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arXiv:1512.00840 · Journal of Modern Optics

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

Moorad Alexanian

Published Nov 13, 2015• 7 citations
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Abstract

We calculate the quantum mechanical, temporal second-order coherence function for a single-mode, degenerate parametric amplifier for a system in the Gaussian state, viz. a displaced–squeezed thermal state. The calculation involves first 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 insures that the temporal second-order coherence function depends only on τ, via τ/t, for the given Gaussian state parameters, Gaussian state preparation time t, and average number n¯ 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.

DOI
10.1080/09500340.2015.1112441

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We calculate the quantum mechanical, temporal second-order coherence function for a single-mode, degenerate parametric amplifier for a system in the Gaussian state, viz. a displaced–squeezed thermal state. The calculation involves first 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 insures that the temporal second-order coherence function depends only on τ, via τ/t, for the given Gaussian state parameters, Gaussian state preparation time t, and average number n¯ 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.
We calculate exactly the quantum mechanical, temporal second-order coherence function for a single-mode, degenerate parametric amplifier for a system in the Gaussian state, viz., a displaced-squeezed thermal state. The calculation involves first the generation of the Gaussian state and subsequent measurements of two photons a time τ≥0τ\geq 0τ≥0 apart. The generation of the Gaussian state by the parametric amplifier insures that the temporal second-order coherence function depends only on τττ.
We use the exact calculation of the quantum mechanical, temporal characteristic function $\ensuremath{\chi}(\ensuremath{\eta})$ and the degree of second-order coherence ${g}^{(2)}(\ensuremath{\tau})$ for a single-mode, degenerate parametric amplifier for a system in the Gaussian state, viz., a displaced-squeezed thermal state, to study the different criteria for nonclassicality. In particular, we contrast criteria that involve only one-time functions of the dynamical system, for instance, the quasiprobability distribution $P(\ensuremath{\beta})$ of the Glauber-Sudarshan coherent or P representation of the density of state and the Mandel ${Q}_{M}(\ensuremath{\tau})$ parameter, versus the criteria associated with the two-time correlation function ${g}^{(2)}(\ensuremath{\tau})$.
We calculate exactly the quantum mechanical, temporal characteristic function χ(η)χ(η)χ(η) for a single-mode, degenerate parametric amplifier for a system in the Gaussian state, viz., a displaced-squeezed thermal state. Knowledge of χ(η)χ(η)χ(η) 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 χ(η)χ(η)χ(η) versus nonclassicality criteria based on the two-time, second-order coherence function g(2)(τ)g^{(2)}(τ)g(2)(τ) and show numerically that the nonclassicality criteria based on χ(η)χ(η)χ(η) does not determine the classical/nonclassical behavior of g(2)(τ)g^{(2)}(τ)g(2)(τ).
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 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|.
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In the coordinate representation of thermofield dynamics, we investigate the thermalized displaced squeezed thermal state which involves two temperatures successively. We give the wavefunction and the matrix element of the density operator at any time, and accordingly calculate some quantities related to the position, momentum and particle number operator, special cases of which are consistent with the results in the literature. The two temperatures have diffenent correlations with the squeeze and coherence components. Moreover, different from the properties of the position and momentum, the average value and variance of the particle number operator as well as the second-order correlation function are time-independent.