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arXiv:1907.08881

A set of nonlinear coherent states for the pseudoharmonic oscillator

Khalid Ahbli, Hamidou Kassogué, Patrick K. Kayupe, Abdelhak Kouraich

Published Jul 20, 2019• 0 citations
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Abstract

We construct two-parameters family of nonlinear coherent states by replacing the factorial in coefficients $z^n/\sqrt{n!}$ of the canonical coherent states by a specific generalized factorial $x_n^{γ,σ}!$ where parameters $γ$ and $σ$ satisfy some conditions for which the normalization condition and the resolution of identity are verified. The obtained family is a generalization of the Barut-Girardello coherent states and those of the philophase states. In the particular case of parameters $γ$ and $σ$, we attache these states to the pseudo-harmonic oscillator depending on two parameters $α,β> 0$. The obtained nonlinear coherent states are superposition of eigenstates of this oscillator. The associated Bargmann-type transform is defined and we derive some results.

DOI
10.48550/arxiv.1907.08881

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Photon-added BarutndashndashndashGirardello coherent states of the pseudoharmonic oscillator
In a previous paper (Popov D 2001 J. Phys. A: Math. Gen. 34 5283) we have constructed the coherent states (CSs) of the pseudoharmonic oscillator (PHO). These states are of the Barut–Girardello type. In the present paper, we have constructed and investigated some properties of the photon-added Barut–Girardello coherent states (PA-BGCSs) of the PHO. These states are not orthogonal but are normalized and satisfy a unity resolution relation (the completeness relation). We have found the analytical form for the positive weight function in the resolution of unity. By using these states, we have calculated some expectation values (the first two powers of the number operator N ), which have evinced some non-classical properties. In order to examine the statistical properties of a PHO gas, which obeys the quantum canonical distribution, the diagonal P -representation of the density operator ρ ( m ) is constructed and the thermal expectation values are calculated. All these quantities are expressed in terms of Meijer G -functions, and so the PA-BGCSs are a new field of application for these functions. Also, the time dependence of the PA-BGCSs is examined. If in the obtained results we put m = 0, we recover all the results from our previous paper.
In this paper we try to introduce the ladder operators associated with the pseudoharmonic oscillator, after solving the corresponding Schrödinger equation by using the factorization method. The obtained generalized raising and lowering operators naturally lead us to the Dirac representation space of the system which is very easier to work with, in comparison to the functional Hilbert space. The SU(1,1) dynamical symmetry group associated with the considered system is exactly established through investigating the fact that the deduced operators satisfy appropriate commutation relations. This result enables us to construct two important and distinct classes of Barut-Girardello and Gilmore-Perelomov coherent states associated with the system. Finally, their identities as the most important task are exactly resolved and some of their nonclassical properties are illustrated, numerically.
In this paper we try to introduce the ladder operators associated with the pseudoharmonic oscillator, after solving the corresponding Schrödinger equation by using the factorization method. The obtained generalized raising and lowering operators naturally lead us to the Dirac representation space of the system which is much easier to work with, in comparison to the functional Hilbert space. The SU (1, 1) dynamical symmetry group associated with the considered system is exactly established through investigating the fact that the deduced operators satisfy appropriate commutation relations. This result enables us to construct two important and distinct classes of Barut—Girardello and Gilmore—Perelomov coherent states associated with the system. Finally, their identities as the most important task are exactly resolved and some of their nonclassical properties are illustrated, numerically.
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