  We calculate exactly the quantum mechanical, temporal characteristic function
$\chi(\eta)$ for a single-mode, degenerate parametric amplifier for a system in
the Gaussian state, viz., a displaced-squeezed thermal state. Knowledge of
$\chi(\eta)$ 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 $\chi(\eta)$ versus nonclassicality criteria based on the two-time,
second-order coherence function $g^{(2)}(\tau)$ and show numerically that the
nonclassicality criteria based on $\chi(\eta)$ does not determine the
classical/nonclassical behavior of $g^{(2)}(\tau)$.

Quantum Physics