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Fidelity decay in trapped Bose-Einstein condensates ∗ G. Manfredi and P.-A. Hervieux Institut de Physique et Chimie des Matériaux, CNRS and Université Louis Pasteur, BP 43, F-67034 Strasbourg, France (Dated: February 3, 2008) The quantum coherence of a Bose-Einstein condensate is studied using the concept of quantum fidelity(Loschmidtecho). Thecondensateisconfinedinanelongatedanharmonictrapandsubjected to a small random potential such as that created by a laser speckle. Numerical experiments show that the quantum fidelity stays constant until a critical time, after which it drops abruptly over a singletraposcillationperiod. Thecriticaltimedependslogarithmicallyonthenumberofcondensed atoms and on the perturbation amplitude. This behavior may be observable by measuring the interference fringes of two condensates evolvingin slightly different potentials. 8 0 PACSnumbers: 03.75.Gg, 05.45.Mt 0 2 Introduction.—Ultracoldatom gaseshavebeen at the thesameinitialconditionintwoslightlydifferentHamil- n center of intensive investigations since the first realiza- tonians,H1 =H0+δH1 andH2 =H0+δH2,whereH0is a J tionofanatomicBose-Einsteincondensate(BEC)inthe the unperturbed Hamiltonian, and δH1,2 are small per- mid-1990s. Applications of BECs include the possibil- turbations characterized by the same amplitude, same 9 2 ity of revisiting standard problems of condensed-matter wavelength spectrum, but different phases. The quan- physics[1],bymakinguseofperiodicopticallatticesthat tumfidelity attime tis then definedasthe squareofthe ] mimic the ionic lattice in solid-state systems. More re- scalarproductofthewavefunctionsevolvingwithH1and ph cently,BECshavebeenusedtostudyquantumtransport H2 respectively: F(t) = hψH1(t) ψH2(t)i2. This proce- in disorderedsystems (another long-standingproblemin dure is sometimesreferredto as the ‘Loschmidtecho’,as - t condensed matter physics), by using a laser speckle to it is equivalent to evolving the system forward in time n a createa disorderedpotential. For instance,itwasshown with H1, then backward with H2, and using the fidelity u that the free expansion of a BEC is restrained, or even to check the accuracy of the time-reversal. q completely suppressed, in the presence of a disordered Virtually all theoretical investigations of the [ potential[2, 3](see also[4]foratheoreticalanalysis),an Loschmidtechoconsiderone-particlesystemsevolvingin 1 effect akin to Anderson localization in solids [5]. a given (usually chaotic) Hamiltonian. Several regimes v These problems are often approached by studying the have been described in the past. For perturbations that 0 interference pattern of two or more BECs that are re- are classically weak but quantum-mechanically strong, 9 leased from the optical trap, expand freely, and even- thefidelitydecayisexponential,witharateindependent 4 tually interact with each other [6, 7, 8, 9]. Experiments on the perturbation and given by the classical Lya- 4 showhigh-contrastmatter-waveinterferencefringes,thus punov exponent of the unperturbed system [14]. This . 1 revealing the coherent nature of Bose-Einstein conden- behavior has been confirmed by numerical simulations 0 sates. Surprisingly,recentexperiments haveshownhigh- [15, 16]. For weaker perturbations, the decay rate is still 8 contrast fringes even for well separated BECs, whose exponential, but perturbation-dependent (Fermi golden 0 : phases are totally uncorrelated [10]. rule regime). For still weaker perturbations, the decay v Becausetheinterferencepatterndependsonthephases is Gaussian (perturbative regime) [15]. For integrable i X ofthecondensates,thefringesshouldbesensitivetoper- systems, other types of decay (notably algebraic) have r turbations that strongly affect the phase, but weakly af- been observed [17]. Finally, a perturbation-independent a fect the motion. Therefore, when two condensates are regime, though with Gaussian decay, was also observed subjected to a randompotential (such as that generated in experiments [18]. by a laser speckle [11, 12]) before interfering, we expect Inapreviouswork[19],wehaveappliedtheconceptof a reduction in the contrast of the interference fringes, Loschmidt echo to a system of many electrons interact- which should depend both on the amplitude of the ran- ing through their self-consistent electric field. The nu- dom potential and on the time during which it has been mericalresultsshowedthatthequantumfidelityremains incontactwiththecondensates. ThepurposeofthisLet- equal to unity until a critical time, then drops suddenly ter is to propose a theoretical procedure to quantify this to much lower values. A similar result was also obtained loss of coherence and to suggest a possible experimental for a classicalsystem of colliding hardspheres [20]. This realization. effect is probably related to the nonlinearity introduced In order to estimate the coherence and stability of a by the interactions between particles. Therefore, BECs quantum system [13], one can compare the evolution of should constitute an ideal arena to determine whether suchbehavioristypicalofmany-bodyquantumsystems. Model.—The dynamics of a BEC is accurately described, in the mean-field approximation, by the Gross- ∗Electronicaddress: [email protected] Pitaevskiiequation(GPE).Weconsideredacigar-shaped 2 condensate, where the transverse frequency of the confining potential is much larger then the longitudinal frequency, ω⊥ ≫ ωz. In this case, a one-dimensional (1D) approximation can be used, and the GPE reads as: ∂ψ ~2 ∂2ψ i~∂t =−2m∂z2+V(z)ψ+g1DNA|ψ|2ψ ≡H0ψ. (1) Here, ∞ |ψ|2dz = 1, N is the number of condensed −∞ A atoms,Rg1D = 2a~ω⊥ is the 1D effective coupling constant, and a is the 3D scattering length. The confining potential containsa small quarticcomponent, which can FIG. 1: Fidelity decay for ∆z = 3Lho, NA = 105, and three be realized optically [21], V(z)= 1mω2(z2+Kz4/L2 ), valuesoftheperturbation: ǫ=10−3,ǫ=10−5,ǫ=10−7 (the 2 z ho where Lho = (~/mωz)1/2 is the harmonic oscillator cdraisthiceadl ltinimeseairnecrfietassoesbtwaiinthedduescirnegasEinqg. (p2e)rwtuirtbhaωtizoTn)=. 4T.8h6e. length. Wechoosetheparametersoftheexperimentdescribed in Ref. [2], where N =105 atoms of 87Rb (a=5.7 nm) A unity until a critical time τ , and then drops rapidly to areconfinedinaquasi-1Dtrapwithω /2π =24.7Hzand C z small values (Fig. 1). The critical time is defined as ω⊥/2π =293 Hz. In the simulations, we normalize time the time at which the fidelity has dropped to 60% of its to ωz−1, space to Lho = 2.16 µm, and energies to ~ωz. maximum value, i.e. F(τ ) = 0.6. Interestingly, the The dimensionless 1D coupling constant is then: gˆ1D = fidelity can be nicely fit byCa Fermi-like curve g1D/(Lho~ωz) = 0.063. The quartic coefficient, which will be crucial to excite a sufficiently complex nonlinear −1 t−τ dynamics, is taken to be K = 0.05. With the above f(t)=(1−f∞) 1+exp C +f∞. (2) (cid:20) (cid:18) T (cid:19)(cid:21) parameters, the half-length of the condensate is roughly 24 µm. Equation (2) reveals the presence of two distinct time The random perturbation δH can be realized in prac- scales: (i) the critical time τ and (ii) T ≪ τ , which tice using a laser speckle [11, 12]. In our simula- C C measures the rapidity of the fidelity decay. The param- tions, we model the random potential through the sum of a large number of uncorrelated waves: δH/~ω = eter f∞ simply reflects the fact that the fidelity cannot z decay to zero, because the system is confined in a finite ǫ Nmax cos(2πz/λ + α ), where ǫ is the amplitude j=Nmin j j region in space. In Fig. 1, ωzT = 4.86 is the same for ofPthe perturbation, λj’s are the wavelengths, and αj’s all three cases and is equal to the oscillation period of are random phases. The smallest wavelength present in a particle trapped in the anharmonic potential V(z), for the random potential is λmin = 2Lho = 4.32 µm, which an initial condition z(0) = ∆z, z˙(0) = 0. This means is consistent with the experimental correlation length, that the fidelity decay occurs over one single oscillation σ = 5 µm [2]. The wavelength spectrum of the pertur- z period. bation (i.e. the values of Nmin and Nmax) affects only weakly the behavior of the fidelity: therefore, we will fo- cus our analysis on the dependence of the fidelity on the amplitude ǫ. In order to compute the quantum fidelity, we proceed as follows: (i) first, we prepare the condensate in its ground state without perturbation; (ii) then, we suddenly displace the anharmonic trap by a distance ∆z (a few micrometers); (iii) finally, we solve numerically the time-dependent GPE (1) with the perturbed Hamil- tonian H0 + δH. Step (iii) is performed for N = 11 uncorrelated realizations of the random potential, thus yielding N evolutions of the wavefunction, ψj(t). We FIG. 2: Fidelity decay for NA =105, ǫ=10−5, and different then use all possible combinations to compute the par- valuesofthedisplacement∆z=2Lho,∆z =3Lho,and∆z= tialfidelitiesFij(t)= hψi(t) ψj(t)i2. Thereareofcourse 4Lho. The steeper curves correspond to larger values of ∆z. M = N!/(N − 2)!2! = 55 independent combinations, Thedashedlines arefitsobtained usingEq. (2),with ωzT = whicharefinallyaveragedtoobtainthequantumfidelity, 4.27 (∆z =4Lho), ωzT =4.86 (∆z =3Lho), and ωzT =5.50 F(t) = 1 M F (t). This averaging procedure al- (∆z=2Lho). M j=1 ij lowed us toPreduce considerably the level of statistical fluctuations. This behavior was confirmed by investigating the de- Results.— Our numerical results showed an unusual pendence of the quantum fidelity on the initial displace- behavior for the quantum fidelity, which stays equal to ment ∆z (Fig. 2). For each value of ∆z, the parameter 3 T appearing in Eq. (2) is taken to be equal to the corresponding oscillation period in the anharmonic potential V(z). The oscillation period decreases with increasing energy (and thus with increasing ∆z) and indeed the fidelity drop becomes steeper for larger displacements. The critical time also slightly increases with decreasing displacement and goes to infinity for ∆z →0. This pre- sumably happens because the dynamics of the condensate becomes too regular when the confinement is harmonic. FIG. 4: Critical time τC (in units of ωz−1) versus number of atoms NA in the condensate, for ∆z = 3Lho and ǫ = 10−5. The straight line is a guide to theeye. tion δH. Changes in ψ add incrementally to each other, butcannottriggerthenonlinearloopobservedintheGP simulations [22]. This behavior is clearly linked to the phases of the wavefunctions and could be tested experimentally by studying the effect of a random potential on the inter- FIG.3: CriticaltimeτC (inunitsofωz−1)versusperturbation ferencepatternoftwocondensates[6,7,8,9]. Apossible amplitude ǫ, for ∆z = 3Lho and NA = 105. The solid line experiment could be performed as follows (see Fig. 5). represents thecurveτC ∼−t0lnǫ, with ωzt0 =3.3. First, a BEC is created in a single-well trap; then, the trapisdeformedintoadouble-wellpotential[9,23],with Figure 3 shows that τ depends logarithmically on the barrier between the wells sufficiently high that the C the perturbation amplitude, i.e. τC ∼ −t0lnǫ, with two condensates cannot tunnel through it. The conden- ωzt0 ≃ 3.3 (this is the straight line depicted in Fig. 3). sates are left in the double trap for a time long enough This is similar to what was obtained for another self- to reach their ground state. A laser speckle is then used consistent model [19], suggesting that such behavior is to create a small random potential of amplitude ǫ and generic for N-body systems, at least in the mean field correlation length σz. If σz ≪d, where d is the distance approximation. The critical time also depends on the betweenthetwoBECs,eachcondensateissubjectedtoa number of condensed atoms N . By decreasing N , τ differentrandompotentialwiththesamestatisticalprop- A A C becomes considerably longer (Fig. 4), and for N → 0 erties. A (i.e. for the linear Schrödinger equation) we have that In order to excite the dynamics, the total double-well τC →∞. Forsufficientlylargecondensates(NA ≥2×104 trap is suddenly shifted by a distance ∆z of the order of forthecaseofFig. 4),thecriticaltimedependslogarith- a few micrometers. The BECs evolve in their perturbed mically on the number of atoms. trap for a certain time t, after which both the trap and Finally, the collapse of the quantum fidelity the random potential are switched off, so that the con- is clearly linked to the phases of the wavefunc- densates can overlap and interfere. We predict that the tions. Indeed, by defining an ‘amplitude fidelity’ contrast of the interference fringes will depend on the 2 F (t) = |ψ ψ |dx (which neglects information time t and onthe perturbation ǫ, in a manner analogous a H1 H2 onthe pha(cid:0)sRes),wehavev(cid:1)erifiedthatFa(t)showsnosign to the quantumfidelity: if t<τC(ǫ), the contrastshould of a sudden collapse when the ordinary fidelity drops. be large, whereas it should drop significantly for times Discussion.—We have shown that the evolutions of largerthanτC. Performingseveralexperiments withdif- two BECsin slightly different Hamiltonians divergesud- ferent perturbation amplitudes and different numbers of denly after a critical time τ . The interaction between condensed atoms should allow one to reproduce qualita- C the atoms is obviously a vital ingredient, as τ → ∞ tivelythelogarithmicscalingsofFigs. 3and4. Accurate C when the coupling constant vanish, i.e. for the linear time-resolvedmeasurementsmightevenreproducethefi- Schrödinger equation. The crucial point is that, for the delity drop time T. GPE, the unperturbed Hamiltonian H0 depends on the Theseresults,togetherwiththoseobtainedforanelec- wave function. When the perturbation induces a small tron gas at low temperature [19], suggest that many- change in ψ, H0 is itself modified, which in turns affects particle systems display a generic sudden decay of the ψ, and so on. Thanks to such a nonlinear loop, the per- quantum fidelity. 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