The equilibrium states of open quantum systems in the strong coupling regime

TL;DR

This paper studies the late-time equilibrium states of open quantum systems strongly coupled to a thermal reservoir, showing they evolve towards the reduced state of the closed system's thermal state, even in non-Markovian regimes.

Contribution

It provides a rigorous analysis of equilibrium states in strong coupling regimes, including exact solutions for harmonic systems and general proofs for broader cases.

Findings

Open systems evolve towards the reduced closed system thermal state.

Multi-time correlations in non-Markovian regimes match those of the thermal state.

Explicit construction of the thermal state at zero temperature.

Abstract

In this work we investigate the late-time stationary states of open quantum systems coupled to a thermal reservoir in the strong coupling regime. In general such systems do not necessarily relax to a Boltzmann distribution if the coupling to the thermal reservoir is non-vanishing or equivalently if the relaxation timescales are finite. Using a variety of non-equilibrium formalisms valid for non-Markovian processes, we show that starting from a product state of the closed system = system + environment, with the environment in its thermal state, the open system which results from coarse graining the environment will evolve towards an equilibrium state at late-times. This state can be expressed as the reduced state of the closed system thermal state at the temperature of the environment. For a linear (harmonic) system and environment, which is exactly solvable, we are able to show in a rigorous way that all multi-time correlations of the open system evolve towards those of the closed system thermal state. Multi-time correlations are especially relevant in the non-Markovian regime, since they cannot be generated by the dynamics of the single-time correlations. For more general systems, which cannot be exactly solved, we are able to provide a general proof that all single-time correlations of the open system evolve to those of the closed system thermal state, to first order in the relaxation rates. For the special case of a zero-temperature reservoir, we are able to explicitly construct the reduced closed system thermal state in terms of the environmental correlations.