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.
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.
It is the aim of this paper to show how to construct á la Perelomov and á la Barut-Girardello coherent states for a polynomial Weyl-Heisenberg algebra. This algebra depends on r parameters. For some special values of the parameter corresponding to r = 1, the algebra covers the cases of the su(1,1) algebra, the su(2) algebra and the ordinary Weyl-Heisenberg or oscillator algebra. For r arbitrary, the generalized Weyl-Heisenberg algebra admits finite or infinite-dimensional representations depending on the values of the parameters. Coherent states of the Perelomov type are derived in finite and infinite dimensions through a Fock-Bargmann approach based on the use of complex variables. The same approach is applied for deriving coherent states of the Barut-Girardello type in infinite dimension. In contrast, the construction of á la Barut-Girardello coherent states in finite dimension can be achieved solely at the price to replace complex variables by generalized Grassmann variables. Finally, some preliminary developments are given for the study of Bargmann functions associated with some of the coherent states obtained in this work.
It is the aim of this paper to show how to construct Perelomov and Barut-Girardello coherent states for a polynomial Weyl-Heisenberg algebra. This algebra depends on r parameters. For some special values of the parameter corresponding to r = 1, the algebra covers the cases of the su(1,1) algebra, the su(2) algebra and the ordinary Weyl-Heisenberg or oscillator algebra. For r arbitrary, the generalized Weyl-Heisenberg algebra admits finite or infinite-dimensional representations depending on the values of the parameters. Coherent states of the Perelomov type are derived in finite and infinite dimensions through a Fock-Bargmann approach based on the use of complex variables. The same approach is applied for deriving coherent states of the Barut-Girardello type in infinite dimension. In contrast, the construction of Barut-Girardello coherent states in finite dimension can be achieved solely at the price to replace complex variables by generalized Grassmann variables. Finally, some preliminary developments are given for the study of Bargmann functions associated with some of the coherent states obtained in this work.
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'.
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.
The most general displaced number 'coherent' states, based on the Heisenberg, su (2) and su (1, 1) Lie algebras symmetries, are constructed. They depend on two parameters, and can be converted into the well-known photon-added, two variable Glauber coherent states and displaced number states respectively, depending on which of the parameters is equal to zero. The relations of the Weyl–Heisenberg algebra guarantee a corresponding resolution of the identity conditions. A discussion of the statistical properties of these states is included. Significant are their squeezing properties, which can be raised by increasing the energy and angular momentum quantum numbers n and m . The maximum squeezing is obtained for B ext = 0. Depending on the particular choice of parameters in the above scenarios, we are able to determine the status of compliance with Poissonian statistics. In the limiting case, we obtain a major result about the non-classical properties of the Glauber minimum uncertainty coherent states. In other words, in addition to the requirement to minimize uncertainty conditions, they carry non-classical features too. Finally, a theoretical framework is proposed to generate them.
A new construction of Perelomov's generalised coherent states (1986) is considered for one-dimensional harmonic oscillators admitting the Heisenberg-Weyl group as invariance Lie group. Exploiting the Niederer maximal kinematical invariance group (1973) for such physical systems, the author deduce further characteristics on the Heisenberg states through the use of the fundamental Perelomov state mod k, k) with k=3/4. The author explicitly gets new normalisation factor and measure for the Heisenberg generalised coherent states. The real Lie algebras so(2, 1) Square Operator h(2), so(2, 1) and h(2) play a prominent role in this study.