Nonlinear optical generation of entangled squeezed states in lossy nonorthogonal quasimodes: an analytic solution

We prove that the density operator for the nonlinearly-generated quantum state of light in the $M$ lossy nonorthogonal quasimodes of a nanocavity system has the analytic form of a multimode squeezed thermal state, where the time-dependence of the squeezing and thermal photon parameters are given by a set of $3M$ coupled differential equations. We apply our approach to a system with two highly nonorthogonal quasimodes and obtain good agreement with simulations using a basis of Fock states. Our approach provides an efficient way to model and optimize the generation of mixed Gaussian cluster states.