Field-quadrature and photon-number variances for Gaussian states

We calculate exactly the quantum mechanical, temporal characteristic function $χ(η)$ for a single-mode, degenerate parametric amplifier for a system in the Gaussian state, viz., a displaced-squeezed thermal state. Knowledge of $χ(η)$ allows only the determination of the time development of arbitrary functions of equal-time products of creation and annihilation photon operators. We calculate, in particular, the fluctuations in photon number, quadrature operators, and quadrature variance. We contrast the very important difference between the nonclassicality criteria based on the one-time characteristic function $χ(η)$ versus nonclassicality criteria based on the two-time, second-order coherence function $g^{(2)}(τ)$ and show numerically that the nonclassicality criteria based on $χ(η)$ does not determine the classical/nonclassical behavior of $g^{(2)}(τ)$.