Equilibration of Quantum Gases

TL;DR

This paper investigates equilibration times for quantum gases of bosons and fermions with weak interactions, revealing that equilibration occurs rapidly with polynomial or linear timescales depending on the measurement type and system dimensionality.

Contribution

It provides the first analytical results showing rapid equilibration timescales for quantum gases under weak interactions, improving upon previous exponential bounds.

Findings

Equilibration times are generally polynomial in particle number N for coarse-grained observables.

Local measurements on lattice systems equilibrate in linear time relative to lattice size.

Fermions initially confined in a box equilibrate in time O(1/N).

Abstract

Finding equilibration times is a major unsolved problem in physics with few analytical results. Here we look at equilibration times for quantum gases of bosons and fermions in the regime of negligibly weak interactions, a setting which not only includes paradigmatic systems such as gases confined to boxes, but also Luttinger liquids and the free superfluid Hubbard model. To do this, we focus on two classes of measurements: (i) coarse-grained observables, such as the number of particles in a region of space, and (ii) few-mode measurements, such as phase correlators and correlation functions. We show that, in this setting, equilibration occurs quite generally despite the fact that the particles are not interacting. Furthermore, for coarse-grained measurements the timescale is generally at most polynomial in the number of particles N, which is much faster than previous general upper bounds, which were exponential in N. For local measurements on lattice systems, the timescale is typically linear in the number of lattice sites. In fact, for one dimensional lattices, the scaling is generally linear in the length of the lattice, which is optimal. Additionally, we look at a few specific examples, one of which consists of N fermions initially confined on one side of a partition in a box. The partition is removed and the fermions equilibrate extremely quickly in time O(1/N).