Repeated and continuous interactions in open quantum systems

TL;DR

This paper studies a finite quantum system interacting with both a heat reservoir and a chain of quantum systems, showing that the system approaches an asymptotic state satisfying the second law of thermodynamics, with explicit results for two-level systems.

Contribution

It introduces a framework combining spectral deformation and repeated interaction techniques to analyze open quantum systems without relying on master equations.

Findings

Systems reach an asymptotic state over time.

Energy exchange occurs between the reservoir and chain despite no direct coupling.

Explicit asymptotic state derived for two-level systems.

Abstract

We consider a finite quantum system S coupled to two environments of different nature. One is a heat reservoir R (continuous interaction) and the other one is a chain C of independent quantum systems E (repeated interaction). The interactions of S with R and C lead to two simultaneous dynamical processes. We show that for generic such systems, any initial state approaches an asymptotic state in the limit of large times. We express the latter in terms of the resonance data of a reduced propagator of S+R and show that it satisfies a second law of thermodynamics. We analyze a model where both S and E are two-level systems and obtain the asymptotic state explicitly (lowest order in the interaction strength). Even though R and C are not direcly coupled, we show that they exchange energy, and we find the dependence of this exchange in terms of the thermodynamic parameters. We formulate the problem in the framework of W*-dynamical systems and base the analysis on a combination of spectral deformation methods and repeated interaction model techniques. We do not use master equation approximations.