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arXiv:1610.08821

Field-quadrature and photon-number variances for Gaussian states

Moorad Alexanian

Published Oct 27, 2016• 0 citations
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Abstract

We calculate exactly the quantum mechanical, temporal characteristic function $χ(η)$ for a single-mode, degenerate parametric amplifier for a system in the Gaussian state, viz., a displaced-squeezed thermal state. Knowledge of $χ(η)$ 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 $χ(η)$ versus nonclassicality criteria based on the two-time, second-order coherence function $g^{(2)}(τ)$ and show numerically that the nonclassicality criteria based on $χ(η)$ does not determine the classical/nonclassical behavior of $g^{(2)}(τ)$.

DOI
10.48550/arxiv.1610.08821

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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 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 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|.
We investigate a free monochromatic electromagnetic field which is the superposition of a squeezed thermal radiation and a coherent one. The main tool in our analysis is the characteristic function that has a Gaussian form. We establish an analytic formula for an arbitrary correlation function, as well as its strong-squeezing limit. Besides the usual quasiprobability densities, the coherent-state, number-state, coordinate, and momentum representations of the density operator are derived. We point out the nonclassical oscillations of the photon-number distribution and find its generating function. Collaterally, displaced thermal states and squeezed thermal states are revisited as nontrivial particular cases. We examine finally the squeezing properties of the field using the distribution functions of the quadratures.
Practically applicable criteria for the nonclassicality of quantum states are formulated in terms of different types of moments. For this purpose the moments of the creation and annihilation operators, of two quadratures, and of a quadrature and the photon number operator turn out to be useful. It is shown that all the required moments can be determined by homodyne correlation measurements. An example of a nonclassical effect that is easily characterized by our methods is amplitude-squared squeezing.