Chaos-induced depletion of a Bose-Einstein condensate
Ralf Wanzenböck, Stefan Donsa, Harald Hofstätter, Othmar Koch, Peter Schlagheck, Iva Březinová
CChaos-induced depletion of a Bose-Einstein condensate
Ralf Wanzenböck, Stefan Donsa, Harald Hofstätter, Othmar Koch, Peter Schlagheck, and Iva Březinová ∗ Institute for Theoretical Physics, Vienna University of Technology,Wiedner Hauptstraße 8-10/136, 1040 Vienna, Austria, EU Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria, EU Département de Physique, University of Liège, 4000 Liège, Belgium, EU (Dated: July 16, 2020)The mean-field limit of a bosonic quantum many-body system is described by (mostly) non-linear equations of motion which may exhibit chaos very much in the spirit of classical particlechaos, i.e. by an exponential separation of trajectories in Hilbert space with a rate given by apositive Lyapunov exponent λ . The question now is whether λ imprints itself onto measurableobservables of the underlying quantum many-body system even at finite particle numbers. Usinga Bose-Einstein condensate expanding in a shallow potential landscape as a paradigmatic examplefor a bosonic quantum many-body system, we show, that the number of non-condensed particlesis subject to an exponentially fast increase, i.e. depletion. Furthermore, we show that the rate ofexponential depletion is given by the Lyapunov exponent associated with the chaotic mean-fielddynamics. Finally, we demonstrate that this chaos-induced depletion is accessible experimentallythrough the visibility of interference fringes in the total density after time of flight, thus openingthe possibility to measure λ , and with it, the interplay between chaos and non-equilibrium quantummatter, in a real experiment. Non-equilibrium quantum many-body systems arenowadays routinely probed in experiments with ultra-cold atoms with unprecedented control over their pa-rameters such as particle number, interaction strength,and external potentials. A plethora of non-equilibriumsystems has been realized to address a wide varietyof physics questions, including Anderson localization ofBose-Einstein condensates (BECs) [1–5], many-body lo-calization in disordered lattices [6, 7], pre-thermalizationof one-dimensional (1D) BECs [8], and quench-dynamicsof spin-model systems [9], to name just a few.At the same time, the theoretical understanding of non-equilibrium quantum many-body systems still lags be-hind that obtained for stationary or equilibrium sys-tems. In particular, the role of “classical” chaos on non-equilibrium quantum many-body systems is currentlysubject of intense scrutiny, see e.g. [10–13]. For a bosonicquantum many-body system, the mean-field limit can beviewed as the classical limit in which the particle creationand annihilation operators lose their quantum propertiesand start to act as classical fields. The mean-field limit,typically involves non-linear (partial) differential equa-tions, and can exhibit chaos with exponential separationin Hilbert space characterized by a positive Lyapunov ex-ponent λ [14, 15]. One of the open questions is, whetherwave chaos, more specifically the positive λ , imprints it-self onto the dynamics of a quantum many-body systemeven at finite particle numbers and how such an imprintcould be measured. Recently, out-of-time-order correla-tors have been suggested as suitable probes for a positive λ (see e.g. [12, 13, 16–19]).In this paper, we find an imprint of chaos on a differ-ent observable within a paradigmatic bosonic system: A ∗ [email protected] quasi 1D BEC initially trapped harmonically and thenreleased to expand in a shallow disordered or periodicpotential. We show that the fraction of non-condensedparticles increases exponentially over time and that theassociated rate is given by the Lyapunov exponent λ ob-tained from mean-field chaos. The depletion occurs ontime scales during which most of the initial interactionenergy is converted into kinetic energy, and comes to ahalt at times close to the so-called scrambling time (orEhrenfest time) [19–21]. We observe chaos-induced de-pletion both in shallow disordered as well as periodic po-tentials showing that the effect is quite general and doesnot rely on the intrinsic randomness of a disordered land-scape.Finally, we demonstrate that the condensate depletionand thus the Lyapunov exponent λ is accessible experi-mentally through the analysis of fluctuations of the to-tal particle density in momentum space. While the con-densed part is coherent and leads to interference fringesin the total density, the non-condensed part is incoher-ent and piles up over time as a non-fluctuating back-ground. Analyzing the interference fringes after time offlight would thus allow to extract experimentally the frac-tion of non-condensed particles as a function of time andcompare to the theoretically obtained λ . Condensate de-pletion thus offers itself as an experimentally accessibleprobe to investigate the role of (mean-field or classical)chaos in non-equilibrium quantum matter.While our findings are generally applicable to bosonicsystems that exhibit mean-field chaos, we pick one spe-cific system already realized experimentally [4] to obtainnumerical results. The initially harmonically trappedquasi 1D BEC of N Rb atoms is released at t = 0 to expand in a shallow potential. As units we use (cid:126) = m = ω = 1 , with ω being the frequency of theinitial longitudinal harmonic trap which amounts to a a r X i v : . [ c ond - m a t . qu a n t - g a s ] J u l e λt d ( ) t [ t ] n incoh ¯ d ( ) FIG. 1. d (2) (Eq. 2) for two initial conditions obtained by lin-ear distortion of the mean-field ground state (black dashed-dotted line), distance function ¯ d (2) (Eq. 4) averaged over thestochastic ensemble (triangles), and fraction of incoherentparticles n incoh (squares). Data shown at integer values of t (except for d (2) ), the lines serve as guides for the eye. The reddashed lines mark an exponential increase with λ = 1 . t − .Number of particles is N = 1 . × , periodic potential V ( x ) = V P cos ( k P x ) used with V P = 0 . e , k P = π/ ξ and ξ the healing length. time unit of t ≈ ms and a space unit of x ≈ . µ m.For the number of atoms, we take N = 1 . × follow-ing [4], as well as larger values, i.e. N = 1 . × and N = 1 . × to investigate the effect of varying N .Describing the quasi-1D system on a mean-field level theGross-Pitaevskii equation (GPE) takes the form i ∂ψ ( x, t ) ∂t = (cid:18) − ∂ ∂x + V ( x ) + g | ψ ( x, t ) | (cid:19) ψ ( x, t ) , (1)where the nonlinearity is g ≈ with the above pa-rameters and the normalization (cid:82) dx | ψ ( x, t ) | = 1 . Thepotential V ( x ) corresponds to the harmonic potential at t = 0 , and to the periodic or disordered speckle potentialat t > with amplitude much smaller than the meanenergy per particle e . This system exhibits chaos on themean-field level [14]: Two wave functions, ψ a ( x, and ψ b ( x, , respectively, initially very close to each otherin Hilbert space as measured by a distance norm, sep-arate exponentially in time until quasi-orthogonality isreached, see Fig. 1. The rate of the exponential growthis given by the Lyapunov exponent λ . For the distancenorm we take d (2) a,b ( t ) = 12 (cid:90) dx | ψ a ( x, t ) − ψ b ( x, t ) | . (2)The Lyapunov exponent λ shows systematic trends as afunction of the parameters of the system: It vanishes inabsence of inter-particle interactions for arbitrary poten-tials, as well as in presence of inter-particle interactions infree space (i.e. without any potential). At fixed period ofthe periodic potential k P , or fixed correlation length σ ofthe speckle potential, it increases both with nonlinearityand the potential amplitude [14], see the supplementalmaterial (SM).To find imprints of chaos on measurable observables of the quantum many-body system with finite N atheory beyond mean-field has to be applied. Themulti-configurational time-dependent Hartree methodfor bosons (see, e.g. [22, 23]), while being in principle ex-act for a sufficient number of orbitals, suffers from the ex-ponentially growing configuration space. For the particlenumbers considered, only two orbitals can be afforded nu-merically [24]. As more than two orbitals are populatedduring the propagation, the MCTHB method entails alarge and not easy to quantify error. We, therefore, re-sort to the truncated Wigner approximation (TWA), seee.g. [25–28] which employs the Wigner representation W for (in general) a many-body density matrix ˆ ρW ( ψ , . . . , ψ M , ψ ∗ . . . , ψ ∗ M ) = 1 π M × (cid:90) dz M Tr (cid:104) ˆ ρe i (cid:80) j ( z ∗ j ˆ ψ † j + iz j ˆ ψ j ) (cid:105) e − i (cid:80) j ( z ∗ j ψ ∗ j − iz j ψ j ) . (3) W can be viewed as a phase-space representation of thequantum many-body state. M is the total number ofmodes in which particles can be created or annihilatedand ˆ ψ † j and ˆ ψ j are the corresponding creation and anni-hilation operators, respectively. In general, particles canbe created or annihilated in an arbitrary single particlemode denoted by j . We choose j to represent a specificpoint in space assuming for simplicity an equidistant spa-tial discretization. We have made sure, however, that thespatial grid is fine enough, i.e. the distance between gridpoints dx < /k max with k max being the largest relevantmomentum in the system, such that we are still in thecontinuum limit.Having W as a function of time at disposal would allowto evaluate all expectation values of symmetrized prod-ucts of creation and annihilation operators. The exactequation of motion for W can be obtained using von Neu-mann’s equation of motion for ˆ ρ . However, it proves tobe intractable, such that approximations have to be in-voked. Within the TWA [25–27], the time evolution of W is sampled stochastically with an ensemble of trajec-tories obeying the GPE, Eq. 1. (The only modificationcomes from the fact that we have to discretize space suchthat the second derivative in Eq. 1 has to be replaced byits second-order finite difference approximation.) It hasbeen shown [21, 29, 30] that this approximation amountsto neglecting non-classical trajectories as well as interfer-ences between distinct trajectories in many-body Hilbertspace. The question then arises at which times do theseneglected effects start to play a role and become non-negligible. For single- or few-particle systems, samplingthe time evolution with classical trajectories is accurateup to the point where an initially maximally localizedstate has spread over the whole system. This time iscalled the Ehrenfest time τ E [31] which is, in presenceof classical chaos, inversely proportional to λ and growslogarithmically with / (cid:126) . This concept can be extendedinto the many-body regime for bosonic systems with (cid:126) be-ing replaced by the effective Planck constant (cid:126) eff (cid:39) /N .Following the lines of [20, 21] we thus assume that ourresults are accurate up to the time τ E = λ log N .The initial conditions within the stochastic ensemble oftrajectories are constructed such as to correctly sam-ple the phase-space distribution of the underlying initialquantum state, which in our case is a BEC at zero tem-perature. The stochasticity of the ensemble comes solelyfrom the sampling of this initial state since Eq. 1 is com-pletely deterministic. We follow [25–27] and constructthe initial wave functions by adding to the mean-fieldground state in the harmonic trap vacuum fluctuationsin form of Gaussian noise (see the SM).The most relevant observables in our case will be the co-herent part of the particle density given by ρ coh ( x j , t ) = |(cid:104) ˆ ψ j ( t ) (cid:105)| , as well as the one-particle reduced density ma-trix (1RDM) D ij ( t ) = (cid:104) ˆ ψ † i ( t ) ˆ ψ j ( t ) (cid:105) . The term “coher-ent" in defining ρ coh ( x j , t ) points to the fact that onlya macroscopically occupied state with a spatially non-random phase will survive the averaging. ρ coh ( x j , t ) cantherefore be associated with the density of condensed par-ticles. Alternatively [32, 33], the condensate state is de-fined through a macroscopic occupation of one eigenstateof the 1RDM. We show in the SM that these two defini-tions of the condensate give practically identical resultsfor the depletion over time such that we use throughoutthe remainder of the paper the term coherent synony-mously to condensed.Within the stochastic ensemble of trajectories, ex-pectation values can be calculated as (cid:104) ˆ ψ j ( t ) (cid:105) = N s (cid:80) N s s =1 ψ s ( x j , t ) , with N s ( (cid:29) ) being the number ofGross-Pitaevskii trajectories ψ s ( x j , t ) within the ensem-ble. To calculate the 1RDM, one has to rewrite (cid:104) ˆ ψ † i ˆ ψ j +ˆ ψ j ˆ ψ † i (cid:105) = 2 (cid:104) ˆ ψ † i ˆ ψ j (cid:105) + Ndx δ ij using the commutator relation [ ˆ ψ j , ˆ ψ † i ] = Ndx δ ij . The term δ ij /dx is the discrete versionof the δ -function for a continuous system, and the factor /N comes from our normalization of the wave functionsof Eq. 1 to one, or equivalently, the creation and an-nihilation operators to /N . The 1RDM is then givenby D ij ( t ) = N s (cid:80) N s s =1 ψ ∗ s ( x i , t ) ψ s ( x j , t ) − Ndx δ ij , andthe total particle density is ρ total ( x j , t ) = D jj ( t ) . Thefraction of coherent particles is determined by n coh ( t ) = (cid:80) j dxρ coh ( x j , t ) . Accordingly, the fraction of incoherentparticles is n incoh ( t ) = 1 − n coh ( t ) . The crucial observa-tion now is that n incoh ( t ) = 1 N s (cid:88) s,r d (2) s,r ( t ) − L N dx = ¯ d (2) ( t ) , (4)which we obtain using Eq. 2, taking into account thatthe norm of the wave functions within the ensembleis /N s (cid:80) s (cid:80) j dx | ψ s ( x j ) | = 1 + L/ N dx with L thelength of the system. (Note that the term L/dx countsthe number of single-particle modes to which vacuumfluctuations have been added.) For a detailed derivation,see the SM. The right-hand side of Eq. 4 is (apart froma constant term) the arithmetic mean over the distancefunction between all pairs of mean-field trajectories, and e λt . . . . . . . . .
91 0 2 4 6 8 10 12 τ E ∝ e λt n incoh (a) d ( ) t [ t ] n totallow-env n incoh (b) n incoh d ( ) (c) t [ t ] n totallow-env n incoh (d) τ E FIG. 2. Left column (a) and (b) for the periodic potentialwith V P = 0 . e and k P = π/ ξ : (a) Fraction of incoherentparticles n incoh for different particle numbers N ( N being . times the number near each curve), and d (2) for the linearlydistorted initial conditions. Red dashed lines correspond toan exponential increase with λ = 1 . t − . (b) Linear plot of(a) including n totallow-env extracted from the total density only,see Fig. 3. The Ehrenfest time τ E = 1 /λ ln N is marked foreach curve. Right column (c) and (d) same as left columnbut for one realization of speckle disorder with V D = 0 . e andcorrelation length σ = 0 . ξ . In (c) the Lyapunov exponentis λ = 1 . t − . Data is shown at integer values of t (exceptfor d (2) ), the lines serve as guides for the eye. we denote it with ¯ d (2) ( t ) .We have explicitly verified the equality of Eq. 4 numeri-cally by independently calculating the arithmetic meanof the distance function, ¯ d (2) ( t ) , and comparing it to n incoh ( t ) , see Fig. 1. The observed rate of exponentialgrowth does not depend on the specific choice of the twoclose initial conditions such that we can clearly associateit with a Lyapunov exponent λ . The equality between ¯ d (2) ( t ) and n incoh ( t ) proves that, if the mean-field limit ischaotic, the fraction of incoherent particles will grow ex-ponentially with a rate given exactly by the mean-field λ .The exponentially fast depletion is thus chaos-induced,or seen from another perspective, measures mean-fieldchaos. Importantly, the exponential increase happens onshorter time scales than τ E , i.e. before effects neglectedwithin the TWA start to play a role.While Fig. 1 depicts the exponential increase for N = 1 . × particles, we see the same exponentialincrease, i.e. the same λ , also for smaller particlenumbers, see Fig. 2. We have varied N while keepingthe nonlinearity g ∝ a s N constant, which amounts toincreasing the scattering length a s by the same factor N is decreased, which preserves the classical phasespace. Indeed, with decreasing N and increasing a s , theBEC naturally shows larger initial depletion, but upon ρ ~ total ρ ~ coh ρ ~ incoh k [1/x ] ρ ~ total ρ ~ coh ρ ~ incoh FIG. 3. Total density ˜ ρ total ( k, t ) (solid), as well as coherent ˜ ρ coh ( k, t ) (filled), and incoherent part ˜ ρ incoh ( k, t ) (dashed) for N = 1 . × at (a) t = 5 t and (b) t = 9 t . The or-ange dots mark the lower envelope of the strong fluctuationsof ˜ ρ total ( k, t ) , and can be used as an accurate estimate for ˜ ρ incoh ( k, t ) . Same potential as in Fig. 1. expansion in the periodic potential, the same λ emerges.We now turn to the question of how the present chaos-induced depletion could be observed in an experiment.We analyze the total particle density in momentum space ˜ ρ total ( k, t ) which is accessible in experiments throughtime-of-flight measurements, see e.g. [34, 35]. Duringthe expansion of the BEC, matter waves start to scatterat the potential landscape preserving initially theirphase coherence. This scattering creates fluctuations inmomentum space with increasingly higher frequenciesas waves originating from points increasingly fartherapart in real space coherently interfere. Ultimately, thedensity exhibits strong fluctuations reaching down toalmost zero density, provided that inelastic scatteringhas been negligible up until this point in time, Fig. 3(a). During inelastic scattering particles lose energy,phase information, and with it, the ability to createinterference fringes. These particles constitute theincoherent part of the density which piles up in formof an almost non-fluctuating background. Using asimple algorithm that determines the lower envelopeof the fluctuations in the total density, we obtain afunctional form very close to ˜ ρ incoh ( k, t ) , see Fig. 3 (b).Interpolating between the points of the lower envelopeand integrating, we obtain n totallow-env , which follows n incoh closely, see Fig. 2 (b). We emphasize that n totallow-env isextracted from the total density only. From Fig. 2 (b)it is obvious that the extraction mechanism will workbest for high particle numbers with a small scatteringlength (e.g., for N = 1 . × two orders of magnitudeof exponential growth can be resolved). For smaller N and correspondingly larger scattering lengths a s theincoherent density starts to pile up before coherentscattering produces sufficiently strong fluctuations inthe coherent part of the density. Therefore, the close L -k L k [1/x ] ρ ~ total ρ ~ coh ρ ~ incoh FIG. 4. Particle density in momentum space at t = 8 t forthe speckle disorder with V D = 0 . e and σ = 0 . ξ averagedover realizations (additional smoothing of the curves hasbeen applied). The vertical lines mark the Landau velocityapproximated by k L = (cid:112) µ ( t ) with µ ( t ) the chemical potentialat time t . association of a non-fluctuating density with ˜ ρ incoh ( k, t ) is broken initially. It becomes, however, more and moreaccurate over time such that, in the experiment, onecould observe the behavior of ˜ ρ incoh ( k, t ) also beyond τ E , where interferences of many-body trajectories notincluded within the TWA become relevant.In order to measure the incoherent fraction of the totaldensity in an experiment it is pivotal to resolve the deepminima of the fluctuations. Half of the distance betweentwo minima is ∆ k (cid:38) . x − . Assuming a linear pixelsize of a CCD camera of µ m the fluctuations couldbe resolved after about ms time of flight. Thepeak amplitude of the fluctuations is ˜ ρ total ( k ) (cid:38) . x leading to ˜ ρ total ( k )∆ k = 1 . × − such that thenumber of particles within each hump is greater than for N = 1 . × and N = 1 . × . Despite thelarge time of flight necessary, we believe that the hereproposed extraction could be realized in state-of-the-artBEC experiments.For the disorder potential, we mostly see the samebehavior as for the periodic potential, see Fig. 2 (c) and(d): n incoh grows exponentially with λ independent ofthe particle number N . Note that we did not performany averages over disorder realizations here. As tothe extraction of the incoherent part of the densityfrom ˜ ρ total ( k, t ) there is one point worth mentioning.Due to the broad spectrum of frequencies the speckledisorder offers, within few time steps, slow particlesstart to be scattered coherently and intertwine withparticles that have lost their coherence through inelastic(i.e. incoherent) scattering near k = 0 . The result isa local maximum in ˜ ρ total ( k, t ) near k = 0 , and localminima near the Landau velocity ± k L due to inelasticscattering out of this momentum, see Fig. 4. Since,however, slow particles scatter from positions in spaceclose to each other, this scattering produces fluctuationswith low frequencies as compared to the fluctuationsobserved for larger k . It is, therefore, impossible toidentify ˜ ρ incoh ( k, t ) near k = 0 based on the fluctuationsof the total density, initially. At later times the localmaximum near k = 0 consists of incoherent particlesonly such that n totallow-env again accurately predicts thevalue of n incoh , see Fig. 2. For N = 1 . × theagreement between n totallow-env and n incoh is accurate onlyafter t (cid:38) t such that we refrained from plotting it.In conclusion, we have shown that a BEC expanding in ashallow periodic or disordered potential is subject to anexponentially growing depletion, and that the depletionis characterized by the “classical" (mean-field) Lyapunovexponent λ . 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