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We use the coordinate Bethe ansatz to study the Lieb-Liniger model of a one-dimensional gas of bosons on a finite-sized ring interacting via an attractive delta-function potential. We calculate zero-temperature correlation functions for seven particles in the vicinity of the crossover to a localized solitonic state and study the dynamics of a system of four particles quenched to attractive interactions from the ideal-gas ground state. We determine the time evolution of correlation functions, as well as their temporal averages, and discuss the role of bound states in shaping the postquench correlations and relaxation dynamics., Comment: v1: Original submission to SciPost Physics. v2: Improvements to text based on referee comments

The problem of understanding how a coherent, macroscopic Bose-Einstein condensate (BEC) emerges from the cooling of a thermal Bose gas has attracted significant theoretical and experimental interest over several decades. The pioneering achievement of BEC in weakly-interacting dilute atomic gases in 1995 was followed by a number of experimental studies examining the growth of the BEC number, as well as the development of its coherence. More recently there has been interest in connecting such experiments to universal aspects of nonequilibrium phase transitions, in terms of both static and dynamical critical exponents. Here, the spontaneous formation of topological structures such as vortices and solitons in quenched cold-atom experiments has enabled the verification of the Kibble-Zurek mechanism predicting the density of topological defects in continuous phase transitions, first proposed in the context of the evolution of the early universe. This chapter reviews progress in the understanding of BEC formation, and discusses open questions and future research directions in the dynamics of phase transitions in quantum gases., Comment: Unedited version of chapter to appear in "Universal Themes of Bose-Einstein Condensation", edited by D. W. Snoke, N. P. Proukakis and P. B. Littlewood (Cambridge University Press). Comments are invited and appreciated

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., Comment: 19 pages, 10 figures. v3: Final version. Typos fixed, and other minor changes

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 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., Comment: 5 pages, 4 figures, plus supplementary material (6 pages, 4 figures). v2: Text shortened, and other minor revisions. To appear in PRL

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-Andre 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 of the 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., Comment: 14 pages, 14 figures, as published

We study the equilibrium correlations of a Bose gas in an elongated three-dimensional harmonic trap using a grand-canonical classical-field method. We focus in particular on the progressive transformation of the gas from the normal phase, through a phase-fluctuating quasicondensate regime to the so-called true-condensate regime, with decreasing temperature. Choosing realistic experimental parameters, we quantify the density fluctuations and phase coherence of the atomic field as functions of the system temperature. We identify the onset of Bose condensation through analysis of both the generalized Binder cumulant appropriate to the inhomogeneous system, and the suppression of the effective many-body T matrix that characterizes interactions between condensate atoms in the finite-temperature field. We find that the system undergoes a second-order transition to condensation near the critical temperature for an ideal Bose gas in the strongly anisotropic three-dimensional geometry, but remains in a strongly phase-fluctuating quasicondensate regime until significantly lower temperatures. We characterize the crossover from a quasicondensate to a true condensate by a qualitative change in the form of the non-local first-order coherence function of the field, and compare our results to those of previous works employing a density-phase Bogoliubov-de Gennes analysis., Comment: 15 pages, 7 figures. v3: Final version. Typos fixed, and other minor changes

We present our views on the issues raised in the chapter by Griffin and Zaremba [A. Griffin and E. Zaremba, in Quantum Gases: Finite Temperature and Non-Equilibrium Dynamics, N. P. Proukakis, S. A. Gardiner, M. J. Davis, and M. H. Szymanska, eds., Imperial College Press, London (in press)]. We review some of the strengths and limitations of the Bose symmetry-breaking assumption, and explain how such an approach precludes the description of many important phenomena in degenerate Bose gases. We discuss the theoretical justification for the classical-field (c-field) methods, their relation to other non-perturbative methods for similar systems, and their utility in the description of beyond-mean-field physics. Although it is true that present implementations of c-field methods cannot accurately describe certain collective oscillations of the partially condensed Bose gas, there is no fundamental reason why these methods cannot be extended to treat such scenarios. By contrast, many regimes of non-equilibrium dynamics that can be described with c-field methods are beyond the reach of generalised mean-field kinetic approaches based on symmetry-breaking, such as the ZNG formalism., Comment: 8 pages. Unedited version of chapter to appear in Quantum Gases: Finite Temperature and Non-Equilibrium Dynamics (Vol. 1 Cold Atoms Series). N.P. Proukakis, S.A. Gardiner, M.J. Davis and M.H. Szymanska, eds. Imperial College Press, London (in press). See http://www.icpress.co.uk/physics/p817.html v2: Added arXiv cross-references

We review c-field methods for simulating the non-equilibrium dynamics of degenerate Bose gases beyond the mean-field Gross-Pitaevskii approximation. We describe three separate approaches that utilise similar numerical methods, but have distinct regimes of validity. Systems at finite temperature can be treated with either the closed-system projected Gross-Pitaevskii equation (PGPE), or the open-system stochastic projected Gross-Pitaevskii equation (SPGPE). These are both applicable in quantum degenerate regimes in which thermal fluctuations are significant. At low or zero temperature, the truncated Wigner projected Gross-Pitaevskii equation (TWPGPE) allows for the simulation of systems in which spontaneous collision processes seeded by quantum fluctuations are important. We describe the regimes of validity of each of these methods, and discuss their relationships to one another, and to other simulation techniques for the dynamics of Bose gases. The utility of the SPGPE formalism in modelling non-equilibrium Bose gases is illustrated by its application to the dynamics of spontaneous vortex formation in the growth of a Bose-Einstein condensate., Comment: 7 pages, 1 figure. Unedited version of chapter to appear in Quantum Gases: Finite Temperature and Non-Equilibrium Dynamics (Vol. 1 Cold Atoms Series). N.P. Proukakis, S.A. Gardiner, M.J. Davis and M.H. Szymanska, eds. Imperial College Press, London (in press). See http://www.icpress.co.uk/physics/p817.html v2: Added arXiv cross-references

In this thesis, we perform investigations into the behaviour of finite-temperature degenerate Bose gases using a classical-field formalism, focussing in particular on the dynamics of quantum vortices in these systems. We demonstrate that the coherence of the classical field can be characterised by its temporal correlations, and discuss how the phase-symmetry-broken averages familiar from mean-field theories emerge from the field trajectories. We show that a finite-temperature condensate containing a precessing vortex in a cylindrically symmetric trap can be realised as an ergodic equilibrium of the classical-field theory, and demonstrate the identification of the rotationally symmetry-broken condensate orbital and core-filling thermal component from the field correlations. We then consider the nonequilibrium dynamics that result when such a precessing-vortex configuration is subjected to a static trap anisotropy which arrests its rotation, and observe novel coupled relaxation dynamics of the condensed and noncondensed components of the field. Finally, we consider the nucleation of vortices in an initially zero-temperature quasi-two-dimensional condensate stirred by a rotating trap anisotropy. We quantify the emergence of a rotating thermal component of the field, which drives the nucleation of vortices from condensate-surface oscillations, and study the relaxation and rotational equilibration of the initially turbulent collection of vortices. We find that thermal fluctuations of the field prevent the vortices from settling into a rigid crystalline lattice in this reduced dimensionality, and that true condensation in the field is completely destroyed by the disordered motion of vortices. We show, however, that the temporal correlations of the field distinguish the quasi-coherent vortex-liquid phase in the trap centre from the truly thermal material in its periphery., Comment: PhD thesis. 276 pages, 49 figures