Analytic solution to the nonlinear generation of squeezed states in a thermal bath

We model squeezed state generation in a lossy optical cavity in the presence of a thermal bath using the Lindblad master equation. We show that the exact solution is a squeezed thermal state, where thermal photons arise both from loss and from the thermal bath. We derive an exact, closed-form solution for the evolution of the quadrature uncertainty arising from pulsed degenerate spontaneous parametric down conversion in the cavity. We apply this solution under different pump conditions and show in detail how the thermal environment reduces quadrature squeezing as well as the second order coherence function.