  The paper deals with the problem of dynamics of externally driven open
quantum systems. Using the path integral methods we found an analytical
expression for time-dependent density matrix of two externally driven coupled
quantum oscillators interacting with different baths of oscillators. It is
shown that at the zeroing of external forces the density matrix becomes
identical to the previously obtained one for freely developing coupled
oscillators. Mean values of observables are computed by using the Hermitian
part of the matrix. All elements of the covariance matrix composed by
coordinates and momenta of two driven coupled oscillators are calculated. The
time-dependent mean values, dispersions and covariances of coordinates of
coupled oscillators at given external forces are numerically studied. It is
shown that the larger the coupling constant the larger is the disturbances of
the second oscillator due to external action on the first oscillator. Coupled
dynamics of forced oscillators at relatively large coupling constant is
demonstrated at different thermodynamic conditions.
