Temporal second-order coherence function for displaced-squeezed thermal states

Abstract

We calculate the quantum mechanical, temporal second-order coherence function for a singlemode, degenerate parametric amplifier for a system in the Gaussian state, viz., a displaced-squeezed thermal state. The calculation involves first the dynamical generation at time t of the Gaussian state from an initial thermal state and subsequent measurements of two photons a time τ ≥ 0 apart. The generation of the Gaussian state by the parametric amplifier ensures that the temporal second-order coherence function depends only on τ , via τ /t, for given Gaussian state parameters, Gaussian state preparation time t, and average numbern of thermal photons. It is interesting that the time evolution for displaced thermal states shows a power decay in τ /t rather than an exponential one as is the case for general, displaced-squeezed thermal states.

Dynamically generated quadrature and photon-number variances for Gaussian states

“…The radiation field is initially in a thermal state ρ0 and a after a preparation time t, the radiation field develops in time into a Gaussian state and so [2] ρG = exp (−i Ĥt/h)ρ 0 exp (i Ĥt/h)…”

“…In a recent work [2], a detailed study was made of the temporal development of the second-order coherence function g (2) (τ ) for Gaussian states-displaced-squeezed thermal states-the dynamics of which is governed by a Hamiltonian for degenerate parametric amplification. The time development of the Gaussian state is generated by an initial thermal state and the system subsequently evolves in time where the usual assumption of statistically stationary fields is not made.…”

“…Nonclassicality were observed [2] for various values of the parameters governing the temporal development of the coherence function g (2) (τ )-such as the coherent parameter α, squeeze parameter ξ, and the mean photon number n of the initial thermal state. Our characterization of nonclassicality was based solely on the coherence function violating inequalities satisfied by the classical correlation functions.…”

“…More recently [3], we dwelt into the notion of nonclassicality based on the characteristic function χ(η) and its two-dimensional Fourier transform to determine the existence or nonexistence of the quasiprobability distribution P (β) of the Glauber-Sudarshan coherent or P representation of the density of state. It was shown [3] that the nonclassicality criteria for the radiation field, based on the one-time function P (β), cannot characterize the * alexanian@uncw.edu classical, quantum mechanical or mixed nature of the dynamical system as attested by the temporal behavior of the two-time g (2) (τ ) function.…”

“…We consider in Sec. II the general Hamiltonian of the degenerate parametric amplifier and present the result for the quantum degree of second-order coherence g (2) (τ ) of Ref. [2].…”