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Physica Scripta

Pair-coherent states of the pseudoharmonic oscillator

Dušan Popov, Vjekoslav D. Sajfert

Published Jul 1, 2009• 4 citations
Publisher

Abstract

In the paper, we have constructed and examined some properties of pair-coherent states of Barut–Girardello kind for two noninteracting subsystems of pseudoharmonic oscillators. The pseudoharmonic oscillator obeys SU(1,1) group symmetry, which was extensively used to study many problems in quantum optics and quantum information theory.
DOI
10.1088/0031-8949/2009/t135/014008

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In this paper we try to introduce the ladder operators associated with the pseudoharmonic oscillator, after solving the corresponding Schro¨dinger 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.
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.
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.
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 the paper we have investigated some properties of the Barut-Giraradello coherent states (BGCS) (the eigenstates of the SU(1,1) lowering generator K-) for the Hamiltonian of the pseudoharmonic oscillator (PHO). By using these states, the diagonal P-representation of the density operator is constructed as a new result for this potential. In addition, we have calculated some thermal expectation values for the quantum canonical gas of the PHOs. The calculations using the BGCS representation seem to be much simpler and easier to program than the corresponding calculations in other representations (e.g. the position representation). At the end of the paper the time dependence of these states is presented.
Abstract We extend recent results on expectation values of coherent oscillator states and SU(2) coherent states to the case of the discrete representations of su(1,1). Systematic semiclassical expansions of products of arbitrary operators are derived. In particular, the leading order of the energy uncertainty of an arbitrary Hamiltonian is found to be given purely in terms of the time dependence of the classical variables. The coherent states considered here include the Perelomov–Gilmore (PG) coherent states. As an important application we discuss the pseudoharmonic oscillator and compare the PG states with the states introduced by Barut and Girardello. The latter ones turn out to be closer to the classical limit as their relative energy variance decays with the inverse square root of energy, while in the former case a constant is approached.