  The paper addresses the problem of relaxation of open quantum systems. Using
the path integral methods we found an analytical expression for 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 covariance 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 on
their initial values. In the weak coupling approach there is good reason to
believe that subsystems asymptotically in equilibrium at their own temperatures
even despite of the arbitrary difference in temperatures within the whole
system.
