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arXiv:1710.11615 · Physical Review A

Quantum quench in a harmonically trapped one-dimensional Bose gas

Mario Collura, Márton Kormos, Pasquale Calabrese

Published Mar 16, 2018• 33 citations
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Abstract

We study the nonequilibrium dynamics of a one-dimensional Bose gas trapped by a harmonic potential for a quench from zero to infinite interaction. The different thermodynamic limits required for the equilibrium pre- and post-quench Hamiltonians are the origin of a few unexpected phenomena that have no counterparts in the translational-invariant setting. We find that the dynamics is perfectly periodic with breathing time related to the strength of the trapping potential. For very short times, we observe a sudden expansion leading to an extreme dilution of the gas and to the emergence of slowly decaying tails in the density profile. The haste of the expansion induces an undertow-like effect with a pronounced local minimum of the density at the center of the trap. At half period there is a refocusing phenomenon characterized by a sharp central peak of the density, juxtaposed to algebraically decaying tails. We finally show that the time-averaged density is correctly captured by a generalized Gibbs ensemble built with the conserved mode occupations.

DOI
10.1103/physreva.97.033609

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We study the nonequilibrium dynamics of a one-dimensional Bose gas trapped by a harmonic potential for a quench from zero to infinite interaction. The different thermodynamic limits required for the equilibrium pre- and post-quench Hamiltonians are the origin of a few unexpected phenomena that have no counterparts in the translational-invariant setting. We find that the dynamics is perfectly periodic with breathing time related to the strength of the trapping potential. For very short times, we observe a sudden expansion leading to an extreme dilution of the gas and to the emergence of slowly decaying tails in the density profile. The haste of the expansion induces an undertow-like effect with a pronounced local minimum of the density at the center of the trap. At half period there is a refocusing phenomenon characterized by a sharp central peak of the density, juxtaposed to algebraically decaying tails. We finally show that the time-averaged density is correctly captured by a generalized Gibbs ensemble built with the conserved mode occupations.
We develop a finite-temperature hydrodynamic approach for a harmonically trapped one-dimensional quasicondensate and apply it to describe the phenomenon of frequency doubling in the breathing-mode oscillations of the quasicondensate momentum distribution. The doubling here refers to the oscillation frequency relative to the oscillations of the real-space density distribution, invoked by a sudden confinement quench. By constructing a nonequilibrium phase diagram that characterizes the regime of frequency doubling and its gradual disappearance, we find that this crossover is governed by the quench strength and the initial temperature rather than by the equilibrium-state crossover from the quasicondensate to the ideal Bose gas regime. The hydrodynamic predictions are supported by the results of numerical simulations based on a finite-temperature ccc-field approach and extend the utility of the hydrodynamic theory for low-dimensional quantum gases to the description of finite-temperature systems and their dynamics in momentum space.
It is widely believed that the stationary properties after a quantum quench in integrable systems can be described by a generalized Gibbs ensemble (GGE), even if all of the analytical evidence is based on free theories in which the pre- and postquench modes are linearly related. In contrast, we consider the experimentally relevant quench of the one-dimensional Bose gas from zero to infinite interaction, in which the relation between modes is nonlinear, and consequently Wick's theorem does not hold. We provide exact analytical results for the time evolution of the dynamical density-density correlation function at any time after the quench and we prove that its stationary value is described by a GGE in which Wick's theorem is restored.
We apply the framework of non-equilibrium quantum thermodynamics to the physics of quenched small-sized bosonic quantum gases in a one-dimensional harmonic trap. We show that dynamical orthogonality can occur in these few-body systems with strong interactions after a quench and we find its occurrence analytically for an infinitely repulsive pair of atoms. We further show this phenomena is related to the fundamental excitations that dictate the dynamics from the spectral function. We establish a clear qualitative link between the amount of (irreversible) work performed on the system and the establishment of entanglement. We extend our analysis to multipartite systems by examining the case of three trapped atoms. We show the initial (pre-quench) interactions play a vital role in determining the dynamical features, while the qualitative features of the two particle case appear to remain valid. Finally, we propose the use of the atomic density profile as a readily accessible indicator of the non-equilibrium properties of the systems in question.
We consider a quantum quench of the trap frequency in a system of bosons interacting through an inverse-square potential and confined in a harmonic trap (the harmonic Calogero model). We determine exactly the initial state in terms of the postquench eigenstates and derive the time evolution of simple physical observables. Since this model possesses an infinite set of integrals of motion that allow its exact solution, a generalized Gibbs ensemble (GGE), i.e., a statistical ensemble that takes into account the conservation of all integrals of motion, can be proposed in order to describe the values of local physical observables long after the quench. Even though, due to the presence of the trap, physical observables do not exhibit equilibration but periodic evolution, such a GGE may still describe correctly their time-averaged values. We check this analytically for the local boson density and find that the GGE conjecture is indeed valid, in the thermodynamic limit.
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.
We investigate finite-size effects in quantum quenches on the basis of simple energetic arguments. Distinguishing between the low-energy part of the excitation spectrum, below a microscopic energy scale, and the high-energy regime enables one to define a crossover number of particles that is shown to diverge in the small quench limit. Another crossover number is proposed based on the fidelity between the initial and final ground states. Both criteria can be computed using ground-state techniques that work for systems larger than full-spectrum diagonalization. As examples, two models are studied: one with free bosons in an harmonic trap whose frequency is quenched and the one-dimensional Bose-Hubbard model that is known to be nonintegrable and for which recent studies have uncovered remarkable nonequilibrium behaviors. The diagonal weights of the time-averaged density matrix are computed, and observables obtained from this diagonal ensemble are compared with the ones from statistical ensembles. It is argued that the ``thermalized'' regime of the Bose-Hubbard model, previously observed in the small quench regime, experiences strong finite-size effects that make a thorough comparison with statistical ensembles difficult. In addition, we show that the nonthermalized regime, emerging on finite-size systems and for large interaction quenches, is not related to the existence of an equilibrium quantum critical point but to the high-energy structure of the energy spectrum in the atomic limit. Its features are reminiscent of the quench from the noninteracting limit to the atomic limit.