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

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.

23 pages, v2-->v3: Revised version: The explanatory section 1.7 has changed and Section 3.2 has been made more explicit

References in corpus (5)

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• A note on the Landauer principle in quantum statistical mechanics
• Overlapping Resonances in Open Quantum Systems
• Towards a construction of inclusive collision cross-sections in the massless Nelson model
• The preparation of states in quantum mechanics
• Diffusion for a quantum particle coupled to phonons in $d\ge 3$
• A Geometric Approach to the Landauer-Büttiker Formula
• Energy Full Counting Statistics and Return to Equilibrium
• Energy conservation, counting statistics, and return to equilibrium