Relaxation of two coupled quantum oscillators to quasi-equilibrium states based on path integrals

This paper addresses the problem of relaxation of open quantum systems. Using the path integral methods, we found an analytical expression for the time-dependent density matrix of two coupled quantum oscillators interacting with different baths of oscillators. The expression for density matrix was found in the linear regime with respect to the coupling constant between selected oscillators. Time dependent spatial variances and covariances were investigated analytically and numerically. It was shown that asymptotic variances in the long-time limit are always in accordance with the fluctuation dissipation theorem despite their initial values. In the weak coupling approach there is good reason to believe that subsystems are asymptotically in equilibrium at their own temperatures despite the arbitrary difference in temperatures within the whole system.