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International Journal of Geometric Methods in Modern Physics

COHERENT STATE OF α-DEFORMED WEYL–HEISENBERG ALGEBRA

Sh. Dehdashti, Ali Mahdifar, R. Roknizadeh

Published Feb 7, 2013• 11 citations
Publisher

Abstract

At first, we introduce α-deformed algebra as a kind of generalization of the Weyl–Heisenberg algebra so that we get the su(2)- and su(1, 1)-algebras whenever α has specific values. After that, we construct coherent states of this algebra. Third, a realization of this algebra is given in the system of a harmonic oscillator confined at the center of a potential well. Then, we introduce two-boson realization of the α-deformed Weyl–Heisenberg algebra and use this representation to write α-deformed coherent states in terms of the two modes number states. Following these points, we consider mean number of excitations (we call them in general photons) and Mandel parameter as statistical properties of the α-deformed coherent states. Finally, the Fubini–Study metric is calculated for the α-coherent states manifold.

DOI
10.1142/s021988781350014x

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At first, we introduce α-deformed algebra as a kind of generalization of the Weyl–Heisenberg algebra so that we get the su(2)- and su(1, 1)-algebras whenever α has specific values. After that, we construct coherent states of this algebra. Third, a realization of this algebra is given in the system of a harmonic oscillator confined at the center of a potential well. Then, we introduce two-boson realization of the α-deformed Weyl–Heisenberg algebra and use this representation to write α-deformed coherent states in terms of the two modes number states. Following these points, we consider mean number of excitations (we call them in general photons) and Mandel parameter as statistical properties of the α-deformed coherent states. Finally, the Fubini–Study metric is calculated for the α-coherent states manifold.
In this paper, we study the one-parameter [Formula: see text]-generalized oscillator algebra [Formula: see text] (which includes the case of the standard Weyl–Heisenberg algebra). This algebra admits representations of finite-dimensional for negative values of the parameter [Formula: see text]. We construct the associated Perelomov like coherent states. We discuss their statistical properties. In particular, we derive the expressions of the Mandel parameter and Husimi function and we discuss the squeezing of the [Formula: see text] coherent states.
The aim of this paper is to construct coherent states à la Perelomov and à la Barut–Girardello for a polynomial Weyl–Heisenberg algebra. This generalized Weyl–Heisenberg algebra, denoted by , depends on r real parameters and is an extension of the one-parameter algebra (Daoud and Kibler 2010 J. Phys. A: Math. Theor. 43 115303) which covers the cases of the su (1, 1) algebra (for κ > 0), the su (2) algebra (for κ < 0) and the h 4 ordinary Weyl–Heisenberg algebra (for κ = 0). For finite-dimensional representations of and , where is a truncation of order s of in the sense of Pegg–Barnett, a connection is established with k -fermionic algebras (or quon algebras). This connection makes it possible to use generalized Grassmann variables for constructing certain coherent states. Coherent states of the Perelomov type are derived for infinite-dimensional representations of and for finite-dimensional representations of and through a Fock–Bargmann analytical approach based on the use of complex (or bosonic) variables. The same approach is applied for deriving coherent states of the Barut–Girardello type in the case of infinite-dimensional representations of . In contrast, the construction of coherent states à la Barut–Girardello for finite-dimensional representations of and can be achieved solely at the price of replacing complex variables by generalized Grassmann (or k -fermionic) variables. Some of the results are applied to su (2), su (1, 1) and the harmonic oscillator (in a truncated or not truncated form). This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.
Interpolating coherent states for HeisenbergndashndashndashWeyl and single-photon SU(1, 1) algebras
New quantal states which interpolate between the coherent states of the Heisenberg–Weyl ( W 3 ) and SU (1, 1) algebras are introduced. The interpolating states are coherent states of a closed and symmetric algebra containing a parameter k which takes values from zero to one. The elements of the algebra are realized in terms of operator-valued functions of creation and annihilation operators. The realization is such that the operators become those of the W 3 algebra when k = 0 and of the SU (1, 1) algebra when k is unity. The overcompleteness of the interpolating coherent states is established by constructing suitable integration measures. Differential operator representations for the elements of the algebra are given along with the relevant spaces of entire functions. A nonsymmetric set of operators to realize the W 3 algebra is presented and the relevant coherent states are studied. Physical properties such as squeezing and photon statistics of the interpolating coherent states are investigated.
In the present article, we build the excitedcoherent states associated with deformed su(1,1) algebra (DSUA), called photon-added deformed Perelomov coherent states (PA-DPCSs). The constructed coherent states are obtained by using an alterationof the Holstein–Primakoff realization (HPR) for DSUA. A general method to resolve of the problem of the unitary operator was developed for these kinds of quantum states. The Mandel parameter is considered to examine the statistical properties of PA-DPCSs. Furthermore, we offer a physical method to generate the PA-DPCSs in the framework of interaction among fields and atoms. Finally, we introduce the concept of entangled states for PA-DPCSs and examine the entanglement properties for entangled PA-DPCSs.