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Journal of Modern Optics

Statistical Properties of a Correlated Squeezed Operator Model

M. Sebawe Abdalla

Published Mar 1, 1993• 7 citations
Publisher

Abstract

The statistical properties of the time evolution operator for the Hamiltonian describing the simultaneous non-degenerate parametric amplifier, and mixing of two modes in the rotator frame is considered. Here we study the effects of this operator on the photon number sum and difference. The Glauber second-order correlation function is calculated, and the squeezing related to the frequency converter and the parametric amplifier has been discussed. The statistical investigations for the quasiprobability distribution functions are carried out.

DOI
10.1080/09500349314550451

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The statistical properties of the time evolution operator for the Hamiltonian describing the simultaneous non-degenerate parametric amplifier, and mixing of two modes in the rotator frame is considered. Here we study the effects of this operator on the photon number sum and difference. The Glauber second-order correlation function is calculated, and the squeezing related to the frequency converter and the parametric amplifier has been discussed. The statistical investigations for the quasiprobability distribution functions are carried out.
In the present work we study the effects of squeezing on coherent states, number states, and on the thermal field states related to the time evolution operator, which is the result of the Hamiltonian describing the simultaneous non-degenerate parametric amplifier with mixing of two modes a and b via a rotation of their polarization. By using the Glauber second-order correlation function we examined the statistical properties of these various squeezed states. The quasi-probabilities of the W Wigner and Q functions are calculated. The Glauber P representation for the squeezed thermal state explicitly shows the limit of its applicability.
In this paper we introduce a new squeeze operator which is a combination of the two single photon squeezing operator and two mode correlated operator. We construct the general coherent-state wave function, and study the effects of this operator on the photon number sum and difference. We find in absence of the squeezing parameters λ1 and λ2 both the sum and the difference are conserved by the two mode correlated operator. The squeezing phenomenon is discussed, and the minimum uncertainty state is found for small value of the squeezing parameter λ. Besides the Glauber second order correlation function, the statistical investigations are carried out for the quasi-probability distribution functions (P-representation, Wigner function and Q-function). The discussion related to the phase space distribution function is given, and the matrix elements of the density operator are calculated. The photocount distribution is also considered.
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 ${g}^{(2)}(\ensuremath{\tau})$.
In the framework of Hamiltonian formalism the quantum statistical properties of an optical field propagating inside a directional coupler operating by nondegenerate parametric amplification are studied. We investigate the effect of switching between the input modes and the outgoing fields from the coupler. Particular attention is paid to two-mode squeezing, the second-order correlation function, quasiprobability distribution functions, and photon-number distribution. Incident number and coherent states are considered. It has been shown that when one of the modes enters the coupler in the Fock state |1 and the other modes are in vacuum states, the coupler can serve as a generator of a coherent state. Furthermore, regimes for the generation and transmission of squeezed and/or sub-Poissonian light are found.