A necessary condition for the thermalization of a quantum system coupled to a quantum bath

TL;DR

This paper derives a necessary condition for the initial state independence in quantum thermalization and demonstrates its violation in a specific spin-bath model, showing initial state dependence of the equilibrium.

Contribution

It introduces a necessary condition for initial state independence in quantum thermalization and applies it to show dependence in a particular spin-bath model.

Findings

The necessary condition for initial state independence is derived.

The condition is violated in a specific spin-bath model.

The equilibrium state depends on the initial state in the considered model.

Abstract

A system put in contact with a large heat bath normally thermalizes. This means that the state of the system approaches an equilibrium state, the latter depending only on macroscopic characteristics of the bath (e.g. temperature), but not on the initial state of the system. The above statement is the cornerstone of the equilibrium statistical mechanics; its validity and its domain of applicability are central questions in the studies of the foundations of statistical mechanics. In the present paper we concentrate on one aspect of thermalization, namely, on the system initial state independence (ISI) of the equilibrium state. A necessary condition for the system ISI is derived in the quantum framework. We use the derived condition to prove the absence of the system ISI in a specific model. Namely, we consider a single spin coupled to a large bath, the interaction being of a specific form. Although the model under consideration is nontrivial enough to exhibit the decoherence and the approach to equilibrium, the derived necessary condition is not fulfilled and thus the equilibrium state depends on the initial state of the spin.