Equilibration and thermalization of the dissipative quantum harmonic oscillator in a non-thermal environment

TL;DR

This paper investigates how a quantum harmonic oscillator reaches equilibrium or thermalization when coupled to a non-thermal environment, providing exact solutions and conditions for thermalization.

Contribution

It offers a comprehensive analysis of equilibration and thermalization conditions for a dissipative quantum harmonic oscillator in non-thermal baths, including exact solutions and hierarchy of thermalization criteria.

Findings

Hierarchy of conditions for thermalization

Relation between asymptotic temperature and initial bath energy distribution

Thermalization depends on initial bath state and system-environment interactions

Abstract

We study the dissipative quantum harmonic oscillator with general non-thermal preparations of the harmonic oscillator bath. The focus is on equilibration of the oscillator in the long-time limit and the additional requirements for thermalization. Our study is based on the exact solution of the microscopic model obtained by means of operator equations of motion, which provides us with the time evolution of the central oscillator density matrix in terms of the propagating function. We find a hierarchy of conditions for thermalization, together with the relation of the asymptotic temperature to the energy distribution in the initial bath state. We discuss the presence and absence of equilibration for the example of an inhomogeneous chain of harmonic oscillators, and illustrate the general findings about thermalization for the non-thermal environment that results from a quench.