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arXiv:2007.04861 · Physical Review Letters

Thermalization of a Trapped One-Dimensional Bose Gas via Diffusion

Alvise Bastianello, Andrea De Luca, Benjamin Doyon, Jacopo De Nardis

Published Dec 10, 2020• 81 citations
arXivHTMLPDF

Abstract

For a decade the fate of a one-dimensional gas of interacting bosons in an external trapping potential remained mysterious. We here show that whenever the underlying integrability of the gas is broken by the presence of the external potential, the inevitable diffusive rearrangements between the quasiparticles, quantified by the diffusion constants of the gas, eventually lead the system to thermalize at late times. We show that the full thermalizing dynamics can be described by the generalized hydrodynamics with diffusion and force terms, and we compare these predictions to numerical simulations. Finally, we provide an explanation for the slow thermalization rates observed in numerical and experimental settings: the hydrodynamics of integrable models is characterized by a continuity of modes, which can have arbitrarily small diffusion coefficients. As a consequence, the approach to thermalization can display prethermal plateau and relaxation dynamics with long polynomial finite-time corrections.

DOI
10.1103/physrevlett.125.240604

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For a decade the fate of a one-dimensional gas of interacting bosons in an external trapping potential remained mysterious. We here show that whenever the underlying integrability of the gas is broken by the presence of the external potential, the inevitable diffusive rearrangements between the quasiparticles, quantified by the diffusion constants of the gas, eventually lead the system to thermalize at late times. We show that the full thermalizing dynamics can be described by the generalized hydrodynamics with diffusion and force terms, and we compare these predictions to numerical simulations. Finally, we provide an explanation for the slow thermalization rates observed in numerical and experimental settings: the hydrodynamics of integrable models is characterized by a continuity of modes, which can have arbitrarily small diffusion coefficients. As a consequence, the approach to thermalization can display prethermal plateau and relaxation dynamics with long polynomial finite-time corrections.
In the hydrodynamics of integrable models, diffusion is a subleading correction to ballistic propagation. Here we quantify the diffusive contribution for one-dimensional Bose gases and find it most influential in the crossover between the main thermodynamic regimes of the gas. Analysing the experimentally measured dynamics of a single density mode, we find diffusion to be relevant only for high wavelength excitations. Instead, the observed relaxation is solely caused by a ballistically driven dephasing process, whose time scale is related to the phonon lifetime of the system and is thus useful to evaluate the applicability of the phonon bases typically used in quantum field simulators.
The dynamics of a weakly interacting Bose gas at low temperatures is close to integrable due to the approximate quadratic nature of the many-body Hamiltonian. While the short-time physics after an abrupt ramp of the interaction constant is dominated by integrable dynamics, integrability is broken at longer times by higher-order interaction terms in the Bogoliubov Hamiltonian, in particular, by Beliaev-Landau scatterings involving three quasiparticles. The two-stage relaxation process is highlighted in the evolution of local observables such as the density-density correlation function: an integrable dephasing mechanism leads the system to a prethermal stage, followed by true thermalization conveyed by quasiparticle collisions. Our results bring the crossover from prethermalization to thermalization within reach of current experiments with ultracold atomic gases.
We study a one-dimensional gas of hard rods trapped in a harmonic potential, which breaks integrability of the hard-rod interaction in a nonuniform way. We explore the consequences of such broken integrability for the dynamics of a large number of particles and find three distinct regimes: initial, chaotic, and stationary. The initial regime is captured by an evolution equation for the phase-space distribution function. For any finite number of particles, this hydrodynamics breaks down and the dynamics becomes chaotic after a characteristic timescale determined by the interparticle distance and scattering length. The system fails to thermalize over the timescale studied (10^{4} natural units), but the time-averaged ensemble is a stationary state of the hydrodynamic evolution. We close by discussing logical extensions of the results to similar systems of quantum particles.
By exploring a phase-space hydrodynamics description of one-dimensional free Fermi gas, we discuss how systems settle down to steady states described by the generalized Gibbs ensembles through quantum quenches. We investigate time evolutions of the fermions which are trapped in external potentials or a circle for a variety of initial conditions and quench protocols. We analytically compute local observables such as particle density and show that they always exhibit power-law relaxation at late times. We find a simple rule which determines the power law exponent. Our findings are, in principle, observable in experiments in a one-dimensional free Fermi gas or Tonk's gas (Bose gas with infinite repulsion).
The relationship between many-body interactions and dimensionality is integral to numerous emergent quantum phenomena.A striking example is the Bose gas, which upon confinement to one dimension (1D) obeys an infinite set of conservation laws, prohibiting thermalization and constraining dynamics.In our experiment, we demonstrate that such a 1D behavior can extend much further into the dimensional crossover toward 3D than expected.Starting from a weakly interacting Bose gas trapped in a highly elongated potential, we perform a quench to instigate the dynamics of a single density mode.Employing the theory of generalized hydrodynamics, we identify the dominant relaxation mechanism as the 1D dephasing of the relevant collective excitations of the system, the rapidities.Surprisingly, the dephasing remains dominant even for temperatures far exceeding conventional limits of one dimensionality where thermalization should occur.We attribute our observations to an emergent Pauli blocking of transverse excitations caused by the rapidities assuming fermionic statistics, despite the gas being purely bosonic.Thus, our study suggests that 1D physics is less fragile than previously thought, as it can persist even in the presence of significant perturbations.More broadly, by employing the exact Bethe ansatz solutions of the many-body system, we facilitate an interpretation of how the emergent macroscopic behavior arises from the microscopic interactions.
In this work we report preliminary results on the relaxational dynamics of one dimensional Bose gases, as described by the Lieb-Liniger model, upon release from a parabolic trap. We explore the effects of integrability and integrability breaking upon these dynamics by placing the gas post-release in an integrability breaking one-body cosine potential of variable amplitude. By studying the post-quench evolution of the conserved charges that would exist in the purely integrable limit, we begin to quantify the effects of the weak breaking of integrability on the long time thermalization of the gas.