Expansion of a quantum gas released from an optical lattice
F. Gerbier, S. Trotzky, S. Foelling, U. Schnorrberger, J. D. Thompson, A. Widera, I. Bloch, L. Pollet, M. Troyer, B. Capogrosso-Sansone, N. V. Prokof'ev, B. V. Svistunov
aa r X i v : . [ c ond - m a t . o t h e r] A ug Expansion of a quantum gas released from an optical lattice
F. Gerbier, ∗ S. Trotzky, S. F¨olling, U. Schnorrberger, J. D. Thompson, A. Widera, I. Bloch, L. Pollet, M. Troyer, B. Capogrosso-Sansone, N. V. Prokof’ev,
5, 6, 7 and B. V. Svistunov
6, 7 Laboratoire Kastler Brossel, ENS, UPMC, CNRS ; 24 rue Lhomond, 75005 Paris, France Institut f¨ur Physik, Johannes Gutenberg-Universit¨at, 55099 Mainz, Germany. Department of Physics, Harvard University, Cambridge, MA 02138, USA. Institut f¨ur Angewandte Physik, 53115 Bonn, Germany Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland Department of Physics, University of Massachusetts, Amherst, MA 01003, USA Russian Research Center “Kurchatov Institute”, 123182 Moscow, Russia (Dated: June 2, 2018)We analyze the interference pattern produced by ultracold atoms released from an optical lattice.Such interference patterns are commonly interpreted as the momentum distributions of the trappedquantum gas. We show that for finite time-of-flights the resulting density distribution can, however,be significantly altered, similar to a near-field diffraction regime in optics. We illustrate our findingswith a simple model and realistic quantum Monte Carlo simulations for bosonic atoms, and comparethe latter to experiments.
PACS numbers: 03.75.Lm,03.75.Hh,03.75.Gg
Experiments with ultracold quantum gases in opticallattices rely heavily on time-of-flight (ToF) expansion toprobe the spatial coherence properties of the trapped gas[1, 2, 3, 4, 5, 6, 7, 8]. When the phase coherence length islarge compared to the lattice spacing, the post-expansiondensity distribution shows a sharp interference patternwith the same symmetry as the reciprocal lattice. Asthe phase coherence length decreases, e.g. , on approach-ing the Mott insulator (MI) transition, the visibility ofthis interference pattern decreases accordingly [1]. Toobtain a more precise understanding beyond this quali-tative description, it is usually assumed that the densitydistribution n ToF ( r ) of freely expanding clouds providesa faithful map of the initial momentum distribution.In this Letter, we point out that, in general, the ToFdistribution differs from the momentum distribution forfinite time-of-flight, the latter being recovered only inthe ”far-field” limit t → ∞ . Practically, the ToF andmomentum distributions become identical after a char-acteristic expansion time t FF = mR l c / ~ , which dependson the particle mass m , the coherence length l c , and thecloud size R prior to expansion. This time scale can beunderstood in analogy with the diffraction of a coherentoptical wave by a periodic grating. Then, the character-istic t FF in the expansion problem exactly correspondsto the Fresnel distance in the diffraction problem. Thefar-field regime is typically reached when the coherencelength is short, for example for a cloud in the MI regime,or a thermal gas well above the critical temperature. Weshow that for phase-coherent samples where a sizeablefraction of the atoms are Bose condensed, the far-fieldcondition is usually not met for typical expansion timesused in current experiments [1, 2, 3, 4, 5, 6, 7, 8]. Exper-imental measurements and quantum Monte-Carlo simu-lations are used to demonstrate that this results in sub- stantial changes in the ToF distribution. We also discussimplications for the interpretation of the ToF images.We consider an ultracold boson cloud released froma periodic trapping potential with cubic symmetry, lat-tice spacing d = λ L /
2, and lattice depth V given inunits of the single-photon recoil energy E R = h / mλ ,where λ L is the lattice laser wavelength. In addition tothe lattice potential, an “external” harmonic potentialis present, due to both the magnetic trap and the opti-cal confinement provided by the Gaussian-shaped latticebeams [4, 9]. This external potential is responsible forthe appearance of a shell structure of alternating MI andsuperfluid regions in the strongly interacting regime.The density distribution after expansion for a time t isusually expressed as a product (see, e.g. , [10]), n ToF ( r ) = (cid:16) m ~ t (cid:17) | ˜ w ( k ) | S ( k ) , with k = m r ~ t , (1)where an envelope function ˜ w is the Fourier transformof the on-site Wannier function w ) and the interferenceterm is S ( k ) = X r µ ,r ν e i k · ( r µ − r ν ) h ˆ a † µ ˆ a ν i . (2)Here the operator ˆ a † µ creates an atom at site r µ . To assessthe validity of the far-field approximation used in Eq. (1),we quickly outline its derivation. Neglecting interactionsduring expansion (see below), the atomic field operatorcan be expressed in Schr¨odinger’s picture as ˆΨ( r , t ) = P r ν W ν ( r , t )ˆ a ν where W ν ( r , t = 0) = w ( r − r ν ). Afterthe cloud is released, the wavefunction W ν evolves in freeflight as W ν ( r , t ) ≈ (cid:0) m ~ t (cid:1) / ˜ w (cid:16) m ( r − r ν ) ~ t (cid:17) e i m ( r − r ν )22 ~ t for ω L t ≫
1, with ω L the oscillation frequency at the bottomof a lattice well. In the limit t → ∞ , the dependence FIG. 1: (a)
Momentum distributions for a one-dimensionallattice with parabolic distribution of the occupation numberscalculated using Eq. (5) (solid line, expansion time t = 20 ms,dashed line: expansion time t = 100 ms, dot-dashed line:expansion time t → ∞ ). (b) Evolution of the peak amplitude A with expansion time t/t FF . The dashed line shows theexpected near-field scaling in one dimension, A ∝ t/t FF . Thenumber of sites is 2 N TF +1 = 61 for (a) and (b). (c) Evolutionof the width of the diffraction peaks with expansion time.The width has been normalized to the separation betweentwo adjacent diffraction peaks for convenience. The circlesshow the experimental measurements and the solid line a fitby a hyperbola ∝ /t , as expected in the near-field. on the initial site position r ν vanishes, and one recoversEq. (1). For finite t , this dependence can be neglected inthe envelope function [11], but not in the phase factor.We thus obtain a generalized interference term S t ( k ) = X r µ ,r ν e i k · ( r µ − r ν ) − i m ~ t ( r µ − r ν ) h ˆ a † µ ˆ a ν i . (3)Note that experimentally one observes a column distri-bution integrated along the probe direction, S ⊥ ( k ⊥ ) = R dk z | ˜ w ( k z ) | S t ( k ). This is included in latter compari-son with experiments, but in the following we base ourdiscussion on Eq. (3) for simplicity.A fruitful analogy can be made with the theory of opti-cal diffraction. The formation of the interference patternresults from the interference of many spherical matterwaves emitted from each lattice site, with phase relation-ships reflecting the initial quantum state of the bosongas. We can exploit this analogy further by defining theequivalent of a Fresnel distance usually introduced in thetheory of optical diffraction to estimate the importance ofthe quadratic phase factor ∝ r µ − r ν . Because the corre-lation function h ˆ a † µ ˆ a ν i suppresses contributions from sitesdistant by more than the characteristic coherence length l c , we can estimate the magnitude of the quadratic phasein Eq. (3) as m ~ t ( r µ − r ν ) ∼ ml c ~ t near the cloud center, and ∼ ml c R ~ t near the cloud edge. Here R the char-acteristic size of the cloud before expansion. The mostrestrictive condition to apply the far-field approximationthus reads t ≫ t FF , with t FF ≈ ml c R ~ . (4)As an example, for a Rb condensate with l c ≈ R ≈ d and a lattice spacing d ≈
400 nm, one finds t FF ≈
100 ms, much larger than typical expansion times t ≈ e .g., in the MI regime), with l c & d , willenter the far-field regime after a few ms. We stress thatthe quadratic Fresnel term is intrinsically non-local, asthe dephasing between two particular points r µ and r ν depends not only on their relative separation but alsoon their absolute positions. Although this has little ef-fect deep in the superfluid or in the MI phase, this castsserious doubts on the validity of a local density approx-imation to compute quantitatively the ToF distributionin regimes where the coherence length is intermediate be-tween the cloud radius and the lattice spacing.To illustrate the influence of Fresnel terms on the in-terference pattern, we consider a 1D lattice with uniformphase and parabolic distribution of the occupation num-bers, h ˆ a † µ ˆ a ν i = c µ c ν with c µ = q − ( µ/N TF ) . The ToFdistribution is given by S t (˜ k ) = 1(2 N TF + 1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N TF X l = − N TF c l e i ˜ kl − iβ l / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (5)with ˜ k = kd , N TF = R /d = 30 the Thomas-Fermi con-densate size in lattice units, and where β = p md / ~ t .The normalization factor (2 N TF + 1) would give thepeak amplitude if the filling factor were uniform. Weplot in Fig. 1a the distributions corresponding to t =20 ms ∼ . t FF which shows a significant broadening ofthe distribution for short time of flight when comparedto the asymptotic result. For longer expansion times t ∼ t FF ∼
100 ms, the far-field approximation is recov-ered to a good approximation.Qualitatively, we expect from dimensional argumentsthat the peak width scales as ( βN TF ) = t FF /t in thenear-field, while approaching a constant value in the far-field. The peak height thus increases as ( t/t FF ) D in D dimensions. This is confirmed by the one-dimensionalcalculation shown in Fig. 1b. This dependence providesa mean to check the importance of near-field effects ex-perimentally. For the measurement, a sample of roughly10 Rb atoms has been been prepared in a three-dimensional optical lattice with a depth V y = 6 E R , andsubsequently released for expansion[14]. After record-ing a series of absorption images for different expansiontimes, the width of the interference peaks was extractedusing a Gaussian fit to the images. We plot the resultsin Fig. 1c, normalized to the separation between twodiffraction peaks for convenience. The data confirms the t FF /t scaling, indicating that the far-field asymptote isnot reached even after the longest expansion time avail-able in the experiment.We now discuss briefly the effect of interactions on theexpansion, and show that this is negligible compared tothe finite ToF effect. When the cloud has just been re-leased from the lattice potential, each on-site wavefunc-tion W µ expands independently with a characteristic ex-pansion time ω − L , until t ≈ t ∗ = p ~ / ( ω L E R ) wherethe wavefunctions expanding from neighboring sites startto overlap. At this time, in the usual situation where ω L t ∗ ≫
1, the local density has dropped dramatically bya factor ( ω L t ) − ≪
1. Hence, the interaction energy con-verts into kinetic energy on the time scale of a few oscilla-tion periods only, and expansion becomes rapidly ballis-tic. The parameter controlling the importance of interac-tions is given by η = U ~ ω L ≈ √ π a s n λ L (cid:16) V E R (cid:17) / , with U being the on-site interaction energy. For typical parame-ters, η is small (for instance η ≈ .
05 for V = 10 E R andthe experimental parameters of [3]). Hence, we expectonly small corrections to the non-interacting picture ofballistic expansion. This has been confirmed using a vari-ational model of the expanding condensate wavefunction[15]. This model predicts that the ”Wannier” envelopeexpands faster as compared to the non-interacting case,which does not affect the interference pattern, and picksup a site-dependent phase factor formally similar to theFresnel term discussed previously, but with a very weakprefactor η ≪ i.e., on approaching the Mott transition and be-yond), we have performed large-scale three-dimensionalquantum Monte Carlo (QMC) simulations accountingfor the external trapping potential using the worm al-gorithm [16, 17] in the implementation of Ref. [18]. Thecalculations were performed for N = 8 × atoms, us-ing exactly the same parameters and system sizes (up to ∼ ) as in the experiments reported in [3]. The simu-lation was done at low constant temperature T = J/k B ,where J is the hopping amplitude. Although simulationsat constant entropy would be closer to the experimentalsituation, the temperature turns out to be approximatelyconstant in this parameter regime [19].The ToF distribution calculated for finite and infiniteexpansion times are shown in Fig. 2. The simulationsconfirm explicitly the analysis made above: the interfer-ence pattern is strongly affected in the superfluid phase, FIG. 2: Results from Quantum Monte Carlo simulations. Onthe left column, we show a horizontal cut through the ToFdistributions for a finite expansion time t = 14 ms (solid line),compared to a cut through the profile calculated for t → ∞ (dashed line). Units for n ⊥ are arbitrary. The insets showdirectly the two-dimensional ToF distributions for t = 14 ms.On the right column, we show the in-trap density profiles forreference. The lattice depths are V = 12 E R (a,d) , 15 E R (b,e) and 17 E R (c,f) , respectively. and the effect becomes less and less pronounced as thelattice depth is increased and the Mott transition crossed.Note finally that the Fresnel phase suppresses the contri-bution from the edges of the cloud, thus favoring the con-tribution of the central region to the ToF pattern. Thisis especially important when superfluid rings surround acentral MI region with lower coherence [20].The interference pattern is often characterized by itsvisibility [3, 4, 5, 6, 7], V = n ToF ( k max ) − n ToF ( k min ) n ToF ( k max ) + n ToF ( k min ) , (6)with the choice k max d = (2 π,
0) and k min d = √ π, π ) tocancel out the Wannier envelope in the division. We firstevaluate the sensitivity of V to the Fresnel phase by plot-ting in Fig. (3) two theoretical ”benchmark” curves as-suming perfect experimental resolution (dashed and dot-dashed lines for t = 14 ms and t → ∞ , respectively). Wefind little difference between the two curves when T /J iskept constant and small. Indeed, the Fresnel terms onlymatter for systems with large coherence length, where thevisibility is by construction very close to unity. We con-clude that a detailed investigation of the superfluid sideof the transition is better achieved by directly measuring
FIG. 3: Visibility of the interference pattern as defined inEq. (6). The dashed and dot-dashed lines show the Quan-tum Monte Carlo result for infinite and finite ( t = 14 ms)expansion times, assuming perfect experimental resolution.The solid line is computed for t = 14 ms accounting for finiteexperimental resolution. Note that the comparison betweenexperiment and simulation is only qualitative, since the sim-ulations were performed at constant temperature T = J/k B while the experiment was not. the ToF distributions, whereas the visibility is well-suitedfor short coherence lengths.We also compare in Fig. 3 the experiments reported in[4] to the predictions of the QMC simulations (solid line).Here, we emphasize that apart from the Fresnel terms, anaccurate comparison requires to account for the experi-mental resolution, which is limited by two effects. First,the signal was obtained by integration over a square boxcentered around the maxima or minima, the integrationarea being ≈ (0 . × π/d ) in momentum units. This iscomparable to a typical peak area, so that the visibilityis calculated from the peak weight, rather than from itsamplitude. Second, the finite resolution of the imagingsystem (about 6 µ m) is not negligible for the sharpestpeaks. Accounting for these two effects when evaluatingthe QMC data, we find good agreement with the exper-imental results. This entails that the experimental dataare compatible with the system remaining at low enoughtemperatures to cross a quantum-critical regime, in con-trast to the analysis made in Refs. [21, 22] which includedneither near-field expansion nor experimental resolution.In conclusion, we have analyzed the interference pat-tern observed in the expansion of a bosonic quantum gasreleased from an optical lattice. We showed that due toan additional Fresnel-like phase appearing for finite timeof flight, the ToF distribution can be markedly differentfrom the momentum distribution for clouds with large co-herence lengths. Conversely, the visibility as calculatedfrom Eq. (6) is rather insensitive to this effect.The Fresnel phase acts as a magnifying lens for thecentral region undergoing a Mott insulator transition bysuppressing the contribution of the outer regions of thecloud when the central density is close to integer filling. This could eventually provide a way to investigate thephysics near the quantum-critical point without ”par-asitic” contributions coming from coexisting superfluidrings.Simulations were ran on the Brutus cluster at ETHZurich. We acknowledge support from IFRAF, ANR(FG), DFG, EU, AFOSR (IB), the Swiss NationalScience Foundation (LP), NSF grant PHY-0653183(BCS,NP,BS) and DARPA (OLE project). ∗ Electronic address: [email protected][1] M. Greiner, O. Mandel, T. Esslinger, T. W. H¨ansch, andI. Bloch, Nature , 39 (2002).[2] T. St¨oferle, H. Moritz, C. Schori, M. K¨ohl, andT. Esslinger, Phys. Rev. Lett. , 130403 (2004).[3] F. Gerbier, A. Widera, S. F¨olling, O. Mandel, T. Gericke,and I. Bloch, Phys. Rev. Lett. , 050404 (2005).[4] F. Gerbier, A. Widera, S. F¨olling, O. Mandel, T. Gericke,and I. Bloch, Phys. Rev. A , 053606 (2005).[5] S. Ospelkaus, C. Ospelkaus, O. Wille, M. Succo, P. Ernst,K. Sengstock, and K. Bongs, Phys. Rev. Lett. , 180403(2006).[6] K. G¨unter, T. St¨oferle, H. Moritz, M. K¨ohl, andT. Esslinger, Phys. Rev. Lett. , 180402 (2006).[7] I. B. Spielman, W. D. Phillips, and J. V. Porto, Phys.Rev. Lett. , 080404 (2007).[8] J. Catani, L. De Sarlo, G. Barontini, F. Minardi, andM. Inguscio, Phys. Rev. A , 011603(R) (2008).[9] M. Greiner, Phd thesis, Ludwig-Maximilian UniversityMunich (2003).[10] P. Pedri, L. Pitaevskii, S. Stringari, C. Fort, S. Burger,F. S. Cataliotti, P. Maddaloni, F. Minardi, and M. In-guscio, Phys. Rev. Lett. , 220401 (2001).[11] Replacing ˜ w “ k = m ( r − r ν ) ~ t ” by ˜ w ` k = m r ~ t ´ requires t ≫ R /dω L , R /d p ω L E R / ~ , or t ≫ λ x = 765 nm for one axis x , and λ y,z = 843 nm for the other two axis. The lattice depthsare chosen such that the tunneling along the x -directionis equal to that along y, z . The experiment is otherwiseidentical to that described in [1].[15] V. M. P´erez-Garcia, H. Michinel, J. I. Cirac, M. Lewen-stein, and P. Zoller, Phys. Rev. Lett. , 5320 (1996).[16] N. V. Prokof’ev, B. V. Svistunov, and I. S. Tupitsyn, Zh.Eksp. Theor. Fiz , 570 (1998).[17] N. V. Prokof’ev, B. V. Svistunov, and I. S. Tupitsyn,Phys. Lett. A , 253 (1998).[18] L. Pollet, K. V. Houcke, and S. M. A. Rombouts, J.Comp. Phys. , 2249 (2007).[19] L. Pollet, C. Kollath, K. V. Houcke, and M. Troyer, NewJ. Phys. , 065001 (2008).[20] G. G. Batrouni, V. Rousseau, R. T. Scalettar, M. Rigol,A. Muramatsu, P. J. H. Denteneer, and M. Troyer, Phys. Rev. Lett. , 117203 (2002).[21] R. B. Diener, Q. Zhou, H. Zhai, and T.-L. Ho, Phys. Rev.Lett.98