Stability of Equilibria with a Condensate

TL;DR

This paper studies the stability of equilibrium states in a quantum Bose gas with a condensate interacting with a quantum dot, demonstrating return to equilibrium under weak coupling and analyzing the system's spectral properties.

Contribution

It extends the concept of Return to Equilibrium to systems with multiple equilibrium states and establishes stability results for a condensate coupled to a quantum dot.

Findings

Return to equilibrium holds in the weak coupling limit.

Equilibrium states depend explicitly on local perturbations.

A new Virial Theorem for Liouville operators is developed.

Abstract

We consider a quantum system composed of a spatially infinitely extended free Bose gas with a condensate, interacting with a small system (quantum dot) which can trap finitely many Bosons. Due to spontaneous symmetry breaking in the presence of the condensate, the system has many equilibrium states for each fixed temperature. We extend the notion of Return to Equilibrium to systems possessing a multitude of equilibrium states and show in particular that a condensate coupled to a quantum dot has the property of Return to Equilibrium in a weak coupling sense: any local perturbation of an equilibrium state of the coupled system, evolving under the interacting dynamics, converges in the long time limit to an asymptotic state. The latter is, modulo an error term, an equilibrium state which {\it depends} in an explicit way on the local perturbation (an effect due to long-range correlations). The error term vanishes in the small coupling limit. We deduce the stability result from properties of structure and regularity of eigenvectors of the generator of the dynamics, called the Liouville operator. Among our technical results is a Virial Theorem for Liouville type operators which has new applications to systems with and without a condensate.