Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

su(1, 1)-Barut–Girardello coherent states for Landau levels

Hossein Fakhri

Published Apr 28, 2004• 27 citations
Publisher

Abstract

It is shown that the Hilbert space corresponding to all the quantum states of the Landau problem can be split in two different ways: as infinite direct sums of the finite- and infinite-dimensional representation subspaces of the Lie algebras su (2) and su (1,1) with finite- and infinite-fold degeneracies, respectively. For each of the Hilbert representation subspaces of the Lie algebra su (1,1), we construct a suitable linear combination of its bases as the Barut–Girardello coherent states.

DOI
10.1088/0305-4470/37/19/007

Related papers

BARUT–GIRARDELLO COHERENT STATES CONSTRUCTED BY Ym m(θ,φ) AND Ym+1 m(θ,φ) FOR A FREE PARTICLE ON THE SPHERE

Alireza Chenaghlou, Hossein Fakhri

Using the realization idea of simultaneous shape invariance with respect to two different parameters of the associated Legendre functions, the Hilbert space of spherical harmonics Yn m(θ,φ) corresponding to the motion of a free particle on a sphere is split into a direct sum of infinite-dimensional Hilbert subspaces. It is shown that these Hilbert subspaces constitute irreducible representations for the Lie algebra u (1,1). Then by applying the lowering operator of the Lie algebra u (1,1), Barut–Girardello coherent states are constructed for the Hilbert subspaces consisting of Ym m(θ,φ) and Ym+1 m(θ,φ).

Modern Physics Letters A•Nov 2, 2004•17 citations

Coherency of su(1,1)-Barut–Girardello type and entanglement for spherical harmonics

Hossein Fakhri, Alireza Dehghani

Barut–Girardello coherent states corresponding to the (l−m)- and (l+m)-integer discrete irreducible representations of su(1,1) Lie algebra are calculated by the spherical harmonics Ylm(θ,ϕ). Their explicit compact forms and also, to realize the resolution of the identity, their corresponding positive definite measures on the complex plane are obtained in terms of the known functions. It is also shown that coherent states of both positive and negative representations separately lead us to construct new representation bases for su(1,1) Lie algebra. Then, it is shown that the su(1,1)-Barut–Girardello coherent states corresponding to two particles containing the spatial parity symmetries of a bipartite quantum system can be entangled in ten different ways.

Journal of Mathematical Physics•May 1, 2009•21 citations

Bosonic andk-fermionic coherent states for a class of polynomial Weyl–Heisenberg algebras

Mohammed Daoud A, M. Kibler

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'.

arXiv:1110.4799•Journal of Physics A Mathematical and Theoretical•May 30, 2012•22 citations

Duality of two types ofSU(1, 1) coherent states and an intermediate type

Alfred W nsche

The two types of SU (1, 1) coherent states of Barut and Girardello and of Perelomov are dual in a sense that the operators in the eigenvalue equation and in the exponentials which create these types of coherent states from the lowest eigenstate (vacuum) form an asymmetric Heisenberg–Weyl algebra. A new type of SU (1, 1) coherent states which takes on an intermediate position between the two dual types of SU (1, 1) coherent states already mentioned is established and investigated. In this new type, the duality of the operators becomes a self-duality and the corresponding operators form a usual Heisenberg–Weyl algebra. Properties of the different SU (1, 1) coherent states are investigated for the realization of SU (1, 1) by one-dimensional quantum-mechanical potential problems leading to a quadratic law of energy-level spacing. Coherent SU (1, 1) phase states are discussed for this realization of SU (1, 1). It is shown that no wavepackets corresponding to any of the SU (1, 1) coherent states can exactly preserve their shape during time evolution.

Journal of Optics B Quantum and Semiclassical Optics•Jun 1, 2003•7 citations

Generalized Grassmann variables for quantum kit (k-level) systems and Barut–Girardello coherent states for su(r + 1) algebras

Mohammed Daoud A, L. Gouba

This paper concerns the construction of su(r + 1) Barut–Girardello coherent states in terms of generalized Grassmann variables. We first introduce a generalized Weyl–Heisenberg algebra A(r) (r≥1) generated by r pairs of creation and annihilation operators. This algebra provides a useful framework to describe qubit and qukit (k-level) systems. It includes the usual Weyl–Heisenberg and su(2) algebras. We investigate the corresponding Fock representation space. The generalized Grassmann variables are introduced as variables spanning the Fock-Bargmann space associated with the algebra A(r). The Barut–Girardello coherent states for su(r + 1) algebras are explicitly derived and their over-completion properties are discussed.

arXiv:1703.00833•Journal of Mathematical Physics•May 1, 2017•2 citations

Generalized coherent states for polynomial Weyl-Heisenberg algebras

M. Kibler, Mohammed Daoud A

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.

arXiv:1201.1811•AIP conference proceedings•Jan 1, 2012•1 citation
It is shown that the Hilbert space corresponding to all the quantum states of the Landau problem can be split in two different ways: as infinite direct sums of the finite- and infinite-dimensional representation subspaces of the Lie algebras su (2) and su (1,1) with finite- and infinite-fold degeneracies, respectively. For each of the Hilbert representation subspaces of the Lie algebra su (1,1), we construct a suitable linear combination of its bases as the Barut–Girardello coherent states.
Using the realization idea of simultaneous shape invariance with respect to two different parameters of the associated Legendre functions, the Hilbert space of spherical harmonics Yn m(θ,φ) corresponding to the motion of a free particle on a sphere is split into a direct sum of infinite-dimensional Hilbert subspaces. It is shown that these Hilbert subspaces constitute irreducible representations for the Lie algebra u (1,1). Then by applying the lowering operator of the Lie algebra u (1,1), Barut–Girardello coherent states are constructed for the Hilbert subspaces consisting of Ym m(θ,φ) and Ym+1 m(θ,φ).
Barut–Girardello coherent states corresponding to the (l−m)- and (l+m)-integer discrete irreducible representations of su(1,1) Lie algebra are calculated by the spherical harmonics Ylm(θ,ϕ). Their explicit compact forms and also, to realize the resolution of the identity, their corresponding positive definite measures on the complex plane are obtained in terms of the known functions. It is also shown that coherent states of both positive and negative representations separately lead us to construct new representation bases for su(1,1) Lie algebra. Then, it is shown that the su(1,1)-Barut–Girardello coherent states corresponding to two particles containing the spatial parity symmetries of a bipartite quantum system can be entangled in ten different ways.
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'.
The two types of SU (1, 1) coherent states of Barut and Girardello and of Perelomov are dual in a sense that the operators in the eigenvalue equation and in the exponentials which create these types of coherent states from the lowest eigenstate (vacuum) form an asymmetric Heisenberg–Weyl algebra. A new type of SU (1, 1) coherent states which takes on an intermediate position between the two dual types of SU (1, 1) coherent states already mentioned is established and investigated. In this new type, the duality of the operators becomes a self-duality and the corresponding operators form a usual Heisenberg–Weyl algebra. Properties of the different SU (1, 1) coherent states are investigated for the realization of SU (1, 1) by one-dimensional quantum-mechanical potential problems leading to a quadratic law of energy-level spacing. Coherent SU (1, 1) phase states are discussed for this realization of SU (1, 1). It is shown that no wavepackets corresponding to any of the SU (1, 1) coherent states can exactly preserve their shape during time evolution.
This paper concerns the construction of su(r + 1) Barut–Girardello coherent states in terms of generalized Grassmann variables. We first introduce a generalized Weyl–Heisenberg algebra A(r) (r≥1) generated by r pairs of creation and annihilation operators. This algebra provides a useful framework to describe qubit and qukit (k-level) systems. It includes the usual Weyl–Heisenberg and su(2) algebras. We investigate the corresponding Fock representation space. The generalized Grassmann variables are introduced as variables spanning the Fock-Bargmann space associated with the algebra A(r). The Barut–Girardello coherent states for su(r + 1) algebras are explicitly derived and their over-completion properties are discussed.
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.