  We show that the light generated via spontaneous four-wave mixing or
parametric down conversion in multiple, coupled, lossy cavities is a multimode
squeezed thermal state. Requiring this state to be the solution of the Lindblad
master equation results in a set of coupled first-order differential equations
for the time-dependent squeezing parameters and thermal photon numbers of the
state. The benefit of this semi-analytic approach is that the number of coupled
equations scales linearly with the number of modes but is independent of the
number of photons generated. With this analytic form of the state, correlation
variances are easily expressed as analytic functions of the time-dependent mode
parameters. Thus, our solution makes it computationally tractable and
relatively straight forward to calculate the generation and evolution of
multimode entangled states in multiple coupled, lossy cavities, even when there
are a large number of modes and/or photons.
