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arXiv:1501.01887

Exact temporal second-order coherence function for displaced-squeezed thermal states

Moorad Alexanian

Published Jan 8, 2015• 0 citations
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Abstract

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 $τ\geq 0$ apart. The generation of the Gaussian state by the parametric amplifier insures that the temporal second-order coherence function depends only on $τ$.

DOI
10.48550/arxiv.1501.01887

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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 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 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).
Abstract We study the cross-correlation between two bosonic modes of a two-mode displaced squeezed thermal state. By calculating the equal-time second-order cross-correlation function, we find that strong anticorrelation between the two bosonic modes can be observed. The mechanism of anticorrelation between the two bosonic modes is the interference between two coherent displacements and a squeeze operation. The optimal values of the two displacements and the squeezing parameter for achieving the minimum value of g ab ( 2 ) ( 0 ) is derived. As an example, we explore the anticorrelation between phonons in a three-mode optomechanical system where two mechanical oscillators are coupled to a single cavity mode. The two-mode displaced squeezed thermal state of the two mechanical oscillators can be achieved if the squeezing amplitude and the mean phonon occupation number of the two mechanical environments are not very large. Furthermore, we find that the two-time second-order correlation function has multiple minima if the frequency difference of the two mechanical oscillators is large. Our results provide us with a tool to generate anticorrelated photons and phonons which may have potential applications in quantum information networks.
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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|.