus, and an electron. It was serendipitously discovered that about 3% of antiprotons stopped in a lowtemperature helium gas target automatically form such metastable atoms. Precision laser spectroscopy of PHe+ is possible by inducing a laser-resonant transition from a metastable antiproton orbit to a neighboring short-lived orbit and by detecting antiproton annihilation events. By comparing the experimental results with the state-of-the-art three-body QED calculations, antiproton to proton mass and charge comparison has been done to a precision of 10 parts per billion, which is currently the most precise test of the CPT symmetry (matter-antimatter symmetry) for baryons.

1. Introduction

1.1. What is antiprotonic helium?

Antiprotonic helium atom @He+)' is a metastable three-body neutral system consisting of an antiproton, an electron and a helium nucleus, first discovered2 at the 12-GeV proton synchrotron at KEK, Japan, and is now being studied in detail by our group, ASACUSA collaboration,a at CERN's antiproton decelerator (AD) facility, using laser-spectroscopic methods. As schematically depicted in Fig. 1, pHef is an exotic helium atom in which one of the two electrons of an ordinary helium atom is replaced by an antiproton. Here, the antiproton is in a highly-excited near-circular orbit *This work is supported by the grant-in-aid for specially promoted research (15002005) of mext japan aASACUSA stands for Atomic Spectroscopy And Collisions Using Slow Antiprotons. 62

63

Figure 1. Antiprotonic helium atom FHef consists of an antiproton, an electron and a helium nucleus.

1 40) while the electron is in the ground state (slightly-polarized towards the opposite side of the antiproton). In the laser spectroscopy of PHe’, we induce a laser-resonant transition from a metastable antiproton orbit (e.g., (n,e)) to a neighboring shortlived orbit (n’, and detect antiproton annihilation events (more to be explained later). The frequency of the ( n , e )+ (n’,l’)transition is hence proportional to the antiproton mass M p and can be written as:

(n

N

N

Mr u =R c P me

(

Zeff (n’,a’) n12

-

zeff:(?

(1)

2

where R is the Rydberg constant, cis the light velocity, is the antiproton reduced mass in atomic unit, and Zeff is the (n- and [-dependent) effective nuclear charge which must be calculated numerically by elaborate threebody QED theories. Eq. 1 shows how we can ‘weigh’ the antiproton by measuring the transition frequencies of j3He’.

1.2. What is pHe+ good for? In addition to being exotic and fun, precision spectroscopy of pHe’ can 1) test three-body QED theories, 2) test CPT (particle-antiparticle mass and charge equality) and 3) contribute to the determination of a fundamental const ant mp. About 10 years ago, when we started our first PHe’ laser spectroscopy attempt, experimental errors on the PHe+ transition frequencies were on the 50ppm, while theoretical errors were about Avth/v order of Ave,,/v 1000ppm. The experimental success3 spurred theoretical efforts, resulting in dramatic improvements. The errors are now typically 10 ppb and are still improving. Meanwhile, the experimental precision has also improved to the similar order4i5i6, and experimental values and theoretical values show good agreement. N

N

64

-mo Figure 2. An anomalous longevity of antiprotons in liquid helium was first discovered at the 12-GeV proton synchrotron a t KEK2.

This good agreement was used to extract the most precise baryonic CPT limit of Imp - mp/mpI< lo-' and IQp QPI / Q p < (particleantiparticle mass and charge e q ~ a l i t y ~ ) ~ . In the near future, the precision of mg may become as good as or even better than that of mp,which is currently knowng to 6mp/mp= 4 . 6 ~ Although pHef can no longer be used to test CPT once we reach this level, we may be able to contribute to the CODATA evaluation of the proton mass (assuming that CPT is not violated at this level).

+

2. Antiprotonic helium atoms

2.1. Serendipitous discovery of pHe+ When a low-energy antiproton is injected into condensed matter, it loses energy by ionization, stops in the target, and annihilates on a nucleus. The whole process typically takes 51ps. It is our common sense that antimatter cannot survive for a long time in ordinary matter, so that we were quite surprised when we observed that about 3 % of antiprotons stopped in liquid helium somehow survive with a mean lifetime of 3ps (Fig. 2)2. We later learned that such antiproton longevity had been theoretically predicted in 196Os1O, but was never experimentally observed until our experiment at KEK. Intrigued by this anomalous longevity phenomenon, we took data at CERN by stopping antiprotons in helium-4 gas, helium-3 gas and solid helium, which all showed longevity. On the other hand, no longevity was observed in other targets such as nitrogen, neon, argon, xenon, etc.'.

-

bThese limits were obtained by combining the laser spectroscopy results, which is roughly proportional t o m p Q Z with the (more precise) antiproton charge-to-mass ratio ( Q p l m p ) measured to 9 X 10Vp1by the TRAP group at CERN LEARE.

65

p4He’atom

VH&+ ion

”2

n =33

-~cc.-cc.-cccc.

2

OJn=4O

--

Capture (n-38)

d

-c.-ccccc.-

-

Radiative Decay lphoton wavoienqlh 300-800nm)

c . -

30

$

-3.4 c .

-3.6

-3.8

Nuclear. Absorption

-ud

-

29

-.

32

31

--

-

-

Metastable state ( z 1 ps) Short-lived state (z-10ns to Ionized state (z ps)

- 10 ps)

28

I

= 30

31

32

33

34

35

36

37

38

39

Figure 3. Level diagram of pHe+ in relation t o that of pHe++. The continuous and wavy bars stand for metastable and short-lived states, respectively, and the dotted lines are for 1-degenerate ionized states.

Analyses of these data combined with theoretical considerations led us to conclude that the longevity must be caused by the formation of antiprotonic helium atoms. 2.2. Remarkable features of pHeS Antiprotonic helium atom is a naturally-occurring antiproton trap which has the following remarkable features1: (1) The atom can ‘store’ an antiproton for more than a microsecond. This ‘longevity’occurs when the antiproton occupies a near-circular ~ and also large e( 2 35). orbit having a large n ( 38) (2) Unlike antihydrogen, it is not at all difficult to make pHe+. Just stop antiprotons in a helium target. Then, about 3% of the stopped antiprotons automatically become trapped in the metastable states. (3) We usually use low-temperature (T 10 K) helium gas as the target. The produced pHe+ atoms collide with the surrounding helium atoms, and are thermalized. Therefore, the antiprotonic helium atoms are already cold. The Doppler width for a typical single-photon transition is about 500 MHz.

-

66

(4) Despite the large M * / m e factor in Eq. 1, n n - 1 transition frequencies are in the optical region around n 40. We have demonstrated already that we can perform laser spectroscopy of PHe'. Note that we are not changing the electronic state as in ordinary laser spectroscopy. We are inducing transitions between different antiproton orbits.

-

Fig. 3 shows an energy level diagram of $He+. The levels indicated by the continuous lines have metastable ( 2 1p) lifetimes and deexcite radiatively, while the levels shown in wavy lines are short lived ( 5 10 ns) and deexcite by Auger transitions to antiprotonic helium ion states (shown in dotted lines). Since the ionic states are hydrogenic, Stark collisions quickly induce antiproton annihilation on the helium nucleus, as indicated in the figure.

3. Laser spectroscopy 3.1. Antiproton decelerator ( A D ) Antiprotons are produced by bombarding a metallic target by 26-GeV protons extracted from CER"s proton synchrotron. Antiprotons are captured in the AD ring at p = 3.57 GeV/c, cooled and decelerated t o p = 100 MeV/c (kinetic energy of 5.3 MeV), and extracted in a short bunch of 100 ns. Each bunch, delivered every 85 s, contains about 2 - 3 x lo7 antiprotons. In our recent experiments, we further decelerate the antiprotons by using a radiofrequency quadrupole decelerator (RFQD: 'inverse linac') to some 100 keV, and stop them in a low-density cryogenic helium target, and a laser beam is injected into the target through a down-stream window, as shown schematically in Fig. 4.

I\

Solenoid magnets Cherenkov detectors

P

I

Quadrupole magnets

I

Quadrupoletriplet

L'er

Cryogenic hellum target

Figure 4.

Experimental layout. Drawing not to scale.6

67

3.2. Resonance detection

Laser spectroscopy of pHe+ works as follows: As shown in Fig. 3, there is a boundary between metastable states and short-lived states. For example, (n, !) = (39,35) is metastable, while (n,!) = (38,34), which can be reached from (39,35) by an El transition, is short lived. Thus, if we use a laser (A = 597 nm in this particular case) to induce a transition from (39,35) to (38,34), (and of course if an antiproton happens to be occupying the (39,35) level at the time of laser ignition), the antiproton is deexcited to the shortlived state, which then Auger-decays to an ionic (ni,li) = (32,31) state within 5 10 ns. The ionic state is then quickly (usually within ps) destroyed by Stark collisions, leading to the nuclear absorption/annihilation of the antiproton. Each annihilation event produces several energetic pions, which can be easily detected with nearly 100 % efficiency using the Cherenkov detectors placed next to the helium target (see Fig. 4). Adding all these together, we expect see a sharp increase in the p annihilation rate in coincidence with the laser pulse. An example is shown in Fig. 5 ((39,35) + (38,34) at 597 nm), which indeed shows a spike at the laser-firing time of 1.8ps, when the laser is on resonance3. This is the first transition we successfully observed in 1993. We used a broad-band ( w 7 GHz) high-power (1 MW/cm2) excimer-pumped N

0.02

E

wg

0.w

0 .j

0.02

.$

:

587.269

0.w

0 .I ............................................

o.a2 0.w

-10 I(37,35) 934) 0 .: .......................

E 0.02

z

1 z

.......................

0.02 587.20

::: 0.w

0

697.30

Wawl.npm (nm)

1

2

9

4

5

lime b)

0 4

Figure 5. First successful observation of laser resonance of antiprotonic helium3, now attributed to the (39,35) -+ (38,34) transition. Left: Observed time spectra of delayed annihilation of antiprotons with laser irradiation of various wavelengths near 597.2 nm. Right: Normalized peak count versus wavelength in the resonance region.

0

OU&

5

(GI&)

Figure 6 . Resonance profiles of antiprotonic helium (left), transition frequencies at various target densities (right)5.

68

Figure 7. Comparisons between experimental result vexp (filled circles with errors) and theoretical predictions Vth obtained assuming mp = mg and Q p = - Q g (squares l 3 and triangles 14).

dye laser for this experiment, resulting in a relative precision of Av,,,/v 50 ppm (5 x Theoretical predictions on the other hand scattered within about 1000 ppm. A breakthrough was made in 1995 by Korobov's first non-relativistic calculation, which improved the accuracy t o some 50 ppm". This was then improved t o 0.5 ppm by including relativistic corrections12. The precisions of theoretical calculations continued t o improve, and they have now reached 5 10 ppb13314. In competition, experimental error bars also continued t o decrease. After the first success, we soon found out that although pHef atoms are fairly stable against frequent collisions with helium atoms, the collisions induce frequency shift and broadening of the resonance lines4. In 2001 we performed a systematic measurement of density shift. As shown in Fig. 6, we measured the resonance centroids at different helium densities and extrapolated to zero densityc. In this way, we reached a precision of 60 ppb in 2001~. Recently, we constructed a radio-frequency quadrupole decelerator (FtFQD)lGwith which we can now decelerate the 5.3 MeV antiprotons extracted from AD t o some 100 keV. This makes it possible to stop antiprotons in a very low density gas target ( w 10l6-l8 atoms/cm3), eliminating the need for the zero-density extrapolation. Fig. 7 shows the present status of N

CIn the resonance profiles (the left panel of Fig. 6), hyperfine structures are visible in some transitions. In fact, each p4He 'level' has a quadruplet structure, due t o the coupling of electron spin, antiproton orbital angular momentum, and antiproton spin. We have studied the pHe+ hyperfine structure by inducing microwave transitions among the hyperfine sublevels15, but I do not have time t o go into details in this talk.

69

the experiment-theory comparison for seven transitions in p4He+ (left) and six transitions in p3He+ (right). Experimental errors include the absolutefrequency calibration uncertainties, and are typically f l O O ppb. The theoretical predictions of Korobov (squares) l 3 and Kin0 (triangles)l4 are mostly within the experimental error bars, but for some transitions there are discrepancies of up to 100 ppb between the two theory values. These differences do not yet affect the final CPT limits, but as the experimental precisions are improved, they will eventually become the dominant error source in the CPT limits deduced from the pHef spectroscopy. In 2004, we started a new series of measurements with a pulse-amplified continuous-wave (CW) laser system, in which the CW laser is stabilized and locked to an optical frequency comb. Chirp compensation and detection are also implemented. Preliminary analyses look very promising, suggesting that we are likely to reach -ppb precisions soon.

-

N

4. Discovery of metastable antiprotonic helium ions

Let us go back again to Fig. 3 and consider the fate of pHef+ ions at very low target densities. The destruction of the pHe++ states usually takes place in a matter of pic0 seconds due to the Stark collisions. If the pH,++ ion is isolated in a vacuum, there are no collisions, and hence the pHe++ states should become metastable (the radiative lifetimes of the circular states around ni 30 is several hundred ns). We therefore expect that the prolongation of pHef+ lifetimes at very low target densities. This is exactly what we recently observed. In the left panel of Fig. 8, we show the annihilation spike produced by inducing the p4He+ transition (n, l ) = (39,35) + (38,34) measured by using the RFQD-decelerated beam at a low target density of 2 x 10l8 atoms/cm3. At this density, the decay

-

2.1

2.2

2.3 2.4 2.5

2.6

2.7

2.8

2.9

3

Elapsed time (p)

Figure 8. Annihilation spike produced by inducing the pHe+ transition ( n , t ) = (39,35) + (38,34), measured at a high target density (a). A prolongation of the tail is observed a t ultra-low densities (b), indicating the formation of long-lived THe++ ions17.

70 '1

1 w o ~ " " I " " ~ " " ~ " " ~

A

-2 K

mpl prea'san (Review of Partick Physics) \ 100

~

-

\

4

c u,

\

m\

10 7

3

'r

D

z

0.1

mp I me predsan (CODATA)

7 \

: h?

\ ! \:

r

. . . . . I . . . . I . . . . I . . , , i .

Figure 9. The mp/meprecisions (CODATA) compared with the mplm, precisions (ASACUSA).

time constant of the laser spike is still consistent with the Auger lifetime of the (38,34) level. However, as shown in the right panel, the shape of the laser-induced spike changes drastically in an ultra-low target density of 3 x 10l6 atoms/cm3. Lifetime prolongation was also observed in the case of antiprotonic helium-3 ions. Systematic measurements of the ion lifetimes at various target densities have been done, which showed that the ionic-level lifetimes get shorter for larger principle quantum numbers. l7

5. Future prospects All CPT-test experiments carried out until now by the ASACUSA collaboration dealt with the pHe+ atoms, heavily relying on the results of (difficult) 3-body QED calculations, which have errors similar to those of the measured laser transition frequencies. The long-lived antiprotonic helium ions p4He++ and p3He++ are quite interesting in this respect, since these are two-body systems and hence are practically free from theoretical errors. This motivates us to perform the laser spectroscopy of PHe++. In principle, this appears possible, since we found that there is up to 50% lifetime difference between the i-jHe++ levels when the principle quantum number of the ion ni is changed by one unit. Hence, if we use a laser to produce an ionic state, and then use another laser (in the UV region) to induce transitions between ni and ni&l,we should be able to observe a slight change in the decay time constant of the laser spike (such as in Fig. 8(b)). This is by no means an easy measurement, but is nevertheless an important one. N

71

At the same time, we will continue to improve the precision of pHe+ (three-body) spectroscopy. We should be able t o reach sub-ppb precision soon, thereby improving the CPT limits on proton-antiproton mass and charge comparison. In order to achieve this, we need continuing efforts of the theory community, so that the existing differences of some 100 ppb in the transition-frequency calculations are diminished. Acknowledgements This work has been carried out working closely with the members of ASACUSA collaboration, in particular, T. Yamazaki, J. Eades, E. Widmann, W. Pirkl, D. Horvath, H.A. Torii and M. Hori. Thanks are also due t o V. I. Korobov and Y. Kin0 for theoretical inputs.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

T. Yamazaki et al., Physics Reports 366,183 (2002). M. Iwasaki et al., Phys. Rev. Lett. 67,1246 (1991). N. Morita e t al., Phys. Rev. Lett. 72,1180 (1994) H.A. Torii et al., Phys. Rev. A 59,223 (1999). M. Hori et aZ., Phys. Rev. Lett. 87 093401 (2001). M. Hori et al., Phys. Rev. Lett. 91 123401 (2003). S. Eidelman et aZ.,. Phys. Lett. B 592,1 (2004). G. Gabrielse et al., Phys. Rev. Lett. 82,3198 (1999). P. J. Mohr and B. N. Taylor, to appear in Review of Modern Physics 76 (2004); also available on web at http://physics.nist.gov/constants. G.T. Condo, Phys. Lett. 9,65 (1964); J.E. Russell, Phys. Rev. Lett. 23,63 (1969). V.I. Korobov, Phys. Rev. A 54,R1749 (1996). V.I. Korobov and D.D. Bakalov, Phys. Rev. Lett. 79,3379 (1997). V.I. Korobov, Phys. Rev. A 67,026501 (2003). Y. Kin0 et al., Nucl. Instr. Methods B 214, 84 (2004). E. Widmann et al., Phys. Rev. Lett. 89,243402 (2002). A. M. Lombardi et al., in Proceedings of the 2001 Particle Accelerator Conference, Chicago, 2001 (IEEE, Piscataway, NJ, 2001), pp. 585-587. M. Hori et al., Phys. Rev. Lett. 94,063401 (2005).

BLOCH OSCILLATIONS OF ULTRACOLD ATOMS: A TOOL FOR METROLOGICAL MEASUREMENTS

s. GUELLATI-KHELIFA, P. CLADE, E. DE MIRANDES,

M. CADORET, C. SCHWOB, F. NEZ, L. JULIEN AND F. BIRABEN Laboratoire Kastler Brossel, Ecole Normale Supe'rieure, CNRS, UPMC, 4 place Jussieu, 75252 Paris Cedex 05, France INM, Conservatoire National des Arts et Me'tiers, 292 rue Saint Martin, 75141 Paris Cedex 03, France// E-mail: [email protected]

In this paper, we show that Bloch oscillations of ultracold atoms in an optical lattice is a strong tool to measure accurately some physical constants. In particular we describe two experimental approaches using a pure and an accelerated vertical standing wave t o perform respectively a determination of the local acceleration of gravity g to ppm precision and of the fine structure constant 01 below 10 ppb.

1. Introduction

Bloch oscillations of atoms in a periodic optical potential can be driven by a constant external force or by accelerating the optical lattice. It is a strong tool to transfer coherently a large number of photon momenta to the atoms '. Combining this method with a velocity sensor based on two 7r-pulse stimulated Raman transitions, we are able to measure, accurately either the external force or the recoil velocity of atoms when they absorb or emit a photon. We have used such a method in a pure vertical standing wave to precisely determine the local acceleration of gravity g. In this case the Bloch period is given by 78 = $ (A is the wavelength of the light and m is the atomic mass). Measuring the Bloch period, we achieve a on the measurement of g. This uncertainty is relative uncertainty of limited by the vibrations of the experimental set-up and the collisions with the background atomic vapor. In a second experiment we have used Bloch oscillations in an accelerated optical lattice to measure the atomic recoil velocity. From this measurement we can deduce the ratio hlm. This ratio 72

73

is related t o the fine structure constant by cx2 =

3:

2Rw AT(X) h --c

Ar(e)

m

where several terms are known with a very small uncertainty: 8 x 10-l' for the Rydberg constant Rw and 4.4 x for the electron relative mass AT(e)6.The Rubidium atomic relative mass A,.(X) is known with a relative uncertainty below 2.0 x 7. The determination of cx using this formula is now limited by the uncertainty of the ratio h/m. 415

2. Experimental set-up

The main experimental apparatus has already been described in reference'. Briefly, s7Rb atoms are captured, from a background vapor, in a u+ - crconfiguration magneto-optical trap (MOT). The trapping magnetic field is switched off and the atoms are cooled to about 3 p K in an optical molasses. After the cooling process, we apply a bias field of w 100 mG. The atoms are then optically pumped into the F = 2 , m F = 0 ground state. The determination of the velocity distribution is performed using a n - n pulses sequence of two vertical counter-propagating laser beams (Raman beams): the first pulse with a fmed frequency bsel, transfers atoms from 5S1/2, IF = 2, m F = 0) state t o 5S1/2, IF = 1,m F = 0 ) state, into a velocity class of about u,/15 centered around (X6,,1/2) - u, where X is the laser wavelength and u,.is the recoil velocity. To push away the atoms remaining in the ground state F=2, we apply after the first n-pulse, a resonant laser beam with the 5S1/2 ( F = 2 ) t o 5P1/2 ( F = 3) cycling transition. Atoms in the state F=l make N Bloch oscillations in a vertical optical lattice. We then perform the final velocity measurement using the second Raman n-pulse, whose frequency is b,,,. The populations ( F = 1 and F = 2 ) are measured separately by using the one-dimensional time of flight technique. To plot the final velocity distribution we repeat this procedure by scanning the Raman beam frequency,,,,S of the second pulse. The Raman beams are produced by two stabilized diode lasers. Their beat frequency is precisely controlled by a frequency chain allowing t o easily switch from the selection frequency (S,,l) to the measurement frequency (b,,,). One of the lasers is stabilized on a highly stable Fabry-Perot cavity and its frequency is continuously measured by counting the beatnote with our two-photon Rb standard The two beams have linear orthogonal polarizations and are coupled into the same polarization maintaining optical fiber. The pair of Raman beams is sent through the vacuum cell.

74

The counter propagating configuration is achieved using a polarizing beamsplitter cube and an horizontal retroreflecting mirror placed above the exit window of the cell. The standing wave used to create the 1-D optical lattice is generated by a Ti:Saphire laser, whose frequency is stabilized on the same highly stable ZFP cavity. This laser beam is splitted in two parts. To perform the timing sequence, each one passes through an acousto-optic modulator to control its intensity and frequency. 3. Bloch oscillations in a vertical standing wave: determination of the local acceleration of gravity

In this first experiment, we study Bloch oscillations in a vertical optical potential under the influence of the acceleration of gravity. We apply a pure vertical standing wave between the two Raman 7r-pulses, during an interrogation time T ~ l ~We ~ h then . study the evolution of the final momentum distribution by changing T ~ l ~ In~ this h . configuration, the atoms fall until they absorb a photon from the upward wave and emit a stimulated photon in the downward wave. The atoms make a succession of A transitions inducing a momentum exchange of 2hk (lc is the wave vector of the standing wave) with the cycling frequency U B = ?$. In the Bloch’s theory, the energy spectrum of the particle presents a band structure (indexed by n) arising from the periodicity of the potential (optical lattice with period d = X/2). The corresponding eigenenergies E,(q) and the eigenstates In, q ) (Bloch states) are periodic functions of the continuous quasi-momentum q , with a period 2k = 27r/d. The quasi-momentum q is conventionally restricted to the first Brillouin zone ] - n / d , 7r/dl. Under the acceleration of gravity, a given Bloch state In, q ( 0 ) ) evolves (up to a phase factor) into the state In, q ( t ) ) according to q ( t ) = q(0)

+ 2k-TBt

(mod 2 7 ~ / d )

The Bloch period TB corresponds to the time required for the quasimomentum to scan a full Brillouin zone. In our experiment, we prepare, first Bloch states around q = 0 (in lattice frame) at the bottom of the fundamental energy band (n = 0) by turning on adiabatically the standing wave (rise time of 300 p ) ,avoiding a population transfer into the higher energy band. We point out that just before turning on the Bloch potential, the selected atoms reach a mean velocity of about 10 v,. In order to compensate this velocity drift, the upward beam’s frequency is shifted by 150 kHz (in the laboratory frame the standing wave is then moving with N

75

a constant velocity of about 10 vr). To use a pure standing wave, we should launch atoms in ballistic atomic-fountain trajectories either from a moving molasses lo or with Bloch oscillations 2 , and turn on the Bloch potential when they reach their summit. After a time T B l o c h , we suddenly switch off the optical potential and measure the final momentum distribution in the first Brillouin zone. In (fig. 1) we report the center of the final velocity distribution as a function of the holding time TBloch. The observed sawtooth shape is the signature of Bloch oscillations (eq. 2). To determine the Bloch

-4.0 -4.2 -4.4 -4.6 -4.8

-5.0 9

10

11

12 TBlcch

65

66

67 68 69

(ms)

Figure 1. The center of the final velocity distribution versus the duration of the standing wave. The dots represent the experimental data and the line the least-square fit performed by fixing the recoil velocity.

period T B , we measure the time interval between the centers of the two extreme slopes of the sawtooth. We extract the value of TB by dividing this time interval by the number of periods. This measurement leads to a determination of the local acceleration of gravity with a relative uncertainty of 1.1 lop6 as ratio and the wavelength X are known with a better accuracy. The linear fit of the experimental data in (fig. 1) is performed by fixing the value of the recoil velocity. The laser beams used to create the optical lattice are detuned by 260 GHz from the 5S1p - 5P3/2resonance line to avoid spontaneous emission, the optical potential depth Vo equals 2.7 ER ( E R= $ is the recoil energy). For this value, when the external acceleration is due to gravity, the transfer of atoms to the higher bands remains insignificant for several periods of Bloch oscillations. However, when we increase the interaction time with the standing wave up to 100 ms, the

2

76

signal is significantly degraded, the loss rate becoming larger than 50%. We have made a rigorous study of these losses and have concluded, that the number of Bloch oscillations, in our experiment, is limited by the collisions with the residual Rb background vapor

4. Bloch oscillation in an accelerated optical lattice: determination of the fine structure constant a In this second experiment we use the same experimental approach to measure the variation of the atomic velocity induced by a frequency chirped standing wave. The acceleration process arises from Bloch oscillations in the fundamental energy band driven by the inertial force seen by the atoms. After N oscillations, we release adiabatically the optical lattice and measure the final velocity distribution which corresponds to the initial one shifted by 2Nv, (z), = h k / m is the recoil velocity). Such measurement leads to a determination of the recoil velocity with an uncertainty uVr = a V / 2 N where uV is the uncertainty on the measurement of the velocity using the two 7r-pulses. The optical lattice is blue detuned by 40 G H z from the one photon transition. It is adiabatically raised (500 p s ) in order to load all the atoms into the first Bloch band. To perform the coherent acceleration, the frequency difference of the two laser beams generating the optical lattice is swept linearly using acousto-optic modulators. Then, the lattice intensity is adiabatically lowered (500 p s ) to bring atoms back in a well defined momentum state. The optical potential depth is 70 E,. With these lattice parameters, the spontaneous emission is negligible. For an acceleration of about 2000 ms-2 we transfer about 900 recoil momenta in 3 ms with an efficiency greater than 99.97% per recoil. To avoid that the atoms from reaching the upper windows of the vacuum chamber, we use a double acceleration scheme: instead of selecting atoms at rest, we first accelerate them using Bloch oscillations and then we make the three steps sequence: selection-acceleration-measurement. This way, the atomic velocity at the measurement step is close to zero. In the vertical direction, an accurate determination of the recoil velocity would require an accurate measurement of the gravity g. In order to get rid of gravity, we make a differential measurement by accelerating the atoms in opposite directions (up and down trajectories) keeping the same delay between the selection and the meaN

77

surement r-pulses. The ratio ti/m can then be deduced from the formula

where (bmeas - 6sel)up/downcorresponds respectively to the center of the final velocity distribution for the up and the down trajectories, NUpldow are the number of Bloch oscillations in both opposite directions, ICB is the Bloch wavevector and kl and kz are the wavevectors of the Raman beams. In fig.2 we present two typical velocity distributions for NuP = 430

0.0

'

'

-1 3.3110 -13.3115 -13.31 20

6, -6 Figure 2.

//

-

13.6145 13.$140 13.$135

'

(MHz)

Typical final velocity distribution for the up and down trajectories.

and Ndown= 460. The center of each spectrum is determined with an uncertainty of 1.7 H z ( w v,/lOOOO) for an integration time of 5 min. Using this double trajectory scheme, we can determine the ratio h l m with an effective recoil number of 2(NuP Ndown)= 1780. The contribution of some systematic effects (energy levels shifts) to dse. or Smea, changes sign when the directions of the Raman beams are exchanged. In order to improve the experimental protocol, we invert alternately the two Raman beams for each trajectory and we take the mean value. Finally one determination of a is obtained with four velocity spectra, leading to a relative uncertainty of about 33 ppb in 20 min integration time. By averaging a set of 72 measurements of the recoil velocity, we are able to get a determination of the fine structure constant a with a relative uncertainty of 4.4 ppb. We are carefully studying the different systematics effects. Actually, we estimate their contributions to (Y-' at 5 ppb. Therefore it could lead to a determination of 0 - l with a relative uncertainty of about 7 ppb.

+

78

Summary We have demonstrated that Bloch oscillation phenomena in an optical lattice has several features for high precision measurements (recoil velocity and acceleration of gravity). We are able to transfer about 1000 photon momenta with a measured efficiency of 99.97% per recoil. To our knowledge, this is the highest number of recoils ever transferred coherently to any physical system. The relative uncertainty on the fine structure constant is close t o the best determination of a based on atomic interferometry 12. The uncertainty on the acceleration of the gravity could be sensitively reduced by increasing the number of cold atoms and improving the vacuum in the cell.

Acknowledgments We thank A. Clairon and C. Salomon for valuable discussions. This experiment is supported in part by the Bureau National de Mktrologie (Contrats 033006) and by the Rkgion Ile de France (Contrat SESAME E1220). The work of E. de Mirandes is in keeping with joint Ph.D thesis between the LENS, (Universitb di Firenze) and Universitk Pierre et Marie Curie and is supported by European community (MEST-CT-2004-503847).

References 1. M. Ben Dahan et al., Phys. Rev. Lett. 76, 4508 (1996). 2. R. Battesti et al., Phys. Rev. Lett. 92, 253001-1 (2004). 3. B. Taylor, Metrologia. 31,181 (1994). 4. C. Schwob et al., Phys. Rev. Lett. 82, 4960 (1999). 5. Th. Udem et al., Phys. Rev. Lett. 79, 2646 (1997). 6. P. Mohr and B.N. Taylor, Rev. Mod. Phys. 77, 1 (2005). 7. M.P. Bradley et al., Phys. Rev. Lett. 83, 4510 (1999). 8. P. Mohr P and B. Taylor Rev. Mod. Phys. 72, 351 2000. 9. P. Clad6 et al., to be publication in Europhys. Lett. 10. A. Clairon et al.,Europhys.Lett 16,(165) 1991. 11. B. de Beauvoir et al., Phys. Rev. Lett. 78, 440 (1997). 12. A. Wicht et al., Physica Scripta T102, 82 (2002).

ASTROPHYSICAL LASER SPECTROSCOPY SVENERIC JOHANSSON Lund Observatory, Lund University, P. 0.Box 43, S-22100 Lund, Sweden VLADILEN LETOKHOV Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow region, 142190, Russia

We present the results of the discovery of optically pumped Astrophysical Lasers (APL) operating on the quantum transitions of the Fen ion and 01 atom in the range 0.8-1.7 pm in gas condensations in the vicinity of Eta Carinae - most luminous and massive variable blue star of our Galaxy.

1. Introduction Stimulated emission concepts proved very useful and productive in astrophysics. This fact was convisingly verified in the microwave region of the spectrum (the discovery of astrophysical masers on a number of simple molecules [l-31). We extends this trend to the visible region along two lines. First, we present the results of our investigations that have, in our opinion, lead to the discovery of optically pumped AstroPhysical Lasers (APL) [4] active in the quantum transitions of the Fe I1 ion [5, 61 and 01 atom [7] in the range 0.8-1.0 pn in the vicinity of Eta Carinae - most luminious and massive star of our Galaxy. The resonance optical pumping of high-lying states in Fe I1 and 01 owes to the accidental coincidence between their absorption lines and the intense HI Lycr and Lyp emission lines, respectively. Secondly, we consider the possibility of measuring the true (sub-Doppler) width of the narrow astrophysical lasers lines by Brown-Twiss-Townes heterodyne correlation interferometry and spectroscopy using spatially-separated telescopes and a diode laser as a local heterodyne coupled by an optical fiber to the telescopes [S]. This can simultaneously ensure a spectral resolution at the level of 1 MHz and angular resolution at the level of 0.001-0.1 arcsec.

79

80

2. The Origin of Population Inversion due to Optical Excitation in Astrophysical Plasmas Consider a four-level scheme (Fig. la) in which the selective excitation of level transition. To better understand the need for selective excitation, consider first the case in which such an atomic system interacts with a radiation field in thermal equilibrium having a temperature of Trd. One should then consider both spontaneous and stimulated emission as well as absorption in the transitions of such a system, as illustrated in Fig. 1. The four-level system, like any atomic system, must reach a population distribution among all four levels following the Boltzmann law for Trd. The 4 always gives rise to population inversion in the 3+2

E

E 4

E

distorted Planck spectrum

BBR

d T w T"En

P

a.

b.

0 P

'

C.

'

d.

Figure 1. (a) Scheme of four-level atom with relevant radiative transitions, showing the radiative interaction between the four-level atom and (b) blackbody radiation (BBR),(c) BBR and PAR radiation absorbed in the 1 4 transition, and (d)distorted BBR with intensity enhancement at the wavelength of the 1-4 transition. (En = energy of level n; Aik, Bit = Einstein coefficients, Trad = radiation temperature, and Tkin= kinetic temperature).

system is in radiative interaction with the resonance frequencies (energy differences) of the equilibrium radiation spectrum (horizontal lines in Fig. 1b) and does not involve any collisions. This fact refers not only to the internal degrees of freedom of the particle, but also to the translational degrees of freedom. This statement requires that only the recoil effect should be taken into account in radiative interactions. Even if some of the quantum transitions only interact very weakly with the radiation field, all the levels will reach a Boltzmann distribution at a rate defined by the strong allowed transitions. For example, let the 3+2 transition in Fig. la

81

be a “semiforbidden“ line with a low transition probability, which is plausible in complex atomic systems. In such a case, the population in state 3 results from the 1 - 4 4 3 sequence of allowed transitions. However, the number of particles accumulated in state 3 will not exceed the equilibrium value, since it will be depleted in the reverse sequence 3+4+1 by stimulated radiative transitions in absorption and spontaneous emission. Even though state 3 is “pseudometastable,” no population inversion can naturally be built up in the 3+2 transition. These basic facts are included here merely to show explicitly that only deviations from the Planck frequency distribution of the radiation can cause level populations to deviate from the Boltzmann distribution and specifically give rise to population inversion in an internal transition. In a Pumping by Accidental Resonance (PAR) process some strong spectral line of one atom (ion) can excite level 4 in another atom (ion) in a selective way at a rate that is high enough to build up a population inversion in the transition 3+2 (Fig. lc). Obviously, the effective spectral temperature, Teflof this pumping radiation must exceed the equilibrium temperature of the atoms (ions) being excited. The equilibrium temperature Tkin in Fig. lc means the kinetic temperature of particles (electrons, ions, etc.) in the astrophysical medium. The rare collisions of these charged particles with the four-level atom (ion) may lead to excitation of the lower level 2 , but it is not significant if Tkin> 1, and then it is called the enhancement factor4. This coupling resembles the interaction - p P(B)6 . B of the magnetic moment p with a magnetic field B, where P(B) accounts for the atomic or molecular structure. It is instructive to compare these two interactions in the case of a free electron with an edm of, say, d, = 5 x exm, just below the present limit. In a 100 kV/cm field the edm energy is so small that it equals the magnetic energy in a field of only 9 x T. Controlling the stray mag129

130

netic field at that level seems close to impossible, especially when applying the electric field. Heavy atoms such as Cs and T1 alleviate this problem by their large enhancement factors. In particular, a ( E ) = -585 for the thallium atom5, which relaxes the necessary field control to the challenging, but achievable fT level. Two magnetic effects are most troublesome. (i) Stray magnetic fields vary both in space and time. (ii) Atoms moving through the large electric field experience a motional magnetic interaction6 - p B . E x v/c2. In both cases the unwanted field components are typically many orders of magnitude larger than fT and heroic efforts were needed to reach the current precision'. 2. Measuring with molecules Heavy polar molecules offer substantial relief from these difficulties7. First, the enhancement factors are generically much larger8 because the electron edm interacts with the polarisation of the charge cloud close to the heavy nucleus. In an atom this polarisation follows from the mixing of higher electronic states by the applied electric field. In a polar molecule, these electronic states are already strongly mixed by the chemical bond and it is only rotational states that have to be mixed by the applied field. Since these are typically a thousand times closer in energy, the molecular enhancement factor is larger. For the YbF molecule used in our experiment, the enhancement factorg is a 21 lo6 at our operating field of 13 kV/cm, which relaxes the requirement on field control to the pT level. There is a second advantage to YbF". Being polar, this molecule has a strong tensor Stark splitting between the Zeeman sublevels, making the applied electric field the natural quantization axis. Hence, the Zeeman shift due to a perpendicular magnetic field is strongly suppressed and the molecule insensitive to the motional field. For our typical operating parameters the motion-induced false edm islo below e.cm, which is negligible.

3. Y b F spin interferometer

-

Our experiment uses a cold, pulsed, supersonic beam of YbF radicals" in a magnetically shielded vertical vacuum chamber 1.5m high. The electronic, rotational and vibrational ground state X 2 C : / , , N = 0, w = 0 is a hyperfine doublet with states F = 0 , l split by 170MHz. We first deplete the F = 1 state by laser excitation of the F = 1 molecules on the aor any system whose tensor Stark splitting greatly exceeds the Zeeman interaction.

131 F=l population (arb.units)

Applied magnetic field (nT) Figure 1. YbF interferometer fringes. Dots: F = 1 population measured by fluorescence. Curve: Calculated fringes with signal scale and offset in both coordinates as free parameters. Variation of fringe visibility is due to the known beam velocity distribution.

X transition. This laser beam is called the pump. An oscillating field, which we call the first beam splitter, then drives the F = 0 molecules into a symmetric coherent superposition of the F = 1 , m = ~ f l states, Al,2 +-

as described later in more detail. Next, parallel dc electric and magnetic fieldsb are applied to introduce a phase shift Aq5 = $ J,'(d, a ( E ) E ( t ) p P(B) B(t))dt between the two superposed states. Here E and B appear as functions of time because they are the fields in the molecular rest frameC. At time T the molecules interact with another oscillating field, the recombining beam-splitter, that couples the symmetric part of the F = 1 coherence back to the F = 0 state. The resulting F = 0 state population exhibits the usual cos2 A4 fringes of an interferometer. We detect the F = 1 population ( 2 ) using fluorescence induced by a probe laser on the A1/2 +- X transition. Figure 1 shows the interference fringes observed in this fluorescence when the magnetic field is scanned.

+

4. The beam splitters We describe two types of beam splitter, both designed t o minimise the distance moved by the molecules whilst they are being split. This is important bThe direction of these fields defines the quantisation axis of our basis. =Variation of field directions can generate geometric phases, neglected here for simplicity.

132

F=l population (arb. units)

170.35

170.4

170.45

170.5

rf pulse frequency (MHz) Figure 2. Lineshape for rf beam splitter. Dots: experiment. Line: eq.1 with 50ps rf pulse length.

as unwanted phase shifts can occur if the ambient fields rotate during splitting. When combined with other imperfections of the apparatus, such a phase can produce a false edm and is therefore undesirable.

4.1. R a d i o frequency splitter The radio frequency beam splitter is an rf magnetic field perpendicular to E along the beam direction. This excites the F = 1 coherent superposition with probability12

where R is the Rabi frequency and S is the detuning of the rf field from resonance. When a pulse of molecules arrives at the centre of the rf loop we subject it to a short rf pulse such that Rt = 7r, i.e. a n-pulse, to induce 100% population transfer at resonance. Figure 2 shows the lineshape measured using 5Ops-long rf pulses, together with a fit to eq. 1, showing quite good agreement. However, the peak transition probability is significantly less than unity. This is due to the beam velocity spread, which makes the gas pulse 5 cm long at the first loop and 10 cm at the second. Since the rf field strength varies along the beam line, this spreading produces a distribution of Rabi frequencies. The effect appears more clearly as a damping of the Rabi oscillations, shown in fig. 3(a), in good agreement with the expected damping from the known magnetic field distribution of the rf loop. The

133

rffield strength (arb. units)

rffield strength (arb. units)

(a)

(b)

F=l population (arb. units)

F=l population (arb. units)

2s

40

~~~

20

30 15

20

10

I0

5 10

20

30

40

rffield strength (arb. units)

(c)

. . : 10

20

30

40

rffieid strength (arb. units)

(dl

Figure 3. Rabi flopping in the rf beam splitter. Units on axes of all the plots are arbitrary but common. The a-pulse occurs at a forward rf power to the loop of around 100mW. The dashed lines in (b) show sections taken for (c) and (d).

maximum fraction of the population transferred to F = 1 is 66%d. The arrival of each YbF pulse at our detector is recorded with 1ps res5 mm olution. This time-resolved data gives us a spatial resolution of at the upper rf loope. This allows us to resolve the inhomogeneous Rabi frequency, as shown in fig.3(b). The plot maps rf transition probability (lighter shade meaning higher probability) versus rf field strength and arrival time at the detector. Contours of constant Rt are clearly seen. At the centre of the loop, corresponding to the dashed line (c) in fig. 3(b), we can drive more than two complete Rabi oscillations, as plotted in fig. 3(c).

-

d A longer rf 7r-pulse increases this fraction by allowing each molecule to sample a longer distance along the beamline, but this is avoided because of the possible systematic errors due to large movements of the molecules. eNote, however, that the molecules move approximately 3cm during the 5 0 p rf pulse.

134

Figure 3(d) shows the much slower Rabi flopping of late-arriving molecules (section (d) of fig. 3(b)) that experienced a weaker rf field. The data follow the expected sinusoidal Rabi-flopping behaviour (lines), demonstrating that the damping in fig. 3(a) is indeed due to rf field inhomogeneity. We are currently working to shorten the rf pulses further so that the molecules move even less distance during the splitting. Preliminary results with new, higher power, rf amplifiers suggest that we can work with pulses less than 10 ps long, corresponding to a beam movement below 6 mm.

4.2. Raman splitter

-

The Raman beam splitter uses two co-propagating 552 nm laser beams with a frequency difference of 170MHz to drive the hyperfine transition. In this case the spatial localisation of the light ensures that the transition occurs in well-defined static fields. An adequate hyperfine transition rate is achieved by tuning the light near the All2 t X transition. At the same time we avoid spontaneous emission from the A l p state as that tends to leave the molecule in a vibrationally or rotationally excited state. We achieve the required detuning by Stark shifting the All2 t X transition. This allows us to use just one dye laser to generate the Raman beams as well as the pump and probe light - a valuable simplification. We apply an electric field of 2.7 kV/cm to detune the transition13 by A = 250MHz, which is 10 natural linewidths. The excitation probability is then well approximated by eq. 1 with the substitution R = RlRz/A, where R1,z are the Rabi frequencies for the two laser fields. Now, b (eq. 1) is the detuning from two-photon resonance and t is the time of flight through the cw Raman splitter. Once again, the velocity spread of the beam makes it impossible to give all the molecules a 7r-pulse. However, the spread is only some 40m/s FWHM on a mean velocity of 580 m/ s, allowing a transition probability of 99.7% to be achieved. Figure4(a) shows the Raman transition lineshape. The width of this line is dominated by Doppler broadening from the transverse momentum spread of the laser beams, an effect that is less well controlled than we would like in our edm measurement. In fig. 4(b) we show the Rabi flopping of the transition, with the intensity held fixed in one beam (the redder one) and varied in the other. Limited laser power restricts us to less than a full cycle.

-

135 F=l population (arb. units)

F=l population(arb. units)

1200

1~~~~ 700

600 500 400

300 1702

600

.J

i

170.4

400 1706

170.8

Laser field strength (arb. units)

171

Raman differencefrequency (MHz)

(a)

(b)

Figure 4. Raman beam splitter. (a) Transition lineshape. (b) Fbbi flopping on the Raman beam splitter transition.

5. Results and outlook

We are now making an electron edm measurement. Our current data set, collected over the last two months is displayed in chronological order in fig. 5. There are 3088 measurements each of which took typically 2 minutes. The small variations in average sensitivity are due primarily to changes in the intensity and stability of the molecular beam and, to a lesser degree, of the laser. In the last two weeks there is a period of sharply increased uncertainty as we start to make checks for systematic errors.

e-edm (e.cm) x lo2’

Figure 5. The current data set. Each of the 3088 points is an independent electron edm measurement over approximately two minutes.

Although we cannot present a value for the edm at this conference, we can make some comments on our data set. Since our last edm measurement lo, gains from a cold, pulsed source1’ and other technical improvements have increased the experimental sensitivity by more than a factor of thirty so that the statistical sensitivity is approximately 10-27e.cm/&. All the data were taken using Raman beam splitters. We are preparing to take new data with the pulsed rf beam splitters once the current systematic checks are complete. As a further control against any

136

false edm we plan to repeat the experiment using CaF molecules. These provide a good null test as CaF is similar to YbF structurally and magnetically, but has 40 times less sensitivity to the electron edm according to the expected Z3 scaling7. We have already performed a trial run and the results look encouraging for making a measurement in the 10-28e.cm range in the near future. Beyond that, we plan to guide and decelerate the bearnl4, to obtain an anticipated further factor of 100 in sensitivity. We acknowledge support from PPARC, EPSRC, the Royal Society and the Cold Molecules Research Training Network of the European Commission. We are indebted to the European Science Foundation for their conference series and to Jon Dyne for expert technical assistance. N

References 1. B.C. Regan et al., Phys. Rev. Lett. 88, 071805 (2002). 2. I.B. Khriplovich and S.K. Lamoreaux, CP violation without strangeness. (Springer, Berlin 1997); Maxim Pospelov and Adam Ritz, arXiv:hep ph/0504231. 3. A.D. Sakharov, Pis’ma ZhETF 5, 32 (1967). [Sov. Phys. JETP Lett. 5, 24 (1967)l; M Dine, A Kusenko, arXiv:hepph/0303065. 4. P.G.H. Sandars, Phys. Lett. 14,194 (1965). 5. Z.W. Liu and H. P. Kelly, Phys. Rev. A 45, R4210 (1992). 6. Karin Sangster et al., Phys. Rev. Lett. 71, 3641 (1993); Phys. Rev. A . 51, 1776 (1995). 7. E. A. Hinds, Physica Scripta T70,34 (1997). 8. P.G.H. Sandars in Atomic Physics 4 ed. G. zu Putlitz, (Plenum, 1975) p.71. 9. M.G. Kozlov and V.F. Ezhov, Phys. Rev. A49, 4502 (1994); M.G. Kozlov, J. Phys. B 30 L607 (1997); A. V. Titov, N. S. Mosyagin, V. F.Ezhov, Phys. Rev. Lett. 77 5346 (1996); H.M. Quiney, H. Skaane, I.P. Grant, J. Phys. B 3 1 L85 (1998)(after correcting for the trivial factor of 2 between s and (T their result becomes 26 GV/cm); F.A. Parpia, J . Phys. B 31 1409 (1998); N. Mosyagin, M. Kozlov, A. Titov, J . Phys. B 31 L763 (1998). 10. J. J. Hudson et al., Phys. Rev. Lett. 89, 023003 (2002). 11. M. R. Tarbutt et al., J. Phys. B 35, 5013 (2002). 12. Norman F.Ramsey, Molecular beams, (Oxford University Press, 1956). 13. P. C. Condylis et al., arXiv:physics/0509084. 14. M.R. Tarbutt et al., Phys. Rev. Lett., 92, 173002 (2004).

Symposium on Cold Atoms and Molecules

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BEC: THE FIRST TEN YEARS CARL E. WIEMAN University of Colorado, Boulder 440 UCB, Boulder, Colorado 80309-0440 Bose-Einstein condensation was first created in a gas on June 5, 1995, thus it was almost exactly ten years old at the date of this conference. In fact, the first public appearance of BEC was made at the ICOLS meeting ten years ago when Eric Cornell presented our results. In this review, I will try to give a brief overview of what has been happening these last ten years. I will concentrate on a discussion of the experiments, but one reason for the rapid excitement and growth in BEC has been the close coupling between experiment and theory. A big reason for this coupling is because the interactions in the BEC have a uniquely simple form for a many body quantum system, and hence, are very tractable theoretically. The implications of this are that the comparison of theory and experiment provides a very rigorous test of theoretical ideas, and the theory has good predictive capabilities that can guide experiments in interesting new directions. This paper will be divided into two sections. The first will provide a very brief historical background on BEC and how we got there, and the second part will look at some examples of the most notable physics that has been done with BECs in the past ten years.

1. Introduction The main challenge of a retrospective review like this is that most of the readers are paying attention to whether their names are mentioned, and if they are not mentioned where they should be, they are so irate they do not pay attention to any of the rest of the article. This ends up with an article that is mostly a tedious long listing of names, but one is still guaranteed to m i s s some. This is particularly true in a field like BEC that has grown so rapidly and so successfully that there have been dozens if not hundreds of significant contributors. So my solution to this challenge is to simply not credit anyone, and discuss nothing but the science. I will say that there are two small exceptions to this: (1) you will see I mention two people who I think are unsung heroes of this field in that their work was very pivotal but was so far ahead of its time they never really received much recognition for it, and (2) I credit the people whose data I use in the figures (note that these choices are often made more by their aesthetic value and ease of procurement rather than their scientific importance, and thus are somewhat weighted more towards work coming out of our group).

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140

The history of BEC starts in 1924 when Bose was working out the BoseEinstein statistics while studying behavior of photons. In 1924 and 1925, Einstein applied these statistics to an ideal gas of atoms and ended up predicting Bose-Einstein condensation. This condensation, of course, is when one has a sample of gas atoms in a container which are sufficiently cooled so that the DeBroglie wavelength of the adjacent atoms start to overlap. This overlap results in a large fraction of atoms going into the ground state of the container, thereby giving them a macroscopic quantum wave function. Einstein never really appreciated the fascinating properties such a quantum wave function would have; it was many decades after his work, starting with the ideas of London proposing that BEC might be connected with the behavior of superfluid liquid helium, that this understanding developed. After a long gap, the story picks up again in 1976 when it was proposed that a gas of spin-polarized hydrogen could form a gaseous Bose-Einstein condensation. In fact, the reasoning behind this was wrong but it was an inspiring idea that got the field started. This idea led to a number of investigators worlung on experiments to try and make BEC in spin-polarized hydrogen starting around 1980. During the early to late 80s, they then carried out many experiments, learning a great deal, and in particular, developed two notable technologies: magnetic trapping of very cold atoms, and the idea of evaporative cooling. This idea of evaporative cooling in which you could actually increase the phase space density by throwing away atoms - a very nonintuitive idea - was created by Harold Hess. He left the field of BoseEinstein condensation well before we achieved it, but his idea was critical in the actual creation and study of Bose condensations. Parallel to this work on attempts in hydrogen to get Bose-Einstein condensation, there was a great deal of work on laser cooling and trapping of atoms starting in 1978 with laser cooling of ions. During the 8Os, there were many developments on laser cooling and trapping of neutral atoms. That brought us to the idea in 1990 of combining technologies to produce Bose-Einstein condensation in a different atom. The key ideas that came from work in spinpolarized hydrogen were: (1) the inspiration to pursue BEC; (2) the ideas of evaporative cooling; and (3) a much better understanding of the difficulties and the physics that hindered the achievement of gaseous BEC. From the laser cooling and trapping side came (1) ways to achieve very low temperatures and very large increases in phase-space density (a factor of 10l6); (2) optical techniques for good diagnostics of what the atoms were doing; (3) a cheap and easy way to get this cooling to high phase space density; and finally (4), an understanding of the photon-induced processes that limited the phase space density in laser cooled trapped samples. Combining these ideas is what led Eric

141

Cornell and me to have the idea in 1990 that maybe hydrogen was not the best atom, and that one would do better by having a hybrid technology where one would use a heavier, fatter, and necessarily much colder atom than hydrogen, but combine both the laser cooling and trapping with magnetic trapping and evaporative cooling. At that time, however, it was far from guaranteed that this would be successful. The primary uncertainty was that there were some key properties; in particular, the collision properties of ultra cold alkali atoms, which were totally unknown. This made it very unclear whether or not evaporative cooling of these laser cooled alkali atoms would work and whether or not the BEC would self destruct as it formed. In fact, there were some fairly prominent critics of this idea, which did make it more difficult to get funding to try to pursue this work. But as we pursued this idea over the years, we collected more experimental data on these collision properties, and it became increasingly apparent that it was likely this idea might work. That also led to a number of competitors getting in and pursuing the same goals. To make a long story short, as you all know, ultimately this combination of laser cooling and trapping of atoms and then magnetic trapping and evaporative cooling in a cell at very low pressure led to the clear observation of Bose-Einstein condensation with the three now-famous peaks showing the condensate arising out of thermal gas (Figure 1)'. This of course started lots of activity, many experimental results, a Nobel Prize, etc.

Figure 1.

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Let’s now jump forward to the current status of the experimental technology. The method for creation of BEC is much the same as in the original work. In an overwhelming fraction of the cases, one still uses laser cooling and trapping, followed by magnetic trapping, and then evaporative cooling. Many different geometries of magnetic traps have been developed, and it is clear from reading the literature that each one of these geometries is clearly the best! The number of atoms in the condensate quickly rose to several times lo6 atoms, but further progress to larger condensates slowed. There are some fairly fundamental reasons for this relating to the basic evaporative cooling process, and if we are to have much larger production rates for condensates, I believe it is going to require another brilliant young physicist like Harold Hess to come up with a fundamentally new approach to making condensates that will get past the barriers that are set by the physics involved with the current approach. Optical traps have become common tools for manipulation of condensates using focused far-offresonance laser beams, and they have found some use in the trapping and evaporation to condensation. BEC has now been achieved in many different atoms including all of the stable alkali atoms, hydrogen, metastable helium, ytterbium, chromium and mixtures of a variety of these different atoms. There seems to be little fundamental limitation on the type of atom. Primarily, it’s an issue of how much work one wants to do on the technology of the lasers needed for the laser cooling and trapping step.

2. Major Physics Advances There have now been literally thousands of papers on BEC looking at enormous number of different aspects of the physics. Here I want to just pick out a few particular examples that are notable in the new physics one has learned through research with BEC. This is of course a somewhat arbitrary list but it is arrived at by a consensus of experts; both of us agreed on the choice. This list includes: (1) looking at the shape and oscillation frequencies of condensates to check the Gross-Pitaevslui (G-P) equation; (2) the behavior of mixtures of different atoms or spin states or condensation; (3) vortices; (4) interference of condensates; (5) BEC in optical lattices; and (6) Feshbach resonance physics. In this necessarily highly abbreviated discussion I will explicitly leave out much of both BEC and lattices and the degenerate Fermi gas spinoff of BEC work, because these subjects are currently of great activity, and there are many other articles in this volume on these topics. The first physics that was done with BEC was to check the G-P equation (Eq. l), and it is interesting from a historical perspective to realize that when we

143

were starting in this field, this equation, which is now always taken as the gospel, had not been checked.

where u =-47zji2a is the self-interaction potential. m Of particular interest in checlung the G-P equation was to look at the self interaction term, which, as noted in the introduction, has a particularly simple form: it involves just a bunch of constants times a single scalar, a, the s-wave scattering length, which depends on the atom. The fiist experiments to check the validity of this equation involved simply measuring the shape of the cloud observed and how it expands, and this was found to match well with the G-P predictions. Much higher precision tests came from collective excitations of the condensate. These collective excitations were excited by poking the cloud with a magnetic force and basically just watching it quiver. One can make precise measurements of the frequency at which it oscillates in and out, and compare them with the predictions of the G-P theory, and they agree extremely well. A second area of research that resulted in a number of somewhat surprising observations and new physics that connects up well with other areas of physics has been looking at the, multi-component condensation, primarily mixtures of

Figure 2.

spin states. Of course, being non identical, a different spin orientation in a magnetic field is a different BEC. The study of the interactions and the controlled and spontaneous couplings of these different components has led to a variety of interesting experiments. Figure 2 gives two examples of this, the first (figure 2.a2) is loolung at condensates in two different F-states, where one can see

144

how the two states separate with the 2 state being squeezed inside a shell of the 1 state that forms around it. When one looks at two different m-levels of the same F-state held in an optical trap (Figure 2b3) even more complex structures emerge. One has the domain structures of the different spins spontaneously separating out. There have been a variety of interesting experiments and theory on the dynamics and behavior of these multi-component systems. Another area of interest has been overlapping two different condensates that are identical species and seeing interference between them. They really do interfere as quantum wave functions. In Figure 3a4 we show that historically well known figure from th MIT group in which they overlapped two condensates and saw clear interference fringes. Figure 3b5, from a few years later, shows how the experimental capabilities have rapidly progressed. It shows very high contrast interference fringes created by moving one condensate across the other. Interference of condensates has now progressed to the point where people are now building rudimentary atom interferometers using BEC and this will no doubt continue to rapidly progress in the future. If interfering two BECs is good, then interfering more of them is better. This was done by loading a condensate into the potential wells created by a onedimensional standing wave. The separate wells then give one several different coherent condensates. When the standing wave is vertical, a series of downward going pulses were observed coming out of the sample. This happens because of

0

05

1

Figure 3. a) First observation of two condensates interfering(MlTBEC group 1997); b) a few years later (NIST BEC group).

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the phase shift between the coherent condensates in the different wells, due to the different gravitational potential energy. This was the first experiment done with the simplest possible optical lattice, but it was only the beginning of very extensive work done in lattices. There continues to be a great deal of work done in condensates in two- and three-dimensional optical lattices, as well as one-dimensional. This work is discussed extensively throughout the talks and poster presentations in this volume, so I won’t attempt to do justice to this field here. Much of the interest arises from the fact that these are extended quantum states with controlled couplings and coherences between the condensates at different lattice sites. One can create some very interesting models of condensed matter systems, accurately observe their behavior, and compare with theoretical predictions. Probably the most well known example of this is the observation of the MOT insulator transition of the Bose-Hubbard model. This was first proposed by theorists and then beautifully seen in experiments. Experiments showed that if the optical wells were shallow, there was coherence between sites and there was one phase. When the optical wells were made deeper, a phase transition occurred where there was no phase coherence between the different sites, and the MOT insulator phase was produced. This particular example and much of the other work on condensates and optical lattices has brought together the atomic and condensed matter physics communities.

a. 1 vortex

b. Stirring with optical beam Multiple vortices.

MIT BEC group

c. many vortices Figure 4.

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The next area of BEC physics is the study of vortices. This is, of course, what one has when you put momentum into a condensate. Because it is an extended quantum state, it can only have angular momentum by transforming the wave function topologically via the creation of a vortex. Studies have proceeded from looking at one vortex, to several, to many. This progression is shown in Figure 4. You see in Figure 4a6 the image of the first vortex that was created using a wave function engineering technique. I have always liked this particular case because it was possible to actually measure the phase of the wave function around the vortex and observe that it went through a two-pi phase shift as required by quantum mechanics. Theorists said for some time that vortices ought to be easily created in Bose condensates, but it took a bit of time to figure out good ways to create and see them. Probably the key idea was the realization that, although the core of the vortex is too small to be optically observed in a stable condensate, if one lets the condensate expand the core expands much more rapidly than the condensate as a whole, and then the core becomes observable. With this idea and the idea of stirring the condensate with optical beams, one can see multiple vortices forming structures as shown in Figure 4b’. As this field progressed further, with optical stirring or spinning magnetic trapping potentials, numerous vortices could be produced, as shown in Figure 4c. There have now been extensive studies on the formation of vortices, the structure of vortex arrays, and the dynamics of the vortex behavior, excitations, etc. This has been a rich area of physics that again connects with condensed matter physics because of the connection to the study of vortices that has been going on for many years in superfluid helium and superconducting systems. The final area of physics I want to discuss is Feshbach resonance physics with BECs. Here one uses magnetic fields to control the interactions in the BEC. Verhaar in 1992 first realized theoretically that cross sections and the scattering length in ultracold atoms could have magnetic field resonances. These are now usually referred to as Feshbach resonances, although perhaps Verhaar resonances would be a more appropriate term. At that time, well before the discovery of BEC, Verhaar realized that if BEC could be created, these resonances would be a valuable tool for controlling the self interactions by allowing one to adjust the value of the scattering length. It was some years before this became possible, but Verhaar’s initial ideas have turned out to be completely correct, and this has become a very useful tool. The basic idea behind a Feshbach resonance is that the atoms have many diatomic bound states and the energy of the bound states particularly the highest vibrationally excited state - is magnetic field dependent. As one changes the magnetic field, one create a condition in which a new bound state appears. Essentially, as a function of magnetic field, one moves a virtual

147

state out of the continuum down through zero binding energy and then it becomes a true molecular bound state. As this new state appears, the scattering length goes through a shift from being very large and attractive as one approaches the resonance to extremely large and repulsive on the other side of the resonance. The scattering length then decreases further away from the resonance, usually going slowly through zero far from the resonance value. Thus, by choosing the magnetic field, one can choose the interaction strength in the condensate. Also, this provides a very weakly bound molecular state near the Feshbach resonance, and so one can also create molecules by coupling into this state. All of these features have now been observed and used in experimental studies. Figure 5as is an illustration of how, as one changes the magnetic field, one can make the scattering length increasingly large. This causes the condensate to expand as shown due to the interaction term in the G-P equation becoming more repulsive. Alternatively, in other experiments, the magnetic field is changed to make the condensate attractive, and this led to observations of such features as the Bosenova collapse and explosion, shown in Figure 5b9. The Feshbach resonances have also been used to create molecular condensates in a variety of ways by jumping suddenly near the resonance and then quickly away to couple the atomic and molecular condensate states, or by ramping adiabatically across the resonance from the side where there is no molecular bound state to where there is a molecular bound state. This couples a large fraction of the atomic sample into the weakly bound molecular state. My group is currently studying these processes in a variety of ways including a new observation that one can be well above the Feshbach resonance and couple with a resonant oscillating magnetic field to create molecular condensates. Probably the most exciting aspect of Feshbach resonance physics is the realization that this control of the interaction in degenerate Fermionic systems can allow Fermionic pairing to form Bose condensation. This exciting area of research is discussed in several articles in this volume.

148

a. making repulsive

b. making attractive

165 G

290 a,

B

550%

3000 a, 8000ao

*i +

Figure 5. Using Feshbach resonances to change BEC interactions.

In summary, ten years of BEC in dilute gases has seen the launch of a new subfield of research that is involved in direct manipulation and observation at the wave function level. There have been thousands of scientific papers generated in this field, there is a strong coupling of experiment and theory, and this field has bridged the atomic/laser physics and condensed matter fields. BEC has gone from an exotic research material to an almost routine production. (By my count, for example, there are now nine machines operating at JILA that can make condensates.) Although there is still much additional interesting science to be done, the field has matured sufficiently that it has started to move into applications. This is likely to be an area of important work in the next ten years.

References 1. E. A. Cornell and C. E. Wieman, Scientijk American 278,26 (1998). 2. D. S. Hall, M. R. Matthews, J. R. Ensher, C. E. Wieman, and E. A. Comell, Phys Rev Lett 81, 1543 (1998). 3. H. J. Miesner, et al., Phys Rev Lett 82,2228 (1999). 4. M. R. Andrews, et a/., Science 275, 639 (1997) 5. J. Simsarian, e t a / . ,Phys Rev Lett 85,2040 (2000). 6. M. R. Matthews, et al., Phys Rev Lett 83,2498 (1999). 7. K. Madison, et al., Phys Rev Lett 84,807 (2000). 8. S. L. Cornish, et al., Phys Rev Lett 85,1795 (2000) 9. E. A. Donley, et al., Nature 412,295 (2001).

OBSERVATION OF BOSE-EINSTEIN CONDENSATION IN A GAS OF CHROMIUM ATOMS *

AXEL GRIESMAIER, JURGEN STUHLER AND TILMAN PFAU 5. Physikalisches Institut Universitat Stuttgart 70550 Stuttgart, Germany E-mail:[email protected] e

We have observed Bose-Einstein condensation of chromium atoms [l], whose large magnetic dipole moment is unique among all species that have been Bosecondensed so far. The arising magnetic forces are of anisotropic and long-range character and therefore introduce a novel type of interaction in the physics of ultra, cold quantum gases. In addition, it is expected that the character of the interaction present in a chromium BEC can be varied from mainly contact t o purely dipolar utilizing one of the recently observed Feshbach resonances in 52Cr-collisions.

1. Introduction Although Bose-Einstein condensation occurs also for non-interacting bosons, the essential properties of degenerate quantum gases depend on range, strength and symmetry of the interactions present. In previously realized Bose-Einstein condensates (BECs) in weakly interacting atomic gases [2, 3, 4, 5 , 6, 7 , 8, 91 , interactions are dominated by short-range isotropic potentials. Such type of interaction is responsible for many interesting phenomena. For an overview see e.g. [lo, 111. Using Feshbach resonances to tune the contact interaction, the collapse and explosion ("Bosenova") of Bose-Einstein condensates has been studied [12], new types of quantum matter like a Tonks-Girardeau gas have been realized [13] and molecular Bose-Einstein Condensates [14] have been produced. Due to the unique electronic structure of the transition metal chromium, one can not only tune the short-range contact interaction in a chromium BEC using one of the recently observed Feshbach resonances [15] but also investigate the effects of long-range anisotropic dipole-dipole interactions. 'This work is supported by the SPP1116 of the German Science Foundation (DFG). 149

150

The spins of the six electrons in the 3d and 4s valence shells of the 7S3 ground state of 52Cr (electronic configuration: [Ar]3d54s1) are aligned. This gives rise to a total electronic spin quantum number of 3 and a magnetic moment as large as p = 6 p g , where p g is the Bohr magneton. This large magnetic moment is responsible for a very strong anisotropic magnetic dipole-dipole interaction (MDDI) between 52Cr atoms. Since the MDDI scales with the square of the magnetic moment, it is a factor of 36 higher for chromium than for alkali atoms. For this reason, dipoledipole interactions which have not yet been investigated experimentally in degenerate quantum gases will become observable in chromium BEC. A well pronounced modification of the condensate expansion depending on the orientation of the magnetic moments has already recently been shown[l6]. It becomes visible when carefully looking a t the expansion of the condensate after release from an anisotropic trap. Depending on the orientation of a homogeneous magnetic offset field on the order of a few Gauss with respect to the geometry of an anisotropic trap, the aspect ratio of the condensate after ballistic expansion is different. The experimentally measured expansion is in excellent agreement with the previously predicted behaviour [17]. Many other phenomena arising from the dipole-dipole interaction like the occurrence of a Maxon-Roton in the excitation spectrum of a dipolar BEC [18] or new kinds of quantum phase transitions [19, 201 have been predicted. These can be investigated experimentally when a Feshbach resonance is used to tune the contact interaction between 52Cr atoms close to zero, thus allowing to realize a regime in which the MDDI is the dominant interaction (dipolar BEC) [21]. Since the MDDI is tuneable [22], too, a degenerate quantum gas with adjustable long- and short-range interactions can be realized. Chromium is also unique with respect to technical applications of degenerate quantum gases. It is a standard mask material in lithographic processes and a well suited element for atom lithography [23]. Chromium has already been used to grow nanostructures on substrates by direct deposition of laser-focused thermal atomic beams [24, 251 and structured doping has been demonstrated by simultaneously depositing a homogeneous matrix material and laser-focused chromium [26]. Using a BEC as a coherent source of atoms (atom-laser) instead of incoherent thermal atomic beams promises t o increase the potential of atom lithography.

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2. Experimental findings

We achieve Bose-Einstein condensation of chromium atoms confined in a crossed optical dipole trap formed by one beam propagating in horizontal and a second one in vertical direction. The detailed preparation scheme will be described further below. The onset of quantum degeneracy becomes visible in absorption images of the releaseded chromium cloud. Figure 1 shows the density profiles of the cloud after 5ms of free expansion. Before

Figure 1. Density distributions from absorption images taken after 5ms of ballistic expansion. a) thermal cloud a t 1.1pK, b) two-component velocity distribution slightly below Tc,c) nearly pure condensate with -50,000 atoms.

releasing the atoms from the trap, they were held at different final powers of the horizontal trapping beam and therefore different temperatures of the atomic cloud. Figure l a ) displays the Gaussian profile of a purely thermal distribution corresponding to a temperature of -1.1 pK, very close t o degeneracy but still above the critical temperature Tc. The two component distribution in b) indicates the presence of a Bose-Einstein condensate. The temperature of the remaining thermal part of the cloud is 625nK and the cloud was released from a trap with a final power of 650 mW in the horizontal beam. Figure 1 c) shows an almost pure condensate with -50,000 atoms and a non-Gaussian distribution at a final power in the horizontal beam of ~ 3 7 0 m W The . critical behavior at the point of emerging degeneracy becomes obvious when we determine the number of atoms in the condensate and the thermal fraction separately for different final laser powers. These numbers can be obtained by fitting a two-dimensional two-component distribution function to the density profiles of the clouds. If the trapping potential is well known, the temperature cloud can be as well calculated from the width of the Gaussian. In figure 2, the fraction of condensed atoms in the total number of atoms ( N o / N )is plotted versus the ratio of the temperature of the remaining thermal part to the critical temperature ( T / T c ) . When we approach the critical temperature from above (T/Tc > l), the kink in the condensate fraction plot marks the onset of Bose-Einstein condensation and provides an experimental value for the critical Temperature

152

Figure 2. Condensate fraction ( N O I N )dependence on temperature relative to the transition temperature of an ideal gas ( T / T g ) ,TZ M 0 . 9 4 F N 1 I 3 .Triangles represent repreB

sent the measured data. Black circles represent the predicted fraction

No = 1 - ($-)3 where TC = Tg 6TFt 6 T g . 6Tg = -0.734N-1/3Tg W is a shift in the critical temperature due to the finite number of atoms and 6TAnt = -1.33--"N1I6Tg takes aHO into account the contact interaction [27]. Here a = 105ao is the chromium scattering length [15],a0 being Bohr's radius, a ~ isothe harmonic oscillator length, T is the temperature of the thermal cloud, w is the geometric and B the arithmetic mean of the trap frequencies. The dashed curve shows the dependence for the ideal gas. Uncertainties in the measurement of the trap frequencies, the determination of the number of atoms and temperature contribute to the errorbars.

+

+

of Teap-700 nK. Based on the trap frequencies, number of atoms and temperature, we have also calculated the expected condensate fraction when finite size effects as well as a correction arising from the contact interaction [27] are taken into account. These expected values are represented by black dots in figure 2 and demonstrate a good agreement of our data with the predicted dependence.

3. Experimental methods Reaching the regime of quantum degeneracy of chromium requires a cooling strategy which is adapted to its special electronic and magnetic properties

153

and to the need to circumvent relaxation processes originating from the dipolar character of the atoms. It makes use of magneto-optical, magnetic and optical trapping techniques. A beam of chromium atoms is generated by a high temperature effusion cell at 1600°C and slowed down by a Zeeman slower. The atoms are continuously loaded into a Ioffe-Pritchard magnetic trap [28] and Doppler-cooled in the fully compressed magnetic trap [29]. With this technique, we prepare about 1.3 x 10' atoms in the energetically highest projection mJ = +3 of the 7S3 ground state. Subsequently, the atoms are cooled by radiofrequency (rf) induced evaporation. Within a rf-ramp of 13s, the atoms are cooled to a phase space density of before increasing loss due to dipolar relaxation[30] causes the rfevaporation to become ineffective and prevents 52Cr from forming a BEC in a magnetic trap. This loss mechanism can be overcome if the atoms are transferred into the energetically lowest Zeeman substate mJ = -3 where energy conservation suppresses dipolar relaxation if the Zeeman splitting is much larger than the thermal energy of the atoms. Because the potential energy of Chromium atoms in states mJ < 0 is lower at higher magnetic fields, they are repelled from regions of low magnetic fields and can not be magnetically trapped. We therefore adiabatically transfer the atoms into an optical dipole trap where the trapping forces are independent of the Zeeman substate. A 20 W fibre laser at 1064nm is used to form an optical trapping potential by splitting up this laser in two beams which can be independently controlled by two acousto-optical modulators (AOMs). The beams are focussed on the center of the magnetic trap and form a steep trapping potential in the region where they cross each other with high trap-frequencies of more than 1kHz in all directions. The setup is shown schematically in figure 3. The stronger horizontal trapping beam has a waist of 30 pm and a power of up to -9 W. Ramping up this beam to its maximum intensity during the final step of the RF ramp, we are able to transfer 1.8 x lo6 atoms into the optical trap. The trapping-potential formed by this beam has a depth of 130 pK and trap frequencies of 1450Hz in radial and 12Hz in axial direction. To purify the polarization and prepare the atoms in the energetically lowest Zeeman state mJ = -3, we optically pump the atoms for 10 ms using 250 pW of blue light from a frequency doubled diode laser system resonant to the ?S3 + ?P3 transition at -427nm. To split up the Zeeman levels, this pumping step is performed at an offset field of 9 G. The efficiency of the transfer is close to 100% and is reflected in a dramatic increase of the lifetime of the trapped gas from 7 s in the mJ = +3 state to >140s in the mJ = -3 state. Keeping

-

154

vertical beam w0=50r1m 0ma* *.5w ,,U -22pK

.

horizontal beam

Figure 3. Schematic setup of the optical dipole-trap. The reader's direction of view is also the direction in which absorption images of thecloud are taken.

lo2

cooling chromium to degeneracy I

'

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'

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'

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-

horizontal beam reduced while dimple ramped up during 1st 5s

optical trap

number of atoms Figure 4. Evolution of the phase-space density during the cooling steps. Phase space density plotted over the number of atoms in the trap.

155

on the offset field during all steps of the preparation scheme, prevents the atoms from thermally redistributing among the other Zeeman states. After the transfer into the optical trap, the power of the trapping lasers is kept at a constant value for 5 s. During this stage of plain evaporation, the number of atoms in the trap drops by -50% and the phase space density increases to lo-*. To increase the local density and the elastic collision rate, a second beam in vertical direction with a waist of 50pm and a power of -4.5 W is ramped up adiabatically during these 5s. About 300,000 atoms are trapped in this crossed trap. Forced evaporation towards the critical temperature for the condensation proceeds now by gradually reducing the intensity of the horizontal beam within 11s. Degeneracy is reached at a remaining power of ~ 8 0 0 m Win the horizontal beam. After holding the atoms in the final trapping potential for 25ms, we switch off both beams simultaneously and let the cloud expand freely for a variable time of flight. The cloud is then detected using a standard absorption imaging technique with a resonant probe beam propagating in the horizontal direction, perpendicular to both trapping beams. For the imaging, the magnetic field is rotated into the probe beam direction just before releasing the atoms from the trap. Figure 4 shows the evolution of the phase-space density of the trapped cloud during the whole preparation process. In conclusion, we have demonstrated the Bose-Einstein condensation in a gas of chromium atoms. This is the first condensation of a transition metal and - due to the large magnetic moment of chromium atoms - offers access to studies of long-range dipole-dipole interaction in degenerate quantum gases. We are able to produce condensates with up to 100.000 condensed atoms which is a very good basis for further experiments. We expect that both short- and long-range interaction can be tuned by external magnetic fields. We also expect that a chromium BEC will have technical applications as a coherent source of atoms in lithographic processes. N

Acknowledgments We thank all members of our atom optics group for their encouragement and practical help. We thank Luis Santos, Paolo Pedri, Stefan0 Giovanazzi, and Andrea Simoni for stimulating discussions. This work was supported by the SPP1116 of the German Science Foundation (DFG).

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References 1. Axel Griesmaier, Jorg Werner, Sven Hensler, Jiirgen Stuhler, and Tilman Pfau. Bose-Einstein condensation of chromium. Phys. Rev. Lett., 94:160401, 2005. 2. M. H. Anderson, J . R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell. Observation of Bose-Einstein condensation in a dilute atomic vapor. Science, 269:198, 1995. 3. K. B. Davis, M.-0. Mewes, M. R. Andrews, N. J . van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle. Bose-Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett., 75:3969, 1995. 4. C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet. Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions. Phys. Rev. Lett., 75:1687, 1995. 5. D. G. Fried, T. C. Killian, L. Willmann, D. Landhuis, S. C. Moss, D. Klepp ner, and T. J. Greytak. Bose-Einstein condensation of atomic hydrogen. Phys. Rev. Lett., 81:3811, 1998. 6. G. Modugno, G. Ferrari, G. Roati, R. J. Brecha, A. Simoni, and M. Inguscio. Bose-Einstein condensation of potassium atoms by sympathetic cooling. Science, 294:1320, 2001. 7. A. Robert, 0. Sirjean, A. Browaeys, J . Poupard, S. Nowak, D. Boiron, and C.I. Westbrook A. Aspect. A Bose-Einstein of metastable atoms. Science, 292:461-464, 2001. 8. Tino Weber, Jens Herbig, Michael Mark, Hanns-Christoph Nagerl, and Rudolf Grimm. Bose-Einstein Condensation of Cesium. Science, 299:232235, 2003. 9. Y. Takasu, K. Maki, K. Komori, T. Takano, K. Honda, M. Kumakura, T. Yabuzaki, and Y. Takahashi. Spin-singlet Bose-Einstein condensation of two-electron atoms. Phys. Rev. Lett., 91(4):040404, 2003. 10. M. Inguscio, S. Stringari, and C.E. Wieman, editors. Proceedings of the International School of Physics Enrico Fermi, Course CXL. 10s Press, Amsterdam, 1999. 11. L. P. Pitaevskii and Sandro Stringari. Bose-Einstein Condensation. Oxford University Press, Oxford, 2003. 12. E. A. Donley, N. R. Claussen, S. L. Cornish, J. L. Roberts, E. A. Cornell, and C. E. Wieman. Dynamics of collapsing and exploding Bose-Einstein condensates. Nature, 412:295, 2001. 13. B. Paredes, A. Widera, V. Murg, 0. Mandel, S. Folling, I. Cirac, G. V. Shlyapnikov, T. W. Hansch, and I. Bloch. Tonks-Girardeau gas of ultracold atoms in an optical lattice. Nature, 429:277, 2004. 14. R. A. Duine and H. T. C. Stoof. Atom-molecule coherence in bose gases. Phys. Rep., 396(3):115-195, June 2004. and references therein. 15. J. Werner, A. Griesmaier, S. Hensler, A. Simoni, E. Tiesinga, J. Stuhler, and T. Pfau. Observation of Feshbach resonances in an ultracold gas of 52Cr. Phys. Rev. Lett., 94:183201, 2005.

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16. Jiirgen Stuhler et al. Magnetostriction in a degenerate quantum gas. in preparation, 2005. 17. S. Giovanazzi, A. Gorlitz, and T. Pfau. Ballistic expansion of a dipolar condensate. J . Opt, B: Quantum Semiclass. Opt., 5:S208-S211, 2003. 18. L. Santos, G. V. Shlyapnikov, and M. Lewenstein. Roton-maxon spectrum and stability of trapped dipolar Bose-Einstein condensates. Physical Review Letters, 90( 25) :250403, 2003. 19. K. G6ra1, L. Santos, and M. Lewenstein. Quantum phases of dipolar bosons in optical lattices. Phys. Rev. Lett., 88(17):170406, 2002. 20. S. Yi, L. You, and H. Pu. Quantum phases of dipolar spinor condensates. Phys. Rev. Lett., 93:040403, 2004. 21. M. Baranov, L. Dobrek, K. G6ra1, L. Santos, and M. Lewenstein. Ultracold dipolar gases - a challenge for experiments and theory. Physica Scripta, T102:74-81, 2002. and references therein. 22. S. Giovanazzi, A. Gorlitz, and T. Pfau. Tuning the dipolar interaction in quantum gases. Phys. Rev. Lett., 89:130401, 2002. 23. M. Oberthaler and T. Pfau. One-, two- and three-dimensional nanostructures with atom lithography. J. Phys.: Condens. Matter, 15:R233, 2003. 24. J. J. McClelland, R. E. Scholten, E. C. Palm, and R. J . Celotta. Laser-focused atomic deposition. Science, 262:877, 1993. 25. Ulrich Drodofsky, Jiirgen Stuhler, Thomas Schulze, Michael Drewsen, Bjorn Brezger, Tilman Pfau, and J”urgen Mlynek. Hexagonal nanostructures generated by light masks for neutral atoms. Appl. Phys. B, 65:755, 1997. 26. T. Schulze, T. Miither, D. Jiirgens, B. Brezger, M. Oberthaler, B. Brezger, T. Pfau, and J. Mlynek. Structured doping with light forces. Appl. Phys. Lett., 1781, 2001. 27. S. Giorgini, L. P. Pitaevskii, and S. Stringari. Condensate fraction and critical temperature of a trapped interacting bose gas. Phys. Rev. A, 54:R4633, 1996. 28. P. 0. Schmidt, S. Hensler, J. Werner, Th. Binhammer, A. Gorlitz, and T. Pfau. Continuous loading of cold atoms into a ioffe-pritchard magnetic trap. J. Opt. B: Quantum Semiclass. Opt., 5:S170-S177, 2003. 29. Piet 0. Schmidt, Sven Hensler, Jorg Werner, Thomas Binhammer, Axel Gorlitz, and Tilman Pfau. Doppler cooling of an optically dense cloud of trapped atoms. J . Opt. SOC.Am. B, 20(5):960-967, 2003. 30. S. Hensler, J. Werner, A. Griesmaier, P. 0. Schmidt, A. Gorlitz, T. Pfau, and K. RzQzewski S. Giovanazzi. Dipolar relaxation in an ultra-cold gas of magnetically trapped chromium atoms. Appl. Phys. B, 77:765-772, 2003.

DISORDERED COMPLEX SYSTEMS USING COLD GASES AND TRAPPED IONS *

A. SEN(DE), u . SEN AND M. LEWENSTEIN~ ICFO-Institut d e Cibncies Fotbniques, E-08094 Barcelona, Spain Institut fur Theoretische Physic, Universitat Hannouer, 0-3016 7 Hannover, Germany. E-mail: [email protected] E-mail: [email protected] E-mail: Maciej.LewensteinQicfo.es V. AHUFINGER Grup d 'Optica. Uniuersitat Autbnoma de Barcelona, E-08193 Bellaterra, Spain. Institut fur Theoretische Physic, Uniuersitat Hannouer, D-90167 Hannouer, Germany. E-mail: Veronica.Ahuj?nger@uab. es

M. PONS Depart. de Fasica Aplicada I. Uniuersidad del Paas Vasco E-20600 Eibar, Spain. E-mail: Marisa.Pons@ehu. es A. SANPERA~ Grup de Fasica Tebrica. Uniuersitat Autbnoma de Barcelona, E-08199 Bellaterra, Spain. Institut fur Theoretische Physic, Universitat Hannouer, D-30167 Hannouer, Germany. E-mail: sanperadifae.es

'This work is partially supported by deutsches forchunggemainshaft SFB407, SPP1078 and S P P l l l 6 , and the Spanish MCYT BFM-2002-02588 and MAT2002-00699 t Also at Institucib Catalana de Recerca i Estudis AvancGats 158

159 We report our research on disordered complex systems using cold gases and trapped ions and address the possibility of using complex systems for quantum information processing. Two simple paradigmatic models of disordered complex systems are here revisited. The first one corresponds t o a short range disordered king Hamiltonian (spin glasses) which can be implemented with a bose-fermi (bosebose) mixture in a disordered optical lattice. The second model we address here is a long range disordered Hamiltonian characteristic of neural networks (Hopfield model) which can be implemented in a chain of trapped ions with appropriately designed interactions.

1. Introduction Complex many body systems are often characterized by structurally simple interactions but complexity arises because the different terms or constrains appearing in the Hamiltonian compete one with another. If the system presents disorder, the Hamiltonian is no longer translational invariant and depends locally on random parameters. When the system is not able to accommodate to all the constrains present in the Hamiltonian it exhibits frustration. This leads to the appearance of exotic phenomena e.g. fractal, hierarchic or ultrametric structures, distinct quantum phase transitioqetc.' Over the last 40 years, disordered and frustrated systems have played a central role in condensed matter physics and have posed some of the most challenging open questions of many body systems. Quenched disorder (i.e., frozen disorder) determines the physics of various phenomena, from transport and conductivity through localization, to percolation, spin glasses, neural networks, high Tc-superconductiviy,etc. The description of such systems is, however, extremelly difficult because it normally requires the averaging over each particular realization of the disorder. Systems which are not disordered but frustrated, lead very often to similar difficulties because often they are, at low temperature, characterized by an enormously large number of low energy excitations. Recently, it has been shown that one can introduce local disorder and/or frustration in ultracold quantum gases in a controlled way using various experimentally feasible methods (for details see e.g.l0 and references therein), ranging from using several incomensurable optical lattices to trap the atoms, or superimposing a speckle pattern in a regular optical lattice, or taking advantatge of Feshbach resonances in fluctuating or inhomogeneous magnetic fields in order to induce a novel type of disorder that corresponds to random, or at least inhomogeneous nonlinear interaction couplings. Thus, different disorder and/or frustrated systems can be conveniently prepared to study e.g. Bose glass2, Anderson localization fermi314

160

onic spin glasses5 or quantum percolation5, kagomb lattices6 among others. We have also recenlty investigated the possibilities offered by trapped ions with engineered interactions718t o model neural networksg. A review of the different phases displayed by ultracold atomic gases in disordered optical lattices can be found in lo. In this contribution we present our approach to the study of both, short and long range, disordered systems. In the former case we focus on a spin glasses model" , i.e. short-range disordered magnetic systems which can be simulated by Bose-Fermi mixtures in random potentials. For the long range interactions we study a neural network model simulated by a chain of trapped ions with appropriate designed interactions. In both cases, we examine the possibilities offered by those systems for quantum information tasks. In spite of the fact that using disordered systems t o perform quantum information processing seems t o be an impossible task, at least two possible advantages arise immediately. First, these systems have typically a large number of different metastable (free) energy minima, as it happens in spin glasses (SG) 12. Such states might be used t o store information distributed over the whole system, similarly to neural network (NN) models 13. The information is thus naturally stored in a redundant way, like in error correcting schemes. Second, in disordered systems with long range interactions, the stored information is robust: metastable states have quite large basins of attraction in the thermodynamical sense. We have shown14 that in both models, short and long range, it is possible to generate entanglement that survives over long times. Moreover, in the neural network model, it is possible to store patterns that can be used as distributed qubits over the whole systems. Since the patterns are robust and act as attractor points in the energy diagram, these qubits can be partially destroyed by noise or any other not desired effect. The free evolution of the systems, however, retrieves the patterns back and makes thus the qubits very robust.

2. Short range disordered systems: Spin glasses

Spin glasses are random disordered systems with competing ferromagnetic and antiferromagnetic interactions, which in dimensions d > 1 present frustration, since it is not possible t o accommodate simultaneously all pairs of spins connected by a ferromagnetic (antiferromagnetic) bond. In the earlier ~ O ' S ,Edwards and Anderson realized that the essential physics of a spin glass does not lay in the details of their microscopic interactions, but rather in the competition between quenched ferro and antiferro interactions. To

161

study the nature of spin glasses they proposed a very simple short range disordered king Hamiltonian, nowadays known as Edwards- Anderson (E-A) model of spin glasses

Here 0; denotes an Ising spin ( f l )at the k-th site, the Jij’s describe nearest neighbors couplings for an arbitrary lattice and h” is a magnetic field along the z-direction. In the E-A model, the Jij couplings are given by independent random variables which have a gaussian probability distribution with mean J = 0 and variance A2. Since interactions are short range, a mean field theory cannot be used15 and, traditionally, one has to rely on replica tricks12 to do the appropriate average over the quenched disorder in order to obtain the free energy F of the system, and derive the thermodynamical properties of the system from F . A formally identical Hamiltonian as the one of Eq.(1) can be derived from the Bose-Fermi (Bose-Bose) Hubbard Hamiltonian’ (BFH,BH) describing a Bose-Fermi (Bose-Bose) mixture in an optical lattice with random disorder:

where b f , b j , f:, f j are the bosonic and fermionic creation-annihilation operators, ni = blbi, mi = f / f i the operator number and pf and p r are the bosonic and fermionic local chemical potentials, respectively. The BFH model describes: i) nearest neighbor (n.n.) boson (fermion) hopping, with an associated negative energy, -TB (-TF);ii) on-site repulsive bosonboson interactions with an energy V ;iii) on-site boson-fermion interactions with an energy U , which is positive (negative) for repulsive (attractive) interactions, and, finally, iv) interactions with the external inhomogeneous potential, with energies pf and p‘. In the limit of equal tunneling for bosons and fermions (TB = TI;.= T ) and a strong coupling regime (T lthe l experiment, the frequency of the dipole trap beams is changed by acousto-optic modulators (AOMs), which are placed in each beam and which are driven by a phase-synchronous digital dual-frequency synthesizer; see Fig. 1. We can realize typical accelerations of a = 10000 m/s2 and thus accelerate the atoms to velocities of up to 5 m/s (limited by the 10 MHz bandwidth of the AOMs) in half a millisecond. Thus, for typical parameters, a 1 mm

195 Target position

8

20

Y

10

. s1 D

s

o

Figure 4. Active position control. a) After transferring a single atom from the MOT into the dipole trap its initial position is determined from an ICCD image and its distance with respect to the target position is calculated. b) The atom is then transported to the target position and its final position is again measured from an ICCD image.

transport takes about 1 ms. At the same time, the displacement of the atoms is controlled to better than the dipole trap laser wavelength since this scheme allows us to control the relative phase of the two trapping laser beams to a fraction of a radian. 3.2. Measuring and Controlling the Atoms ' Positions

If one wants to take advantage of the optical conveyor belt transport above in order to place atoms at a predetermined position, the atoms' initial position along the dipole trap axis has to be known with a high precision. This can be achieved by recording and analyzing an ICCD fluorescence image of the trapped atoms. We have shown that by fitting the corresponding fluorescence peaks with a Gaussian, the atoms' position can be determined with a 41150 nm precision from an ICCD image with 1 s exposure time.g Furthermore, we have demonstrated that by means of our optical conveyor belt technique, we can place an atom at a predetermined position along the dipole trap axis with a 41300 nm a c c ~ r a c y Such . ~ a position con-

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Distance d between atoms [hDT/21 Figure 5 . Cumulative distribution of separations between simultaneously trapped atoms inside the standing wave potential. The discreteness of the atomic separations due to the standing wave potential is clearly visible.

trol sequence is exemplified in Fig. 4. After loading one atom from the MOT into the dipole trap, its position has a f 5 pm uncertainty, corresponding to the diameter of the MOT. We determine the atom’s initial position from a first ICCD fluorescence image and calculate its distance L from the desired target position. The atom is then transported to this target position and the success of the operation is verified by means of a second ICCD image. In order to measure the distance between two simultaneously trapped atoms, we determine their individual positions as above. From one such measurement, their distance can thus be inferred with a precision of fi x 150 nm. This precision can even be further increased by taking more than one image of the atom pair and by averaging over the measurements obtained from these images. Now, since the atoms are trapped inside a periodic potential, their distance d should be an integer multiple of the standing wave period: d = ~ X D T / ~see ; Fig. 2 a). This periodicity is clearly visible in Fig. 5, where the cumulative distribution of atomic separations is given when averaging over more than 10 distance measurements for each atom pair. The resolution of this distance measurement scheme is f 3 6 nm, much smaller than the standing wave period. We directly infer this value from the width of the vertical steps in Fig. 5. This result shows that we can determine the exact number of potential wells separating the simultaneously trapped atoms.g 3.3. Two-Dimensional Position Manipulation

A single standing wave optical dipole trap allows to shift the position of a string of trapped atoms as a whole in one dimension along the dipole

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Figure 6. Rearranging a string of three atoms using two perpendicular standing wave dipole traps. See text for details.

trap axis using the optical conveyor belt technique presented above. If one seeks t o prepare strings with a predefined spacing or to rearrange the order of a string of trapped atoms, however, a two-dimensional manipulation of the atomic positions is required. For this reason, we have set up a second standing wave dipole trap, perpendicular t o the first one, which acts as optical tweezers and which allows us t o extract atoms out of a string and to reinsert them a t another predefined position. Figure 6 shows a first preliminary result towards this atom sorting and distance control scheme. We start with a string of three randomly spaced atoms which has been loaded from the MOT into the horizontal (conveyor belt) dipole trap . In Fig. 6 a), the string has already been shifted such that the rightmost atom is placed at the position of the vertical (optical tweezers) dipole trap. This atom is then extracted with the vertical dipole trap and, after shifting the remaining two atoms along the horizontal dipole trap, we place it 15 pm t o the left of the initially leftmost atom of the string; see Figs. 6 b)-d). Repeating this procedure a second time, we prepare a string of three equidistantly spaced atoms, where the order of the string has been modified according t o (1,2,3) +. (3,1,2) +. (2,3,1); see Figs. 6 e)-h).

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4. Conclusions

We have shown that a string of cesium atoms, trapped inside a standing wave dipole trap, can realize a quantum register. Using the atomic hyperfine states for encoding coherent information, we have demonstrated all basic register operations: initialization, selective addressing, coherent manipulation, and state-selective detection of the individual atomic states.6 We have furthermore demonstrated a high level of control of the atoms’ external degrees of f r e e d ~ mWe . ~ have measured the absolute and relative positions of the atoms along the dipole trap with a submicrometer accuracy. This high resolution allows us to measure the exact number of potential wells separating simultaneously trapped atoms in our 532 nm-period standing wave potential and to transport an atom t o a predetermined position with a suboptical wavelength precision. Finally, using a second dipole trap operated as optical tweezers, we have obtained first results towards an active control of the atoms’ relative positions within the string. This will allow us to prepare strings with a preset interatomic spacing and to rearrange the order of atoms within the string at will. The presented techniques are compatible with the requirements of cavity QED and controlled cold collision experiments. In our laboratory, we now actively work towards the implementation of such experiments in order t o realize quantum logic operations with neutral ground state atoms.

Acknowledgements This work was supported by the Deutsche Forschungsgemeinschaft and by the EC (IST/FET/QIPC project “QGATES”). I. D. acknowledges funding from INTAS. D. S. acknowledges funding by the Deutsche Telekom Stiftung.

References 1. N. Davidson et al., Phys. Rev. Lett. 74,1311 (1995). 2. S. Kuhr et al., Phys. Rev. Lett. 91,213002 (2003). 3. L. You, X. X. Yi, and X. H. Su, Phys. Rev. A 67,032308 (2003). 4. 0. Mandel et al., Nature 425, 937 (2003). 5. S. Kuhr et al., Science 293,278 (2001). 6. D. Schrader et al., Phys. Rev. Lett. 93,150501 (2004). 7. W. Alt, Optik (Jena) 113,142 (2002). 8. D. Haubrich et al., Europhys. Lett. 34,663 (1996). 9. I. Dotsenko et al., Phys. Rev. Lett. 95, 033002 (2005). 10. S. Kuhr et al., Phys. Rev. A,,accepted (2005); e-print quant-ph/0410037, 11. Y . Miroshnychenko et al., Opt. Express 11,3498 (2003).

BOSE-EINSTEIN CONDENSATES STUDIED WITH A LINEAR ACCELERATOR

CH. BUGGLE, J. LEONARD; w. VON KLITZING~AND J. T. M. WALRAVEN FOM Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands and Van der Waals-Zeeman Institute of the University of Amsterdam, Valckenierstraat 65/67, 1018 X E The Netherlands E-mail: [email protected]

We present a stand-alone interference method for the determination of the s- and d-wave scattering amplitudes in a quantum gas. Colliding two ultracold atomic clouds we observe the halo of scattered atoms in the rest frame of the collisional center of mass by absorption imaging. The clouds are accelerated up to energies at which the scattering pattern shows the interference between the s- ( 1 = 0) and d- ( I = 2) partial waves. With computerized tomography we transform the images to obtain the angular distribution, which is directly proportional to the differential cross section. This allows us to measure the asymptotic phase shifts of the s- and d-wave scattering channels. The method does not require knowledge of the atomic density. It allows us to infer accurate values for the s- and dwave scattering amplitudes from the zero-energy limit up to the first Ramsauer6 coefficient as theoretical Townsend minimum using only the Van der Waals c input. For the "Rb triplet potential, the method yields an accuracy of 6%.

1. Introduction The scattering length, the elastic scattering amplitude in the zero-energy limit, is a key parameter in the theoretical description of quantum gases.l The scattering length a determines the kinetic properties of these gases as well as the bosonic mean field. Its sign is decisive for the collective stability of the Bose-Einstein condensed state. Near scattering resonances, pairing behavior and three-body lifetime can also be expressed in terms of a. As a consequence, the determination of the low-energy elastic scattering *present address: Universit6 de Strasbourg, Institut de Physique et Chimie des Materiaux. tpresent address: IESL - FORTH, Vassilika Vouton, 711 10 Heraklion, Greece. 199

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properties is a key issue to be settled prior to further investigation of any new quantum gas. Over the past decade the crucial importance of the scattering length has stimulated important advances in collisional physics.2 In all cases except hydrogen the scattering length has to be determined experimentally as accurate ab initio calculations are not p ~ s s i b l eAn . ~ estimate of the modulus la1 can be obtained relatively simply by measuring kinetic relaxation times.4 In some cases the sign of a can be determined by such a method, provided p or d-wave scattering can be neglected or accounted for the~retically.~ These methods have a limited accuracy since they rely on the knowledge of the atomic density and kinetic properties. Precision determinations are based on photo-association,6 ~ibrational-Raman,~ and Feshbach-resonance s p e c t r o s ~ o p yor , ~ a~ combination ~ of those. They require refined knowledge of the molecular structure in ground and excited electronic states2 In this contribution we describe a new method to determine the scattering length by studying the halo of scattered atoms after the collision of two ultracold atomic clouds." We compare our results with related work.

2. The accelerator We start our experiments by loading about one billion 87Rb atoms into a magneto-optical trap (MOT). After optical pumping into the fully stretched 15'&p, F = 2, m F = 2 ) hyperfine level, the atoms are transferred into a Ioffe-Pritchard trap (21 x 477 Hz) with an offset field of Bo = +0.9 G . We pre-cool the atomic cloud to about 6 p K using forced radio-frequency (RF) evaporation. To prepare for a collision experiment, the cloud is split in two parts by applying a rotating magnetic field and ramping Bo down to a negative value B,. This results in two Time-averaged Orbiting Potential (TOP) traps loaded with atoms.13 By RF-evaporative cooling we reach Bose-Einstein condensation with about lo5 atoms in each cloud and a condensate fraction of 60%. By switching off the T O P fields and ramping Bo back t o positive values we linearly accelerate the clouds until they collide with opposite horizontal momenta at the location of the trap center. The collision energy E = 12p~BFI = h2k2/rn (with p~ the Bohr magneton and m the mass of 87Rb) can be varied from 138 pK to 1.23 mK with an overall uncertainty of 3% (RMS). Approximately 0.5 ms before the collision we switch off the trap. A few ms later a halo of scattered atoms is observed by absorption imaging (see Fig. 1).

-

20 1

Figure 1. Left: Scattering halo of two a7Rb condensates for collision energy E / k e = 138(4) pK (mostly s-wave scattering), measured 2.4 ms after the collision; Right: idem but measured 0.5 ms after a collision at 1230(40) p K (mostly d-wave scattering). The field of view of the images is 0.7 x 0.7mm2. N

3. Data analysis

As the atoms are scattered by a central field, the scattering pattern must be axially symmetric around the (horizontal) scattering axis (z-axis). This allows a computerized tomography transformation to reconstruct the radial density distribution of the halo in cylindrical coordinate~,'~

Here p = (z2+y2)lI2 is the radial coordinate and f i 2 ( K ~z ,) the 1D Fourier transform along the 2-direction of the optical density with respect to z ; Jo (e) is the zero-order Bessel function. From the radial density distribution of the halo we obtain the angular scattering distribution, which (for gas clouds much smaller than the diameter of the halo) is directly proportional to the differential cross section

(e) = 27r (e) + f (T - 8)12.

(2) Here, the Bose-symmetrized scattering amplitude is given by a summation over the even partial waves,

C (21 + l ) e i q ~ ~ ( c o ssei n) q .

f (e) + f (7r - e) = ( 2 / l ~ )

(3)

l=ewen

Given the small collision energy in our experiments, only the s- and d-wave scattering amplitudes contribute,

+ fs

0) = (2/k)eZqosin70

fs

(6)

fd

(e) + fd (T e) = (2/k)(5/2)eiq2(3cos2e - 1) s i n ~ p .

(7r -

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Therefore, the differential cross section is given by a quadratic expression,

with u = (sinqz/sinqo) ( 3 ~ 0 ~ ~ 1). 13 As suggested by Eq. (4), we make a parabolic fit to the measured angular distribution plotted as a function of (3 cos2B - 1). This yields directly a pair of asymptotic phase shifts [qrp(k),qyp(k)]- defined modulo 7r corresponding to the two partial waves inv01ved.l~The absolute value of a(e) depends on quantities that are hard to measure accurately (like the atom number) so we leave it out of consideration. We rather emphasize that the measurement of the phase shifts allows a complete determination of the (complex) s- and d-wave scattering amplitudes at a given energy. 4. Determination of the energy dependence The radial wavefunctions corresponding to scattering at different (low) collision energies and different (low) angular momenta should all be in phase at small interatomic distances. l6 This so-called accumulated phase is common to all low-energy wave functions and is extracted by a least-square fit to the full data set {qrp(k),qyp(k)}. In practice, we use the experimental phase shifts q r p ( k ) and qyp(k) as boundary conditions to integrate inwards the Schrodinger equation h2d2X(r)/dr2 p2(r)x(r) = 0 for given E and 1, and obtain the radial wavefunctions x(r)/r down to radius Tin = 20ao. Here, p 2 ( r ) = rn ( E - V ( r ) )- h21(Z 1)/r2, where V ( r )T -Cs/r6approximates the tail of the interaction potential. At radius 20a0, the motion of the atoms is quasi-classical and the accumulated phase can be written as @ ( r )T arctan[p(r)/(hilnx/ir)]. The distance 20ao is small enough for @(Ti,) to be highly insensitive to small variations in E or 1 but also large enough for the -C6/r6part of the interaction potential to be dominant over the full range of integration.16 With a least-square procedure we establish the best value Qopt(rin)= 1.34 f (T x 0.025) for the accumulated phase at 20ao. Here the error bar reflects the experimental accuracy and not the systematic error related to the choice of c6.The d-wave scattering resonance results in a sudden variation of qypwith the collision energy in the vicinity of that resonance (see Fig. 2a).17 This imposes a stringent condition on the optimization of QOpt and constrains its uncertainty. We emphasize that aopthas no physical significance but is valuable as a boundary condition to integrate the Schrodinger equation back outwards to compute q ( k ) for any desired (low) value of k and 1. Fig. 2 shows the

+

+

203

Figure 2. a) d-wave and b) s-wave phase shifts versus collision energy in pK; s-d interference is only observed in the gray areas. The circles are the results of the parabolic fits for individual images. The full black lines is calculated from the accumulated phase aOpt optimized from all data points. The grey lines show the influence of the uncertainty of &(n x 0.025) on aOpt. The vertical dotted line indicates the condition 70 = 7 2 . The first s-wave Ramsauer-Townsend minimum is found at ERT = 2.1(2) mK.

resulting phase shifts for collision energies up t o 5 mK. The first RamsauerTownsend minimum in the s-wave cross section is found a t collision energy E R T / ~ = B 2.1(2) mK. The solid dots represent the qfXP(ki) obtained from the parabolic fits for individual images. The three open circles correspond t o measurements for which the sign of the phase shifts could not be established. lo Refinements to the procedure may account for multiple scattering effects as well as the presence of a non condensed fraction. Note that this procedure does not require knowledge of the density of the clouds, unlike the stimulated raman detection scheme.l8 Knowing the phase shifts, we can infer all low-energy scattering properties. In particular, the total elastic scattering cross section is given by

(T

(0) sin 0dB = (87r/k2) l=ewen

(21

+ 1)sin2 q.

(5)

204

1

10 100 Colision Energy ( pK)

1000

Figure 3. swave (dashed line), d-wave (dotted line) and total (full black line) elastic cross sections (in cm’) versus collision energy (in pK), computed from the optimized accumulated phase @.opt as determined in this work. The gray lines are the total elastic cross sections, obtained from aoptf (T x 0.025).

Note that the total cross section, unlike the differential cross section, does not contain interference terms. Our results are shown in Fig. 3. The (asymmetric) d-wave resonance emerges pronouncedly at 300(70) pK with an approximate width of 150 pK (FWHM). Most importantly, the scattering length follows from the k 4 0 limiting behavior, vo(k 4 0) = -ka. We find a = +102(6) ao, whereas the state-of-the-art value is a = 98.99(2) ao.19 5. Comparison with related work

Comparison of our results with the precision determinations shows that our method is fairly accurate, although it only relies on the input of the c 6 coefficient. We used the value Cs = 4.698(4) x lo3 a.u..19 In the present case, one does not need to know CS to this accuracy. Increasing c 6 by 10% results in a 1 %-change of our computed scattering length. Clearly, the systematic error in GOpt accumulated by integrating the Schrodinger equation inward with a wrong c 6 largely cancels when integrating back outward. However, in the case of a s-wave resonance other atomic species may reveal a stronger influence of ‘26 on the calculated scattering length. Simple numerical simulations show that the value of c 6 becomes critical only when the (virtual) least-bound state in the interaction potential has an extremely small (virtual) binding energy (less than level spacing). Hence our method should remain accurate in almost any case.

205

We point out that our method do not require the use of Bose-Einstein condensed ultracold clouds. However, the use of condensates is practical as they allow high energy resolution and analysis of the largest possible window of scattering angles. At the University of Otago similar collision experiments were done with thermal clouds of 87Rbin the 152&/2,F = 2, m F = 2) state.ll In these experiments the differential and total cross section were measured directly and found to be in good agreement with theory. At this ICOLS conference the Otago/Nist team reported interferometric observation of p-wave scattering between non-identical bosons by colliding a cloud of 87Rb atoms in the 152S1/2,F = 1,m F = -1) state with a 87Rb cloud in the 152S1/2,F = 2 , m F = 1) state.20 We finally point to the relation between the halos observed in our experiments and those observed by dissociation of 87Rb-dimers near a Feshbach resonance at the Max Planck Institute for Quantum Optics (MPQ) in Garching.12 In these experiments interference was observed between dissociation halos from the s- and d-wave channels by Feshbach tuning the dissociation energy to the value corresponding to the d-wave resonance. Like in our experiments the final state pair wave functions can be written in the dissociation case as

*(r, t) = g(T, t ) [eivofiYt

+ eivzf i ~ , ~ ( e ,) ]

(6)

+

where POand 0 2 (with PO 02 = 1) are the branching ratios for the s- and d-wave channels, respectively and 770 and 772 set their relative asymptotic phase. For the collision experiments the branching ratios depend on the phase shifts qo and 772 and on the decomposition of the incident plane waves into partial waves, and are given by

In the dissociation experiments there is no incident wave, but the phase difference 772 - 70 remains well defined and agrees with our experiments. Therefore, the difference between the two experimental situations shows up as a difference in branching ratios. Acknowledgments The authors acknowledge valuable discussions with S. Kokkelmans, T. Voltz, S. Diirr, and G. Rempe. This work is part of the research programme of the ‘Stichting voor Fundamenteel Onderzoek der Materie (FOM)’, supported by the ‘Nederlandse organisatie voor Wetenschappelijk

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Onderzoek (NWO)’. JL acknowledges support from a Marie Curie IntraEuropean Fellowship (MEIF-CT-2003-501578).

References 1. See e.g. L. Pitaevskii and S. Stringari, Bose-Einstein condensation, Clarendon Press, Oxford 2003; C.J. Pethick and H. Smith, Bose-Einstein condensation in dilute gases, Cambridge University Press, Cambridge 2002. 2. J. Weiner, V.S. Bagnato, S. Zilio, P.S. Julienne, Rev. Mod. Phys. 71,l (1999). 3. D.G. Friend and R.D. Etters, J . Low Temp. Phys. 39,409 (1980); Y.H. Uang and W.C. Stwalley, J . d e Phys. 41, C7-33 (1980). 4. C. R. Monroe et al., Phys. Rev. Lett. 70,414 (1993); S. D. Gensemer et al., Phys. Rev. Lett., 87, 173201 (2001). 5. G. Ferrari et al., Phys. Rev. Lett. 89, 53202 (2002); P. Schmidt et al., Phys. Rev. Lett. 91, 193201 (2003). 6. D. Heinzen in: Proceedings of the international School of Physics, Enrico Fermi, M. Inguscio, S . Stringari and C. Wieman (Eds.), 10s Press, Amsterdam 1999. 7. C. Samuelis, E. Tiesinga, T. Laue, M. Elbs, H. Knckel and E. Tiemann, Phys. Rev. A 63, 12710 (2001). 8. C. Chin, V. Vuletic, A. J. Kerman, and S. Chu, Phys. Rev. Lett. 85, 2717 (2000); P. J. Leo, C. J. Williams, and P. S. Julienne, Phys. Rev. Lett. 85, 2721 (2002). 9. A. Marte, T. Volz, J . Schuster, S. Drr, G. Rempe, E. G. M. van Kempen, and B. J . Verhaar, Phys. Rev. Lett. 89, 283202 (2002). 10. Ch. Buggle, J . Leonard, W. von Klitzing and J.T.M. Walraven, Phys. Rev. Lett. 93 173202 (2004) 11. N.R. Thomas, N. Kjaergaard, P.S. Julienne, and A.C. Wilson, Phys. Rev. Lett. 93 173201 (2004). 12. T. Voltz, S.Durr, N. Syassen, G. Rempe, E. van Kempen, S. Kokkelmans, cond-mat /04 10083. 13. T.G. Tiecke, M. Kemmann, Ch. Buggle, I. Schvarchuck, W. von Klitzing and J.T.M. Walraven, J. Opt. B 5, S119 (2003); see also N. R. Thomas, A. C. Wilson, and C. J. Foot, Phys. Rev. A 65, 063406 (2002). 14. M. Born and E. Wolf, Principles of Optics, 7th (expanded) Edition, Cambridge University Press, Cambridge 1999. 15. This procedure breaks down in the marginal case 70 = 72,where the expression in the square brackets in Eq. (4) becomes phase-shift independent. 16. B. Verhaar, K. Gibble, and S. Chu, Phys. Rev. A 48, R3429 (1993); G.F. Gribakin and V.V. Flambaum, Phys. Rev. A 48, 546 (1993). 17. H.M.J.M. Boesten, C.C. Tsai, J.R. Gardner, D. J. Heinzen, B.J. Verhaar, Phys. Rev. A 55, 636 (1997). 18. R. Legere and K. Gibble, Phys. Rev. Lett. 81, 5780 (1998). 19. E.G.M. van Kempen, S.J.J.M.F. Kokkelmans, D.J. Heinzen, and B.J. Verhaar, Phys. Rev. Lett. 88, 93201 (2002). 20. A.S. Mellish, N. Kjaergaard, A.C. Wilson and P.S. Julienne, This meeting, Poster 97.

ELECTRIC FIELD SPECTROSCOPY OF ULTRACOLD POLAR MOLECULAR DIMERS JOHN L. BOHN AND CHRIS TICKNOR JILA, NIST and Department of Physics, University of Colorado Boulder, CO 80309, USA We propose a novel kind of electric field spectroscopy of ultracold, polar molecules. Scattering cross sections for these molecules will exhibit a quasi-regular series of resonance peaks as a function of applied electric field. In this contribution, we derive a simple approximate formula for F(n), the electric field F at which the nth resonance appears.

The topic of this contribution does not strictly belong at the International Conference on Laser Spectroscopy, I suppose, since it has nothing whatever to do with lasers. However, I do intend to discuss a kind of spectroscopy that is peculiar to polar molecules in an ultracold environment. It turns out that polar molecules at low temperatures respond really strongly to electric fields, particularly when the molcules collide. So, by monitoring the scattering cross section as a function of electric field, you can try to read out information on the dimer formed by the collision partners. In what I discuss here, I consider the effect of a dc field; you could think of it as a kind of “dc laser spectroscopy,” if you like. Consider a pair of molecules that approach one another in an ultracold environment, with essentially zero relative kinetic energy. The actual scale of this energy is assumed to be something like milliKelvin, or perhaps tens of microKelvin; it depends on whose experiment you’re talking about [I]. Considering that ten microKelvin corresponds to something like 200 MHz of energy, a cold collision event could in principle probe resonant structure between the colliding partners on about this scale. This is plenty enough resolution to distinguish between different ro-vibrational states, for example. Well, there’s an obvious problem with using these collisions for spectroscopy. The collision energy, within this temperature-defined width, is always the same, namely, zero. This seems to imply that you could probe one very tiny slice of the collisional spectrum really well, but at the expense of 207

208

everything else going on in the complex. In conventional spectroscopy, this would not be a problem, since you could bring in photons of a desired energy and raise or lower the energy of the collision partners to any other energy level you want to probe. (In the context of cold collisions, this is in fact done frequently, and is known as photoassociation [2].) There is an alternative way to proceed, however. Rather than driving the molecules to a bound state at a different energy, you could instead shift the energy of the bound state until it coincides with the energy of the molecules, i.e., zero. This is not completely crazy, or at least it’s not without precedent. For years now experimenters have been using magnetic-field Feshbach resonances in alkali atoms to make resonant states coincide with the threshold of the scattering continuum. This has been a very fruitful tool for engineering the mean-field interaction energy of ultracold gases, but it is something else, too. For a realistic two-body interaction potential to reproduce such a resonance at the correct field is quite a severe constraint on the potential [ 3 ] . The point is, field variation of cold collision cross sections can really be a useful probe of the interaction, i.e, a spectroscopic tool.

n

0

1000

2000

3000

Electric Field (V/cm)

4000

1 5000

Figure 1. Elastic scattering of SrO molecules at a collision energy of 10-”K, as a function of electric field. This scattering produces a surprisingly regular series of resonances.

The effect of an electric field on polar molecules is likely to be much more profound, since electric forces are generically stronger than magnetic forces in molecules. And sure enough: in Figure 1 I show the electric-field-dependent elastic scattering cross section of ultracold SrO molecules. These molecules are assumed to be in their electronic (‘E), vibrational (v=O), and rotational (J=O)

209

ground states, and to collide at an energy of lo-’’ K. This figure is reproduced from our recent study of electric field resonances [4], and the details of the scattering model are explained there. The main point of figure 1 is that the cross section is dominated by a strikingly regular series of resonances. In Ref. [4] we explain that these resonances originate in the purely long-range dipole-dipole interaction between the molecules. Roughly, the field can change the degree of polarization of the molecules, hence their dipole-dipole interaction. Ref. [4] also shows that the exact position and spacing of the resonances carries information about the “short-range” physics, where the molecules actually collide and probe their potential energy surface. (Also appearing in this figure, although far less prominent, are various Feshbach resonances that describe excitation to higherlying rotational and fine-structure states. Obviously they will also carry information about the interaction that excited these degrees of freedom.) This discussion should sound vaguely familiar. The backbone of atomic spectroscopy is the Rydberg series. Any atom with a singly-excited electron has a spectrum of energy levels that scales as -1ln’. Well, no, of course that’s not true. In fact, the spectrum of such an atom scales as -ll(n-p)’, where p is a quantum defect [5]. The role of the quantum defect, as all good spectroscopists know, is to encode the electron’s interaction with all the other electrons in the parent ion. The entire series of energy levels is described by a simple formula that is flexible enough to apply to any atom. What I would like to do here is to derive an analogue of the Rydberg formula for an electric-field spectrum like the one in Figure 1. In other words, I would like an analytic expression F(n) that gives the electric field value F at which the nth resonance is observed. This will not be the world’s greatest derivation, but it will get the basic physical ideas right. Let’s first note the following. In spectra such as that in Fig.1 that I’m considering, the collision energy is nearly zero, and it is the electric field that is being scanned. The resonant state therefore extends to infinite intermolecular separation R. In this case, the relevant interaction potential is described by the largest-R part of the adiabatic potential energy curve. Well, we’ve discussed the form of this curve previously [6]. The lowest potential, the one that caries the resonances, is mostly s-wave in character (i.e., it has partial wave angular momentum Z=O). But the dipole-dipole interaction vanishes by symmetry for swaves, so the only effect of the dipolar interaction is to mix s-waves with nearby d-waves (Z=2). It turns out that this means two things. First, in zero field the effective interaction scales as - C, 1 R 6 , Second, in nonzero field, the interaction quickly

210

turns over to a - C, / R 4 behavior, where the effective C4 coefficient is given by 161

Here 7 = 2 p / ~ r e p r e s e n t sthe electric field F in units of the critical field F ~ = A that / ~ separates ~ the quadratic and linear Stark effect regimes; A is the energy splitting between the given rotational level and the next higher one; and rn, is the reduced mass of the collision partners. Equation (1) includes only the perturbation due to the 2=2 partial wave, but you can easily extend it to higher partial waves, if you like. Notice that the value of C4saturates as you go to the high-field limit F >> F, . Well, of course: once the molecules are polarized, the field can’t do much more to change the situation. (I do not consider here fields large enough to distort the electronic wave functions, which would be another story.) Since I’m talking here about field-dependent resonances, I will concentrate on the field-dependent l/R4 part of the interaction. At zero collision energy, the WKB phase for this potential is easily calculated. To identify a bound state, you set this phase equal to an integral multiple of z

Here I have imposed an arbitrary small-R cutoff radius, a. And it’s a good thing, too, since the integral would diverge if I were to let R 0 . But anyway, the potential is clearly not proportional to l/R4 all the way to small R, and besides, whatever goes on for Rca is going to be accounted for in the quantum defect. Now you could just invert (2) to find the resonant field at which the nth bound state appears. As a fitting formula, this leaves something to be desired, however, since the result would depend explicitly on the unphysical cutoff radius a. A better use of the quantization condition would be to exploit the saturation behavior, and to write 2 77 - a , (3) I++? i m where cm=fip2mr/m$is the approximate total number of bound states in the potential in the limit of large electric field. In other words, when the field is large, q + 00 and n + E-. Having made this correction, then you could then invert (3) to find the set of resonant field values F(n). Oh no, wait, I’m sorry, there is one more thing. In practice, it’s more useful to append the quantization condition (3) to read

+

211

In this expression, no/ nmrepresents the additional part of the phase shift arising from the short-range part of the potential. Here is our analogue of the quantum defect. Finally, when you do actually invert the expression (4) to find the resonant fields, you get

-

where n, = n, -no ,and F' is expected to be on the order of Fo. Does this work? Surprisingly, yes. Look at Figure 2, where I have plotted the resonant fields F(n) versus n for both the SrO example in Fig.1, and for RbCs. In both cases I have fit to the formula (3,and shown the results as solid lines. The fit is not half bad, and clearly gets the general form of the spectrum correct.

n

E

0

3

25 L

n Figure 2.Electric field values F(n) at which the nth resonance occurs, as seen in close-coupled calculations of SrO (squares) and RbCs (triangles) cold collisions. Solid lines represent fits to the formula in Eqn. (5).

A few remarks are in order. First, the actual value of n is pretty arbitrary, since I have no idea how many bound states there really are. In fact, I have anchored the calculation in the final number of bound states after the field has

212

saturated the interaction. Therefore, it actually makes more sense to count the resonances “backwards,” from n, down. (Note that n, need not be an integer, of course.) This is actually a typical way to count the very most weakly-bound states of potentials [7]. Second, the assumption that the long-range potential scales with R as l/R4 is not true at zero field. Thus, in order to get a suitable fit for the RbCs spectrum, I had to ignore the first four calculated resonances. Third, by no means is ( 5 ) intended to be a rigorous result. Rather, it is a simple and comfortable way to characterize the series in a simple formula. It should, for example, provide an estimate of how many resonances to expect for a given molecule in a given field range. Plus, it strongly suggests that there should exist a simple formula. Someday somebody should go back and find the correct formula in a more rigorous way.

Acknowledgments This work was supported by the National Science Foundation, and by a grant from the W. M. Keck Foundation.

References [ l ] J. Doyle, B. Friedrch, R. V. Krems, and F. Masnou-Seeuws, Euro. Phys. J. D 31, 149 (2004). [2] J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, Rev. Mod. Phys. 71, 1 (1999). [3] N. R. Claussen, S. J. J. M. F. Kokkelmans, S. T. Thompson, E. A. Donley, and C . E. Wieman, Phys. Rev. A 67,060701 (2003). [4] C . Ticknor and J. L. Bohn, physics/0506104, to appear in Phys. Rev. A. [5] H. Friedrich, Theoretical Atomic Physics (Berlin: Springer,1998). [6] A. V. Avdeenkov and J. L. Bohn, Phys. Rev. A 66,052718 (2002). [7] R. J. LeRoy and R. B. Bernstein, J. Chem. Phys. 52, 3689 (1970).

OPTICAL PRODUCTION OF ULTRACOLD POLAR MOLECULES

D. DEMILLE~,J. M. SAGE, AND s. SAINIS Department of Physics Yale University New Haven, CT 06520, USA E-mail: [email protected] T. BERGEMAN Department of Physics and Astronomy S U N Y Stony Brook Stony Brook, N Y 11794, USA We present recent work resulting in the production of ultracold, polar RbCs molecules in their vibronic ground state. The production process consists of several steps: photoassociation of laser-cooled atoms; radiative stabilization of the resulting molecules; identification of the resulting population distribution; and laserstimulated state transfer to the vibrational ground state. We discuss the properties of the resulting sample of of X I C f ( v = 0) molecules, with a view t o the future directions of these experiments.

1. Introduction

There is a growing interest in methods to produce samples of ultracold, polar molecules (UPMs),' which could provide access to new regimes in many phenomena. The large electric polarizability of a polar molecule can be used to engineer tunable, strong, and anisotropic interactions between molecules. At ultracold temperatures, the molecules could be trapped and coherently manipulated over long times. These features make UPMs attractive as qubits for quantum computation,2 as building blocks for novel many-body ~ y s t e m s ,and ~ for the study of chemistry in the ultracold regime.4 The large polarizability also enhances the visibility of parity-violating effects such as the electric dipole moment of the electron and nuclear anapole moment^.^ Hence, the prospect of high-precision spectroscopic measurements with UPMs is very attractive as a means of testing models of nuclear 213

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and particle physics. There remains no demonstrated method for directly cooling molecules to the ultracold (T ’5 p M 1.3 D) and have a translational temperature of 100 pK. The distribution of both vibrational and rotational levels within the X’C+ state is also quite narrow, so the resulting sample of XIC+ state molecules can be described as cold in all degrees of freedom. Details of these experiments are given in our earlier paper;14 here we give only a brief outline of the techniques and primary results. N

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2. Overview

Figure 1 gives an overview of the methods we use to produce and detect UPMs. Rb and Cs atoms are collected in a dual-species magneto-optic trap.

a3C+

I I

( v = 37)

Figure 1. Formation and detection processes for ultracold ground state RbCs. (a) Colliding atom pairs are excited into bound RbCs* molecules, which (b) decay prominently into the a3C+(w = 37) state. (c) Metastable a(w = 37) molecules are excited to level i, then (d) stimulated into the XIC+(w = 0) state. The molecules are detected directly by (e) driving them back to the original excited level and (f) ionizing them.

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A pair of colliding, ultracold Rb and Cs atoms is photoassociated into an electronically excited molecular level.16 This RP = 0- level decays rapidly into various, highly-excited vibrational levels of the long-lived a3C+ state, with branching fraction of 7% into the v = 37 level.6 After a period of PA, a resonant laser pulse (“pump” pulse) transfers these a3Cf(v = 37) molecules to an intermediate, electronically excited state (2). The population of state i can be monitored by applying an intense laser pulse (“ionization” pulse), with a frequency chosen to selectively form a RbCs’ molecular ion. We detect these ions using time-of-flight mass spectroscopy. N

1

2

3

Time of flight (ps) Figure 2. Time of flight mass spectrum showing delayed detection of re-excited X I C f ( v = 0) state molecules. The undelayed peaks arise from accidental multi-photon ionization caused by the pump and/or dump lasers.

To produce X ( w = 0) molecules, a second tunable laser pulse (“dump” pulse), arriving just after the pump pulse, resonantly drives molecules in state i t o the X ( v = 0) (or, for diagnostic purposes, = 1) state. Transfer to the X ( v = 0 , l ) states results in a depletion of the i state population immediately after the dump pulse, which can be monitored by applying the ionization pulse just after the dump pulse. In order to verify transfer to the X state, a third pulse (“re-excitation” pulse), identical in frequency to the dump, drives these stable X I C f ( v = 0 , l ) molecules back into state i, where they are detected via ionization as before. The re-excitation pulse arrives after a delay of 100 ns, designed to insure both that the population of state i has decayed completely, and that the time-of-flight signal of ions from the X state is clearly distinguishable from the earlier signals arising from accidental multi-photon ionization by the pump and/or dump pulses. The delayed ion signal is monitored as the frequency of the identical dump and re-excitation pulses is scanned; a peak indicates that molecules have been N

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resonantly transferred to the X'C+(v = 0 , l ) states by the dump pulse, then back to the i state by the re-excitation pulse. Fig. 2 shows a time-offlight mass spectrum with the dump/re-excitation laser on resonance. The definitive signatures of UPM production are resonances in the re-excitation spectrum a t the exact frequencies predicted by earlier spectroscopy of the a3C+, and X I C + levels.'' We have seen such resonances for several different intermediate states i, and in all cases the observed lines are within the uncertainty in the predictions. Results for a scan of the dump/re-excitation laser over the range between the predicted resonances for transfer t o the X ( v = 0) and (v = 1) levels is shown in Fig. 3. This shows clearly the desired peaks, with no other spectral lines appearing over the entire range. i,6717

I

I

. 13545

13560

13575

13590

Dump laser frequency (cm-l) Figure 3. Spectral signatures of the production of XIC+(v = 0 , l ) state molecules. Delayed ion signals (corresponding to re-excitation of the X state population) are plotted vs. the frequency of the dump/re-excitation laser. The results shown are obtained using the intermediate state z located 9786.10 cm-' above the initial a3C+(v = 37) state. The dotted lines indicate the predicted dump/re-excitation laser frequency for the desired transition. All data is averaged over 100 pulses.

The overall efficiency of the SEP process is -6%. This may be compared t o the maximum possible efficiency for a two-step SEP process of 25% (50% in each step, taking an average over random final phases for the Rabi flopping experienced in different parts of ensemble). The lower value we observe is consistent with our expectations, because of the fluctuating, sparse comb-like spectral structure of the multiple longitudinal-mode pulsed lasers used here.lg

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3. Properties of the sample of X I E + ( v = 0) molecules

As discussed in the introduction, an ideal UPM source would provide a sample of molecules that is ultracold in all degrees of freedom -not only translationally, but also rotationally and vibrationally. For long trapping times, it is also likely necessary to prepare the molecules in their absolute rovibrational ground state X ( v = 0, J = 0), to avoid losses due to inelastic collisions. Hence, it is important to understand the rotational and vibrational state distribution of UPMs produced with our technique. Our XICf molecular sample is almost entirely in the w = 0 vibrational state. This is a consequence of our choice of an intermediate state i which is primarily of triplet character. Specifically, we used states i in the manifold of levels associated with the overlapping s1 = 1 electronic states c3Cf, B'II, and b311. By choosing levels well below the minimum of the adiabatic B state potential, we ensure that state i is of predominantly triplet (c/b) character, with only a small admixture of BIII1 due to off-diagonal spin-orbit coupling. Because singlet-triplet transitions are forbidden, the probability of uncontrolled spontaneous decay of state i to the XIC+ state (which has very pure singlet character) is thus extremely small. However, the B'II component of state i is large enough to allow efficient stimulated emission to the X state, using moderate laser power. To quantify the resulting vibrational distribution, we used our earlier analysis of the mixed c / b / B state structure17 to calculate F'ranck-Condon factors, and hence branching ratios, for decay of the i states. While these calculations are only qualitative (due to incomplete knowledge of the relevant state wavefunctions), they indicate that the total population in all other vibrational levels of the XICf state is only 1%of the population driven into the w = 0 level in our experiments. We verified our F'rankCondon calculations qualitatively, by measuring the pulse energy required to power-broaden the i - X transition. The results are consistent with the for the observed i - X trancalculated oscillator strengths of 1- 10 x sitions. Thus, we conclude that the vibrational distribution in the X state is indeed narrowly peaked in the vibronic ground state. For completeness, we note that there remains a substantial background of a3Cf molecules in our samples. This arises both from the original step of PA/radiative stabilization (untransferred molecules) and from the spontaneous decay of the residual i state population that is not stimulated down to X(w = 0) (partially transferred molecules). The latter component leads to a small population of more deeply-bound vibrational levels of the a3Cf

-

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state, but in fact we predict that 75% of the i state population decays to dissociated atomic pairs. Thus, the remaining population in all levels of the a3C+ state is typically a few times larger than the X ( v = 0) population in our experiments. The rotational and hyperfine state distribution of the X(w=O) molecules is determined by selection rules in each step, and by the laser spectral resolution. Hyperfine structure (hfs) is unresolved in all stages of the process, and hence the nuclear spin degrees of freedom are unconstrained. Our results for the rotational distribution are consistent with population of only two low-lying rotational levels ( J = 1 and J = 3); note that the splitting between these levels is approximately the same as the average spectral width of the pulsed lasers used for the SEP. N

4. Future directions

Both the purity and the production efficiency of our ground-state sample can be improved by using transform-limited lasers for the state transfer. Using a stimulated Raman adiabatic passage (STIRAP) technique,20 the a(v=37) to X(v=O) transfer efficiency could approach 100%. In addition, such pulses would have sufficient spectral resolution to populate a single rotational level (in particular, the J = 0 level). Although hfs is too small to resolve in the XICf state, selective population of a single hfs level may also be possible by the use of spin-polarized atoms and hfs-resolved transitions for the PA and initial state transfer steps. To study the physical phenomena of most interest to us, two further steps will be necessary. First, we must trap the molecules as they are formed, in order to enable long storage and manipulation times. Second, we must separate the ground-state molecules from the atoms and vibrationallyexcited a3C+ molecules that also remain in the trap. This is necessary because collisions with these other species will likely lead to rapid loss of ground-state molecules from the trap. We believe the trapping will be straightforward. Far-detuned optical dipole traps (such as the QUEST21) can trap atoms,22 vibrationally-excited r n o l e c u l e ~ ,and ~ ~ X ( w = 0) molecules.24 Excited molecules formed from atoms in such a trap could be efficiently transferred to their absolute rovibrational ground state as described here. The number of trapped UPMs produced in this way will depend on the initial atomic conditions, the rate of collisional quenching of the excited molecules, etc. However, the efficiency of the population-transfer process demonstrated here makes us optimistic

220

that large numbers of UPMs will be available. We have also devised a procedure for removing all unwanted states and species from such a trap, leaving only the ground-state RbCs molecules. Atoms can be easily removed by resonant light pushing. Our procedure for removing vibrationally-excited states relies on the fact that groundstate RbCs has a dramatically larger DC electric polarizability than the vibrationally-excited states. l1 The small value of the molecular electric dipole moment in the high vibrational states of RbCs (particularly in the a3Cf level) arises primarily because the Rb and Cs atoms are spatially far apart in these high-lying levels (as is evident from a semi-classical picture of motion near the turning point of the potential). An additional suppression arises from the small numerical value of the permanent dipole moment in the weakly-bound a3Cf potential (this appears to be a general property of the pure van der Waals triplet ground states of bi-alkali atoms). Together, these effects yield a crude prediction that the excited-state polarizabilities are more than 100 times less than that of the X ( v = 0, J = 0) state. We will take advantage of this difference in polarizability by arranging a trap configuration in which only ground-state RbCs is held against the force of gravity, by a suitable electric field gradient. Our proposed method for purification of RbCs X(v=O, J=O) molecules is shown schematically in Fig. 4. Here we describe the proposed protocol in some detail. Rb and Cs molecules are trapped in a COz laser QUEST in vertical, 1-D lattice configuration with the trapping laser retroreflected in order to create a standing wave. A pulse of photoassociating laser light is followed immediately by short STIRAP pulses driving population into the X I C + ( v = 0, J = 0) state. Following population of the ground state, voltages will be applied to a pair of electrodes positioned horizontally on either side of the trapped sample. The resulting electric field has negligible effect on the atoms and vibrationally-excited molecules trapped in the system. However, for the strong-field seeking, highly polar X I C f ( v = 0, J = 0) RbCs molecules, the field gradient provides an additional confining force in the vertical direction. (The transverse force is negligible compared to the horizontal confinement force of the focused QUEST beam.) In order to release the unwanted particles, we will next convert the QUEST from a 1-D lattice to a pure dipole trap. This can be accomplished by rotating a waveplate in the retro-reflected beam path, SO that the reflected beam has polarization orthogonal to that of the incoming beam. Particles not subject to the Stark potential will simply fall out of this shallow dipole trap, due to the effect of gravity. However, the Stark

221

v=O.J=O

\

I

Both configurations: all species transversely confined by optical potential from tight laser focus

Figure 4. Schematic of proposed protocol for creating pure samples of ultracold X ( w = 0, J = 0) RbCs molecules.

potential can be made sufficiently steep t o hold against gravity for groundstate RbCs molecules. With reasonable technical parameters, the trapped RbCs molecules will occupy approximately the same vertical extent as was originally loaded into the 1D lattice QUEST, leading to minimal heating of the sample during the purification process. After a brief delay to allow the vibrationally-excited molecules to exit, the remaining trapped sample should be quite pure and experiments on the ground-state RbCs molecules can begin.

5 . Conclusions

In summary, we have produced ultracold polar RbCs molecules in their ground vibronic state. The optical transfer technique used here should be applicable to other heteronuclear bi-alkali molecules, including those produced by Feschbach resonance techniques2' Translational temperatures limited only by atomic cooling methods should be achievable, and with available technology population of a single rovibronic state with high purity should be possible. This opens a route to the study and manipulation of polar molecules in the ultracold regime.

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Acknowledgments

We acknowledge support at Yale from NSF Grant DMR0325580, t h e David and Lucile Packard Foundation, and the W.M. Keck Foundation; and at Stony Brook from NSF grant PHY0354211 a n d the US. Office of Naval Research. We t h a n k A.J. Kerman for crucial contributions to earlier stages of this work. References 1. J. Doyle et al., Eur. Phys. J . D31, 149 (2004). 2. D. DeMille, Phys. Rev. Lett. 88,067901 (2002). 3. M. A. Baranov et al., Phys. Rev. A66, 013606 (2002); K. Goral, L. Santos, and M. Lewenstein, Phys. Rev. Lett. 88,170406 (2002). 4. E. Bodo, F. A. Gianturco, and A. Dalgarno, J . Chem. Phys. 116, 9222 (2002), and references therein; R.V. Krems, in Recent Research Developments in Chemical Physics, Vol. 3 pt. 11, p. 485 (2002). 5. M. Kozlov and L. Labzowsky, J . Phys. B28,1933 (1995); J . J. Hudson et al., Phys. Rev. Lett. 89,023003 (2002); D. DeMille, Bull. Am. Phys. SOC.49,97 (2004). 6. A. J. Kerman et al., Phys. Rev. Lett. 92,153001 (2004). 7. M. W. Mancini et al., Phys. Rev. Lett. 92,133203 (2004). 8. C. Haimberger et al., Phys. Rev. A70,021402(R) (2004). 9. D. Wang et al., Phys. Rev. Lett. 93,243005 (2004). 10. S. Inouye et al., Phys. Rev. Lett. 93,183201 (2004); C.A. Stan et al., Phys. Rev. Lett. 93,143001 (2004). 11. S. Kotochigova, P. S. Julienne, and E. Tiesinga, Phys. Rev. A68, 022501 (2003); M. Aymar, and 0. Dulieu, J . Chem. Phys. 122,204302 (2005); S. Kotochigova, private communication. 12. V . A. Yurovsky et al., Phys. Rev. A62, 043605 (2000). 13. T. Mukaiyama et al., Phys. Rev. Lett. 92,180402 (2004). 14. J. Sage, S. Sainis, T . Bergeman, and D. DeMille, Phys. Rev. Lett. 94,203001 (2005). 15. G. Igel-Mann et al., J . Chem. Phys. 84,5007 (1986). 16. A. J. Kerman et al., Phys. Rev. Lett. 92,033004 (2004). 17. T . Bergeman et al., Eur. Phys. J . D31, 179 (2004). 18. C. E. Fellows et al., J. Mol. Spectrosc. 197,19 (1999). 19. Th. Weber, E. Riedle, and H. J. Neusser, J . Opt. SOC.Am. B7,1875 (1990). 20. G. He et al., J . Opt. SOC.Am. B7,1960 (1990); R. Sussman, R. Neuhauser, and H. J. Neusser, J. Chem. Phys. 100,4784 (1994). 21. T. Takekoshi, J.R. Yeh, and R.J. Knize, Opt. Commun. 114,421 (1995). 22. T. Takekoshi and R.J. Knize, Opt. Lett. 21,77 (1996). 23. T. Takekoshi, B. M. Patterson, R. J. Knize, Phys. Rev. Lett. 81,5105 (1998). 24. B. Friedrich and D. Herschbach, Phys. Rev. Lett. 74,4623 (1995); P. Braun and A. Petelin, Sou. Phys. JETP 39,775 (1974). 25. W. C. Stwalley, Eur. Phys. J. D31,221 (2004).

THERMODYNAMICS AND MECHANICAL PROPERTIES OF A STRONGLY-INTERACTING FERMI GAS

J. E. THOMAS, J. KINAST, AND A. TURLAPOV Physics Department, Duke University, D u r h a m , NG 27708-0305, USA E-mail: [email protected]. edu We study the properties of an optically-trapped, strongly-interacting Fermi gas of 6Li atoms, near the center of a broad Feshbach resonance. We observe a transition in the heat capacity, by precisely adding energy to the gas and measuring an empirical temperature that is based on the spatial profiles of the cloud. Recent theory, using a pseudogap formalism, enables the first temperature calibration, and interprets the transition as the onset of superfluidity in a strongly-attractive Fermi gas. We also measure the empirical temperature dependence of the radial breathing mode in the same regime. The frequency remains near the hydrodynamic value, while the damping rate reveals a clear transition in behavior near the predicted superfluid transition temperature. We consider quantum viscosity as a cause of damping and show that the predicted magnitude is consistent with observations, but the predicted scaling in the total atom number is not.

1. Introduction

Since the first observation of a degenerate, strongly-interacting Fermi gas [l],the field of interacting Fermi gases has made spectacular progress. Strongly-interacting Fermi gases are produced in an optical trap [2], by using a magnetic field to tune to a Feshbach resonance, where the zero energy scattering length is large compared to the interparticle spacing [l-41. By tuning below resonance, Bose-Einstein condensates (BECs) of molecular dimers have been produced from a two-component strongly-interacting Fermi gas [5-81. In contrast to studies of stable molecular BEC’s, which are produced below resonance, the study and proof of superfluidity just above resonance, in the strong Cooper pairing or strongly-attractive regime, has been less straightforward. Over the past two years, however, substantial evidence for superfluidity has been obtained in a variety of experiments. Macroscopic measurements provide evidence for superfluidity in a 223

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strongly-attractive Fermi gas, and provide important information on the equation of state of this universal quantum system. Evidence for superfluid hydrodynamics has been obtained in observations of anisotropic expansion after release of the cloud [1,8] and in studies of the temperature and magnetic field dependence of the frequency and damping of collective modes [9-121. Measurement of the heat capacity [13] and collective mode damping [12] as a function of empirical temperature reveal transitions in behavior, close to the predicted superfluid transition temperature [13-161. Recently, the observation of vortices [17] in a strongly-attractive Fermi gas has provided what appears to be a definitive proof of superfluidity. Microscopic studies of strongly-attractive Fermi gases have concentrated on the detection and probing of fermionic atom pairs. Pairs were first observed by projection onto a molecular BEC [18,19]. The pair binding energy has been probed in measurements of the pairing gap, by radiofrequency spectroscopy [20] and by modulating the interaction strength [21]. In the region of a Feshbach resonance, the pair wavefunction can contain both a dominant triplet contribution in the open collision channel, and a much smaller singlet contribution from the closed molecular channel [22,23]. Recently, molecular spectroscopy in the singlet manifold has been used to probe the molecular amplitude of the fermionic atom pairs and the superfluid order parameter throughout the Feshbach resonance region [23].

2. Preparation of a Strongly-Interacting Fermi Gas

We prepare a highly degenerate, strongly-interacting Fermi gas of 6Li. This is accomplished using evaporation of an optically-trapped, 50-50 mixture of spin-up/down states at 840 G, just above the center of a broad Feshbach resonance [1,9,11-131. To reduce the temperature, we do not employ a magnetic sweep from a BEC of molecules, in contrast to several other groups [8,10,17,19,20,24]. Instead, we evaporate directly in the strongly attractive, unitary regime: We simply exploit the large collision cross section and the rapid vanishing of the heat capacity with decreasing temperature, which is especially effective in the superfluid regime. These properties make the gas easier to cool. We find that the spatial profiles of the coldest clouds assume nearly the shape of a zero-temperature Thomas-Fermi profile, as expected on general grounds for a unitary Fermi gas at zero temperature [l,25,261. In the forced evaporation, the depth of COz laser optical trap is reduced to 1/580 of its maximum value, and then recompressed to 4.6% of

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the maximum trap depth for most of the experiments. From the trap frequencies measured under these conditions and corrected for anharmonicity, we obtain: w l = = 27r x 1696(10) Hz, w,/wz( = 1.107(0.004), and X = w , / w l = 0.045. Then, a = ( W , W ~ W , ) ~ / ~= 27~x 589(5) Hz is the mean oscillation frequency. For most of the data reported, the total number of atoms is N = 2.0(0.2) x lo5. The corresponding Fermi temperature at the trap center for a noninteracting gas is TF = (3N)1/3E/kB 2~ 2.4pK, small compared to the final trap depth of U o / k ~ = 35pK (at 4.6% of the maximum trap depth). The coupling parameter of the strongly-interacting gas at B = 840 G is k F a 2 -30.0, where h k = ~ dis the Fermi momentum, and a = a ( B ) is the zero-energy scattering length estimated from the measurements of Bartenstein et al. [27]. 3. Tools for Thermodynamic Measurements

Equilibrium thermodynamic properties of the trapped gas, as well as dynamical properties, can be measured as functions of the temperature or of the total energy. The temperature is changed by adding energy t o the gas at fixed total atom number and fixed magnetic field, starting from the lowest temperature samples. In the following, we describe first a method for precisely adding a known energy t o the gas. Then we describe a method for associating an empirical temperature with the spatial profile of the gas, and a temperature calibration method using theoretically predicted spatial profiles [28]. 3.1. Precision Energy Input

Energy is added to the gas by abruptly releasing the cloud and then recapturing it after a short expansion time t h e a t . During the expansion time, the total kinetic and interaction energy is conserved. When the trapping potential U ( x ) is reinstated, the potential energy of the expanded gas is larger than that of the initially trapped gas, increasing the total energy. After waiting for the cloud to reach equilibrium, the sample is ready for subsequent measurements. After recapture, the increase in the total energy, A E , is given by

where no is the initial spatial distribution, and n is the spatial distribution after the expansion time t h e a t . To determine the energy input, we need to know no and understand the expansion dynamics of the cloud.

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Experimentally, we find that the spatial profile of a trapped, highly degenerate, strongly-interacting Fermi gas closely resembles that of an ideal noninteracting, Fermi gas [l]. This can be understood using arguments based on universal thermodynamics: For a unitary zero temperature gas, the net effect of the interactions is equivalent to changing the bare mass to an effective mass [13], m* = m/(l p), where /3 is the ratio of the interaction energy to the local Fermi energy at T = 0 which is a universal many-body parameter [l,25,291. The equation of state then yields a zero temperature, Thomas-Fermi profile, n o ( x ) , for which the Fermi radii are altered from those of a noninteracting gas by a factor of (1 ,B)1/4 [25]. Hence, the initial spatial distribution is well determined, as long as we start with a very low temperature cloud. For such a low temperature cloud, the local chemical potential is just 1 + ,B times the local Fermi energy for a noninteracting gas. In this case, after release from the trap, the gas is expected to expand hydrodynamically and anisotropically [30], as observed in our experiments [l].For the short expansion times used in our experiments, where 0 5 t h e a t 5 450ps, the axial z dimension remains nearly constant, so that the spatial profile after expansion is just

+

+

where bx ( t )and by ( t )are hydrodynamic expansion factors which are readily determined [l,301. For a harmonically trapped cloud which is initially at nearly zero temperature, the total energy is close to that of the ground state, which is 314 of the Fermi energy per particle. The change in the effective mass alters the Fermi energy of the interacting gas by a factor of from that of the noninteracting gas. Hence, the ground state energy is

4~~B~FJ1SP.

E~ = 3

(3)

Then, the energy after expansion and recapture is given by

Eq. 4 has a simple physical interpretation. After release from a harmonic trap, and subsequent recapture after a time t h e a t , the potential energy in each transverse direction is increased as the square of the expansion factors,

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b, and b,. The total potential energy is half of the total energy, since the unitary gas obeys the virial theorem for an ideal gas at all temperatures, as shown below. Hence, the initial potential energy in each direction is 1/6 of the total energy. Note that using Eq. 1, the corrections to the energy change arising from trap anharmonicity are readily determined [13]. 3.1.1. Virial Theorem Eq. 4 enables a test of an important prediction of universal thermodynamics [31]: A strongly-interacting Fermi gas must obey the virial theorem for an ideal gas. The experimental confirmation of the virial theorem also tests the consistency of the assumptions leading to Eq. 4: The determination of the ground state energy relies on the assumption of universal thermodynamics, while the expansion factors are computed by assuming hydrodynamic expansion in the radial dimension, as observed at low temperatures [l]. The virial theorem follows from the form of the pressure, P(n, T), which can depend only on the local density n and temperature T, so that P = $ E ( x ) , where E(X) is the local energy density, i.e., the sum of the local kinetic and interaction energies [31,32]. Balance of the pressure and trapping forces in a harmonic potential requires that N ( V ) = (3/2) Jd3x P(x),where (V) is the average potential energy per particle. Using J d3x&(x)= E - N ( U ) ,one obtains [32]

E N ( U ) = -. (5) 2 This result is remarkable: A trapped, strongly-interacting, unitary gas (containing superfluid pairs, noncondensed pairs, and unpaired fermions) should obey the virial theorem for an ideal noninteracting gas. Since ( U ) c( (x2), the measurement of the mean square transverse radius ( x 2 )of the trapped cloud as a function of theat should reveal linear scaling with the total energy, calculated using Eq. 4. Fig. 1 shows the results of the measurements, which confirm the virial theorem predictions within a few percent [32]. In the experiments, the gas is evaporatively cooled to the lowest temperature and then the energy is increased as described above. For each value of t h e a t , the final energy E is calculated according to Eq. 4. For each El the gas is released and the transverse radius of the cloud is measured after a fixed expansion time of 1 ms. The observed linear scaling of ( x 2 )with the calculated E confirms the virial theorem prediction. F’urther, the linear scaling (within 2%) shows that the expansion factor (for imaging after 1 ms) is nearly independent

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3.0 i

-

2.5

2.0

I

A h

s

Nx 1.5

2 NA

X

v

1.0

0.5

0.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

EIE,,- 1 Figure 1. Verification of the virial theorem prediction: (z2)/(z2(0)) versus E/Eo for a unitary gas of 6Li, showing linear scaling. Here (z2)is the measured transverse mean square size. E is the total energy, calculated using Eq. 4. Eo and ( ~ ~ ( 0denote )) the ground state values.

of the total energy over the range of energies, and hence temperatures, studied. We have measured the scaling of the transverse dimension over a wide range of expansion times at our lowest temperatures. We find that it is closely hydrodynamic. Since the expansion factor for the imaging is constant, it must be hydrodynamic over the range of temperatures studied.

3.2. Empirical Temperature Measurement Measurement of temperature in a noninteracting or weakly interacting Fermi gas is readily accomplished by fitting a Thomas-Fermi (T-F) distribution to the spatial profile of the cloud either in the trap, or after ballistic expansion, which alters the profile by a scale factor [1,30].We normally integrate the measured column density of the expanded cloud over the axial dimension, and obtain the spatial profile in one transverse dimension, ~ T F [ Zoz, ; T / T F ] The . spatial profile is taken t o be a function of two parameters, the Fermi radius oz,i.e., the cloud radius at zero temperature, and the reduced temperature T/TF, i.e., the ratio of the Boltzmann temperature T t o the trap Fermi temperature for a noninteracting gas, T F .

229

One can consider u, = & ~ T ~ / ( r n w ; ) to set the length scale of the spatial profile and T/TF as a shape parameter. At low T/TF, the shape approaches a zero temperature T-F profile, c( ( 1 - 2 ? / 0 2 ) ~ / ~ , while at high T/TF, the profile approaches a Maxwell-Boltzmann shape 0: exp [ -rnw2z2 / ( 2 1 c ~ T ) = I exp [-(x 2/uE)(TF/ T ) ]. In the latter case, only the product of T/TFand u2 appears. Hence, for determination of the reduced temperature, it is convenient to determine the Fermi radius from the lowest temperature data, and then to hold this radius constant, i.e., to take u, = c, N'/6 in subsequent measurements at higher temperature, where c, is held constant. In this way, the reduced temperature T/TF is uniquely correlated with (and can be used to parametrize) the shape of the spatial profile. For a unitary gas, the spatial profile is not precisely known, and there are no simple analytical formulae except at T = 0, where the equation of state assures that the shape of the cloud must take the zero temperature T-F form, with 0, + u;, where u; = u, (1 ,B)1/4, as discussed above [25].We obtain ,B by comparing the transverse radius of the trapped cloud for the interacting gas with that of the noninteracting gas [25].For the noninteracting gas, we use either the calculated u, or the radius measured after ballistic expansion. For the interacting gas, we obtain uk after hydrodynamic expansion for 1 ms. We find that ,l3 = -0.49(0.04) (statistical error only) [13]. Similar results are obtained by measurements of the axial dimension of the trapped cloud without expansion [24]and by direct measurements of the interaction energy [33].The measurements are in reasonable agreement with predictions [34,35].Discrepancies between the measurements may arise from the sensitivity of ,B to the precise location of the Feshbach resonance [24],which in 6Li has been most recently measured by radiofrequency spectroscopy methods [27]. Although the spatial profile of a unitary gas is not precisely known, we observe experimentally that the shape is closely approximated by a T-F profile for a noninteracting gas. Further, recent theoretical predictions of the spatial profile [28] show that the shape is nearly of the T-F form at all temperatures, as a consequence of the existence of noncondensed pairs. Hence, to provide a parametrization of the spatial profiles, we define an empirical reduced temperature ? = (T/TF)fit,and take the one dimensional spatial profile of the cloud to be of the form T Z T F ( Z ;u;, F ) . In general, the empirical reduced temperature does not directly determine the reduced temperature T / T F . However, at T = 0, the T-F shape is exact, so that ? = 0 coincides with T/TF = 0. Hence, the procedure

+

230

for determining oh from the data at very low temperature, where P 0, is consistent, i.e., we take oh = ck N116,where ck is a constant. Further, at sufficiently high temperature, the profile gas must have a ] exp[-(z2/ah2) (1/?)]. Maxwell-Boltzmann form IXexp[-mw2z2/( 2 k ~ T )= This determines the natural reduced temperature scale, T,,t , rn

+

which follows from the interacting gas Fermi radius, oh = ox( 1 /3)114, and the definition of the noninteracting gas Fermi radius ox.The empirical temperature scale is therefore exact at T = 0 and at high temperature, for a fixed interacting gas Fermi radius (which determines 0). To calibrate p more generally, we fit profiles of the form n ~ ~ a;, ( z?) ; to the spatial profiles predicted as a function of T/TF using a pseudogap formalism [13,28].The value of o; is determined from the lowest temperature theoretical profile, and p is determined for all of the predicted profiles. The full empirical temperature calibration is shown in the inset in Fig. 3. If the natural temperature were the correct scale at all T , then one would expect T/TF = T m = 0.71 p. Remarkably, above the predicted superfluid transition temperature, where T,/TF = 0.29, i.e., for 2 0.45, the natural temperature scale is in close agreement with predictions [13], even though noncondensed pairs are believed to exist up to at least T / T F N 0.6, and are present in the predicted profiles. However, below the transition, for 0 < T 5 0.29, i.e., for 0 5 p 5 0.45, we find that there is a systematic deviation: Here, T/TF = O.54i?'l3, and the natural temperature scale underestimates the reduced temperature [13].This is reasonable, since the energy of the unitary gas, and hence the mean square cloud size, increases as a higher power of TITp than quadratic. 4. Heat Capacity

The techniques of precision energy input and empirical temperature measurement provide a method for exploring the heat capacity [13,26] of a strongly-interacting Fermi gas. In the experiments, the 6Li gas is cooled to very low temperature, 5? = 0.04, by forced evaporation at 840 G, just above the center of the Feshbach resonance, as described above. Then, the gas is heated by adding a known energy. Finally, the gas is released from the trap and allowed to expand for 1 ms. As observed above, the gas expands hydrodynamically by a scale factor, so that the shape of the expanded cloud

23 1

closely approximates that of the trapped cloud, enabling a determination of rl;.

10

8

2

O I 0

I

0.5

I

1

N

I

I

1.5

2

T Figure 2. Total energy versus temperature. For each heating time t h e a t , the temperature parameter F is measured from the cloud profile, and the total energy E(theat) is calculated from Eq. (4) in units of the ground state energy Eo. Circles: noninteracting Fermi gas data; Diamonds: strongly-interacting Fermi gas data. Upper solid curve: predicted energy versus reduced temperature for a noninteracting, trapped Fermi gas, Eideal(?)/Eideal(0); Lower solid curve: predicted energy versus ? for the unitary case. No temperature calibration is applied since F x Fnat over the broad temperature range shown. Note that the lowest temperature point (solid square) is constrained to lie on the upper noninteracting gas curve.

Fig. 2 shows that the reduced energy of the gas, EIEo, scales with empirical temperature = (T/Tp)/i,in much the same way as that of an ideal, noninteracting, Eermi gas. However, closer examination reveals that the low temperature data is better fit by a power law in than by the ideal gas scaling. The same data on a log - log plot shows a transition in behavior [13,26]. By using the temperature calibration, and replotting the raw data as in Fig. 3, we find that the transition occurs at TITF = 0.27, in very good agreement with the prediction for the superfluid transition, TJTF = 0.29 [13]. We also find that the scaling of the energy with temperature is in very good agreement with the predictions.

232

10

7

1

Lu" Am al

kc

z0.1 0.01

0.1

1

Figure 3. Energy input versus temperature from Fig. 2 after temperature calibration, on a log - log scale. The strongly-interacting Fermi gas shows a transition in behavior near T/TF = 0.27. Circles: noninteracting Fermi gas data; Diamonds: strongly-interacting Fermi gas data; Lower (upper) solid curves: prediction for a unitary (noninteracting), trapped Fermi gas, calculated at trap depth &/EF = 14.6 as in the experiments; Dashed line: best fit power law 9 7 . 3 ( T / T ~ )to~ the . ~ ~calibrated unitary data for T/TF 5 0.27. The inset shows the calibration curve, which has been applied to the unitary data (diamonds). The diagonal dashed line in the inset represents T/TF = m F . Here EO= E(T = 0 ) , and EF = keTF is the noninteracting gas Fermi energy.

By fitting a power law in T/TF to the data above and below the transition temperature, we obtain analytic approximations to the energy E(T/TF), from which the heat capacity is calculated using C = (BE/BT)N,a,where the number N and trap depth U are constant in the experiments. For T/TF 5 0.27, we obtain E/Eo - 1 = 9 7 . 3 ( T / T ~ while for T/TF 2 0.27, E/Eo - 1 = 4 . 9 8 ( T / T ~ ) l ,By ~ ~differentiating . the energy in each region with respect t o T , we find that the heat capacity exhibits a jump at the transition temperature, comparable in size t o that expected for a transition between a superfluid and a normal fluid [13]. The appearance of a transition in the behavior in the heat capacity, i.e., in the behavior of the energy versus temperature, is model-independent, as

233

it appears in the empirical temperature data, E(T)/Eo,without calibration [26]. However, the estimate of the transition temperature T c / T ~ and the magnitude of the jump in heat capacity are a model-dependent, since the temperature estimates are obtained by calibration using the theoretical spatial profiles.

5. Collective Oscillations

As the determination of the transition temperature in the crossover regime is of great. interest, we look for corresponding transitions in the mechanical properties of the gas. In this section, we describe our comprehensive measurements of the temperature dependence of the frequency and damping of the radial breathing mode [12]. The temperature is increased by adding energy to the gas. Then, the empirical temperature is measured from the spatial profiles of the released cloud as described above. The radial breathing mode is excited by releasing the gas from the trap for a short time and then recapturing the cloud. In contrast to the method used to add energy, the gas is not allowed to thermalize, and the expansion time, 2 5 p , is so short that the energy increase is negligible. After recapture, the cloud is allowed to oscillate in the trap for a variable time, thold after which it is released and imaged as described above. The mean square width of the cloud oscillates at a frequency w. The oscillation amplitude decays at a rate 1/r. To determine w and r , we fit a damped sinusoid to the mean square width as a function of t h o l d . From the measured values of the trap oscillation frequencies, we predict the radial breathing frequency for a noninteracting gas wno,int = 2w, = 2 . 1 0 ~ 1and the hydrodynamic frequency for a strongly interacting (uni= 1 . 8 4 ~ 1 .Here, we give the precise hydrodynamic fretary) gas, quency, taking into account that wy # w,. The result differs slightly from the value = 1 . 8 3 ~ obtained 1 when w, = wg. Fig. 4 shows the measured frequency w in units of w l , as a function of temperature. Remarkably, after correction for anharmonicity, the frequency is very close to the hydrodynamic value, and far from the collisionless (ballistic) value over the entire range of temperatures explored. This behavior suggests that the gas oscillates under conditions which are close to locally isentropic [32]. In contrast to the frequency, the damping rate, Fig. 5 shows a transition in behavior at N 0.5. For empirical temperatures in the range 0 5 '!i 5 0.5, the data is well fit by a line (0.998 correlation coefficient), while above

234

Figure 4. Radial breathing mode frequency w versus empirical reduced temperature ?. Open circles-measured frequencies; Black dots-after correction for anharmonicity using a finite-temperature Thomas-Fermi profile t o estimate (2’). The dot-dashed line is the for our trap parameters. The dashed unitary hydrodynamic frequency WH = 1.84~1, line at the top of the scale is the frequency 2 w , = 2 . 1 0 ~ 1observed for a noninteracting gas at the lowest temperatures.

0.5, the damping rate behaves quite differently, exhibiting non-monotonic behavior. The value of p = 0.5 lies just above the predicted superfluid transition temperature, where ?? N pnat is a good approximation. Using Eq. 6, we find that = 0.5 corresponds to T/TF = 0.35. This is quite close to the value measured for the transition in the heat capacity, TITF = 0.27, and is consistent with recent predictions, Tc/TF = 0.29 [15,26], Tc/TF = 0.31 [14], and Tc/TF = 0.30 [16]. The damping rate also appears to have a plateau and a further increase near p = 1.0, i.e., T/TF = 0.71, close to the region where the pairing gap is comparable to the collective mode quantum, tiw. This behavior may arise from the breaking of weakly bound pairs in this temperature region.

5.1. Quantum Viscosity In a unitary Fermi gas, there is a natural unit of viscosity, 77 which is determined by the interparticle spacing, L. Viscosity has dimensions of momentum/area. Hence, 77 N h/L3 0: fin, where n is the local density.

235

T

1.5

Figure 5 . Temperature dependence of the damping rate for the radial breathing mode of a trapped 6Li gas at 840 G, showing a transition in behavior. The dashed line shows fit to a line which extrapolates close to zero at zero temperature.

Since 7) 0: ti, we consider this scale as the natural unit of quantum viscosity. Following Gelman et al. [36],we take 7) = atin,

(7)

where a is a dimensionless constant. It is instructive to determine a from the lowest damping rates observed in measurements of the breathing mode. For the axial mode measured by the Innsbruck group [lo], the axial damping ratio is found to be very small, ~/(w,T,) = 1.5 x lov3. This corresponds to observed axial damping times of several seconds, since w, = 21r x 22.5 Hz. For the radial breathing mode measured by the Duke group [11,12], the damping ratio is ~ / ( W L T L ) = 1.3 x lo-', corresponding to damping times of up to seven milliseconds. Similar results are obtained by the Innsbruck group [lo]. To determine a,we introduce a shear viscosity pressure tensor term [37] into the hydrodynamic equations [32]. For low damping, the gas can be assumed to oscillate under nearly isentropic conditions [32]. In this case,

236

the local stream velocity components are of the form ui = xi &/bi, where b i ( t ) is a scale factor [30,38]. The spatial derivatives of the stream velocity, which determine the shear pressure, are therefore spatially independent, and the gradient of the viscosity determines the spatial dependence of the shear pressure. The equations of motion for the bi are readily solved and yield for the radial mode,

and for the axial mode,

1

-- W,T,

QX 5 (3NX)1/3m'

16 -

(9)

Eq. 9 predicts that the axial damping ratio is smaller than that of the radial mode by a factor of X = w , / w l l5,l6.The fundamental mechanism responsible for many of these spin phenomena is a coherent collisional process in which the spin of each colliding particle is changed while the total magnetization is preserved. Refs.l4>l5>I6 observe in particular population oscillations a t low magnetic fields in 87Rb, both in the upper ( F = 2) and lower ( F = 1) hyperfine manifolds. As pointed out in Ref.17, the coherent nature of these oscillations remains as yet unproven. Although spin oscillations are observed in theoretical studies at zero t e m p e r a t ~ r e ~real ? ~ ,life multi-component systems are produced at finite temperatures. As a result, many effects come into play, including thermalization by contact with the thermal cloud18 or phase fluctuations in elongated geometrieslg. Here we summarize our recent work20’21,where we study this microscopic collision process in an ensemble of isolated atom pairs localized to lattice sites of a deep optical lattice. We observe oscillations between twoparticle Zeeman states, coupled by the spin-changing interaction in the upper F = 2 manifold of 87Rb. We show that for a broad range of parameters this dynamics can be described by a Rabi-type model, with which we find excellent agreement. This proves the coherent nature of the spin oscillations in this system, intrisically protected against decoherence effects. We stress finally that this is quite different from “usual” single-particle Rabi oscillations driven by an external rf-field. Here the coherent coupling is mediated by direct collisions and occurs between two-particle states. Furthermore, our system allows for a precise measurement of the coupling parameters for spin changing collisions.

2. Theory To illustrate the experimental situation, we consider a pair of 87Rb atoms, localized in the vibrational ground state of a deep trapping potential. The atom pair is described by a spin wavefunction for two atoms in the upper hyperfine ground state with spin 2. For simplicity we abbreviate the (non-symmetrized) two particle states as ( m l ,m2), where the mi’s denote the projection of each atom’s angular momentum on the z- axis. In a

24 1

collision between the two atoms forming the pair, the projection of the total angular momentum on the quantization axis is conserved, even in a finite magnetic field, due to the symmetry of the underlying interaction hamiltonian under rotation The interaction thus couples an initial state I4i) 3 lml,m2) to a final state lq5j) = Im3,m4),provided the total magnetization is conserved, i. e. 1,3,4,697.

ml+

m2 = m3

+ m4.

(1)

Furthermore, s-wave collisions between spin f = 2 bosons are characterized by three scattering lengths U F for the collision channels with total angular momentum F ( F = 0,2, 4)3. ForS7Rb,predicted values23are uo = 87.93 U B , a2 = 91.28 U B , and a4 = 98.98 U B , where U B is the Bohr radius. The matrix element R i j of the interaction hamiltonian between states I&) and l + j ) can be written as

where 4 0 is the vibrational ground state wavefunction, M is the atomic mass and Au is a weighted difference of the U P ' S that depends on the specific values of the magnetic quantum numbers. For example, in the case I&) = l0,O) and 142) = I 1,-l), one finds Au = (-7uo - 5u2 12u4)/35. Due to the localization in the vibrational ground state and to the constraint of a conserved magnetization, for a given initial state only a few final two-particle states participate in the spin evolution. In the simplest cases, only one final state is available (see below). The atom pair then oscillates between initial and final state at the effective Rabi frequency s2if = &R27 f 62 according to the familiar sinusoidal law (see e.g. 2 4 ) . The Rabi model is parameterized by the coupling strength R i j discussed above, and the detuning 6if = 60 6 ( B 2 )between the initial and final states, where B is the value of the static external magnetic field. As the magnetization is conserved, the initial and final states experience the same first order Zeeman shift, which has no influence on the spin dynamics. However, the second order Zeeman shifts are different and introduce the B2-dependent detuning 6(B2).The constant detuning 60 originates from the difference in interaction energies in the initial and final states.

+

+

+

+

3. Experimental sequence

In our experiment we do not produce a single atom pair, but rather an ensemble of such pairs by driving an ultracold atomic sample in the Mott

242

insulator regime. As described elsewhere (see, e.g., Ref. 22), we prepare a Mott-insulator of around 2 x lo5 87Rb atoms in the = 1,mf = -1) state in a combined optical lattice plus magnetic trap potential. In this work, we use a typical lattice depth of 40 E,. Here E, = h2/2MX2 is the single photon recoil energy, and X = 840nm the lattice laser wavelength. The overall harmonic confinement of the system leads to the creation of Mott-shells with different filling factors25. For our trapping parameters and our atom number, approximately half of the atoms are in the central region with two atoms per site. The remaining atoms are distributed in a surrounding shell with one or zero atoms per site, which do not participate in the spin dynamics. To study spin dynamics, we subsequently switch the magnetic trap off to load the sample in a pure optical lattice. In order to preserve spin polarization of the atoms, a homogeneous magnetic field of approximately 1.2G is maintained. The spin dynamics is initialized by transferring the sample into either = 2 , mf = 0 ) or = 2 , m f = -l), and the magnetic field is subsequently ramped to a final value between 0.2G and 2 G . After time evolution for a variable time t the optical trap is switched off. In order to spatially separate the different magnetic substates, a magnetic gradient field is switched on during the first 3ms of time-of-flight (TOF) The population Nm, of each magnetic sublevel mj is then detected after 7ms TOF with standard absorption imaging.

If

If

If

192,14,15,16.

4. Spin dynamics

4.1. Sample initially polarized in IF = 2 , m = ~ -1) In the case where we start with atom pairs in the i#i) = 1 - 1,-1) state, only the state l O , - 2 ) can be reached via spin-changing collisions due to the constraint of conserved magnetization. The dynamics can therefore be described by a two level system even for low magnetic fields. This is shown in Fig. 1 for the case of B = 0.6 G and a lattice depth of 40 E,. The relative populations in Fig. 1 have been calculated as N-1/NtOt for I - 1,-1) and (No N-2)/Nt,t for 10, -2), where Ntot is the total atom number. In order to describe the oscillations we slightly modify the Rabi-like model to account for the observed damping,

+

243

with TiYif being the damping rate. The measured population in m f = 0, -2 can be written as

No

+ N-2

=n Pf, (4) Ntot where n M 0.5 is the fraction of atoms localized in doubly occupied lattice sites, as introduced above. A fit to the data using Eq. (3) allows to determine the oscillation frequency and the damping rate.

f 53

n o

r 9 cn

E,

0.5

36

5.a 'E N d, .c a

0

10

0

20

30

Hold time t (ms)

Figure 1. Spin dynamics of atom pairs localized in an optical lattice at a magnetic field of B = 0.6 G. Here, the atoms are initially prepared in I4i) = I - 1, -1) and coherently oscillate between I&) and l4f) = I - 2 , O ) . Shown are the populations in rnf = 0 (top) and m f = 0 plus rnf = -2 (bottom), together with a fit to a damped sine.

4.2. Sample initially polarized in IF = 2 , m = ~ 0)

We now consider the case where we start with both atoms in lq5i) = 10,O). This state couples to l4f) = I 1,-1) and Jq5ft) = I 2, -2). The coupling constant for l$i) ++ l4fj) is calculated to be two orders of magnitude smaller than for Iq5i) c) ldf), and can be neglected. However, a two-step coupling channel I4i) cf 14f) H I4y) is also possible, with comparable coupling constants for each step. Although present at low magnetic field (seel4~l6 and Fig. 2b), this second step l4f) cf lcjf,) is increasingly suppressed as the magnetic field is increased due to its large detuning. For B > 0.6G, the system mostly oscillates between l0,O) and 1 1,-1) (see Fig. 2a).

+

+

+

244

For lower magnetic field, the secondary process becomes noticeable and the oscillation proceeds in a three-level, rather than two-level system. Due t o limited space, the three-level model will be presented elsewhere21. Here we simply perform a Fourier transform of the measured spin oscillations, where peaks appear at the characteristic frequencies of the system. From the position of those peaks we extract the effective Rabi frequencies for each step, which we analyze in the next section.

1

0.5

0 10 Hold time t (ms)

0

20

1

0.5

-

0

0

20

10

Hold time t (ms)

Figure 2. Spin dynamics of atom pairs localized in an optical lattice at a magnetic field of B = 0.6 G. The atoms are initially prepared in the initial state I+i) = lO,O), from which the final states /df) = I 1,-1) and l+y) = 1 2, -2) are reachable (see text). It turns out that occupancy of the latter state is supressed above a magnetic field of 0.6 G , as shown in (a). However, if the magnetic field is below this value the population in m j = f 2 increases and the atom pair now behaves as a three-level system (b). Shown are (from top to bottom) the populations in rnj = 0, m f = 3x1 and m f = f 2 together with a fit to a damped sine in (a).

+

+

245

5. Oscillation frequency For these two initial two-particle states )O,O) and 1 - 1,-l), spin oscillations have been observed for various magnetic fields up to 2 G, corresponding to different detunings. The measured oscillation frequency, plotted in Fig. 1 and 2, are well fitted to the expected behaviour of the effective Rabi frequency W ( B )with varying detuning (solid lines in Fig. 3). As the magnetic field dependance is known, this fit allows to extract the bare coupling strength Rif and zero-field detuning 50, related to known combinations of the characteristic scattering lengths a F in the way explained above. This measurement represents a high precision test of spin-dependent scattering properties, and can be used in particular to model potentials for ultracold collisions between two Rb atoms. Using the values quoted in section 2 for the O F ’ S , theoretical values for Rif and 60 can be calculated for our trapping conditions and directly compared t o experiments. We find a general agreement in fig. 3 between the measured oscillation frequencies (circles) and the calculated ones, based on the predictions of Ref. 23 for the a F ’ s (dashed lines). More details on this comparison will be given in a forthcoming publication 21. 6. Conclusion

In summary, we have observed coherent spin dynamics between two-particle states in the upper hyperfine ground state of 87Rb due to spin changing collisions. The observation of high contrast Rabi-type oscillations make this system a promising starting point for quantum information purposes. In this work we have demonstrated a method to create an array of entangled atom pairs with intrinsic robustness against magnetic field fluctuations. This is the first step towards the creation of pair-correlated atomic beams as proposed in Refs. Another intriguing question is the evolution of quantum correlations upon melting the Mott-insulator. A possible outcome would be a non-local condensate of Bell-like pairs delocalized over the entire cloud. This highly entangled state could be distinguished from a coherent superposition of condensates through counting statistics. This work was supported by the DFG, the European Union (OLAQUI) and the AFOSR. 9210.

References 1. D. M. Stamper-Kurn and W. Ketterle, in Coherent mutter waves, edited by

R. Kaiser, C. Westbrook, and F. David (Springer NY, 2001); arXiv:cond-

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Detuning S(B2) (Hz)

I

0.0

.

I

.

0.4

I

.

1

.

0.8

I

.

I

.

1.2

l

.

I

.

I

1.6

.

I

.

I

2.0

Magnetic field B (G) Figure 3.

Oscillation frequency of spin dynamics versus magnetic field for the case (hollow circles) and I - 1, -1) ++ 10, -2) (filled circles). The solid lines are fits to the expected behaviour of the effective Rabi frequency (3). The upper curve has been offset by 200 Hz for clarity. The dashed lines are the expected oscillation frequencies for the lattice parameters used here, using a set of calculated scattering lengths. The error bars are typically on the order of a few percent. l0,O)

++

I + I,-1)

mat/0005001 (2000). 2. M. D. Barrett, J. A. Sauer, and M. S. Chapman, Phys. Rev. Lett. 87,010404 (2001). 3. T.-L. Ho, Phys. Rev. Lett. 81,742 (1998). 4. T. Ohini and K. Machida, J. Phys. SOC.Jpn. 67,1822 (1998). 5. M. Ueda and M. Koashi, Phys. Rev. A 65,063602 (2002). 6. C. K. Law, H. Pu, and N. P. Bigelow, Phys. Rev. Lett. 81,5257 (1998). 7. H. Pu, C. K. Law, S. Raghavan, J. H. Eberly, and N. P. Bigelow, Phys. Rev. A 60,1463 (1999). 8. Q. Gu, K. Bongs, and K. Sengstock, Phys. Rev. A 70,063609 (2004). 9. H. P u and P. Meystre, Phys. Rev. Lett. 85,3987 (2000). 10. L.-M. Duan, A. Sorensen, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 85, 3991 (2000). 11. E. Demler and F. Zhou, Phys. Rev. Lett. 88,163001 (2002). 12. A. Imambekov, M. Lukin, and E. Demler, Phys. Rev. A 68,063602 (2003). 13. J. J. Garcia-Ripoll, M. A. Martin-Delgado, and J. I. Cirac, Phys. Rev. Lett. 93, 250405 (2004). 14. H. Schmaljohann, M. Erhard, J. Kronjager, M. Kottke, S. van Staa, L. Cacciapuoti, J. J. Arlt, K. Bongs, and K. Sengstock, Phys. Rev. Lett. 92, 040402 (2004).

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15. M.-S. Chang, C. D. Hamley, M. D. Barrett, J. A. Sauer, K. M. Fortier, W. Zhang, L. You, and M. S. Chapman, Phys. Rev. Lett. 92, 140403 (2004). 16. T. Kuwamoto, K. Araki, T. Eno, and T. Hirano, Phys. Rev. A 69, 063604 (2004). 17. W. Zhang, D. L. Zhou, M.-S. Chang, M. S. Chapman, and L. You, Phys. Rev. A 72,013602 (2005). 18. M. Erhard, H. Schmaljohann, J. Kronjager, K. Bongs, and K. Sengstock, Phys. Rev.A. 70, 031602 (2004). 19. J. Mur-Petit, M. Guilleumas, A. Polls, A. Sanpera, M. Lewenstein, K. Bongs, and K. Sengstock, cond-mat/0507521. 20. A. Widera, F. Gerbier, S. Folling, 0. Mandel, T. Gericke, and I. Bloch, cond-mat/0505492. 21. A. Widera, F. Gerbier, S. Folling, 0. Mandel, T. Gericke, and I. Bloch (in preparation). 22. S. Folling, F. Gerbier, A. Widera, 0. Mandel, T. Gericke, and I. Bloch, Nature 434, 481 (2005). 23. E. G. M. van Kempen, S. J. J. M. F. Kokkelmans, D. 3. Heinzen, B. J. Verhaar, Phys. Rev. Lett. 88, 093201 (2002); S. Kokkelmans (private communication). 24. C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum mechanics (Wiley VCH, 1977). 25. D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. 81,3108 (1998).

INHIBITION OF TRANSPORT OF A BOSE-EINSTEIN CONDENSATE IN A RANDOM POTENTIAL

J. A. RETTER~,A. F. V A R O N ~D. , CLEMENT^, M. HUGBART~, P. BOUYER~, L. SANCHEZ-PAL EN CIA^ , D. GANGARDT~, G. V. SHLYAPNIKOV273,AND A. ASPECT' Laboratoire Charles Fabry de l'lnstitut d'optique, Universite' Paris-Sud X I , 91403 Orsay cedex, France. Laboratoire d e Physique The'orique et Modbles Statistiques, Universite' Paris-Sud X I , 91405 Orsay cedex, fiance. Van der Waals-Zeeman Institute, University of Amsterdam, Valckenierstraat 65/67, 1018 X E Amsterdam, The Netherlands. We observe the suppression of the 1D transport of an interacting elongated BoseEinstein condensate in a random potential with a standard deviation small compared to the typical energy per atom, dominated by the interaction energy. Numerical solutions of the Gross-Pitaevskii equation reproduce well our observations. We propose a scenario for disorder-induced trapping of the condensate in agreement with our observations1.

1. Introduction Coherent transport of waves in disordered systems is a topic of primary importance in condensed-matter physics, for example in the description of normal metallic conduction, superconductivity and superfluid flow, and has relevance also to optics and acoustics2. The presence of disorder can lead to intriguing and non-intuitive phenomena such as Anderson localization3, percolation dynamics4, and disorder-driven quantum phase transitions to Bose-glass5 or spin-glad phases. The main difficulty in understanding quantum transport arises from the subtle interplay of scattering on the potential landscape, interferences and interparticle interactions. Due to a high degree of control and measurement possibilities, dilute atomic BoseEinstein condensates in optical potentials have proved an ideal system in which to revisit traditional condensed matter problems, and recent theoretical works discuss disorder-induced phenomena in this context7. Apart from the (undesired) effect of a rough potential on trapped cold atoms on atom 248

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chipss, there have been few experiments on BEC in random potentialsg?l0. In this experiment', we study the axial expansion of an elongated BEC in a magnetic guide, in the presence of a disordered potential generated by laser speckle. We observe that a weak disordered potential inhibits both the expansion and the centre-of-mass (COM) motion of the condensate. 2. Laser Speckle - a disordered potential for atoms Laser speckle" is the random intensity pattern produced when coherent laser light is scattered from a rough surface. Such patterns arise from interferences between wavefronts coming from different scattering sites. In the Fraunhofer limit, a speckle pattern does not depend on the details of the scattering surface, but follows a well-defined statistical distribution. The intensity distribution is exponential P ( I ) = exp ( - I / o I ) with the intensity standard deviation equal to the local mean intensity: 01 = (I(z)). The typical distance Az between speckle 'grains' can be characterised by the half-width of the autocorrelation function. For a circular aperture, this is an Airy function, with the first zero located at AZ = 1.22Xl/D, where X is the laser wavelength, 1 the distance from the scattering plate to the observation plane, and D the aperture diameter at the scattering surface.

.

D

GI

Scattenng Plate

t

* I~ R T F

2LTF

-A2

Figure 1. Left: Optical setup used to create the random speckle potential. The condensate is at the focal point of the lens system, with its long axis oriented perpendicular to the page. Right: Example of speckle intensity profile, with condensate to scale. The speckle potential is effectively 1D for the trapped condensate.

In our experiment] a blue-detuned laser beam is shone onto the atoms through a scattering plate, as shown in Fig. 1. The beam is derived from a tapered amplifier, injected by a free-running diode laser at X 780nm and fibre-coupled to the experiment. The out-coupled beam is expanded and then focused onto the condensate, the fibre out-coupler and lenses being mounted on a single small optical bench, aligned perpendicular to the long axis of the BEC. Inserting the scattering plate in the position shown projects an optical speckle potential onto the atoms, with an intensity dis-

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tribution I(r) which is simply the Fourier transform of the phase distribution at the scattering plate. The scattered beam has a total power of up to 150mW and diverges to an rms radius of 1.83mm at the condensate. The mean intensity (the gaussian envelope) of the beam can be assumed constant over the region where the atoms are trapped. To calibrate the speckle pattern, the optical set-up is removed from the BEC apparatus and the intensity distribution observed on a CCD camera at the same distance 1 as the atoms. Taking images with various beam apertures D , we verify the exponential intensity distribution P ( I ) ,and compute the autocorrelation function to obtain the grain size Az. Taking into account the modulation transfer function of the camera12, we find that the measured grain size follows the prediction. For our setup, 1 = 140(5) mm and D = 25.4(1) mm, giving Az = 5.2(2) pm. In this experiment, we produce condensates with an aspect ratio of 100, typical Thomas-Fermi halflength LTF = 150 pm and radius RTF= 1.5 pm. The trapped BEC occupies about 45-50 minima along its length, but experiences an almost constant potential in the radial directions. (Along the axis of the laser beam, the typical length scale of the speckle grains is much longer, N Az2/X = 35 pm.) The speckle potential is therefore effectively one-dimensional (1D) for the atoms: RTF< AZ p of the speckle potential. The continually evolving contributions of the lowdensity wings of the BEC are responsible for the fluctuations of L observed in the simulation for a v 2 0 . 2 ~even ~ once the core of the wavefunction is localized. At intermediate distances (Fig. 3b) , the Thomas-Fermi approximation is no longer valid, and the density profile exhibits time-dependent modulations with a length scale intermediate of and Az. To understand the role of the disordered potential in this trapping scenario, it is useful to compare this situation with that of a periodic potential, with a lattice spacing AZ and depth VO= 2av. Our model predicts similar behaviour in the central region, but differs in the low density wings, where the condensate would continue to expand due to tunnelling between the lattice sites. In the lattice, the condensate fragments when the central density reaches the value no z Vo/glD, independent of the lattice spacing. In the case of the disordered potential, this final density depends on the statistical distribution of the optical potential. We can calculate the probability of finding two speckle peaks of a given height within a given distance, which when combined with the condensate expansion dynamics leads to the following estimate for the final peak density: no = 1 . 2 5 ( a v / g l ~l)n ( 0 . 4 7 L ~ ~ l A This formula, dependent on both a v and Az, is in good agreement with our numerical findings and will be the subject of future experimental work17.


50%. For investigations on entangling the internal electronic state of the ion and the cavity mode we can use the single photon emission scheme28. The static qubit can be encoded in superpositions of either Sl/2 and D5/2 states or of the Zeeman sublevels of the S1/2 ground state. Driving a Raman passage transfers part of such a superposition to an excitation of the cavity mode, i.e. a transfer from a basis {IS),ID)}or {IS),IS')} to the photon Fock basis {lo), 11)). For our parameters, the Raman process works coherently with a 70% probability, i.e. in 7 out of 10 cases there is no spontaneous emission during the Raman passage which could destroy the coherence of the process. We therefore anticipate that the planned experiments should yield almost deterministic entanglement between atomic and photonic states. The experimental scheme starts with a preparation of a certain superposition state of the ion. After emission of a photon, entanglement between the internal electronic states and the photon state has to be verified. We propose to convert the {lo), 11)) photon basis into a time-bin basis (t1,tz) where the photon state is encoded in two well defined time intervals3'. In this scheme, one has to drive two Raman passages starting from the two electronic levels in which the superposition state is encoded. The cavity output is coupled into an optical fiber interferometer where a fiber switch

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directs the first pulse into a long fiber arm and the second pulse into a short fiber arm, with a length difference equal t o the pulse separation. Both fiber arms are then recombined in a beamsplitter. If one fiber arm contains a phase shifter one can observe interference effects. Entanglement between atomic states and photonic states can now be verified by preparation of superpositions of atomic states with different phases, transfer and observation of interference fringes depending on the phase shift in one interferometer arm. The long duration of the photon wavepackets here requires a fiber length of more than 10 km, certainly a challenging task. However, if the cavity coupling can be increased by new techniques (e.g. by the combination of miniaturized ion traps and fiber cavities) the requirements for the experiment are easier to fulfill: both the repetition rate of emission increases and the required fiber length decreases such that the signal to noise ratio for correlation counts increases. Acknowledgments We gratefully acknowledge support by the European Commission (CONQUEST (MRTN-CT-2003-505089) and QGATES (IST-2001-38875) networks), by the ARO (No. DAAD19-03-1-0176), by the Austrian Fonds zur Forderung der wissenschaftlichen Forschung (FWF, SFB15), and by the Institut fur Quanteninformation GmbH. T.K. acknowledges funding by the Lise-Meitner program of the FWF. C. Russo acknowledges support by Funda@o para a Cihcia e a Tecnologia (Portugal) under the grant SFRH/BD/6208/2001. References 1. E. Schrodinger, Proceedings of the Cambridge Philosophical Society 31, 555 (1935). 2. S.J. Freedman and J.F. Clauser, Phys. Rev. Lett. 28, 938 (1972). 3. A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. 47,460 (1981). 4. H. Weinfurter, J . Phys. B: At. Mol. Opt. Phys., 38,S579 (2005). 5. T. Yu and J.H. Eberly, Phys. Rev. B 66, 193306 (2002). 6. W.H. Zurek, Physics Today 44,36 (1991). 7. H. Haffner et a1,Appl. Phys. B 81,151(2005). 8. M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, Cambridge, 2000). 9. S. Haroche and J.M. Raimond, Physics Today 49,51 (1996). 10. J.M. Raimond, M. Brune, and S. Haroche, Rev. Mod. Phys. 73,565 (2001). 11. C.A. Sackett et al., Nature 404,256(2000). 12. Z. Zhao et al., Nature 430,54(2004).

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M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A 223,l(1996). B.M. Terhal, Phys. Lett. A 271,319(2000). M. Bourennane et al., Phys. Rev. Lett. 92,087902 (2004). W. Diir, G. Vidal, and J.1. Cirac, Phys. Rev. A 62,062314 (2000). A. Zeilinger, M.A. Horne, and D.M. Greenberger, NASA Conf. Publ. 3135, pp 73-81 (1992). 18. C.F. Roos et al., Science 304,1478(2004). 19. C.H. Bennett and D.P. DiVincenzo, Nature 404,247(2000). 20. H. Buhrman, W. van Dam, P. Heryer, and A. Tapp, Phys. Rev. A 60, 2737( 1999). 21. J. Joo, J. Lee, J. Jang, and Y.-J. Park, arXiv:quant-ph/0204003; J . Joo, Y.-J. Park, J. Lee, J. Jang, and I. Kim, J. Korean Phys. SOC.46,763(2005). 22. H. Haffner et al., submitted to Nature, (2005). 23. C.F. Roos et al., Phys. Rev. Lett. 92,220402 (2004). 24. A S . Parkins, P. Marte, P. Zoller, 0. Carnal, and H.J. Kimble Phys. Rev. A 51, 1578 (1995). 25. J.I. Cirac, P. Zoller, H.J. Kimble, and H. Mabuchi Phys. Rev. Lett 78,3221 (1997). 26. G.R. Guthohrlein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, Nature 414,49 (2001). 27. A.B. Mundt et al.,Phys. Rev. Lett 89,103001 (2002). 28. C. Maurer, C. Becher, C. Russo, J. Eschner, and R. Blatt, New J . Phys. 6, 94 (2004). 29. B.B. Blinov, D.L. Moehring, L.-M. Duan, and C. Monroe, Nature 428,153 (2004). 30. J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, Phys. Rev. Lett 82,2594 (1999). 31. F. Schmidt-Kaler et al., Appl. Phys. B 77, 789 (2003). 32. D. Kielpinski et al., Science 291,1013 (2001). 33. R. Ozeri et al., Phys. Rev. Lett. 95,030403 (2005). 34. J.J. Bollinger, D.J. Heinzen, W.M. Itano, S.L. Gilbert, and D.J. Wineland, IEEE Trans. Instrum. Meas. 40,126 (1991). 35. F. Schmidt-Kaler et al., J . Phys. B: At. Mol. Opt. Phys. 36,623 (2003). 36. H.J. Briegel, and R. Raussendorf, Phys. Rev. Lett. 86,000910 (2001). 37. A. Sen(De), U. Sen, M. Wiesniak, D. Kaszlikowski, and M. Zukowski, Phys. Rev. A 68,062306 (2003). 38. D.M. Greenberger, M. Horne, and A. Zeilinger, in Bell’s Theorem, Quantum theory, and conceptions of the universe, edited by M. Kafatos (Kluwer Academic, Dordrecht, 1989). 39. Z. Hradil, J. RehGek, J. Fiuraek, and M. Jeiek, Lect. Notes Phys. 649, 59(2004). 40. W. Diir and J.I. Cirac, Phys. Rev. A 61,042314 (2000). 41. W.K. Wootters, Phys. Rev. Lett. 80,2245(1998). 42. W. Diir and J.I. Cirac, Journal of Physics A: Mathematical and General 34, 6837( 2001). 13. 14. 15. 16. 17.

QUANTUM CONTROL, QUANTUM INFORMATION PROCESSING, AND QUANTUM-LIMITED METROLOGY WITH TRAPPED IONS

D. J. WINELAND, D. LEIBFRIED, M. D. BARRETT*, A. BEN-KISH~ J. C. BERGQUIST, R. B. BLAKESTAD, J. J . BOLLINGER, J. BRITTON, J. CHIAVERINI, B. DEMARCO~D.HUME, w. M. ITANO, M. JENSEN, J. D. JOST, E. KNILL, J. KOELEMEIJ, C. LANGER, W. OSKAY, R. OZERI, R. REICHLE, T. ROSENBAND, T. SCHAETZ! P. 0. SCHMIDT: AND S. SEIDELIN National Institute of Standards and Technology, Boulder CO 80305-3328 E-mail: djwQboulder.nist.gov

We briefly discuss recent experiments on quantum information processing using trapped ions a t N E T . A central theme of this work has been t o increase our capabilities in terms of quantum computing protocols, but we have also applied the same concepts to improved metrology, particularly in the area of frequency standards and atomic clocks. Such work may eventually shed light on more fundamental issues, such as the quantum measurement problem.

1. Introduction

In 1995, Ignacio Cirac and Peter Zoller described how an ensemble of trapped ions could be used to implement quantum information processing (QIP).’ Several experimental groups throughout the world have pursued this basic idea, and although a useful device still does not exist, iontrappers are optimistic that one can eventually be built. In part, this is because the ion-trap scheme can satisfy the basic requirements for a quantum computer as outlined by DiVincenzo’: (1) a scalable system of well defined qubits, (2) a method to reliably initialize the quantum system, (3) long coherence times, (4) existence of universal gates, and (5) an efficient *present address: Physics Department, University of Otago, New Zealand +present address: Technion, Haifa, Israel $present address: Physics Dept., Univ. of Illinois §present address: Max Planck Inst. for Quantum Optics, Garching, Germany qpresent address: Inst. for Experimental Physics, University of Innsbruck, Austria 393

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measurement scheme. Most of these requirements have been demonstrated, and straightforward, albeit technically difficult, paths to solving the remaining problems exist. In this paper, we summarize recent trapped-ion QIP experiments carried out at NIST, but note that similar work is currently being pursued at Aarhus, Barcelona, Garching (MPQ), Innsbruck, LANL, London (Imperial), Ontario (McMaster), Michigan, MIT, Oxford, Siegen, Sussex, Teddington (NPL), and Ulm. We describe how the system might be scaled up by use of an array of interconnected trap zones and cite experimental implementation of algorithms that utilize the basic elements of this scheme. We then summarize efforts devoted to construction of traps by use of methods that are suitable for large-scale fabrication. We briefly discuss how QIP methods might be used in metrology, and finally suggest how QIP studies might eventually shed light on fundamental issues of decoherence.

2. QIP with multiplexed ion trap arrays

Although large numbers of ions can be cooled into regular arrays in single traps, many of the practical N-ion gates ( N 2 2), such as the original Cirac/Zoller two-ion gate,’ require addressing of individual ions and single modes (or a very small number of modes) of ion motion. Individual ion addressing can be accomplished with focused laser beams as long as the ions aren’t too close together (or equivalently, as long as the mode frequencies are not too high). Mode addressing is usually accomplished by spectrally isolating the mode(s) of interest (out of 3N possible modes). This has the consequence that when the number of trapped ions becomes large, the mode spectrum becomes so dense that spectral isolation becomes impractical. Although the group at Innsbruck has successfully implemented a number of interesting algorithms on multiple ions in single trap zones by using focused laser beams for individual qubit addressing (see their contribution to these proceedings), as the number of ions increases further, and increased gate speeds (proportional to mode frequencies) become more important, such addressing will become more difficult. Therefore, many groups are considering a multiplexed system of trapping zones where only a small number of ions are confined in the zones that are used for implementing gates. The sharing of quantum information between zones might be accomplished by moving ion qubits between zone^,^'^ by moving an information-carrying “head” ion between zones,5 by coupling separated ions with photons as an intermediary,6 or by probabilistically

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creating entangled pairs of separated ions via light coupling, which then act as a computational resource to be used later.7 2.1. QIP in a linear ion trap array

As a first step towards multiplexing, we have used a six-zone linear array that is an extension of the three-zone trap reported earlier.8 Recent experiments with this device have included demonstrations of quantum telep~rtation,~ quantum error correction,1° quantum-dense coding,ll and the quantum Fourier transform. l 2 These experiments required that entanglement between ions was preserved when the ions were located in different zones. Referring to Fig. 1, entanglement was created in zone A, and the ions were sent to zone S for separation. Electrode S is relatively narrow to facilitate separation of a single group of ions into subgroups by inserting a potential wedge between selected ions. For example, in the teleportation experiment on 'BeS ions,g three ions could be separated into a group of two which were delivered to zone A, with the third ion delivered to zone B. We optimized the separation to minimize the heating of the ions delivered to zone A. With a separation time of 200 ps, the ions could be separated without error. The axial center-of-mass motion of ions in zone A (frequency 3 MHz) experienced a kinetic energy increase corresponding to about l quantum, the stretch mode had gained negligible kinetic energy, and the axial motion in zone B gained about 10 quanta. In the future, traps with much smaller internal dimensions should enable shorter separation times with negligible heating, due t o the higher motional frequencies and sharper separation potential wedge features. However, with all other parameters held constant, smaller dimensions will aggravate ion heating13 and sympathetic cooling will likely be required to maximize gate f i d e l i t ~ . ~ ? ~ Other recent experiments in these traps (that did not require multiple zones) included investigations of spontaneous emission decoherence during Raman transitions14 and a long-lived ( 7 1 , 7 2 > 10 s) qubit memory based on first-order magnetic field-insensitive transitions. l5 N

2.2. FzLture ion trap arrays

For manipulating very large numbers of ions with high gate speeds, it appears that new types of trap construction methods, including twodimensional layouts, will be required. Since (two-qubit) gate speed is proportional to the ions' motional frequencies, which are in turn proportional

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Figure 1. Photograph of one wafer of a six-zone linear trap array. Two of these wafers, properly spaced, comprise the trap as described in [8]. The lower part of the figure shows gold traces (approximately 3 pm thick) deposited onto an alumina substrate (lighter color). The upper figure is an expanded view of the boxed section shown below. For the wafer shown, an R F potential (- 200 V at N 150 MHz) is applied to the upper (continuous) electrode. “Control” potentials are applied to the eight segmented electrodes. Varying the potentials on these electrodes in a coordinated way enables ions to be moved between the six zones located above the electrodes labeled L,1,2,A,S, and B. Zone L is the “Loading” zone, whose width is relatively large to increase the capture volume for beryllium atoms (emitted from a thermal source) that are ionized (by electron impact) in this area. In most of the algorithms demonstrated using this trap, zones A and B (“Alice” and “Bob”) were used to manipulate the internal states of qubits (with laser beams overlapping those zones). (Traps constructed by M. D. Barrett and J. D. Jost)

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to (electrode dimensions)p2, we would like to implement traps with dimensions smaller than those of the traps indicated in Fig. 1. Such gold coated alumina electrode s t r u c t ~ r e shave , ~ ~a~size ~ ~limitation ~ from the fact that the laser-machined cuts in the wafers are limited to a width of around 20 CLm. To overcome this limitation, it should be possible to take advantage of MEMS fabrication techniques, where significantly smaller structures can be fabricated. If this is done, we must of course worry about ion motional heating, which increases as the electrodes become ~ma1ler.l~ An obvious construction material would be silicon; however, with typically available substrates, RF loss at the trap drive frequency appears to be a problem. At NIST we constructed a single-zone two-layer trap of the type described in Ref. [8] whose electrodes were made of commercially available boron-doped silicon (Fig. 2). In this apparatus, we trapped and laser cooled 24Mg-t ions. Electrode features as small as 5 pm were defined by use of photolithography and industry standard silicon deep reactive ion etching (DRIE Bosch process). Structural support and spacing of the electrodes was provided by a borosilicate glass thermally matched to silicon and attached to the electrodes by anodic bonding.17 Such an approach is applicable to the fabrication of many-zone large-scale traps including planar traps (below) since the number of processing steps does not increase with the number of zones in the array. In a different approach, the University of Michigan group has built a two-layer trap with GaAs electrodes and AlGaAs insulators" and observed trapping of Cd+ ions.a A three-layer geometry4 has been implemented for Cd+ ions16 and geometries that would optimize the separation of ions into separate groups have been studied.lg Sandia researchers have fabricated arrays of very small (- 1 pm) three-dimensional trap structuresI2' which also might be configured for QIP. Borrowing from the groups pursuing magnetic waveguide traps for neutral atoms, linear traps based on electrodes confined to a surface might also be considered.21 Such "planar" traps would be relatively easy to fabricate on a large scale and would permit on-board electronics beneath the electrode surface.b In addition to finding a way to construct large-scale trap arrays, a way to multiplex laser beams must be sought. It might be possible to use miniature

aC.Monroe, Univ. of Michigan, private communication bR. Slusher, Lucent, private communication

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Figure 2. The photograph shows a singlezone two-layer trap of the type described in Ref. [8] The bottom part of the figure shows a schematic of the trapping region for one of the trap electrode wafers, which are fabricated from boron-doped silicon. Laser cooled 24Mg+ ions have been confined in this trap (constructed by J. Britton, NIST).

steerable mirrors based on MEMS technology for this purpose.CvdMiniature, large-solid-angle photon detectors (possibly without optics) located very near trapping zones may be essential for highly parallel detection as required in error correction.

3. QIP applied to metrology In the Time and Frequency Division of NIST, we have been interested in applying the methods of QIP to metrology, in particular, to improve the =O.Blum Spahn, Sandia National Labs, private communication. J. Kim, Duke University, private communication.

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signal-to-noise ratio in spectroscopy and atomic clocks. For this purpose, we take advantage of entanglement. The improvement obtained from “spinsqueezed” ~ t a t e s , ~where ~ , ~the ~ operator * ~ ~ of the effective mean spin vector is measured, has been demonstrated for two ions.25 We have also demonstrated the gain in signal-to-noise ratio with certain states in combination with other operators such as the variance and par it^.^^^^^ More recently, we have extended a two-ion phase gate27 to implement a form of Ramsey spectroscopy where each of the two conventional Ramsey 7112 pulses are replaced with a rotation and one-step phase gate.28 Starting with all ions in the state I I), the first modified “7r/2” pulse generates a generalized GHZ state” or “Schrodinger-cat” state of the form ’[11)11-1)2...l-1)~+e~Plf)llf)2. -.l f ) ~ ] . Jz During the Ramsey free-precession interval T , the relative phase of the two components of the wavefunction ,B = N(w0 - w)T, where w is the frequency of the probe oscillator and wo is the resonance transition frequency, advances N times faster than that of a single atom. This is the main reason for the increase in spectroscopic resolution. After application of the two modified Ramsey pulses, the equivalent net spin vector is measured in the I I), 17) basis. In ideal circumstances, all ions are measured to be in either all I t) states with probability Pr = [1+cos N(w0 - w ) T ] or all I 1) states with probability Pl = 1 - Pt, cases that are relatively easy to distinguish. Although the fringes occur N times faster, the gain in signal-to-noise ratio is limited to fi compared to the case of N unentangled particles, because the N unentangled particles yield a signal from N separate systems, where the “projection” noise3’ averages down as N-1/2. Although the experimentally observed gain was limited to less than fi, we were able to demonstrate a signal-to-noise ratio better than could be obtained in a perfect experiment on unentangled ions, first on three ions28 and more recently on up to six entangled ions.31 QIP might also be used to improve detection. In one application relevant for frequency standards, it was shown that transitions in a LLclock” ion can be detected in a simultaneously-trapped “logic” ion by mapping the internal state of the clock ion onto the logic ion (with elementary quantum logic operations) where it is easily detected.32 In a more general context, detection sensitivity of quantum systems can be improved in certain situations by use of elementary quantum logic operations on the system to be measured, in conjunction with ancilla particles that are also measured. l1

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4. QIP and the “measurement problem” By the measurement problem, we mean the difficulty that arises because we live in a world that predicts definite outcomes (e.g., bits in our PCs are either 0 or 1)whereas quantum mechanics alone, in general leaves the world in superposition states. In addition to the simple collapse postulate, many attempts have been made to resolve the problem with ideas that include concepts such as “many worlds,” decoherence theory, an as-of-yet unseen collapse mechanism, or simply that the theory of quantum mechanics is only a computational tool that allows prediction of classical outcomes (for a recent review, see for example the paper by Leggett33). Given the unresolved state of affairs on the measurement problem, it seems interesting to press the issue experimentally - that is, can we realize larger and larger entangled superposition states that begin to approach our more macroscopic world where such states aren’t observed? The paper by Leggett suggests one measure for approaching the classical world in which the number of elementary particles involved in a superposition state is of primary i m p ~ r t a n c e .At ~ ~this stage, since we really don’t know what the important parameters are, we might cook up alternative measures that play more to the strengths of atomic physics and quantum optics. With atomic ions, we can emphasize the aspects of entanglement and duration. For example, we might take as a figure of merit the product of the number of particles in a GHZ state (since its phase sensitivity is N-fold larger than that of a single particle) times the duration of the state. A start in this direction is that a six particle approximation to a GHZ state was observed to last longer than approximately 50 ps.31 Note also that superpositions of the (phase-insensitive) Bell states Q& = 4 1)11T ) 2 f l t)lI 1 ) 2 ) have been observed to last for durations exceeding 5 s in gBef ions15 and even longer for Ca+ ions (see the paper by the Innsbruck group in these proceedings). Whatever your favorite measure is, it seems likely that as the quest to make a large-scale QIP machine progresses, states that look more and more like Schrodinger’s cat will be produced - or not, if some fundamental source of decoherence is discovered!

Acknowledgments We thank E. Donley and S. Jefferts for comments on the manuscript. This work was supported by the US National Security Agency (NSA) and the Advanced Research and Development Activity (ARDA) under contract number MOD-7171.05. The work was also supported, in part, by the US Office

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of Naval Research (ONR) and NIST. This manuscript is a publication of NIST and is not subject t o U. S. copyright. References 1. J. I. Cirac and P. Zoller, “Quantum Computation with Cold, Trapped Ions,” Phys. Rev. Lett. 74, 4091-4094 (1995). 2. D. P. DiVincenzo, in Scalable Quantum Computers, S . L. Braunstein, H. K. Lo, and P. Kok, eds., (Wiley-VCH, Berlin, 2001), pp. 1-13. 3. D. J . Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, “Experimental issues in coherent quantum-state manipulation of trapped atomic ions,” J. Res. Nat. Inst. Stand. Tech. 103, 259-328 (1998). 4. D. Kielpinski, C. Monroe, and D. J. Wineland, “Architecture for a large-scale ion-trap quantum computer,” Nature 417, 709-711 (2002). 5. J. I. Cirac and P. Zoller, “A scalable quantum computer with ions in an array of microtraps,” Nature 404, 579-581 (2000). 6. R. G. DeVoe, “Elliptical ion traps and trap arrays for quantum computation,” Phys. Rev. A 58, 910-914 (1998). 7. L. M. Duan, B. B. Blinov, D. L. Moehring, and C. Monroe, “Scalable trapped ion quantum computation with a probabilistic ion-photon mapping,” Quant. Inform. Comp. 4, 165-173 (2004). 8. M. A. Rowe et al., ‘Transport of quantum states and separation of ions in a dual RF ion trap,” Quant. Inform. Comp. 2, 257-271 (2002). 9. M. D. Barrett et al., “Deterministic quantum teleportation of atomic qubits,” Nature 429, 737-739 (2004). 10. J . Chiaverini et al., “Realization of quantum error correction,” Nature 432, 602 - 605 (2004). 11. T. Schaetz, M. D. Barrett, D. Leibfried, , J. B. J. Chiaverini, W. M. Itano, J. D. Jost, E. Knill, C. Langer, and D. J. Wineland, “Enhanced quantum state detection efficiency through quantum information processing,” Phys. Rev. Lett. 94, 010501-1-4 (2005). 12. J. Chiaverini et al., “Implementation of the semiclassical quantum Fourier transform in a scalable system,’’ Science 308, 997-1000 (2005). 13. Q. A. Turchette et al., “Heating of trapped ions from the quantum ground state,” Phys. Rev. A 61,063418-1-8 (2000). 14. R. Ozeri et al., “Hyperfine coherence in the presence of spontaneous photon scattering,” Phys. Rev. Lett. 95, 030403-1-4 (2005). 15. C. Langer et al., “Long-lived qubit memory using atomic ions,” Phys. Rev. Lett. 95, 060502-1-4 (2005). 16, L. Deslauriers, P. C. Haljan, P. J. Lee, K. A. Brickman, B. B. Blinov, M. J. Madsen, and C. Monroe, “Zero-point cooling and low heating of trapped l1’Cdf ions,” Phys. Rev. A 70, 043408-1-5 (2004). 17. D. Kielpinski, Ph.D. thesis, Univ. Colorado, Dept. of Physics, Boulder, 2001. 18. J. J. Madsen, W. K. Hensinger, D. Stick, J. A. Rabchuk, and C. Monroe, “Planar ion trap geometry for microfabrication,” Appl. Phys. B 78, 639-651 (2004).

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19. J. P. Home and A. M. Steane, “Electric octopole configurations for closelyspaced ion traps,” quant-ph/0411102 (2004). 20. M. G. Blain, L. S. Riter, D. Cruz, D. E. Austin, G. Wu, W. R. Plass, and R. G. Cooks, “Towards the hand-held mass spectrometer: design considerations, simulation, and fabrication of micrometer-scaled cylindrical ion traps,’’ Int. J. Mass Spect. 236, 91-104 (2004). 21. J. Chiaverini, R. B. Blakestad, J . Britton, J. D. Jost, C. Langer, D. Leibfried, R. Ozeri, and D. J . Wineland, “Surface-electrode architectre for ion-trap quantum informaion processing,” Quant. Inform. Comp. 5 , 419-439 (2005). 22. D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore, and D. J . Heinzen, “Spin Squeezing and Reduced Quantum Noise in Spectroscopy,” Phys. Rev. A 46, R6797-R6800 (1992). 23. M. Kitagawa and M. Ueda, “Squeezed spin states,” Phys. Rev. A 47, 51385143 (1993). 24. D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, “Squeezed Atomic States and Projection Noise in Spectroscopy,” Phys. Rev. A 50, 6788 (1994). 25. V. Meyer, M. A. Rowe, D. Kielpinski, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental demonstration of entanglement-enhanced rotation angle estimation using trapped ions,” Phys. Rev. Lett. 86, 58705873 (2001). 26. J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optimal Frequency Measurements with Maximally Correlated States,’’ Phys. Rev. A 54, R4649-R4652 (1996). 27. D. Leibfried et al., “Experimental demonstration of a robust, high-fidelity geometrical two ion-qubit phase gate,’’ Nature 422, 412-415 (2003). 28. D. Leibfried, M. D. Barrett, T. Schatz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled states,” Science 304, 1476-1478 (2004). 29. D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger, “Bell’s Theorem Without Inequalities,’’ Am. J. Phys. 58, 1131-1143 (1990). 30. W. M. Itano, J . C. Bergquist, J . J. Bollinger, J. M. Gilligan, D. J. Heinzen, F. L. Moore, M. G. Raizen, and D. J. Wineland, “Quantum projection noise: population fluctuations in two-level systems,” Phys. Rev. A 47, 3554-3570 (1993). 31. D. Leibfried et al., in preparation (2005). 32. P. 0. Schmidt, T. Rosenband, C. Langer, W. M. Itano, J. C. Bergquist, and D. J. Wineland, “Spectroscopy using quantum logic,” Science 309, 749-752 (2005). 33. A. J. Leggett, “Testing the limits of quantum mechanics: motivation, state of play, prospects,” J. Phys.: Condens. Matter 14, R415-R451 (2002).

THE ATOMIC SPIROGRAPH: ATOMIC WAVE FUNCTION AND LASER PULSE SHAPE MEASUREMENTS FROM COHERENT TRANSIENTS

B. CHATEL, A. MONMAYRANT, B. GIRARD Laboratoire de Collisions, Agre'gats et Re'activite' (CNRS UMR 5589), I R S A M C Uniuersite' Paul Sabatier, 31 062 Toulouse CEDEX, fiance E-mail:[email protected] Coherent transients result from the excitation of a two-level system by an ultrashort chirped pulse A sequence of two measurements provides a direct access to the excited state wave function from which the pulse electric field can be retrieved.

1. Introduction

The effect of laser pulse shape on a quantum system is related to the nature of the interaction. For a linear response of the system (one-photon transition in the weak field regime), the final outcome depends only on the spectral component at the resonance frequency and is therefore independent of the pulse shape, and particularly of the spectral phases '. This explains for instance why signals equivalent to wave-packet interferences could be observed with incoherent light as well as with ultrashort pulses '. However, the temporal evolution towards the final state may depend strongly on the pulse shape. A straightforward illustration of this statement is the nonresonant interaction which leads to transient excitation of the system, but to no final excitation. In the absence of predesigned control mechanisms only a closed loop scheme may be employed to find efficient pulse shapes 5 : The outcome of many different shapes is fed back into an algorithm that iteratively optimizes the excitation shape without insight into the physical mechanism that is triggered by a particular shape. In contrast the effect of shapes on small systems can be systematically studied within an open-loop This open-loop approach is well adapted to these systems for scheme which theoretical predictions are reliable. It consists of reaching a specific goal (manipulation of the temporal response of a system excited by a light 314

697t879.

403

404

pulse) without any experimental feed-back. Physical analysis of the process allows one to predetermine the theoretical pulse shape which leads to the desired result. It is then implemented experimentally. In this report, we describe manipulation of Coherent Transients(CT) in an open loop approach. These CT are oscillations in the excited state population resulting from the interaction between a two-level system and a weak chirped pulse. Their high sensitivity provides a new scheme for quantum state measurement and electric field reconstruction. 2. Observation of coherent transients

The CT result from the excitation of a two-level system (19) towards ) .1 with a chirped pulse E ( t )of carrier angular frequency wo close to resonance (WO N w e g ) .

If)

The transient excited state population is probed towards the level in real time by a second ultrashort pulse Eprobe(t)which is Fourier transform limited and short compared to the characteristic features of E ( t ) .Its frequency is close t o the probe resonance (wf,).The fluorescence arising from the state is then recorded as a function of the pump-probe delay 7 . In the following, the probe pulse provides access to the temporal evolution of the population in le), produced by the pump beam. The result of the interaction is described by the first order perturbation theory, and the fluorescence is proportional to

If)

S(7) = laf(7)l2

(1)

+m

o;

Is_,

E p r o b e ( t - 7) exp(iwfe(t

-

/2

7))ae(t)dt

with t

a,(t) =

L

E(t’)exp(iw,,t’)dt’

(2)

In the case of a simply chirped pulse E ( t ) , a quadratic phase appears in the integral giving a,(t) (Eq. 2), leading to oscillations of the probability Jaf(7)I2as already demonstrated lo. To illustrate the method, an experiment has been performed in an atomic Rb vapor. The Rb (5s - 5p (P1/2))transition (at 795 nm) is resonantly excited with the pump pulse sequence. The transient excited state population is probed “in real time” on the (5p - (8s, 6d)) transitions with an ultrashort pulse produced by a home-made NOPA (607 nm, 25 fs). Figure 1 shows the experimental results.

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Figure 1. Experimental Coherent Transients on Rb ( 5 ~ - 5 p at ~ /X~= 795 nm), for a chirp of -8. lo5 fs2 (dots) and the corresponding simulation obtained by numerical resolution of the Schrodinger equation (solid line) lo. Inset : Theoretical excited state amplitude drawn in the complex plane.

Another way to explain the CT phenomenon is to examine the behavior of a, ( t )in the complex plane as displayed in the inset of Fig. 1. The probability amplitude follows a double spiral starting from the origin. Three regions can be distinguished. The two spirals result from contributions before (I) and after (111) resonance. The intermediate region (11) corresponds to the passage through resonance. It provides the main contribution to the population. The two spirals, although similar, have totally different consequences. The first one (I) winds round the origin with an increasing radius. The resulting probability increases thus slowly and regularly and remains small. After resonance (111), a second spiral winds round the asymptotic value leading to strong oscillations of the population. 3. Control of coherent transients with simple spectral shapes

The Coherent Transients described in the previous section result directly from interferences between resonant and non-resonant excitation paths. They are thus strongly sensitive to small modifications of the electric field driving these paths. In this section, we demonstrate how a spectral phase

406

shift of 7r applied at a frequency w, reached after resonance has a striking effect on the CT. The experimental demonstration has been performed using both a low resolution (128 pixels SLM)’ and a high resolution (640 pixels SLM)ll pulse shapers. Figure 2 presents the comparison between initial CT and inverted CT with a 7r step applied 1 pixel after resonance with the low resolution pulse shaper, measured experimentally and calculated by first

Figure 2. Shaped CT with a low resolution pulse shaper 9,12. Experimental (dotted) and numerical (solid) results: chirped pump pulse (black), and .rr-step shaped chirped pulse (gray). Here 4’’ = -9.4105fs2 corresponding to N 21ps. Here Xstep = 2.rrc/w,tep = 795.23 nm.

order perturbation theory. The agreement is excellent. The inverted CT behave as expected. The sensitivity of shaped CT to the position of the step is huge. It is thus well suited as a test of the sensitivity of high resolution pulse shapersll. Moreover since atomic transitions can be found in a very broad spectral range, this method provides a tool to observe shaping effects in spectral domains where usual methods cannot be used. 4. Reconstruction of quantum state and laser pulse from

Coherent Transients The high sensitivity of Coherent Transients to slight modifications of the laser pulse12 opens possibilities to use CT measurements as a characterization of the laser pulse. This can be achieved through a measurement of the temporal evolution of the quantum state created by the unknown pulse.

407

4.1. Reconstruction of the quantum state We present here a general method able to determine any pulse shape. One limitation of CT resides in the fact that only the part of the pulse after resonance leads to oscillations which can be used to determine the shape. Another difficulty is that probabilities are measured whereas probability amplitudes are necessary to measure the quantum state and deduce the electric field. To overcome these limitations, a new excitation scheme with two pump pulses is used (see Fig. 3). Two measurements are performed, each with a two pulse sequence with a well defined phase relationship E s h a p e d ( t ) = &(t) eieE2(t) where E l ( t ) and &(t) are two replica of the unknown pulse generated by splitting the same initial pulse E ( t ) and adding an additional spectral phase t o E2(t). These can be obtained either with a Michelson-type interferometer (with a dispersive rod in one arm) or with a pulse shaper. The first pulse E l ( t ) creates an initial population in the excited state. The second pulse E2(t) is strongly chirped and sufficiently delayed by 712 so that it does not overlap with the first pulse. This second pulse creates a population in the excited state which interferes with the initial population created by the first pulse. Thus, oscillations due to CT occur on the whole duration of the second pulse. The final state population during the second pulse can be written as

+

+ eisaf,2(T12)12 = l a f , l ( + 4 2 + laf,2(712)I2

laf,e(t)I2= l a f , 1 ( + 4 +me

{aj,,(+oo)af,2(712)eie}

(3)

laf,1(+~0)1~ can be obtained from the plateau reached between the two pulses. The last two contributions in Eq. 3 depend on 712 and on the second pulse. For a second pulse of smaller peak power, the crossed term is dominant so that the response is mostly linear with respect to the second pulse. A first measurement for t9 = 0 gives Re a~,,(+co)a~,2(7-12)}. A second measurement for 0 = 7r/2 brings the complementary part

{

{

(+W)af,2(712)}. For pulse intensities of comparable magnitude, Im the system of nonlinear equations resulting from both measurements can be solved to extract af~(7-12)13. By derivation one obtains

from which E ~ ( wcan ) be deduced provided that the reference pulse is known and short enough. Finally, knowing the extra phase added to gen-

408

erate the pulse sequence, E ( w ) and thus E ( t ) is obtained. A phase and

u El (4 three-level system

lnknOWn

pulse

Figure 3. Set-up Principle: The unknown pulse is sent into a pulse shaper programmed t o generate a sequence of two pulses. An extra phase can be added. An optional glass rod is added in the front of the set-up. The probe is a pulse shorter than the unknown pulse. Inset: Excitation scheme.

amplitude 640 pixels SLM pulse shaper l1 is used to generate the pump pulse sequence by applying a complex transmission in the spectral domain:

H ~ ( w=) (1

+ exp[i(f?+ +'(w

-

wo)

+ +"(w

-

~0)~/2]}/2

(5)

where wo is the carrier frequency of the pump pulse. The first pulse in the pump sequence is identical to E ( t ) . The second one is strongly chirped with 4" = -2.105 fs2 in order to produce CT, and delayed by 4' = 6 ps. An extra phase factor f? can be added. In a first experiment, E ( t ) is Fourier limited. Two recordings are performed for f? = 00 and f? = 80 + ~ / 2(with f?,, = - 0 . 2 ~ ) as shown in Fig. 5a). The main difference with previous experiments is the preparation of a coherent superposition of le) and 19) by the first pulse. Then the second -strongly chirped- pulse produces large oscillations during its whole duration. These oscillations can be seen as beats between the atomic dipole (which behaves as a local oscillator) and the electric field from the second pulse, as in heterodyne detection. By combining the two measurements, it is therefore possible to retrieve fully the temporal evolution of the excited state probability amplitude due to the second pulse. Figure 5b displays the reconstructed excited state probability amplitude in the complex plane. The expected Cornu spiral lo is observed. It can also be seen in a 3D plot (Fig 4) to have a direct view of the temporal evolution. Several examples of quantum phase measurements of states created by ultrashort pulses are based on interferences between an unknown wave function and a "reference" wave function. These wave functions are created by '7"

409

Figure 4. 3D spirale. the vertical axis represents the time. The Cornu spirale is the evolution of the amplitude of probability of the excited state during the field-driven.

a sequence of two ultrashort pulses (an unknown pulse and a reference pulse). The quantum state created by the unknown pulse is deduced either by time- and frequency- integrated fluorescence measured as a function of the delay15,or by measuring the population of each eigenstate for different values of the relative phases16, or the amplitude of fluctuations when the delay is randomly fluctuating17. In another approach, the dispersed fluorescence emitted by an oscillating nuclear wave packet in a diatomic molecule was recorded as a function of time". In this case, the fluorescence wavelength - position relationship is derived from the Fkanck-Condon principle.

In all these examples, the quantum state is first prepared and then measured in a second step. In the work reported here, the quantum state is measured during the interaction with the unknown laser pulse. Its evolution is thus recorded in real time. 4.2.

Reconstruction of the Electric-Field

Considering now that the probe pulse is well-known, one can retrieve the complete phase of the electric field. &(t) obtained by simple derivation of a,,z(t), is displayed on Fig. 6 . The temporal amplitude and phase are represented. As a comparison, the exact theoretical temporal phase applied

410

I . I . I . , . I . , . I . I . I . I . I . I . I 0 1 2 3 4 5 6 7 8 9 1 0 1 1

-1

Re[a.%l

7 (PSI

Figure 5 . a) Experimental Coherent Transients resulting from the excitation of the atom by a FT limited pulse (at time T = 0) followed by a chirped pulse (centered a t T = 6ps), for two different relative phases 0 = 0 and 7 ~ / 2between the two pulses. b) Probability amplitude a f , z ( ~reconstructed ) from the two measurements presented in a) and displayed in the complex plane. The Cornu spiral appears clearly.

by the pulse shaper is shown (dashed line) without any other adjustment than the offset. The agreement is excellent. The quadratic phase added by the pulse shaper is perfectly retrieved.

(PSI Figure 6. Reconstructed phase (dots), theoretical phase (dashed) and amplitude (dashdotted) of Ez(T).

In a second set of experiments 14, the dispersion of a glass rod (4" = 20500 fs2) inserted in the pump beam is measured. CT are monitored with and without the rod in the pump beam. This dispersion is sufficiently small so that E l ( t ) and &(t) do not overlap. Experimental results in the spectral domain are presented on Fig. 7. Figure 7a shows the spectral phase of & ( w ) with and without the dispersive rod. Their difference is plotted on Fig. 7b together with the value calculated from the rod coefficients. The

411

agreement is excellent on the spectral domain where the intensity (dashed line) is significant.

Figure 7. Electric field reconstruction. a) Spectral Phase retrieved obtained with (gray) and without (black) rod. b) diamonds, difference in spectral phase between the phase obtained with and without the rod. The expected spectral phase (solid line) and the spectral intensity (spectrum)(dashed line) are also shown.

5 . Conclusion

In this work we have demonstrated how coherent transients can be used to study the interaction between an atomic system and a short resonant shaped pulse. For a known pulse shape, we can monitor the temporal evolution of the wave function and thus all the information on the behavior of the quantum system during its interaction. For a known atomic system, we can reverse the process and use the same data to fully characterize the exciting pulse with unusual properties.

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This pulse shape measurement has been demonstrated here on a simple example. As a difference t o most other approaches, the scheme presented here is linear with respect t o the unknown pulse and requires an external reference pulse which does not need t o be in the same spectral range. This method provides interferometric-like terms but the two spectral shears remove the main constraints of usual interferometric techniques : same spectral domain, interferometric control of delay, temporal step small enough t o resolve fringes. References 1. M. A. Bouchene, V. Blanchet, C. Nicole, N. Melikechi, B. Girard, H. Ruppe, S. Rutz, E. Schreiber and L. Woste, Eur. Phys. J. D 2, 131 (1998). 2. R. R. Jones, D. W. Schumacher, T. F. Gallagher and P. H. Bucksbaum, J. Phys. B 28, L405 (1995). 3. R. S. Judson and H. Rabitz, Phys. Rev. Lett. 68,1500 (1992). 4. W. S. Warren, H. Rabitz and M. Dahleh, Science 259,1581 (1993). 5. A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle and G. Gerber, Science 282, 919 (1998). 6. D. Meshulach and Y. Silberberg, Nature 396,239 (1998). 7. H. U. Stauffer, J. B. Ballard, Z. Amitay and S. R. Leone, J. Chem. Phys. 116,946 (2002); J. B. Ballard, H. U. Stauffer, Z. Amitay and S. R. Leone, J. Chem. Phys. 116,1350 (2002). 8. N. Dudovich, D. Oron and Y. Silberberg, Phys. Rev. Lett. 88, 123004 (2002). 9. J. Degert, W. Wohlleben, B. Chatel, M. Motzkus and B. Girard, Phys. Rev. Lett. 89,203003 (2002). 10. S. Zamith, J. Degert, S. Stock, B. De Beauvoir, V. Blanchet, M. A. Bouchene and B. Girard, Phys. Rev. Lett. 87, 033001 (2001). 11. A. Monmayrant and B. Chatel, Rev. Sci. Instr. (submitted). 12. W. Wohlleben, J. Degert, A. Monmayrant, B. Chatel, M. Motzkus and B. Girard, Appl. Phys. B (accepted). 13. A. Monmayrant, B. Chatel and B. Girard, Phys. Rev. Lett. in preparation, (2005). 14. A. Monmayrant, B. Chatel and B. Girard, Opt. Lett. submitted, (2004). 15. C. Leichtle, W. P. Schleich, I. S. Averbukh and M. Shapiro, Phys. Rev. Lett. 80, 1418 (1998). 16. X. Chen and J. A. Yeazell, Phys. Rev. A 56,2316 (1997). 17. T. C. Weinacht, J. Ahn and P. H. Bucksbaum, Nature 397,233(1999). 18. T.J. Dunn, I. A. Walmsley and S. Mukamel, Phys. Rev. Lett. 7 4 , 8 8 4 (1995).

CONTROLLED PHOTON EMISSION AND RAMAN TRANSITION EXPERIMENTS WITH A SINGLE TRAPPED ATOM

M. P. A. JONES, B. DARQUIE, J. BEUGNON, J. DINGJAN, s. BERGAMINI, Y. SORTAIS, G. MESSIN, A. BROWAEYS AND P. GRANGIER Laboratoire Charles Fabry de l’lnstitut d’Optique BGt. 503 Centre Universitaire 91403 Orsay, France E-mail:Matt. [email protected] We present recent results on the coherent control of an optical transition in a single rubidium atom, trapped in an optical tweezer. We excite the atom using resonant light pulses that are short (4ns) compared with the lifetime of the excited state (26 ns). By varying the intensity of the laser pulses, we can observe an adjustable number of Rabi oscillations, followed by free decay once the light is switched off. To generate the pulses we have developed a novel laser system based on frequency doubling a telecoms laser diode at 1560nm. By setting the laser intensity to make a T pulse, we use this coherent control to make a high quality triggered source at the of single photons. We obtain an average single photon rate of detector. Measurements of the second-order temporal correlation function show almost perfect antibunching at zero delay. In addition, we present preliminary results on the use of Raman transitions to couple the two hyperfine levels of the ground state of our trapped atom. This will allow us to prepare and control a qubit formed by two hyperfine sub-levels.

1. Introduction In order to use a particular physical system for quantum computation, it is necessary to be able to perform single-qubit operations such as rotations, and two-qubit operations such as controlled-not gates. These two basic building blocks can then be concatenated to realise any other desired logical operation. Neutral atoms have been proposed as a candidate physical system for quantum information processing. For example, the alkali atoms have two hyperfine levels in the ground state which can be used to make a qubit with very long coherence times. Single-qubit operations can be realised by using microwaves to drive the hyperfine transition directly, or by using a two413

414

photon Raman transition. Recently, addressable single-qubit operations have been successfully demonstrated on a “quantum register” of trapped atoms using microwaves’. So far, individually addressed two-qubit gates have not been demonstrated with neutral atoms. Deterministic gates generally require a strong interaction between the particles that are used to carry the physical qubits2, such as the Coulomb interaction between trapped ions. Promising results have been obtained on entangling neutral atoms using cold controlled collisions in an optical lattice3, but the single-qubit operations are difficult to perform in such a system. Another approach is to bypass the requirement for a direct interaction between the qubits, and use instead an interference effect and a measurement-induced state projection to create the desired ~ p e r a t i o n An interesting recent development of this idea is to use photon detection events for creating entangled states of two atom^^^^^*^^. This provides “conditional” quantum gates, where the success of the logical operation is heralded by appropriate detection events. Such schemes can be extended to realize a full controlled-not gate, or more generally to implement conditional unitary operations6y10. These proposals require the controlled emission of indistinguishable photons by the two atoms. In this paper we describe our recent experiments’’ on triggered singlephoton emission from a single rubidium atom trapped at the focal point of a high-numerical-aperture lens (N.A. = 0.7). We show that we have full control of the optical transition by observing Rabi oscillations. Secondly, we describe preliminary results on the use of Raman transitions to couple the two ground state hyperfine levels of the trapped atom with a view to performing single-qubit operations.

2. Trapping single atoms We trap the single rubidium 87 atom at the focus of the lens using a fardetuned optical dipole trap (810 nm), loaded from an optical molasses. The same lens is also used to collect the fluorescence emitted by the atom (Fig. l),which is then detected using a photon counting avalanche photodiode. The overall detection and collection efficiency for the light emitted from the atom is measured to be 0.60f0.04%. A crucial feature of our experiment is the existence of a “collisional blockade” m e c h a n i ~ m ’ ~which ~ ’ ~ allows only one atom at a time to be stored in the trap: if a second atom enters the trap, both are immediately ejected. In this regime the atom statistics are

415

sub-Poissonian and the trap contains either one or zero (and never two) atoms, with an average atom number of 0.5. The experimental apparatus is described in more detail in referencesl27l3.

Figure 1. Schematic of the experiment. The same lens is used t o focus the dipole trap and collect the fluorescence light. The fluorescence is separated by a dichroic mirror and imaged onto two photon counting avalanche photodiodes (APD), placed after a beamsplitter (BS). The insert shows the relevant hyperfine levels and Zeeman sub-levels of rubidium 87. The cycling transition is shown by the arrow.

3. Triggered single photon emission We excite the atom with 4ns pulses of laser light, resonant with the S1/2, F = 2 -+ P3/2, F' = 3 transition at 780.2nm (see insert in figure 1). These pulses are much shorter than the 26ns lifetime of the upper state. The pulsed laser beam is &-polarized with respect t o the quantisation axis which is defined by a 4.2G magnetic field. The trapped atom is optically pumped into the F = 2, m F = +2 ground state by the first few laser pulses. It then cycles on the F = 2, m F = +2 + F' = 3, mb = +3 transition, which forms a closed two-level system emitting g+-polarized photons. Impurities in the polarisation of the pulsed laser beam with respect to the quantisation axis, together with the large bandwidth of the exciting pulse (250 MHz), result in off-resonant excitation t o the F' = 2 upper state, leading t o possible de-excitation t o the F = 1 ground state. To counteract this, we add a repumping laser resonant with the F = 1 + F' = 2 transition. We have checked that our two-level description is still valid in the presence of the repumper by analyzing the polarisation of the emitted single photonsll.

416

Figure 2.

Schematic of the pulsed laser system. FC: Fibre Coupler

To generate these short laser pulses, we have developed a novel laser s y ~ t e n i shown ~ ~ , in figure 2, based on frequency doubling pulses at the 1560 nm telecoms wavelength. The pulses are generated using an electrooptic modulator to chop the output of a continuous-wave diode laser. A commercial fibre amplifier is used to boost the peak power of the pulses prior to the doubling stage. The laser, modulator, and amplifier are all standard telecommunications components. The frequency doubling is performed in a single pass using a heated periodically-poled lithium niobate (PPLN) crystal. We monitor the centre frequency using the fluorescence in a rubidium vapour cell, and tune the system using the temperature of the diode laser. The repetition rate of the source is 5 MHz, and we obtain peak powers of up to 12W.

Rabi oscillations For a two-level atom and exactly resonant square light pulses of fixed duration T , the probability for an atom in the ground state to be transferred to the excited state is sin2(RT/2), the Rabi frequency R being proportional to the square root of the power. Therefore the excited state population and hence the fluorescence rate oscillates as the intensity is increased. To observe these Rabi oscillations, we illuminate the trapped atom with the laser pulses during 1 ms. We keep the length of each laser pulse fixed at 4ns, with a repetition rate of 5 MHz, and measure the total fluorescence rate as a function of the laser power. The Rabi oscillations are clearly visible on our results (see Fig. 3a). From the height of the first peak and the calibrated detection efficiency, we derive a maximum excitation efficiency

417

per pulse of 95 f 5%. The reduction in the contrast of the oscillations at high laser power is mostly due to fluctuations of the pulsed laser peak power. This is shown by the theoretical curve in Fig. 3a, based on a simple two-level model. This model shows that the 10% relative intensity fluctuations that we measured on the laser beam are enough to smear out the oscillations as observed. The behaviour of the atom in the time domain can be studied by using time resolved photon counting techniques to record the arrival times of the detected photons following the excitation pulses, thus constructing a time spectrum. By adjusting the laser pulse intensity, we observe an adjustable number of Rabi oscillations during the duration of the pulse, followed by the free decay of the atom once the laser has been turned off. The effect of pulses close to 7 r , 27r and 37~are displayed as inserts on Fig. 3a.

Single photon emission In order to use this system as a single photon source, the laser power is set to realize a 7r pulse. To maximise the number of single photons emitted before the atom is heated out of the trap, we use the following sequence. First, the presence of an atom in the dipole trap is detected in real-time using its fluorescence from the molasses light. Then, the magnetic field is switched on and we trigger an experimental sequence that alternates 115 ps periods of pulsed excitation with 885 ps periods of cooling by the molasses light. After 100 excitation/cooling cycles, the magnetic field is switched off and the molasses is turned back on, until a new atom is recaptured and the process begins again. On average, three atoms are captured per second under these conditions. The average count rate during the excitation is 9600 s-l, with a peak rate of 29000 s-'. To characterize the statistics of the emitted light, we measure the second order temporal correlation function, using a Hanbury Brown and Twiss (HBT) type set-up. This is done using the beam splitter in the imaging system (Fig. l ) , which sends the fluorescence light to two photon-counting avalanche photodiodes that are connected to a high-resolution counting card in a start-stop configuration (resolution of about 1 ns). The card is gated so that only photons scattered during the periods of pulsed excitation are counted, and the number of coincidence events is measured as a function of delay. The histogram obtained after 4 hours of continuous operation is displayed in Fig. 3b, and shows a series of spikes separated by the period of the excitation pulses (200 ns). The l / e half width of the peaks is 27 f 3 ns,

418

in agreement with the lifetime of the upper state. After correction for a small flat background due to stray laser light and dark counts, the integrated residual area around zero delay is 3.4% f 1.2% of the area of the other peaks. This is due to a small probability for the atom to emit a photon during the pulse, and be re-excited and emit a second photon. From our calculation'', the probability for the atom to emit two photons per pulse is 0.018.

1""!"'""'""'""'"""1

0 10 20 30 0 10 20 30 0 10 20 30

500

400

i

8

3oo

8 200 100 0

0

a)

50

200 250 Y'lnteneity (arb. un.) 100

150

300

b)

0 500 loo0 1500 Delay (no)

-1000-500

Figure 3. a) Total count rate as a function of average power for a pulse length of 4 ns. Rabi oscillations are clearly visible. The inset shows the time resolved fluorescence signal for n, 27r and 3n pulses. b) Histogram of the time delays in the HBT experiment. The peak at zero delay is absent, showing that the source is emitting single photons.

4. Raman transitions

- towards a qubit

To drive Raman transitions between the two hyperfine levels, two phasecoherent laser beams with a frequency difference equal to the hyperfine transition frequency are required. In our experiment we use the dipole trap itself as one of the Raman beams. The large detuning minimises problems due to spontaneous emission during the Raman pulse. Due to the very high intensity at the waist of the dipole trap, high two-photon Rabi frequencies can still be obtained. The second Raman beam is generated using two additional diode lasers that are phase-locked to the dipole trap by injection locking. The 6.8 GHz frequency offset is generated by modulating the current of one of the diode lasers at 3.4 GHz.

419

0

5

10

15 20 25 30 35 Pulse length microseconds

b) Figure 4. a) Fluorescence as a function of detuning between the Raman beams with an applied magnetic field of 4.2 G. Zero detuning corresponds to the hyperfine splitting in zero applied field. Two of the possible transitions between magnetic sublevels are shown: F = 1 , m F = -1 + F = 2 , m F = -2 (left) and F = 1 , m F = 0 -+ F = 2 , m F = -1 (right). b) Fluorescence a s a function of the duration of the Raman pulse.

To drive Raman transitions for qubit rotations, we superimpose the second Raman beam with the dipole trap beam. As the beams are copropagating, this makes the transition insensitive to the external state of the atom. The beam is linearly polarised orthogonal to the dipole trap. With the quantisation axis defined by a 4.2 G magnetic field as described above, this means that we can drive n/a+ or r / a - transitions. We perform spectroscopy of the transitions as follows. The atom is prepared in the F = 1 hyperfine level by switching off the repump light 1ms before the molasses light. This process populates all of the magnetic sublevels. Then we transfer the atom to the F = 2 hyperfine level by pulsing on the second Raman beam. The population in the F = 2 level is detected using the fluorescence from a resonant probe beam. By measuring the fluorescence as a function of the frequency difference between the two Raman beams we obtain spectra such as those shown in figure 4a. The width of these peaks is limited by the length of the Raman pulse in each case. We have also made a preliminary observation of Rabi oscillations. The detuning between the Raman lasers was set resonant with the F = 1,m F = -1 -+ F = 2, m F = -2 transition, and we measured the population in F = 2 as a function of the pulse length. The results are shown in figure 4b. The measured Rabi frequency is 52 = 2n x 65 kHz, with a power of only 60nW in the second Raman beam. As we have 10mW available, we should be able to attain Rabi frequencies in the MHz range.

420

5. Conclusions and outlook We have realized a high quality source of single photons based on the coherent excitation of an optically trapped single atom. Preliminary results on using Raman transitions to couple the two hyperfine ground states of the trapped atom have also been obtained. In the future, we would like to extend these experiments to several atoms. In previous work we have shown that we can create two-dimensional arrays of dipole traps separated by distances of several microns, each containing a single atom15. Our goal is t o see whether we can realise single-qubit and two-qubit operations in such arrays.

Acknowledgments This work was supported by the European Union through the IST/FET/QIPC project QGATES and the Research Training Network CONQUEST. M. Jones was supported by EU Marie Curie fellowship MEIFCT-2004-009819

References 1. D. Schrader et al., Phys. Rev. Lett. 93, 150501 (2004). 2. P. Zoller, J.I. Cirac, L. Duan, J.J. Garcia-Ripoll, in Les Houches 2003: Quantum entanglement and information processing, D. EstBve, J.-M. Raimond, J. Dalibard, Eds. (Elsevier, Amsterdam, 2004), pp 187-222. 3. 0. Mandel et al., Nature 425,937 (2003). 4. E. Knill, R. Laflamme, G.J. Milburn, Nature 409, 46 (2001). 5. J.P. Dowling, J.D. Franson, H. Lee, G.J. Milburn, Quantum Information Processing 3, 205 (2004). 6. I. Protsenko, G. Reymond, N. Schlosser, P. Grangier, Phys. Rev. A 66, 062306 (2002). 7. C. Simon, W.T.M. Irvine, Phys. Rev. Lett. 91, 110405 (2003). 8. L.-M. Duan, H.J. Kimble, Phys. Rev. Lett. 90, 253601 (2003). 9. S. D. Barrett and P. Kok,Phys. Rev. A 71,060310(R), (2005) 10. Y.L. Lim, A. Beige, L.C. Kwek, in preparation (available at http://arXiv.org/abs/quant-ph/0408043). 11. B. Darquib et al., Science, 309, 454 (2005)). 12. N. Schlosser, G. Reymond, I. Protsenko, P. Grangier, Nature 411, 1024 (2001) 13. N. Schlosser, G. Reymond, P. Grangier, Phys. Rev. Lett. 89, 023005 (2002). 14. J. Dingjan et al., quant-ph/0507238 15. S . Bergamini et al., J . Opt. SOC.Am. B 21, 1889, (2004)

ION TRAP NETWORKING: COLD, FAST, AND SMALL

D. L. MOEHRING, M. ACTON, B. B. BLINOV, K.-A. BRICKMAN, L. DESLAURIERS, P. C. HALJAN, W. K. HENSINGER, D. HUCUL, R . N. KOHN, JR., P. J . LEE, M. J . MADSEN, P. MAUNZ, S. OLMSCHENK, D. STICK, M. YEO, C. MONROE FOCUS Center and University of Michigan, Department of Physics, USA J. A. RABCHUK Western Illinois University, Department of Physics, USA A large-scale ion trap quantum computer will require low-noise entanglement schemes and methods for networking ions between different regions. We report work on both fronts, with the entanglement of two trapped cadmium ions following a phase-insensitive Molmer-Sorensen quantum gate, the entanglement between a single ion and a single photon, and the development of advanced ion traps at the micrometer scale, including the first ion trap integrated on a semiconductor chip. We additionally report progress on the interaction of ultrafast resonant laser pulses with cold trapped ions. This includes fast Rabi oscillations on optical S-P transitions and broadband laser cooling, where the pulse laser bandwidth is much larger than the atomic linewidth. With these fast laser pulses, we also have developed a new method for precision measurement of excited state lifetimes.

1. Ion Entanglement 1.1. Local i o n entanglement Laser-addressed trapped ions with qubits embedded in long-lived internal hyperfine levels hold significant advantages for quantum information applications1i2. A critical issue is the robust generation of scalable entanglement. Trapped-ion entangling gates mediated by phonons of the collective ion motion are susceptible to various forms of noise - qubit and motional decoherence, impure initial conditions, and technical issues associated with the optical Raman lasers driving the gate'. Robust schemes for gates based on spin-dependent forces have been p r o p ~ s e d ~and > ~ ex> perimentally implemented718 that, for example, relax the purity requirement on the initial motional state of the ions. We have recently realized a 42 1

422

Mdmer-Smensen entangling gate. for pairs of trapped l1'Cdf ions using an implementation proposed in Refs. 10-11 that reduces sensitivity to optical phase drifts through appropriate Raman beam setup and reduced sensitivity to magnetic field fluctuations through the use of magnetic-field insensitive qubit states. The gate interaction is based on a bichromatic field of first Raman sidebands coupling to a collective vibrational mode of the trapped ions [Fig. l(a)]. This realizes a nonlinear qubit interaction H @az, which can couple 1.1.1) to 111) to produce an entangled Bell state @ = h(l.1.1) A full evaluation of the entangled state including a quantitative measure of the entanglement requires access to the full density matrix, in particular all the off-diagonal coherences. Quantum state t ~ m o g r a p h y l ~isl used ~ ~ to reconstruct the density matrix for all four entangled Bell states. Repeated preparation of a target state followed by projective measurements of the two ion-qubits in nine different bases {ai @ aj;i , j = 1,2,3} is performed for 200 shots per measurement basis. Maximum likelihood estimation is used to fit the data to a density matrix constrained to a physical form. An example of a reconstructed density matrix for the entangled state created with the Mdmer-Sorensen scheme is shown in Figure l(b). The ideal state is created with fidelity -0.80. One measure of entanglement that is directly calculable from the density matrix for two qubits is Wooters' entanglement of formation E ranging from zero for a separable state to one for a maximally entangled one14. The experimental value for the state in Figure l(b) is E=0.65. The tomographic reconstruction of the entanglement demonstrates universal two-qubit control, which can be directly applied to simple algorithms. One such algorithm is Grover's searching algorithm which searches an unsorted database quadratically faster than any known classical search15. A common analogy is to be given a phone number but no name to go with the number16. For N entries in the phone book, the worst case scenario classically requires N queries, and on average requires N/2 queries. However, if the correlation between name and phone number is encoded with quantum bits, the name can be found after only about queries. The key to the algorithm is in the oracle query which, using entangled superposition, checks if an input state is a solution to the problem. Contingent upon being a solution, the state is marked by flipping the sign of its amplitude. Following the oracle are several quantum operations that amplify the weighting of the marked state independent of which state is marked. After a prescribed number of iterations the marked state accumulates nearly all

- Fa,

+ Irr))

423

of the weight and is revealed following a m e a ~ u r e m e n t l ~For . N >> 1, the marked element appears with high probability after approximately 7 r m / 4 iterations, and for the special case of N = 4 elements, a single query provides the marked element with unit probability. Re(p)

(b)

0.5-

I$t) F

Figure 1. (a)The Molmer-Sorensen entangling gate is based on a bichromatic field composed of red (rsb) and blue (bsb) Raman sidebands coupling ion spin to collective vibrational levels {In)}. Typical gate interaction time to generate the entangled state @ = -&(lTT) 111))is loops. (b)Tomographic density matrix reconstruction of the entangled state @ created with fidelity -0.8.

+

Using the Molmer-Sorensen entangling gate described above this algorithm was carried out for a four element search space (n=2 qubits). The algorithm starts by creating an equal superposition of input states with a global single qubit 7r/2 rotation. Next the oracle query is applied. This is ) . . I with the state carried out by first swapping the target marked state, ILL). The Molmer-Sorensen entangling gate is then used in conjunction with single qubit rotations t o create a controlled-z gate. This gate takes the 111) state t o -111) and leaves the other states unchanged. Then the Ixx) and 111) states are swapped back. This results in the target marked state acquiring a minus sign, Ixx) -+ -(zx).Another global single qubit rotation followed by a final entangling gate complete the algorithm by amplifying the weighting of the marked state. Upon measurement the marked state is recovered with 60(2)% probabilityls. The classical counterpart is a simple shell game: suppose a marble is hidden under one of four shells, and after a single query the location of the marble is guessed. Under these conditions the best classical approach gives an average probability of success: PCl = 1/4 3/4(1/3) = 50%, (1/4 of the time the query will give the correct location of the marble while 3/4 of the time a guess must be made amongst the three remaining choices with 1/3 probability of choosing the

+

424

correct location). If Grover’s algorithm is used, the answer to the query would always result in a successful ‘guess’ of the marble’s location. Experimentally the marked state was recovered with a probability, averaged over the four markings, of 60(2)%, surpassing the performance of any possible classical search. 1.2. Probabilistic remote ion entanglement

In the ongoing remote ion entanglement experiment, single ll1Cd+ ions are trapped in two different ion traps spaced by approximately one meter. Each ion is initially excited to a state with multiple decay channels and single photons are emitted from each decaying ion. Along a certain emission direction selected by an aperture, each photon’s polarization is entangled with particular hyperfine ground states in its de-excited parent ion19>20. Current progress in the entanglement of the ions through joint detection of the photons21 includes the use of ultrafast laser pulses for single photon generation [Fig. 3(a)] and precise mode-matching of the photons from each ion onto a beamsplitter. Even though the photon emission is a probabilistic process, unlike other probabilistic sources such as two-photon downconversion, this atom-atom entanglement is heralded by the joint detection of the photons, and can be subsequently used for further applications, including scalable quantum computing architecture^^^?^^. 2. Ultrafast Laser Interactions 2.1. Ultrafast laser cooling

Narrow-band lasers are essential tools for the precise control of internal and external states of cold atoms, from laser cooling of atomic motion to the formation of optical lattices and dipole traps. However, very little has been achieved concerning the control of cold trapped atoms with ultrafast lasers, notwithstanding recent theoretical proposals on ultrafast quantum gates using trapped i o n ~ ~ As ? ~a ~first . step toward realization of these proposals, we have observed picosecond Rabi oscillations between the S and P levels of singly ionized cadmium. Additionally, we have directly observed trapped ion laser cooling from broadband pulses25. The extent of cooling is determined by directly measuring the spatial localization of the ion within the trap, and the most efficient cooling occurs when the center frequency of the laser is tuned roughly one bandwidth (-420 GHz) below the atomic resonance. When cooled by the pulsed laser, ions are

425

localized to a few microns corresponding to a temperature of a few Kelvin, allowing clear images of ion crystals even in the face of large amounts of rf micromotion [Fig. 21.

Figure 2. (a) Images of a single trapped ion taken at various pulsed laser detunings, 6/27r, indicated at the bottom. The pulsed laser beam direction in each image is diagonal from lower-left corner to upper-right corner. (b) Crosssections of the images in (a) along the vertical direction. The solid lines are Gaussian fits to the data. (c) An image of a narrow-band laser-cooled ion localized t o -30 nm, with its crossection and Gaussian fit plotted in (d).

2 . 2 . Precision lifetime measurements

These fast transitions to the excited state have also allowed for a new method for precision lifetime measurements26. This method, designed especially to eliminate common systematic errors, involves selective excitation of a single trapped ion to an excited state (lifetime of order nanoseconds) by a fast laser pulse (length of order one picosecond). Arrival of the spontaneously emitted photon from the ion is correlated in time with the excitation pulse, and the distribution of time delays from many such events provides the information for the excited state lifetime. By using this technique, we are able to eliminate prevalent systematic errors such as pulse pileup, radiation trapping, flight from view, sub/superradiance, non-selective excitation and/or detection, and potential effects from applied light during the measurement interval. With uncertainties of less than 0.4%, these results are among the most precise measurements of atomic state lifetimes to date [Fig. 3(b)]. Furthermore, this technique has the potential to achieve N 100 ppm precision by eliminating the remaining technical systematic effects.

426

;120 5

100

.-2

80

0

8

(a) g‘zb correlation measurement

(b) Lifetime data

60

*.

2

40

=a,

20

5 2

0 -36

-24 -12 0 12 24 Detection -time delay (ns)

36

0

2

4

6 8 1 0 1 2 Time (ns)

Figure 3. (a) g(2) correlation function. An absence of a peak at time t = O (coincident detection) shows that the ion is a single photon source. The peak is not fully extinguished because of background photons scattered off a nearby electrode (as evidenced by the asymmetry in the peak). Between pulses, the curves do not go all the way to zero since the peaks are only 1 2 ns apart whereas the excited state lifetime is ~3 ns. (b) Excited state. The open circles show the data used state lifetime histogram data for the 5p2P1/2 to extract the excited state lifetime: 3.148 f0.011 ns.

3. Trap Fabrication

3.1. Shuttling ions i n two-dimensional arrays Trapping and s h ~ t t l i n g trapped ~ ~ ~ ions ~ ~ in ~ ~complex ~ multi-zone trap structures is critical for scaling the trapped ion quantum computer. We have demonstrated an 11-zone linear ion trap consisting of 49 electrodes in a three-layer geometry, and have shuttled cold Cd+ ions between several zones [Fig. 4(a)]. This trap features a “T”-junction and we have demonstrated reliable shuttling around the corner. Furthermore, we demonstrated controlled swapping of ion positions within a linear crystal by forcing the ions to undergo a “three-point turn” through the junction. This trap topology may be a fundamental building block toward implementing complex entanglement algorithms on an ion trap quantum computer. 3.2. Variable electrode micron-scale needle trap

We report the successful operation of a novel ion trap geometry formed with two needle-like electrodes mounted on linear translation stages, allowing for the trap electrode spacing to be varied in-situ over a range of separations between 20-1000 mm [Fig. 4(b)]. The variable electrodes may allow for the systematic study of a host of ion trap properties at the micron scale, such as electrode surface noise and ion heating. The results of this study may impact the design and construction of future ion trapping apparatus relevant to quantum information applications, and the open geometry of

427

Figure 4. (a) T-junction trap array. The magnified inset shows the trapping array including the junction. (b) Single Cd+ ion trapped between needles (picture of needle electrodes added to scale). (c) Scanning electron microscope image of monolithic GaAs semiconductor linear ion trap. (d) Image of a single trapped Cd+ ion along a view perpendicular to the chip plane.

the trap may be suitable for trapped-ion cavity QED. 3.3. Scalable integrated chip trap

Scaling ion trap quantum computing requires micro-fabrication methods that allow the production of large scale arrays. We report the successful operation of a micron-scale integrated chip ion trap fabricated from epitaxially-grown GaAs/AlGaAs layers and shaped with chemical and dry etching techniques. Development of this fabrication process may allow scaling elementary ion trap quantum processors to a large number of qubits. Figure 4(c) shows the ion trap and Figure 4(d) shows a single Cd+ ion inside the chip trap. 4. Acknowledgments

This work is supported by the U.S. National Security Agency and Advanced Research and Development Activity under Army Research Office contract and the National Science Foundation ITR program.

428

References 1. D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, J. Res. Nat. Inst. Stand. Tech., 103,259 (1998). 2. B. B. Blinov, D. Leibfried, C. Monroe, and D. J. Wineland, Quant. Inf. Proc., 3(1), l(2004). 3. K. Molmer and A. S~rensen,Phys. Rev. Lett., 82,1835 (1999). 4. E. Solano, R. L. de Matos Filho, and N. Zagury, Phys. Rev. A , 59, R2539 (1999). 5. G. J. Milburn, S. Schneider, and D. F. V. James, Fortschr. Phys., 48, 801 (2000). 6. J. J. Garcia-Ripoll, P. Zoller, and J. I. Cirac, Phys. Rev. Lett., 91, 157901 (2003). 7. C. A. Sackett et al., Nature, 404,256 (2000). 8. D. Leibfried et al., Nature, 422,412 (2003). 9. A. Sorensen and K. Mdmer, Phys. Rev. A , 62,022311 (2000). 10. P. C. Haljan, K.-A. Brickman, L. Deslauriers, P. J . Lee, and C. Monroe, Phys. Rev. Lett., 94,153602 (2005). 11. P. J. Lee et al., quant-ph/0505203. 12. J. B. Altepeter, D. F. V. James, and P. G. Kwiat, Lecture Notes in Physics, 649,113 (2004). 13. C. F. Roos et al., Phys. Rev. Lett., 92,220402 (2004). 14. W. K. Wootters, Phys. Rev. Lett., 80, 2245 (1998). 15. L. K. Grover, Phys. Rev. Lett., 79,325 (1997). 16. G. Brassard, Science, 275,627 (1997). 17. M. Boyer, G. Brassard, P. H ~ y e r ,and A. Tapp, Fortschr. Phys., 46, 493 (1998). 18. K.-A. Brickman et al., In Preperation. 19. B. B. Blinov, D. L. Moehring, L.-M. Duan, and C. Monroe, Nature 428,153 (2004). 20. D.L. Moehring, M. J. Madsen, B. B. Blinov, and C. Monroe, Phys. Rev. Lett. 93,090410 (2004). 21. C. Simon and W. T. M. Irvine, Phys. Rev. Lett. 91,110405 (2003). 22. L.-M. Duan, B. B. Blinov, D. L. Moehring, and C. Monroe, Quantum Inf. Comput. 4,165 (2004). 23. L.-M. Duan and R. Raussendorf, quant-ph/0502120 (2005). 24. L.-M. Duan, Phys. Rev. Lett. 93,100502 (2004). 25. B. B. Blinov et al., quant-ph/0507074 (2005). 26. D. L. Moehring et al., quant-ph/0505111 (2005). 27. M. D. Barrett et al., Nature 429,737 (2004). 28. D. Kielpinski, C. Monroe, and D. J . Wineland, Nature 417,709 (2002). 29. M. A. Rowe et al., Quantum Inf. Comput. 2,257 (2002).

Author Index

A Acton, M. 421 Adams, C.S. 307 Ahufinger, V. 158 Alnis, J. 38 Alt, W. 189 Amthor, T. 264 Amy-Klein, A. 104 Ashworth, H.T. 129 Aspect, A. 248, 299 Auffeves, A. 371

BordB, Ch.J. 104 Bouyer, P. 248 Boyd, M.M. 14 Boyd, R.W. 123 Briaudeau, S. 104 Brickman, K.-A. 421 Britton, J. 393 Browaeys, A. 413 Brune, M. 371 Buggle, Ch. 199

C Cadoret, M. 72 Cancio Pastor, P. 52 Chardonnet, C. 104 Chatel, B. 403 Chek-Al-Kar, D. 381 Chiaverini, J. 393 Chwalla, M. 381 CladC, P. 72 ClBment, D. 248 Close, J.D. 256 Condylis, P.C. 129 Corney, J.F. 167 Courteille, Ph.W. 291

B Banaszek, K. 360 Barrett, M.D. 393 Barwood, G.P. 317 Becher, C. 381 Benhelm, J. 381 Ben-Kish, A. 393 413 Bergamini, S. Bergeman, T. 213 Bergquist, J.C. 393 Bertet, P. 371 Beugnon, J. 413 Biraben, F. 72 Blakestad, R.B. 393 Blatt, R. 381 Blatt, S. 14 Blinov, B.B. 421 239 Bloch, I. Bohn, J.L. 207 Boirin, D. 299 Bollinger, J.J. 393

D DarquiB, B. 413 Daussy, C. 104 72 De Mirandes, E. De Natale, P. 52 Deh, B. 291 Deiglmayr, J. 264 Demarco, B. 393 429

430

DeMille, D. 213 Deslauriers, L. 421 Dingjan, J. 413 Dotsenko, I. 189 Drummond, P.D. 167 Diir, W. 381

E Eikema, K.S.E. Esslinger, T.

94 283

F Farkas, D. 112 Fendel, P. 38 Figl, C. 256 Fischer, M. 38 239 Falling, S. Foreman, S.M. 14 Forster, L. 189 G Gabrielse, G. 112 Gangardt, D. 248 Gerbier, F. 239 Gerginov, V. 337 Gericke, T. 239 Gill, P. 317 403 Girard, B. Giusfredi, G. 52 Gleyzes, S. 371 Gohle, Ch. 38 Grangier, P. 413 Griesmaier, A. 149 Griffin, P.F. 307 Guellati-KhClifa, S. 72 Guhne, 0. 381 Guinet, M. 104 Gunter, K.J. 283 Gunther, S. 291

H Hachisu, H.

327

Haffner, H. 381 Haljan, P.C. 421 Hall, B.V. 275 Hall, J.L. 3 Hannaford, P. 275 Hansch, T.W. 38 Hansel, W. 381 Haroche, S. 371 Hayano, R.S. 62 Hensinger, W.K. 421 Hermier, Y. 104 Herrmann, M. 38 Higashi, R. 327 Hijlkema, M. 178 Hinds, E.A. 129 Hogervorst, W. 94 Hollberg, L. 337 Holzwarth, R. 38 Hong, F.-L. 327 Hoppeler, R. 299 Hosaka,K. 317 Hough, J . 25 Hu, H. 167 Huang, G. 317 HUCUI,D. 421 Hudson, J.J. 129 Hugbart, M. 248 Hume, D. 393

I Ido, T. 14 Inguscio, M. Itano, W.M.

52 393

J Jensen, M. 393 Jeppesen, M. 256 Johansson, S. 79 Jones, M.P.A. 413 Jones, R.J. 14

431

Jost, J.D. Julien, L.

393 72

K Katori, H. 327 Khudaverdyan, M. 189 Kienberger, R. 349 Kinast, J. 223 Kishimoto, T. 327 Kitching, J. 337 Klein, H.A. 317 Knappe, S. 337 Knill, E. 393 Koelemeij, J. 393 Kohl, M. 283 Kohn Jr., R.N. 421 Kolachevsky, N. 38 Korber, T. 381 Krachmalnicoff, V. 52 Krausz, F. 349 Kreuter, A. 381 Kuhn, A. 178

L Lancaster, G.P.T. 381 Langer, C. 393 Lea, S.N. 317 421 Lee, P.J. Leibfried, D. 393 Leonard, J. 199 Lepeshkin, N. 123 Letokhov, V. 72 Lewenstein, M. 158 Ludlow, A.D. 14 Lui, X.-J 167

M Macleod, S.G. 307 Madsen, M.J. 421 371 Maioli, P. Mandel, 0. 239

Marcassa, L.G. 264 Margolis, H.S. 317 Marian, A. 14 Marzok, C. 291 Maunz, P. 178, 421 Mazzotti, D. 52 Meschede, D. 189 Messin, G. 413 Meunier, T. 371 Miroshnychenko, Y. 189 Moehring, D.L. 421 Moll, K. 14 Monmayrant, A. 403 Monroe, C. 421 Monz, T. 381 Moritz, H. 283 Morrison, A.K. 256 Murr, K. 178

N Nez, F. 72 Nogues, G. 371 Notcutt, M. 3, 14 Nussmann, S. 178

0 Olmschenk, S. 421 oskay,w. 393 Ozeri, R. 393

P Perrin, A. 299 Pfau, T. 149 Phillips, E.S. 381 Pinkse, P.W.H. 178 Piredda, G. 123 Pons, M. 158 Potvliege, R.M. 307 Puppe, T. 178

R Rabchuk, J.A.

421

432

Raimond, J.M. 371 Rapol, U.D. 381 Rauschenbeutel, A. 189 Reetz-Lamour, M. 264 Reichle, R. 393 Rempe, G. 178 Retter, J.A. 248 Riebe, M. 381 Robins, N.P. 256 Robinson, H.G. 337 ROOS,C.F. 381 Rosenband, T. 393 Russo, C. 381

S Sage, J.M. 213 Sainis, S. 213 Sanchez-Palencia, L. 248 Sanpera, A. 158 Sauer, B.E. 129 Schaetz, T. 393 Scharnberg, F. 275 Schellekens, M. 299 Schmidt, P.O. 393 Schmidt-Kaler, F. 381 Schori, C. 283 Schrader, D. 189 Schuster, I. 178 Schweinsberg, A. 123 Schwindt, P.D.D. 337 Schwob, C. 72 Seidelin, S. 393 Sen, A. 158 Sen, U. 158 Shah, V. 337 Shi, Z. 123 Shin, H. 123 Shlyapnikov, G.V. 248 Sidorov, A. 275 Silber, C. 291

Silberhorn, C. 360 Singer, K. 264 Sortais, Y. 413 Stick, D. 421 Stoferle,T. 283 Stowe, M. 14 Stuhler, J. 149 Syassen, N. 178 Szymaniec, K. 317

T Takamoto, M. Tarbutt, M.R. Thomas, J.E. Thorpe, M. Ticknor, C. Turlapov, A.

327 129 223 14 207 223

U Ubachs, W. Udem, Th. U’ren, A.B.

94 38 360

V VBron, A.F. 248 Viana Gomes, J. 299 von Klitzing, W. 199 W Walmsley, I.A. 360 Walraven, J.T.M. 199 Weatherill, K.J. 307 Weber, B. 178 Weidemiiller, M. 264 Westbrook, C.I. 299 Whitlock, S. 275 Widera, A. 239 Wieman, C.E. 139 Wineland, D.J. 393 Witte, S. 94 Y Ye, J. 3, 14

433

Yeo, M.

421

Z 14, 112 Zelevinsky, T . 123 Zerom, P.

360 Zhang, L. Zimmermann, C. Zimmermann, M. Zinkstok, R.Th.

291 38 94