Quasi-equilibrium relaxation of two identical quantum oscillators with arbitrary coupling strength

TL;DR

This paper derives an analytical model for the time evolution of two coupled quantum oscillators interacting with separate thermal reservoirs, revealing how coupling strength and temperature differences affect their stationary variances.

Contribution

It provides an explicit analytical expression for the density matrix of coupled oscillators out of equilibrium using path integral methods, highlighting the effects of coupling and temperature differences.

Findings

At low coupling, variances reach steady states independent of initial conditions.

Differences in reservoir temperatures lead to deviations from equilibrium variances.

Strong coupling causes divergences in variances, indicating instability.

Abstract

The paper deals with the problem of open systems out of equilibrium. An analytical expression for time-dependent density matrix of two arbitrary coupled identical quantum oscillators interacting with separate reservoirs is derived using path integral methods. The temporal behavior of spatial variances and of covariance from given initial values up to stationary values is investigated. It was shown that at comparatively low coupling strengths the asymptotic variances in the long-time limit achieve steady states despite on initial conditions. Stationary values of variances differ from the case of total equilibrium due to their coupling simultaneously with thermal reservoirs of different temperatures. The larger the difference in temperatures of thermal baths, the larger is the difference of the stationary values of variances of coupled oscillators comparing with values given by the fluctuation dissipation theorem. At strong couplings the variances have divergent character. Otherwise, in the weak coupling limit the asymptotic stationary variances are always in accordance with the fluctuation dissipation theorem despite of the difference in temperatures within the whole system.