Structure of Quantum Mean Force Gibbs States for Coupled Harmonic Systems

TL;DR

This paper derives the exact structure of quantum mean force Gibbs states for coupled harmonic oscillators interacting with multiple thermal baths, revealing how bath coupling effects decay with distance and exploring the ultrastrong coupling limit.

Contribution

It provides a nonperturbative, exact formulation of the MFG state for coupled quantum harmonic oscillators using path integrals, including covariances and ultrastrong coupling analysis.

Findings

Bath coupling effects decay exponentially with distance from the boundary.

Exact MFG state can be computed nonperturbatively for coupled harmonic systems.

Ultrastrong coupling limit connects to recent quantum system results.

Abstract

An open quantum system interacting with a heat bath at given temperature is expected to reach the mean force Gibbs (MFG) state as a steady state. The MFG state is given by tracing out the bath degrees of freedom from the equilibrium Gibbs state of the total system plus bath. When the interaction between the system and the bath is not negligible, it is different from the usual system Gibbs state obtained from the system Hamiltonian only. Using the path integral method, we present the exact MFG state for a coupled system of quantum harmonic oscillators in contact with multiple thermal baths at the same temperature. We develop a nonperturbative method to calculate the covariances with respect to the MFG state. By comparing them with those obtained from the system Gibbs state, we find that the effect of coupling to the bath decays exponentially as a function of the distance from the system-bath boundary. This is similar to the skin effect found recently for a quantum spin chain interacting with an environment. Using the exact results, we also investigate the ultrastrong coupling limit where the coupling between the system and the bath gets arbitrarily large and make a connection with the recent result found for a general quantum system.