We have studied the effects of a disordered optical potential on the
transport and phase coherence of a Bose-Einstein condensate (BEC) of 7Li atoms.
At moderate disorder strengths (V_D), we observe inhibited transport and
damping of dipole excitations, while in time-of-flight images, random but
reproducible interference patterns are observed. In-situ images reveal that the
appearance of interference is correlated with density modulation, without
complete fragmentation. At higher V_D, the interference contrast diminishes as
the BEC fragments into multiple pieces with little phase coherence.Comment: 4 pages, 5 figures, distortions in figures 1 and 4 have been fixed in
  version 3. This paper has been accepted to PR

C. Welford

D. Dries

J. Hitchcock

J. W. Goodman

M. Junker

P. W. Anderson

R. G. Hulet

Yong P. Chen

English

arXiv

Crossref

Phase coherence and superfluid-insulator transition in a disordered Bose-Einstein condensate

We have studied the effects of a disordered optical potential on the transport and phase coherence of a Bose-Einstein condensate (BEC) of L7i atoms. At moderate disorder strengths (VD), we observe inhibited transport and damping of dipole excitations, while in time-of-flight images, random but reproducible interference patterns are observed. In situ images reveal that the appearance of interference is correlated with density modulation, without complete fragmentation. At higher VD, the interference contrast diminishes as the BEC fragments into multiple pieces with little phase coherence

Chen, Yong P.

Hitchcock, J.

Dries, D.

Junker, M.

Welford, C.

Hulet, R.G.

Rice University Research Repository

We have studied the effects of a disordered optical potential on the transport and                 phase coherence of a Bose-Einstein condensate (BEC) of Li-7 atoms. At moderate                 disorder strengths (V-D), we observe inhibited transport and damping of dipole                 excitations, while in time-of-flight images, random but reproducible interference                 patterns are observed. In situ images reveal that the appearance of interference is                 correlated with density modulation, without complete fragmentation. At higher V-D,                 the interference contrast diminishes as the BEC fragments into multiple pieces with                 little phase coherence

Chen, Y. P.

Hulet, R. G.

Purdue E-Pubs

PhysicsPhysics Research PublicationsPurdue University Year Phase coherence and superfluid-insulatortransition in a disordered Bose-EinsteincondensateY. P. Chen J. Hitchcock D. DriesM. Junker C. Welford R. G. HuletThis paper is posted at Purdue e-Pubs.http://docs.lib.purdue.edu/physics articles/736Phase coherence and superfluid-insulator transition in a disordered Bose-Einstein condensateYong P. Chen,1,2,* J. Hitchcock,1 D. Dries,1 M. Junker,1 C. Welford,1 and R. G. Hulet11Department of Physics and Astronomy and Rice Quantum Institute, Rice University, 6100 Main Street, Houston, Texas 77005, USA2The Richard E. Smalley Institute for Nanoscale Science and Technology, Rice University, 6100 Main Street, Houston, Texas 77005, USAReceived 26 October 2007; published 31 March 2008We have studied the effects of a disordered optical potential on the transport and phase coherence of aBose-Einstein condensate BEC of 7Li atoms. At moderate disorder strengths VD, we observe inhibitedtransport and damping of dipole excitations, while in time-of-flight images, random but reproducible interfer-ence patterns are observed. In situ images reveal that the appearance of interference is correlated with densitymodulation, without complete fragmentation. At higher VD, the interference contrast diminishes as the BECfragments into multiple pieces with little phase coherence.DOI: 10.1103/PhysRevA.77.033632 PACS numbers: 03.75.Hh, 03.75.Kk, 64.60.CnThe behavior of a superfluid or a superconductor in thepresence of disorder is of fundamental interest. A superfluidcan flow without friction around obstacles, and a supercon-ductor can have zero resistance despite material defects. Onthe other hand, disorder is able to localize particles, resultingin an insulating state 1. Experimentally, disorder-inducedsuperfluid/superconductor to insulator transitions SIT havebeen probed in many systems, including superfluid helium inporous media 2, thin-film and granular superconductors3,4, and random Josephson junction arrays 5. While manybelieve that such a SIT is a quantum phase transition drivenby quantum fluctuations, it remains a central task to under-stand exactly how the superfluid/superconducting order pa-rameter, which consists both an amplitude and a phase, maybe destroyed with increasing disorder. Numerous fundamen-tal questions remain, such as the nature of the insulator, thefate of phase coherence throughout the transition, and thepossibility of intermediate metallic phases 3,6,7.Cold atoms, with their intrinsic cleanliness coupled withremarkable controllability of physical parameters, haveemerged as exceptional systems to study various condensedmatter problems. Recently, several experiments 8–11 havestudied 87Rb condensates in random optical potentials andobserved, for example, damping of collective excitations 8and inhibition of expansion 9–11 due to disorder. Anotherexperiment 12 has examined a Bose-Einstein condensateBEC in an incommensurate quasirandom optical lattice inorder to investigate a possible “Bose-glass” phase 13. Ex-periments with disordered atomic quantum gases may pro-vide unique insights into disordered quantum systems andmay uncover a rich variety of quantum phases 14.Here we report experiments on a BEC of interacting 7Liatoms subject to a well-controlled disordered potential.While we corroborate previous transport measurements8–11, we have also probed the ground-state density distri-bution and phase coherence of the disordered BEC by per-forming both in situ and time-of-flight TOF imaging. Whiledisorder inhibits transport of the BEC, reproducible TOF in-terference patterns are observed for intermediate disorderstrengths VD, reflecting an underlying phase coherence in thedisordered BEC. At stronger VD, the interference contrastdiminishes as the BEC fragments into a “granular” conden-sate, which is expected to have no phase coherence.We create an optically trapped BEC of 7Li atoms in themJ=−1 /2, mI=3 /2 state at a bias magnetic field B720 Gusing procedures essentially similar to those describedin 15. The bias field tunes the scattering length as via aFeshbach resonance to a large positive value 200a0, wherea0 is the Bohr radius, enabling efficient evaporative cooling15,16. The optical dipole trap is produced by focusing a1030 nm Yb:YAG laser beam to a Gaussian 1 /e2 intensityradius of 34 m. In the following, we label the axial direc-tion along which the optical trap beam propagates as z andthe two radial directions as x and y. At the final trap depth,the radial trap frequencies are x=y2180 Hz and theaxial trap frequency is z23.6 Hz. Part of the axialconfinement is contributed by a residual curvature from thebias magnetic field. At this trap depth, the BEC contains5105 atoms with no discernible thermal component. Thecorresponding chemical potential h1 kHz, where h isPlanck’s constant.To create the disordered potential, we pass another beam,derived from the same laser as used for the optical trap,through a diffusive plate. This results in laser speckle 17similar to that used in 8,10. This beam propagates along thex direction. Because of the optical dipole force, the speckle’sspatially varying intensity pattern gives rise to a disorderedpotential Vx ,y ,z for the atoms. Figure 1a shows a two-dimensional 2D cross-section V0,y ,z as imaged by acamera, while the main panel shows an axial cut Vz=V0,0 ,z of the disordered potential through the center x=y=0 of the atomic cloud. The disorder strength VD is pro-portional to the intensity of the speckle laser and is definedas the standard deviation of the mean of Vz. The disordercorrelation length along the z direction, z, is defined byfitting the autocorrelation function VzVz+z	 toe−2z2/z2Fig. 1b; we find z15 m, which is muchgreater than the condensate healing length of 1 m. Sincexy =z 10,17 and the radial Thomas-Fermi radius ofthe BEC is less than 10 m, to a good approximation theBEC is subject to a 1D disordered potential Vz=V0,0 ,zalong its axial direction.*Current address: Purdue University, 525 Northwestern Ave.,West Lafayette, IN 47907, USA.PHYSICAL REVIEW A 77, 033632 20081050-2947/2008/773/0336324 ©2008 The American Physical Society033632-1After producing a pure BEC to within our resolution, thedisordered potential is ramped on in 100 ms 18 to a spe-cific value of VD, which we express in terms of . The BECis then held in the disordered and trap potentials while vari-ous experiments are performed, as described below. The finalstate is probed by absorption imaging also along the x di-rection of the atomic cloud released from the disordered andoptical trap potentials or by in situ phase-contrast imaging19.We first describe data characterizing the transport proper-ties of the disordered BEC. In one type of measurement, weslowly ramp on a magnetic field gradient along the z direc-tion, which offsets the center of the harmonic trap from z=0 to z=−d. The disordered potential is kept stationary dur-ing the trap offset. As shown in Figs. 2a and 2b, withoutdisorder the center of the atomic cloud follows the trap. Forintermediate disorder strength the cloud lags behind the newtrap center and is stretched, as shown in Fig. 2c. For stron-ger disorder, as shown in Fig. 2d, the cloud is pinned at itsinitial position and does not respond to the offset. We plot thecloud center as a function of VD for four different offsetdistances d in Fig. 2. The cloud is pinned and does not re-spond to trap offset for VD0.8, within our experimentaltime scale and uncertainty.We have also studied the effect of disorder on the dipoleoscillation of the trapped BEC Fig. 3. In contrast to theslow offset experiment discussed above, here the harmonictrap center is abruptly 2 ms offset and the cloud evolves inthe presence of the shifted trap and stationary disorder for avariable time before being released for imaging. Without dis-order, the condensate undergoes undamped dipole oscilla-tions in the trap with frequency z. Even disorder as small asVD0.1 , however, damps these oscillations. Increasingthe disorder strength to VD0.4  causes the dipole motionto be overdamped, until when VD, the cloud becomespinned at its initial position to within our experimental reso-lution.In addition to transport measurements, we have performedin situ 20 and TOF imaging on the disordered BEC, yield-ing information on both its density and phase. Figures 4aand 4b show in situ and corresponding TOF images taken 8ms after releasing the cloud. For intermediate VD Figs. 4band 4h striking random fringes, which we interpret as mat-ter wave interference, develop after sufficiently long TOFexpansion of the BEC following release from the opticalpotentials. We note that at such VD, the corresponding in situ0 200 400 600 800 1000040080012001600z (µm)V/h(Hz)(a)-20 -10 0 10 20∆z (µm)〈V(z+∆z)V(z)〉(arb.)(b)VDFIG. 1. Color Typical 1D disordered potential Vz. a False-color surface plot of the disordered speckle potential Vy ,z. Thepixel size corresponds to 2.2 m in the plane of the atoms. bAutocorrelation function of Vz. The line is a fit to e−2z2/z2, withina scaling factor. As expected for fully developed speckle, the inten-sity follows a negative exponential probability distribution.0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60200400600800100012001400d1400 µm900 µm500 µm300 µmCloudCenter(µm)VD / µ(a)(c)(d)(b)dFIG. 2. Color Effect of disorder on slow transport of the BEC.The trap center is offset to the left for a distance d over a timeinterval of 700 ms. a Initial state of the BEC, without disorder oroffset. The center of this initial cloud defines the reference position“0.” b–d Result of offsetting the trap by d1400 m, with thecloud subjected to a disordered potential of various VD: b VD=0,c VD0.3 , and d VD. Images a–d are obtained byabsorption imaging taken with a 1-ms TOF after the cloud isabruptly released from the disordered and optical trap potentials.Main panel shows the center by weight of the cloud as a function ofVD, for several offsets d. VD and  have uncertainties of 20%.0 300 600 9000200400600-4000400VD / µ0.370.581.03Evolution Time (ms)(b)VD / µ00.080.160.24CloudCenter(µm)(a)FIG. 3. Color Effect of disorder on dipole excitation of thecondensate. Data show the cloud center as a function of time fol-lowing a sudden axial offset of the trap for various disorderstrengths VD shown in two separate panels for clarity. The linesare fits to damped dipole oscillations.CHEN et al. PHYSICAL REVIEW A 77, 033632 2008033632-2images Figs. 4a and 4g are consistent with the cloudstill being connected. By increasing VD above , the disor-dered BEC becomes fragmented as shown in the in situ pro-file, Fig. 4i, and the fringe contrast observed in TOF dimin-ishes, as shown in Fig. 4j. To quantify the dependence ofthe TOF fringe pattern on VD, we plot both the TOF and insitu contrast versus VD / in Fig. 5. The TOF fringe contrastpeaks at VD0.5 , while the contrast in the in situ imagesincreases up to the highest VD investigated.Similar, but less well resolved fringes in TOF imageshave been reported previously 8,11. Our highly elongatedBEC facilitates a systematic and quantitative study of thefringes. We find that the positions of these irregularly spacedfringes are reproducible in repeated measurements, evenwith different holding times of the BEC in the disorderedpotential. During preparation of this paper, we became awareof a closely related paper 21, which also reports reproduc-ible fringes in TOF images of a disordered BEC. The repro-ducibility suggests that these fringes are unlikely due tosome initial before release phase fluctuations 22 in thedisordered BEC or due to interference of a few BECs sepa-rated by high potential barriers with no well-defined relativephase between them, as in either of these scenarios the fringepositions are not expected to be reproducible.Numerical simulations based on the Gross-Pitaevskiiequation 11,21. have shown that such interference can oc-cur when moderate disorder-induced density fluctuationscause different parts of a phase-coherent BEC to expandwith different velocities and overlap. As VD is increased from0, the contrast of the interference is expected to increase,consistent with the data shown in Fig. 5 at small VD. WhenVD becomes sufficiently large, however, the BEC fragmentsinto multiple pieces, as is seen in the in situ image, Fig. 4i.An array of randomly spaced condensates should not pro-duce visible interference in TOF 23,24, again consistentwith our observation of diminishing fringe contrast at highVD Fig. 4j.At high VD, we observe completely inhibited transport,indicating a transition to an insulator. This was seen in pre-vious transport experiments, where the nature of the insulatoris inferred to be that of a fragmented BEC 8–10. Our in situimages give the first direct observation of fragmentation of ahighly disordered BEC. Because of the exponentially sup-pressed Josephson tunneling between fragments due to highpotential barriers, the fragmented BEC is expected to havelarge phase fluctuations and no phase coherence 25. Thisregime is analogous to a granular superconductor 3,4,which is an insulator without global phase coherence.Dipole oscillations are damped at moderate VD, as shownin Fig. 3, and in agreement with previous observations 8.Excitations such as solitons, vortices, and phase slips havebeen suggested 27 as possible mechanisms giving rise todamping. Our in situ as well as TOF data at moderate VDsuggest a disordered BEC that is still connected not yetfragmented and phase coherent in order to give a reproduc-ible interference pattern. Such a state may provide insightinto the physics of “homogenous” disordered superconduct-ing films, where there has been tremendous interest regard-ing the form of the order parameter, possibilities of metallicstates, and the fate of phase coherence and phase stiffness28 during the SIT 3,7. Damped dipole oscillations werealso recently studied for a BEC in a periodic potential 29 inthe context of a possible “Bose metal” phase 7.Strong repulsive interactions are believed to precludeAnderson localization AL 11,25 in a BEC such as oursand those in previous experiments where the condensatehealing length is small in comparison to z 8–11. Interac-tions, however, can be significantly reduced in 7Li due to-500 0 500 -500 0 500tTOF = 8 mstTOF = 0 ms(a) (b)Columndensity(arb.)z (µm)( i )(g)(e)(c)( j )(h )( f )(d )z (µm)FIG. 4. Color In situ profiles and matter wave interference. aIn situ image for VD0.5  and b TOF image for VD0.5 .The TOF 2D image uses a different color scale from the one usedfor the in situ 2D image, and the displayed aspect ratio of the in situimages is reduced by 50% in order to enhance the transverse sizefor visibility. c–j In situ column-density profiles on left and cor-responding TOF profiles on right: c,d VD=0; e,f VD0.3 ; g,h VD0.5 ; and i,j VD1.0 . Solid red linesare fits to Thomas-Fermi distributions. z22.8 Hz for datashown in Figs. 4 and 5.0.00 0.25 0.50 0.75 1.000.00.20.40.60.81.0VD / µ8 ms TOFIn-situContrastFIG. 5. Color online Contrast of the matter wave interferencepattern in 8-ms TOF images solid diamonds and contrast of in situdensity fluctuations open circles as a function of VD /. The con-trast is extracted from the central axial cut of the column density asits mean deviation from the Thomas-Fermi fit normalized by thepeak value of the fit.PHASE COHERENCE AND SUPERFLUID-INSULATOR … PHYSICAL REVIEW A 77, 033632 2008033632-3an extremely gradual zero crossing in the B dependence ofas 15,16, creating more favorable conditions to observeAL. Such experiments are currently underway. A similar zerocrossing has been identified in 39K 26.Financial support of this work was provided by NSF,ONR, the Welch Foundation Grant No. C-1133, and theKeck Foundation. We thank A. Aspect, L. Sanchez-Palencia,and T. A. Corcovilos for helpful discussions.1 P. W. Anderson, Phys. Rev. 109, 1492 1958; P. A. Lee and T.V. Ramakrishnan, Rev. Mod. Phys. 57, 287 1985.2 J. D. Reppy, J. Low Temp. Phys. 87, 205 1992.3 A. M. Goldman and N. Markovic, Phys. Today 5111, 391998; Y. Dubi, Y. Meir, and Y. Avishai, Nature London449, 876 2007.4 I. S. Beloborodov, A. V. Lopatin, V. M. Vinokur, and K. B.Efetov, Rev. Mod. Phys. 79, 469 2007.5 E. Granato and J. M. Kosterlitz, Phys. Rev. B 33, 6533 1986;M. Y. Choi, J. S. Chung, and D. Stroud, ibid. 35, 1669 1987.6 M. P. A. Fisher, Phys. Rev. Lett. 65, 923 1990.7 P. Phillips and D. Dalidovich, Science 302, 243 2003.8 J. E. Lye, L. Fallani, M. Modugno, D. S. Wiersma, C. Fort, andM. Inguscio, Phys. Rev. Lett. 95, 070401 2005.9 C. Fort, L. Fallani, V. Guarrera, J. E. Lye, M. Modugno, D. S.Wiersma, and M. Inguscio, Phys. Rev. Lett. 95, 1704102005.10 D. Clément, A. F. Varon, M. Hugbart, J. A. Retter, P. Bouyer,L. Sanchez-Palencia, D. M. Gangardt, G. V. Shlyapnikov, andA. Aspect, Phys. Rev. Lett. 95, 170409 2005; D. Clémentet al., New J. Phys. 8, 165 2006.11 T. Schulte, S. Drenkelforth, J. Kruse, W. Ertmer, J. Arlt, K.Sacha, J. Zakrzewski, and M. Lewenstein, Phys. Rev. Lett. 95,170411 2005.12 L. Fallani, J. E. Lye, V. Guarrera, C. Fort, and M. Inguscio,Phys. Rev. Lett. 98, 130404 2007; J. E. Lye, L. Fallani, C.Fort, V. Guarrera, M. Modugno, D. S. Wiersma, and M. Ingus-cio, Phys. Rev. A 75, 061603R 2007.13 M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S.Fisher, Phys. Rev. B 40, 546 1989; W. Krauth, N. Trivedi,and D. Ceperley, Phys. Rev. Lett. 67, 2307 1991.14 B. Damski, J. Zakrzewski, L. Santos, P. Zoller, and M. Lewen-stein, Phys. Rev. Lett. 91, 080403 2003; A. Sanpera, A. Kan-tian, L. Sanchez-Palencia, J. Zakrzewski, and M. Lewenstein,ibid. 93, 040401 2004.15 K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G.Hulet, Nature London 417, 150 2002.16 L. Khaykovich et al., Science 296, 1290 2002.17 J. W. Goodman, Speckle Phenomena in Optics Roberts,Greenwood Village, CO, 2007.18 We have checked that ramping on of the disordered potentialdoes not produce significant heating by ramping the disorderedpotential back to zero to recover a pure condensate.19 C. C. Bradley, C. A. Sackett, and R. G. Hulet, Phys. Rev. Lett.78, 985 1997.20 The in situ images are taken immediately within 5 s afterthe cloud is released from the optical trap.21 D. Clément, P. Bouyer, A. Aspect, and L. Sanchez-Palencia,preceding paper, Phys. Rev. A 77, 033631 2008.22 S. Dettmer et al., Phys. Rev. Lett. 87, 160406 2001.23 Z. Hadzibabic, S. Stock, B. Battelier, V. Bretin, and J. Dali-bard, Phys. Rev. Lett. 93, 180403 2004.24 L. Fallani, C. Fort, J. E. Lye, and M. Inguscio, Opt. Express13, 4303 2005.25 L. Sanchez-Palencia, D. Clement, P. Lugan, P. Bouyer, G. V.Shlyapnikov, and A. Aspect, Phys. Rev. Lett. 98, 2104012007; L. Sanchez-Palencia, Phys. Rev. A 74, 053625 2006;P. Lugan, D. Clement, P. Bouyer, A. Aspect, M. Lewenstein,and L. Sanchez-Palencia, Phys. Rev. Lett. 98, 170403 2007.26 G. Roati, M. Zaccanti, C. D’Errico, J. Catani, M. Modugno, A.Simoni, M. Inguscio, and G. Modugno, Phys. Rev. Lett. 99,010403 2007.27 T. Paul, P. Schlagheck, P. Leboeuf, and N. Pavloff, Phys. Rev.Lett. 98, 210602 2007.28 P. W. Anderson, Basic Notions of Condensed Matter PhysicsWestview, Boulder, CO, 1997.29 D. McKay, M. White, M. Pasienski, and B. DeMarco, e-printarXiv:0708.3074.CHEN et al. PHYSICAL REVIEW A 77, 033632 2008033632-4

Phase coherence and superfluid-insulator transition in a disordered Bose-Einstein
            condensate

https://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=2523&amp;context=physics_articles

Phase Coherence and Superfluid-Insulator Transition in a Disordered
  Bose-Einstein Condensate

Phase Coherence and Superfluid-Insulator Transition in a Disordered Bose-Einstein Condensate

Abstract

Similar works

Full text

Available Versions

Crossref

Rice University Research Repository

Purdue E-Pubs