Coupled microcavities with dissipation

We analyze a system of two coupled cavities, where one of them is leaky, namely, is interacting with the environment. We derive the master equation for the system, and show that the relaxation term, in the case of strongly coupled cavities, is not simply the standard one, which is obtained by neglecting the interaction between the two cavities, when calculating the effect of the bath. The relaxation term is calculated here by first diagonalizing the Hamiltonian of the coupled modes of the cavities, and then it is properly evaluated by eliminating the degrees of freedom of the reservoir, maintaining only the energy-conserving terms. Only by following this prescription one can guarantee that the coupled system approaches thermal equilibrium. We study both the steady-state solutions of the master equation, and the time evolution of the system from given initial conditions.