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arXiv:1702.01729 · Physical review. B./Physical review. B

Quenches from bosonic Gaussian initial states to the Tonks-Girardeau limit: Stationary states and effects of a confining potential

Alvise Bastianello, Mario Collura, Spyros Sotiriadis

Published May 23, 2017• 19 citations
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Abstract

We consider the general problem of quenching an interacting Bose gas from the noninteracting regime to the strongly repulsive limit described by the Tonks-Girardeau gas with the initial state being a Gaussian ensemble in terms of the bosons. A generic multipoint correlation function in the steady state can be described fully in terms of a Fredholm-like determinant suitable both for a numerical and for an analytical study in certain limiting cases. Finally, we extend the study to the presence of a smooth confining potential showing that, in the thermodynamic limit, the time evolution of the two-point function can be mapped to a classical problem.

DOI
10.1103/physrevb.95.174303

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We consider the general problem of quenching an interacting Bose gas from the noninteracting regime to the strongly repulsive limit described by the Tonks-Girardeau gas with the initial state being a Gaussian ensemble in terms of the bosons. A generic multipoint correlation function in the steady state can be described fully in terms of a Fredholm-like determinant suitable both for a numerical and for an analytical study in certain limiting cases. Finally, we extend the study to the presence of a smooth confining potential showing that, in the thermodynamic limit, the time evolution of the two-point function can be mapped to a classical problem.
We consider the non-equilibrium dynamics of a gas of impenetrable bosons released from a harmonic trapping potential to a circle. The many-body dynamics is solved analytically and the time dependence of all the physically relevant correlations is described. We prove that, for large times and in the thermodynamic limit, the reduced density matrix of any subsystem converges to a generalized Gibbs ensemble as a consequence of the integrability of the model. We discuss the approach to stationary behavior at late times. We also describe the time dependence of the entanglement entropy which attains a very simple form in the stationary state.
The nonequilibrium dynamics of integrable systems are highly constrained by the conservation of certain charges. There is substantial evidence that after a quantum quench they do not thermalize but their asymptotic steady state can be described by a generalized Gibbs ensemble (GGE) built from the conserved charges. Most of the studies on the GGE so far have focused on models that can be mapped to quadratic systems, while analytic treatment in nonquadratic systems remained elusive. We obtain results on interaction quenches in a nonquadratic continuum system, the one-dimensional (1D) Bose gas described by the integrable Lieb-Liniger model. The direct implementation of the GGE prescription is prohibited by the divergence of the conserved charges, which we conjecture to be endemic to any continuum integrable systems with contact interactions undergoing a sudden quench. We compute local correlators for a noninteracting initial state and arbitrary final interactions as well as two-point functions for quenches to the Tonks-Girardeau regime. We show that in the long time limit integrability leads to significant deviations from the predictions of the grand canonical ensemble, allowing for an experimental verification in cold-atom systems.
Abstract. We consider the non-equilibrium dynamics of a gas of impenetrable bosons released from a harmonic trapping potential to a circle. The many body dynamics is solved analytically and the time dependence of all the physically relevant correlations is described. We prove that, for large times and in the thermodynamic limit, the reduced density matrix of any subsystem converges to a generalized Gibbs ensemble as a consequence of the integrability of the model. We discuss the approach to the stationary behavior at late times. We also describe the time-dependence of the entanglement entropy which attains a very simple form in the stationary state. 1.
Aiming at studying the emergence of Non-Equilibrium Steady States (NESS) in quantum integrable models by means of an exact analytical method, we focus on the Tonks-Girardeau or hard-core boson limit of the Lieb-Liniger model. We consider the abrupt expansion of a gas from one half to the entire confining box, a prototypical case of inhomogeneous quench, also known as "geometric quench". Based on the exact calculation of quench overlaps, we develop an analytical method for the derivation of the NESS by rigorously treating the thermodynamic and large time and distance limit. Our method is based on complex analysis tools for the derivation of the asymptotics of the many-body wavefunction, does not make essential use of the effectively non-interacting character of the hard-core boson gas and is sufficiently robust for generalisation to the genuinely interacting case.
Describing finite-temperature nonequilibrium dynamics of interacting many-particle systems is a notoriously challenging problem in quantum many-body physics. Here we provide an exact solution to this problem for a system of strongly interacting bosons in one dimension in the Tonks-Girardeau regime of infinitely strong repulsive interactions. Using the Fredholm determinant approach and the Bose-Fermi mapping, we show how the problem can be reduced to a single-particle basis, wherein the finite-temperature effects enter the solution via an effective ``dressing'' of the single-particle wave functions by the Fermi-Dirac occupation factors. We demonstrate the utility of our approach and its computational efficiency in two nontrivial out-of-equilibrium scenarios: collective breathing-mode oscillations in a harmonic trap and collisional dynamics in the Newton's cradle setting involving real-time evolution in a periodic Bragg potential.
We investigate the Bose gas with repulsive or attractive interactions between atoms in the scheme of Bethe Ansatz equation in a hard wall trap. Three typical quantum phases in the current research of 1D interacting cold atoms are clarified in terms of the energy spectrum, single particle density distribution and two-particle correlation function. We identify two matching points in the phase diagram, i.e. the TG and STG gas show the same profiles at the strongly interacting point −1/γ=0-1/γ=0−1/γ=0, and in the weakly interacting limit γ=0γ=0γ=0 the ground states TG and BS join to each other smoothly.