Non-classicality criteria: Glauber–Sudarshan P function and Mandel parameter

Abstract

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 non-classicality criteria, which is independent of the displacement parameter , based on the Glauber–Sudarshan quasiprobability distribution and the classical/non-classical behaviour of the Mandel parameter, which depends strongly on . We find a phase transition as a function of such that at the critical point , , as a function of , goes from strictly classical, for , to a mixed classical/non-classical behaviour, for .