Abstract
We study a system of two coupled oscillators (the A oscillator) each of the oscillators linearly interacts with its own heat bath consisting of a set of independent harmonic oscillators (the B oscillators). The initial state of the A oscillator is taken to be coherent while the B oscillators are in thermal states. We analyze the time-dependent state of the A oscillator which is a two-mode Gaussian state. By making use of Simon's separability criterion we show that this state is separable for all times. We consider the equilibrium state of the A oscillator in detail and calculate its Wigner function.