We investigate the relaxation dynamics of the integrable Lieb-Liniger model of contact-interacting bosons in one dimension following a sudden quench of the collisional interaction strength. The system is initially prepared in its noninteracting ground state and the interaction strength is then abruptly switched to a positive value, corresponding to repulsive interactions between the bosons. We calculate equal-time correlation functions of the nonequilibrium Bose field for small systems of up to five particles via symbolic evaluation of coordinate Bethe-ansatz expressions for operator matrix elements between Lieb-Liniger eigenstates. We characterize the relaxation of the system by comparing the time-evolving correlation functions following the quench to the equilibrium correlations predicted by the diagonal ensemble and relate the behavior of these correlations to that of the quantum fidelity between the many-body wave function and the initial state of the system. Our results for the asymptotic scaling of local second-order correlations with increasing interaction strength agree with the predictions of recent generalized thermodynamic Bethe-ansatz calculations. By contrast, third-order correlations obtained within our approach exhibit a markedly different power-law dependence on the interaction strength as the Tonks-Girardeau limit of infinitely strong interactions is approached.

In general, isolated integrable quantum systems have been found to relax to an apparent equilibrium state in which the expectation values of few-body observables are described by the generalized Gibbs ensemble. However, recent work has shown that relaxation to such a generalized statistical ensemble can be precluded by localization in a quasiperiodic lattice system. Here we undertake complementary single-particle and many-body analyses of noninteracting spinless fermions and hard-core bosons within the Aubry-André model to gain insight into this phenomenon. Our investigations span both the localized and delocalized regimes of the quasiperiodic system, as well as the critical point separating the two. Considering first the case of spinless fermions, we study the dynamics ofthe momentum distribution function and characterize the effects of real-space and momentum-space localization on the relevant single-particle wave functions and correlation functions. We show that although some observables do not relax in the delocalized and localized regimes, the observables that do relax in these regimes do so in a manner consistent with a recently proposed Gaussian equilibration scenario, whereas relaxation at the critical point has a more exotic character We also construct various statistical ensembles from the many-body eigenstates of the fermionic and bosonic Hamiltonians and study the effect of localization on their properties. PACS number (s): 03.75.Kk, 05.70.Ln, 02.30.Ik, 05.30.Jp 0(7)= lim ^ f r'^oo X' Jo O{X) ^aa = {Ô)DE, (3)which defines the expectation value of Ô in the DE [5]. The DE involves as many constraints as the dimension ofthe manybody Hilbert space (the overlaps of the initial state with the eigenstates of H), which grows exponentially with system size. By contrast, for models that can be mapped to noninteracting Hamiltonians, the GGE involves a number of constraints that is only polynomially large in the size ofthe system [19]. It may, therefore, appear surprising that the predictions of the GGE

We demonstrate the role of interactions in driving the relaxation of an isolated integrable quantum system following a sudden quench. We consider a family of integrable hard-core lattice anyon models that continuously interpolates between noninteracting spinless fermions and strongly interacting hard-core bosons. A generalized Jordan-Wigner transformation maps the entire family to noninteracting fermions. We find that, aside from the singular free-fermion limit, the entire single-particle density matrix and, therefore, all one-body observables relax to the predictions of the generalized Gibbs ensemble (GGE). This demonstrates that, in the presence of interactions, correlations between particles in the many-body wave function provide the effective dissipation required to drive the relaxation of all one-body observables to the GGE. This relaxation does not depend on translational invariance or the tracing out of any spatial domain of the system. One-dimensional (1D) quantum systems exhibit two features unfamiliar in the three-dimensional world. The first is the breakdown of the strict distinction between bosonic and fermionic particle statistics [1,2], and the second is the prospect of integrability in the presence of interactions [3]. Integrable models have been of particular interest as they can be studied using exact analytic and computational approaches to gain insights into strongly correlated quantum systems [4]. A recent surge of interest in the nonequilibrium dynamics of these systems [4][5][6][7][8] has been motivated by the failure of some quasi-1D systems in cold-atom experiments [9,10] to relax to states consistent with conventional statistical mechanics.A paradigmatic model in this realm is that of lattice hardcore bosons (HCBs), which is integrable by virtue of an exact mapping via the Jordan-Wigner transformation to a system of noninteracting spinless fermions (SFs) [11]. Rigol et al. [12] showed that, following an abrupt change of Hamiltonian parameters (quantum quench), certain single-particle properties of HCBs such as site and momentum occupations relax to stationary distributions that are not consistent with the predictions of conventional statistical ensembles but can be described by a generalized Gibbs ensemble (GGE). The GGE is obtained by maximizing the entropy subject to the constraints that the mean values of the conserved quantitieŝ I l that make the system integrable are fixed to their values in the initial state. This yields the density matrixwhere the Lagrange multipliers λ l are such that Trfρ GGEÎl g ¼ hÎ l i I , with hÁ Á Ái I denoting an expectation value taken in the initial (prequench) state of the system and the partition function Z GGE ¼ Trfexpð− P l λ lÎl Þg. The validity of the GGE for various classes of observables has now been verified for the relaxed states following quenches of HCBs in a number of distinct geometries [12][13][14][15] and in a range of other integrable systems [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. However, the role of interactions ...

We use the coordinate Bethe ansatz to exactly calculate matrix elements between eigenstates of the Lieb-Liniger model of one-dimensional bosons interacting via a two-body delta-potential. We investigate the static correlation functions of the zero-temperature ground state and their dependence on interaction strength, and analyze the effects of system size in the crossover from few-body to mesoscopic regimes for up to seven particles. We also obtain time-dependent nonequilibrium correlation functions for five particles following quenches of the interaction strength from two distinct initial states. One quench is from the noninteracting ground state and the other from a correlated ground state near the strongly interacting Tonks-Girardeau regime. The final interaction strength and conserved energy are chosen to be the same for both quenches. The integrability of the model highly constrains its dynamics, and we demonstrate that the time-averaged correlation functions following quenches from these two distinct initial conditions are both nonthermal and moreover distinct from one another.

The classical-field formalism has been widely applied in the calculation of normal correlation functions, and the characterization of condensation, in finite-temperature Bose gases. Here we discuss the extension of this method to the calculation of more general correlations, including the so-called anomalous correlations of the field, without recourse to symmetry-breaking assumptions. Our method is based on the introduction of U(1)-symmetric classical-field variables analogous to the modified quantum ladder operators of number-conserving approaches to the degenerate Bose gas, and allows us to rigorously quantify the anomalous and non-Gaussian character of the field fluctuations. We compare our results for anomalous correlation functions with the predictions of mean-field theories, and demonstrate that the nonlinear classical-field dynamics incorporate a full description of many-body processes which modify the effective mean-field potentials experienced by condensate and noncondensate atoms. We discuss the role of these processes in shaping the condensate mode, and thereby demonstrate the consistency of the Penrose-Onsager definition of the condensate orbital in the classical-field equilibrium. We consider the contribution of various noncondensate-field correlations to the overall suppression of density fluctuations and interactions in the field, and demonstrate the distinct roles of phase and density fluctuations in the transition of the field to the normal phase.

We present a quantum-mechanical treatment of the mechanical stirring of Bose-Einstein condensates using classical field techniques. In our approach the condensate and excited modes are described using a Hamiltonian classical field method in which the atom number and ͑rotating frame͒ energy are strictly conserved. We simulate a T = 0 quasi-two-dimensional condensate perturbed by a rotating anisotropic trapping potential. Vacuum fluctuations in the initial state provide an irreducible mechanism for breaking the initial symmetries of the condensate and seeding the subsequent dynamical instability. Highly turbulent motion develops and we quantify the emergence of a rotating thermal component that provides the dissipation necessary for the nucleation and motional damping of vortices in the condensate. Vortex lattice formation is not observed, rather the vortices assemble into a spatially disordered vortex liquid state. We discuss methods we have developed to identify the condensate in the presence of an irregular distribution of vortices, determine the thermodynamic parameters of the thermal component, and extract damping rates from the classical field trajectories.

We consider a finite-temperature Bose-Einstein condensate in a quasi-two-dimensional trap containing a single precessing vortex. We find that such a configuration arises naturally as an ergodic equilibrium of the projected Gross-Pitaevskii equation, when constrained to a finite conserved angular momentum. In an isotropic trapping potential, the condensation of the classical field into an off-axis vortex state breaks the rotational symmetry of the system. We present a methodology to identify the condensate and the Goldstone mode associated with the broken rotational symmetry in the classical-field model. We also examine the variation in vortex trajectories and thermodynamic parameters of the field as the energy of the microcanonical field simulation is varied.

The coherence properties of degenerate Bose gases have usually been expressed in terms of spatial correlation functions, neglecting the rich information encoded in their temporal behavior. In this article we show, using a Hamiltonian classical-field formalism, that temporal correlations can be used to characterize familiar properties of a finite-temperature degenerate Bose gas. The temporal coherence of a Bose-Einstein condensate is limited only by the slow diffusion of its phase, and thus the presence of a condensate is indicated by a sharp feature in the temporal power spectrum of the field. We show that the condensate mode can be obtained by averaging the field for a short time in an appropriate phase-rotating frame, and that for a wide range of temperatures, the condensate obtained in this approach agrees well with that defined by the Penrose-Onsager criterion based on one-body (spatial) correlations. For time periods long compared to the phase diffusion time, the field will average to zero, as we would expect from the overall U(1) symmetry of the Hamiltonian. We identify the emergence of the first moment on short time scales with the concept of U(1) symmetry breaking that is central to traditional mean-field theories of Bose condensation. We demonstrate that the short-time averaging procedure constitutes a general analog of the "anomalous" averaging operation of symmetry-broken theories by calculating the anomalous thermal density of the field, which we find to have form and temperature dependence consistent with the results of mean-field theories.