Role of interactions in time-of-flight expansion of atomic clouds from optical lattices
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] A ug Role of interactions in time-of-flight expansion of atomic clouds from optical lattices
Joern N. Kupferschmidt , and Erich J. Mueller Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853-2501, USA and Dahlem Center for Complex Quantum Systems and Institut f¨ur Theoretische Physik,Freie Universit¨at Berlin, 14195 Berlin, Germany
We calculate the effect of interactions on the expansion of ultracold atoms from a single site of anoptical lattice. We use these results to predict how interactions influence the interference patternobserved in a time of flight experiment. We find that for typical interaction strengths their influenceis negligable, yet that they reduce visibility near a scattering resonance.
PACS numbers: 67.85.-d, 03.75.Dg, 03.65.Vf, 37.10.Jk
I. INTRODUCTION
One of the most important probes of cold atom sys-tems is time-of-flight imaging. Turning off all trappingpotentials, a cloud of cold atoms expands for tens of mil-liseconds, and an absorption image is taken. In the farfield limit, the resulting image can be directly interpretedas the momentum distribution of the original cloud, if in-teractions among the atoms can be neglected during theexpansion. Here we critically evaluate the validity of ne-glecting such interactions during the expansion from anoptical lattice.The question of how to interpret time-of-flight imagesis crucial. These images have been used, for example,to distinguish the Mott insulating and superfluid phases[1, 2]. They have also been proposed as a tool to de-tect vortices in rotating condensates [3], and are a crucialcomponent of more sophisticated probes such as modu-lation spectroscopy [4] and Bragg/Raman spectroscopy[5].Interactions between cold neutral atoms are parame-terized by the s -wave scattering length a , which is typi-cally on the order of 5 − λ/ λ/ ≈ λ is the wavelength of the laser used to create the opti-cal lattice [1]. The scattering length can, however, be-gin to approach the size σ r of the atomic states in onewell. For example σ r ≈ V ∼ E R , where E R = ¯ h (2 π ) / (2 mλ ) isthe recoil energy of the lattice. Thus when a few par-ticles occupy a single site, their interactions are signifi-cant [7]. Experiments have measured the resulting energyshifts [8], and recently used them to study atom numberstatistics [9]. While these on-site interactions are impor-tant, by the time the wave-packets have expanded enoughto overlap with neighboring sites, interactions are greatlyattenuated.Hence in our analysis we include interactions betweenatoms expanding from the same site, but neglect all inter-site interactions. Thus we are able to investigate whetherinteractions during the initial expansion period affect theinterference image. Our estimate of the role of interac-tions is a lower bound; there may be further interaction effects during later stages of the expansion. Most impor-tantly, interference effects could lead to strongly inter-acting high density regions at intermediate times [10].Within our approximation, the density profile of themany-body system during time of flight depends only onthe t = 0 wavefunction, and the time dependence of acluster of particles expanding from a single site. In thenext Sec. II we consider the expansion from a single site.Time-of-flight interferometry is considered in Sec. III. Asa numerical example we consider the expansion of a two-dimensional, harmonically trapped cloud forming a su-perfluid in Sec. IV. We summarize our results in Sec. V.In 2008, Gerbier et al. [11] reported the results of avery similar calculation, however they gave very few de-tails. More recently, Fang, Lee, and Wang [12] reporteda complementary investigation, where they used a trun-cated Wigner approximation to investigate the role ofinteractions during time-of-flight expansion. Restrictingthe expansion to one dimension (1D), they consideredthe dynamics of 10 atoms released from a 10-site opticallattice. As we discuss in section II C, interactions play amuch larger role in 1D expansion than in 3D, and Fang etal. consequently found nearly a factor of two attenuationof the central Bragg peak compared to the noninteractinggas. Using very similar parameters, we find that interac-tions during 3D expansion only lead to a 5% reductionin the amplitude of the central Bragg peak. II. SINGLE SITE EXPANSIONA. Statics
An optical lattice is typically modeled as a potentialof the form V ( x, y, z ) = V [sin ( kx ) + sin ( ky ) + sin ( kz )] (1)where k = 2 π/λ . Near the local minima one mayapproximate the sinusoidal as a harmonic potential V eff = mω r /
2, with small oscillation frequency ω r =(¯ hk /m ) p V /E R . The single-particle ground state inthis potential is a Gaussian φ ,i ( r ) = 1( πσ ) / exp (cid:20) − ( r − r i ) σ (cid:21) , (2)where σ = ¯ h/mω r . We model the interaction among theparticles as a contact interaction,ˆ H int = g Z d r ˆ ψ † ( r ) ˆ ψ † ( r ) ˆ ψ ( r ) ˆ ψ ( r ) , (3)where g = 4 π ¯ h a/m .We are concerned about how these interactions modifythe few-body wavefunction on a single site, and how thisinfluences the time-of-flight expansion. As long as a ≪ σ r , the effects of interactions are captured by a Gaussianvariational ansatzΦ N ( { r α } ) = e iξ N φ cm ( r cm ) Y α<β φ N ( r α − r β ) (4) φ cm ( r ) = 1( πσ ) d/ exp (cid:20) − (cid:18) N σ − iβ N (cid:19) r (cid:21) φ N ( r ) = 1( πσ N ) d/ exp (cid:20) − (cid:18) N σ N − iN β N (cid:19) r (cid:21) where we have introduced the center of mass r cm =(1 /N ) P Nα =1 r i and generalized to arbitrary spatial di-mension d . For N = 1, Eq. (4) reduces to the harmonicoscillator wavefunction φ ( r ). We will use this wavefunc-tion as a time-dependent variational ansatz to describethe dynamics, hence we have introduced the parameters β and β N which are nonzero only if the cluster is ex-panding or shrinking.For convenience we will work with dimensionless quan-tities, using units where ¯ h = m = λ/ B. Dynamics
One produces a variational estimate of the dynamicsby minimizing the action S = Z dt Z Y α d d r α (cid:26) i (cid:20) φ ∗ ∂φ∂t − (cid:18) ∂φ ∗ ∂t (cid:19) φ (cid:21) (5) − φ ∗ − X α (cid:18) ∂ ∂ r α + r α σ r (cid:19) + X α<β gδ (3) ( r α − r β ) φ . We use the trial wave-function in Eq. (4), for which thespatial integrations can be performed analytically, andallow all variational parameters to be arbitrary functionsof time. A similar approach has been used to describethe role of interactions in the dynamics of a harmon-ically trapped BEC, where the atom number is muchlarger [13]. Minimizing the action leads to a second orderdifferential equation for the width σ N , σ N ∂ σ N ∂t = 1 − σ N σ r + N g (2 π ) d/ σ − dN , (6) β N = 12 σ N ∂σ N ∂t . The center-of-mass width σ obeys the same equation,but with g = 0. Note that in the noninteracting limit all N dependence drops out. The sole contribution from theoptical lattice is the term σ N /σ r . During time-of-flightexpansion, the optical lattice as well as the harmonictrapping potential are removed, and this term no longerappears in the equations of motion.At time t = 0 we set β = 0, and take σ to be given bythe static solution with ∂ t σ = 0. Analytic solutions tothe resulting algebraic equation can only be found when d = 2, where σ N (0 , d = 2) = r N g π σ r . (7)The center-of-mass width is simply σ = σ r in all dimen-sions.In two dimensions we can analytically integrate theequations of motion, σ ( t ) = p σ r + t /σ r (8) σ N ( t, d = 2) = (cid:18) N g π (cid:19) / σ ( t ) , (9) β ( t ) = β N ( t, d = 2) = 12 tσ r + t . (10)The expressions for σ and β apply in all dimensions. C. Phase accumulation
The phase of the expanding cluster is crucial for de-termining the observed interference pattern. In terms of σ N , one finds ∂ξ N ∂t = − N d σ (11) − N − " dσ N − dσ + ( d + 2) N g (cid:18) πσ N (cid:19) d/ . The contribution in square brackets arises from the in-teractions. Interactions increase the width of the initialstate, which reduces the contribution of the kinetic en-ergy and retards the phase relative to the noninteractingexpansion. This should be contrasted with the contri-bution from the potential energy as well as the directinterparticle interactions themselves, which increase theenergy and advance the phase. To determine the netsign of the interaction correction is thus not straightfor-ward. In particular, σ N is generally larger than σ andthe quantity in square brackets does not have a definitesign.We produce a rough estimate of the phase accumu-lated by replacing σ N with σ in this expression. Theinteraction contribution to the phase will then scale as ξ int ∝ Z t dt σ ( t ) d = σ r arcsinh (cid:16) tσ (cid:17) d = 1arctan (cid:16) tσ (cid:17) d = 2 tσ r √ σ + t d = 3 (12)Whereas the contribution is logarithmically divergentin the one-dimensional case, it very quickly reaches afinite value in the two-dimensional as well as the three-dimensional case. This indicates that the influence ofinteractions is confined to the very beginning of the time-of-flight expansion, t < ∼ σ , essentially corresponding tothe time required for the cluster to expand to less thantwice its initial size, where σ ( t ) ≈ /
2. (Recall, we areusing units where lengths are measured in terms of thelattice spacing and times, up to numerical constants, inunits of the inverse recoil energy.) Typically, this meansthat for d = 2 or d = 3 interactions become irrelevantwell before the clusters overlap. Conversely, interactionsbetween clusters can not be neglected during d = 1 ex-pansion.The non-interacting contribution to the phase is ξ ( t ) = − N d tσ ( t ) . (13)In two dimensions, where we have analytic expressionsfor σ N ( t ), we further find ξ N ( t, d = 2) = − " N − r N g π arctan tσ ( t ) . (14)The fact that interactions only modify the prefactor is areflection of the scaling symmetry of the expanding cloudin d = 2. III. TIME-OF-FLIGHT IMAGES
Having calculated the expansion dynamics of a singlecluster of particles, we now explore the consequences forthe atom density seen in a time-of-flight expansion exper-iment. Neglecting correlations between sites, we assumethat the initial state can be written as a generalizationof the standard Gutzwiller Ansatz, | Ψ i = N s O i =1 ∞ X n =0 f i,n Z d n r Φ n ( { r α − R i } ) |{ r α }i ! (15)where i runs over all N s lattice sites R i , n is the numberof particles on a given site, and |{ r α }i = 1 √ n ! ˆ ψ † ( r ) ˆ ψ † ( r ) . . . ˆ ψ † ( r n ) | i . (16)The state is normalized when the norm of the f -vectoris one, ∞ X α =0 | f i,α | = 1 . (17)The n -particle wavefunction on site i , Φ n ( { r α − R i } ), isgiven by Eq. (4). Within our approximation, where we neglect interac-tions between atoms on different sites, the time evolutionof Eq. (15) simply amounts to separately time evolvingeach cluster, as described in Sec. II. The resulting densityprofile is h Ψ | ˆ ψ † ( r ) ˆ ψ ( r ) | Ψ i = N s X i =1 X n n | f i,n | | φ cn ( r − R i ) | (18)+ N s X i =1 X k = i X n √ nf ∗ i,n f i,n − φ c ∗ n,n − ( r − R i ) ! × X m √ mf k,m f ∗ k,m − φ cm,m − ( r − R k ) ! with | φ cn ( r ) | = Z Π nα =2 d r α | Φ n ( r , r , . . . , r n ) | (19) φ cn,n − ( r ) = Z Π nα =2 d r α Φ n ( r , r , . . . )Φ ∗ n − ( r , . . . ) . In the noninteracting limit both contractions reduce tothe noninteracting single particle wavefunction, so that | φ cn ( r ) | = | φ ( r ) | φ cn,n − ( r ) = φ ( r ) . (20)In this case one can write a more readily interpretableexpression for the density [3], h Ψ ni | ˆ n ( r ) | Ψ ni i = | φ ( r ) | (cid:2) ( N − N c ) + | Λ( r ) | (cid:3) (21)Λ( r ) = N s X i =1 α i e − iβ ( r · R i − R i ) . (22)Here N is the total number of particles in the lattice. | φ ( r ) | is a simple gaussian with width σ ( t ) / √
2. Cor-rections to the featureless gaussian peak, N c and Λ( r ),signal the presence of superfluid order in the system. N c is the condensed number of particles, whereas α i is theexpectation value of the annihilation operator on site i and thus the superfluid order parameter in the system. α i = h ˆ a i i = X n √ nf i,n f ∗ i,n − (23) N c = N s X i =1 | α i | (24)Gerbier et al. [11] have pointed out that in Eq. (22) above,for experimentally relevant expansion times on the orderof tens of milliseconds, it is necessary to keep the Fresnellike terms quadratic in R i . In the absence of the Fresnelterms, the shape of Bragg peaks is simply the Fouriertransform of the superfluid order parameter.Here we go beyond the approximations in Eqs. (20)-(24), and include the effects of interactions on the ex-pansion. These interactions have two effects. First theybroaden each of the expanding clusters. This broadensthe incoherent background, but it also reduces the con-trast of the Bragg peaks. This latter effect occurs becauseof the reduced overlap between the expanding clusterswith different numbers of particles. Second, the inter-actions introduce a nonlinear phase difference betweenthe different particle number clusters. This dephasingfurther reduces the contrast of the Bragg peaks.The broadening of the incoherent background is quan-tified by | φ cN ( r ) | = 1( πσ N, eff ) d/ exp[ − r σ N, eff ] , (25)where σ N, eff = ( N − σ N + σ N . (26)Clearly σ N, eff > σ , reflecting the larger size of the inter-acting cluster.The influence of interactions on the Bragg peaks isquantified by identifying the difference from the nonin-teracting wavefunction φ ( r ), φ cn,n − ( r ) = φ ( r ) e iδξ N | δN N | exp[ r δs N ] . (27)The overall phase δξ N , the width ( σ − − δs ) − / , as wellas the corresponding prefactor | δN N | , affect the peaks.The expressions for each of these terms are complicated,with δξ N = ξ N − ξ N − + arg[ δN N ] − ξ (28) δN N = (cid:18) σ N σ N − σ N + σ N − − iσ N σ N − ( β N − β N − ) (cid:19) ( N − d/ × (cid:20) N σ N σ (2 N − σ N + σ − iσ σ N ( β N − β ) (cid:21) d/ (29) δs N = N − N σ N (cid:2) σ N − σ − iσ σ N ( β − β N ) (cid:3) × N σ N + ( N − σ (1 − iσ N β N )(2 N − σ N + ( N − σ − iσ σ N [( N − β N − β ] . (30)In the noninteracting case the only phases contribut-ing to the interference come from the terms f i,n f ∗ i,n − [cf. Eq. (23)]. Here there are additional contributions asgiven by Eq. (28). IV. NUMERICAL EXAMPLE
To illustrate our results, we consider a two dimensionalharmonically trapped gas of Rb in optical lattices with V = 10 . , , E R , yielding ν r ≈ . , . , . a = 5 , . , FIG. 1: (Color online) Initial evolution of the widths of dif-ferent number states N = 1 through 10 for a = 50nm and V = 5 E R . The smallest widths are reached for N = 1 (lowestcurve), the largest for N = 10 (highest curve). Interactionscause states with larger particle number to broaden and ex-pand faster. Times are measured in units of τ = m ( λ/ / ¯ h ,which is approximately 0 . Rb. The range plottedis much shorter than a typical time-of-flight experiment. Allsubsequent expansion is ballistic. U/ t ≈ .
5, well on the superfluid site of the superfluid-Mott transition [14]. Adjusting the chemical potential µ in the center of the trap to obtain the same total numberof particles thus yields identical initial states. To find theinitial f i,n ’s of Eq. (15) we solve the discrete variationalGutzwiller problem, minimizing h Ψ G | H L | Ψ G i , with H L = − t X h ij i (ˆ a † i ˆ a j + ˆ a † j ˆ a i ) + X i U ˆ n i (ˆ n i − − µ ˆ n i (31) | Ψ G i = Y i X n f i,n (ˆ a † i ) n √ n ! | vac i , (32)where h i, j i denotes nearest neighbor sites, ˆ a i annihi-lates a boson at site i , ˆ n i = ˆ a † i ˆ a i , and t and U areextracted from the non-interacting Wannier wavefunc-tions [14]. The corrections to U from using the many-body wavefunctions on each site are very small at thislattice depth, and the corrections to t are at most 10%[7]. We take the expansion to be three dimensional, treat-ing the individual wells as spherically symmetric.We produce initial conditions by solving Eq. (6) withthe conditions ∂ t σ = 0 and β = 0. Starting from theseinitial conditions, we numerically integrate Eqs.˜(6) and(11). We then plot the densities, Eq. (18).Figure 1 shows the time evolution of the widths σ N ofthe clusters expanding from sites with different particlenumbers. As one can see, and as discussed in Sec. II, theexpansion very quickly becomes ballistic.Figure 2 shows the time evolution of the phase differ-ences − δξ N , see Eq. (28). One sees that when a = 50nmthe phase difference between clusters of different particlenumbers are on the order of 2 π , and hence the interfer- FIG. 2: (Color online) Overall phase difference betweenwavefunctions with particle occupation differing by one [seeEq.(28)], plotted for a = 50nm and V = 5 E R . Interactions donot affect the lowest (blue) curve, corresponding to n = { , } ,but progressively affect the higher pairs n = { , } , ..., { , } ,yielding the largest effect for n = { , } , the top (violet)curve. The effect is approximately linear in N . ence pattern will be influenced by the interactions. Fortypical Rb parameters, a = 5nm, the phase differenceis correspondingly smaller.Figure 3 shows cuts through illustrative density imagesalong the lattice direction after a 100ms time-of-flight.This time was chosen to minimize distortions from Fres-nel terms [11]. Weak interactions, a = 5nm, have neg-ligable effect on the image. While stronger interactions a = 50nm begin to reduce the amplitudes of the interfer-ence peaks, the peaks remain clearly visible.Comparing expansion images at fixed U/ t and fixedtime of flight results in an interesting structure. In orderto have the same U/ t , the initial wavefunctions in thestronger interacting case must be larger, resulting in aslower initial expansion. Therefore in Fig. 3 we see thatthe central peak is larger for stronger interactions, whilethe satellite peak is smaller. Using our intuition from thenoninteracting expansion, one can think of this effect asbeing due to the envelope of the Bragg peaks which fallsoff on a scale inversely proportional to the size of the ini-tial Wannier states. In the inset of Fig. 3 we normalizeout this effect by multiplying with the inverse envelope,( πσ ( t )) / exp[ r /σ ( t )]. Taking this normalization intoaccount, interactions reduce the amplitude of the inter-ference peak. The reduction is about 5% for a = 5nm,rising to 33% for a = 50nm. As shown in the inset ofFig. 3, the reduction is greater for the central Braggpeak than for the first satellite peak.There are a number of ways of increasing the impor-tance of the interactions during time of flight. For ex-ample, changing the geometry of the lattice sites influ-ences how the cloud expands and how long interactionsremain relevant: the expansion from needle shaped sitesis predominantly in the x - y plane, and the 2D scalingin Eq. (12) approximately holds. Additionally we havestudied what happens when one suddenly increases the xxxx FIG. 3: (Color online) Density of a nonrotating atomic cloudexpanded for 100ms, with distance d measured from its centeralong a lattice direction. The interference peaks are clearlyvisible and remain so even in the strongest interacting limitconsidered here. Plotted are a = 0nm (blue), 5nm (violet),15 . d , butbottom to top in the central peak. The inset shows the cen-tral and the first satellite peak with adjustment for differentexpansion velocities, as described in the main text. scattering length while releasing the atoms from the op-tical lattice. This allows one to independently controlthe size of the initial Wannier states and the scatteringlength. In the expansion shown in Fig. 3, where t/U wasfixed while changing a , the initial Wannier states werelarger when a was made larger.Starting from the equilibrium state with a = 5nmand V = 10 . E R , we investigate the expansion for a = 15 . a = 50nm. Although we do not show theresults here, we find that the suppression of the centralinterference peak is roughly a factor of 1 . V. SUMMARY AND DISCUSSION
We have considered the effect of two-particle interac-tions on the time of flight images of cold atoms on opticallattices. We show that on-site interactions can be impor-tant for these images, but argue that one can neglect theinteractions between atoms on different sites.We find that even if interactions are increased by afactor of ten from their normal strength, no qualitativelynew features appear in the time of flight images. How-ever, the quantitative size of the peaks is sensitive tothe interactions. Given the wide tunability achievableby employing Feshbach resonances [15], it is conceivablethat experiments can study the role of interactions dur-ing time of flight. The analysis presented here will failwhen the scattering length a becomes comparable to thesize of the Wannier state σ r .Conceptually it is worth noting that in many electronicmesoscopic systems a situation markedly different fromthe one here is encountered. There the dynamics is de-termined by impurity scattering or scattering off systemboundaries, and at low temperatures are not affected bythe interaction. Consequently there exists a regime inwhich interactions effectively only add additional phasesto the relevant propagation amplitudes. Such a regimeis not identifiable in the system we considered here. In-stead, we find that whenever the interactions producerelevant phases, they also perturb the dynamics. VI. ACKNOWLEDGEMENTS
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