Statistical properties of a two-photon cavity mode in the presence of degenerate parametric amplifier

A Hamiltonian model that includes two-photon interaction with a two-level atom and a degenerate parametric amplifier is considered. By invoking a canonical transformation an exact solution of the wave function in the Schrödinger picture is obtained. The result presented in this context is employed to discuss the purity, the entropy squeezing, and the variance squeezing, in addition to the normal squeezing. It has been shown that the existence of the second harmonic generation leads to reduction in the squeezing amount for all quadrature variances and we found that as the value of the coupling parameter λ2 increases the squeezing phenomenon gets more apparent. Further we have also considered the Q-function as an example of the quasi-probability distribution