  Continuous-variable (CV) entanglement is a valuable resource in the field of
quantum information. One source of CV entanglement is the correlations between
the position and momentum of photons in a two-mode squeezed state of light. In
this paper, we theoretically study the generation of squeezed states, via
spontaneous parametric downconversion (SPDC), inside a two-mode lossy cavity
that is pumped with a classical optical pulse. The dynamics of the density
operator in the cavity is modelled using the Lindblad master equation, and we
show that the exact solution to this model is the density operator for a
two-mode squeezed thermal state, with a time-dependent squeezing amplitude and
average thermal photon number for each mode. We derive an expression for the
maximum entanglement inside the cavity that depends crucially on the difference
in the losses between the two modes. We apply our exact solution to the
important example of a microring resonator that is pumped with a Gaussian
pulse. The expressions that we derive will help researchers optimize CV
entanglement in lossy cavities.
