Field-quadrature and photon-number variances for Gaussian states

TL;DR

This paper provides an exact calculation of the quantum characteristic function for Gaussian states in a parametric amplifier, analyzing photon and quadrature fluctuations and contrasting nonclassicality criteria based on different quantum functions.

Contribution

It offers a precise derivation of the temporal characteristic function for Gaussian states and compares nonclassicality criteria based on this function versus second-order coherence.

Findings

Exact calculation of the quantum characteristic function for Gaussian states.

Demonstration that nonclassicality criteria based on $ ext{chi}( exteta)$ do not necessarily imply nonclassicality of $g^{(2)}( au)$.

Numerical contrast between different nonclassicality criteria.

Abstract

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 $g^{(2)}(\tau)$.