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arXiv:1206.5499

Husimi's Q-function isotonic oscillator in a generalized negative binomial states representation

Zouhaı̈r Mouayn

Published Jun 24, 2012• 0 citations
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Abstract

While considering a class of generalized negative binomial states, we verify that the basic minimum properties for these states to be considered as coherent states are satisfied. We particularize them for the case of the Hamiltonian of the isotonic oscillator and we determine the corresponding Husimi's Q-function. This function may be used to determine a lower bound for the thermodynamical potential of the Hamiltonian by applying a Berezin-Lieb inequality

DOI
10.48550/arxiv.1206.5499

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While considering a class of generalized negative binomial states, we verify that the basic minimum properties for these states to be considered as coherent states are satisfied. We particularize them for the case of the Hamiltonian of the isotonic oscillator and we determine the corresponding Husimi's Q-function. This function may be used to determine a lower bound for the thermodynamical potential of the Hamiltonian by applying a Berezin-Lieb inequality
Displaced negative-binomial mixed states are introduced and their properties are studied. It is shown that they may be used to construct a resolution of the identity operator and, therefore, that they can be used as a ``basis'' in the Hilbert space. Generalized P and Q representations with respect to these states are introduced and their properties are discussed. The pure SU(1,1) coherent states discussed by Perelomov are considered in the two-mode harmonic oscillator Hilbert space and it is shown that the partial trace with respect to one of the modes leads to negative-binomial mixed states. Using this observation, the formalism of thermo-field-dynamics is generalized in a corresponding negative-binomial field dynamics.
The generalized hypergeometric coherent states (GHCSs) have been introduced by Appl and Schiller (1). In the present paper we have extended some considerations about GHCSs to the mixed (thermal) states and applied, particularly, to the case of pseudoharmonic oscillator (PHO). The Husimi's Q distribution function and the diagonal P - distribution function, in the GHCSs representation, have been deduced for these mixed states. The obtained distribution functions were used to calculate thermal averages and to examine some nonclassical properties of the generalized hypergeometric thermal coherent states (GHTCSs), particularly for the PHO. We have also deflned and calculated the thermal analogue of the Mandel parameter and the thermal analogue of the second-order correlation function. By particularizing the parameters p and q of the hypergeometric functions, we recover the usual Barut-Girardello coherent states and their main properties for the PHO from our previous paper (9). All calculations are performed in terms of the Meijer's G-functions (2), which are related to the hypergeometric functions. This manner provides an elegance and uniformity of the obtained results and so the GHCSs become a new fleld of application for these functions. Moreover, this mathematical approach can be used also for other kind of coherent states (e.g. Klauder-Perelomov, Gazeau-Klauder or nonlinear coherent states (10), (12)).
In this paper we examine some of the statistical properties of the negative binomial states (NBSs) that are a superposition of the number of states with appropriately chosen coefficients, but on the basis of Fock-vectors, which correspond to the pseudoharmonic oscillator. These states have the coherent states’ behaviour not only for the harmonic limit. We examine the expectation values of the integer powers of the number operator N , which is useful when calculating Mandel's parameter for these states. Depending on the value of this parameter, we can determine the statistical behaviour of the NBSs, where these states are: sub-Poissonian, Poissonian and supra-Poissonian. Meijer's G -functions formalism was used in the calculation.
We used the numerical method to study nonclassical properties of the even and odd generalized coherent states and superposition states of generalized coherent states for an isotonic oscillator. The following was shown. (1) The quantum statistical properties of the even and odd generalized coherent states are very different from those of the usual even and odd coherent states, and the N th-order ( N = 2 m + 1, m = 0,1,2,...) squeezing and sub-Poisson distribution appear alternately for both the even and odd generalized coherent states in some ranges of z = |β| 2 . The weaker the isotonic oscillator potential, the narrower the ranges. (2) The superposition states of generalized coherent states may exhibit the N th-order squeezing effect too, and this kind of higher-order squeezing effect appears periodically.