'Return to equilibrium' for weakly coupled quantum systems: a simple polymer expansion

TL;DR

This paper introduces an elementary polymer expansion method to analyze the return to equilibrium in weakly coupled quantum systems, providing sharper results than previous complex-analytic approaches.

Contribution

It presents a new polymer expansion approach that simplifies analysis and yields improved results for the return to equilibrium in weakly coupled quantum systems.

Findings

Weakly coupled quantum systems approach a unique invariant state.

The invariant state is perturbatively close to the Markov approximation.

Results are sharper than previous methods employing complex deformations or Mourre theory.

Abstract

Recently, several authors studied small quantum systems weakly coupled to free boson or fermion fields at positive temperature. All the approaches we are aware of employ complex deformations of Liouvillians or Mourre theory (the infinitesimal version of the former). We present an approach based on polymer expansions of statistical mechanics. Despite the fact that our approach is elementary, our results are slightly sharper than those contained in the literature up to now. We show that, whenever the small quantum system is known to admit a Markov approximation (Pauli master equation \emph{aka} Lindblad equation) in the weak coupling limit, and the Markov approximation is exponentially mixing, then the weakly coupled system approaches a unique invariant state that is perturbatively close to its Markov approximation.