Density-Operator Theory of Harmonic Oscillator Relaxation

The reduced density operator for a driven electromagnetic field mode relaxing to a bath of oscillators is formally found in closed form. In the limit of weak coupling, the density operator which maximizes the entropy subject to the constraints of a given average energy and average electric and magnetic field remains invariant under interaction with the bath. The quantum characteristic function is found in the weak-coupling case and agrees with previous results. The method may be generalized to many modes relaxing to a bath of oscillators.