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

We calculate exactly the quantummechanical, temporalWigner quasiprobability density for a singlemode, 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 α, based on the Glauber–Sudarshan quasiprobability distribution P(β) and the classical/non-classical behaviour of the Mandel QM(τ ) parameter, which depends strongly on α. We find a phase transition as a function of α such that at the critical point αc , QM(τ ), as a function of τ , goes from strictly classical, for |α| < |αc |, to a mixed classical/non-classical behaviour, for |α| > |αc |. ARTICLE HISTORY Received 25 April 2017 Accepted 25 August 2017