Dynamically generated quadrature and photon-number variances for Gaussian states

We calculate exactly the quantum mechanical, temporal Wigner quasiprobability density for a single-mode, degenerate parametric amplifier for a system in the Gaussian state, viz., a displaced-squeezed thermal state. The Wigner function allows us to calculate the fluctuations in photon number and the quadrature variance. We contrast the difference between the nonclassicality criteria, which is independent of the displacement parameter $\alpha$, based on the Glauber-Sudarshan quasiprobability distribution $P(\beta )$ and the classical/nonclassical behavior of the Mandel $Q_M(\tau )$ parameter, which depends strongly on $\alpha$. We find a phase transition as a function of $\alpha$ such that at the critical point ${\alpha }_c$, $Q_M(\tau )$, as a function of $\tau$, goes from strictly classical, for $\mathrm{\mid }\alpha \mathrm{\mid }<\mathrm{\mid }{\alpha }_c\mathrm{\mid }$, to a mixed classical/nonclassical behavior, for $\mathrm{\mid }\alpha \mathrm{\mid }>\mathrm{\mid }{\alpha }_c\mathrm{\mid }$.