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 Mathematical Physics

Coherent states associated with the continuous spectrum of noncompact groups

M. Hongoh

Published Oct 1, 1977• 15 citations
Publisher

Abstract

A new, explicit formula is obtained for the coherent states associated with a continuous spectrum of the noncompact group SU(1,1). The method is based on a simple and unified algebraic approach. We briefly discuss its relations to the generalized coherent states of Barut and Girardello, and of Perelomov.
DOI
10.1063/1.523152

Related papers

SU(1,1) coherent states and associated Wick symbol calculus

Shunlong Luo

We display the relationship of Perelomov and Barut–Girardello coherent states associated with the discrete series of SU(1,1) by establishing the interwining operators. Both Perelomov and Barut–Girardello coherent states are deformations of Glauber coherent states of Heisenberg–Weyl group, but they approximate the latter from different sides and exhibit a certain duality. We also develop a symbol calculus of SU(1,1) algebra in terms of Barut–Girardello coherent states (Perelomov coherent states are not suitable for this purpose) and rederive the disentanglement (Baker–Campbell–Hausdorff) formula for SU(1,1) generators by the symbol calculus and parameter differentiation technique.

Journal of Mathematical Physics•Jul 1, 1997•10 citations

SU(1,1) Nonlinear Coherent States

B. Mojaveri, Alireza Dehghani

The idea of construction of the nonlinear coherent states based on the hypergeometric- type operators associated to the Weyl-Heisenberg group [J:P hys:A 45(2012) 095304], are generalized to the similar states for the arbitrary Lie group SU(1, 1). By using of a discrete, unitary and irreducible representation of the Lie algebra su(1, 1) wide range of generalized nonlinear coherent states(GNCS) have been introduced, which admit a resolution of the identity through positive definite measures on the complex plane. We have shown that realization of these states for different values of the deformation pa- rameters r = 1 and 2 lead to the well-known Klauder-Perelomov and Barut-Girardello coherent states associated to the Irreps of the Lie algebra su(1, 1), respectively. It is worth to mention that, like the canonical coherent states, GNCS possess the temporal stability property. Finally, studying some statistical characters implies that they have indeed nonclassical features such as squeezing, anti-bunching effect and sub-Poissonian statistics, too.

arXiv:1212.6888•Dec 31, 2012•0 citations

Coherent States for Kronecker Products of Non-Compact Groups: Formulation and Applications

Bindu A. Bambah, G. S. Agarwal

We introduce and study the properties of a class of coherent states for the group SU(1,1) X SU(1,1) and derive explicit expressions for these using the Clebsch-Gordan algebra for the SU(1,1) group. We restrict ourselves to the discrete series representations of SU(1,1). These are the generalization of the `Barut Girardello' coherent states to the Kronecker Product of two non-compact groups.The resolution of the identity and the analytic phase space representation of these states is presented. This phase space representation is based on the basis of products of `pair coherent states' rather than the standard number state canonical basis. We discuss the utility of the resulting `bi-pair coherent states' in the context of four-mode interactions in quantum optics.

arXiv:quant-ph/9604025•Apr 23, 1996•0 citations

Aspects of coherent states of nonlinear algebras

T. Shreecharan, K. V. S. Shiv Chaitanya

Various aspects of coherent states of nonlinear su(2) and su(1, 1) algebras are studied. It is shown that the nonlinear su(1, 1) Barut–Girardello and Perelomov coherent states are related by a Laplace transform. We then concentrate on the derivation and analysis of the statistical and geometrical properties of these states. The Berry's phase for the nonlinear coherent states is also derived.

arXiv:1005.5607•Journal of Mathematical Physics•Dec 1, 2010•2 citations

Coherent States for Generalized Laguerre Functions

Jellal, A

We explicitly construct a Hamiltonian whose exact eigenfunctions are the generalized Laguerre functions. Moreover, we present the related raising and lowering operators. We investigate the corresponding coherent states by adopting three different methods which are due to $(i)$ Barut-Girardello, $(ii)$ Gazeau-Klauder and $(iii)$ Perelomov associated to Lie algebra $su(1,1)$. Resolution of unity and overlapping properties are examined in each method.

•Sep 4, 2001•16 citations

Generalized Coherent States Associated with the Cλ-Extended Oscillator

C. Quesne

Two new types of coherent states associated with the Cλ-extended oscillator, where Cλ is the cyclic group of order λ, are introduced. The first ones include as special cases both the Barut-Girardello and the Perelomov su(1,1) coherent states for λ = 2, as well as the annihilation-operator coherent states of the Cλ-extended oscillator spectrum generating algebra for higher λ values. The second ones, which are eigenstates of the Cλ-extended oscillator annihilation operator, extend to higher λ values the paraboson coherent states, to which they reduce for λ = 2. All these states satisfy a unity resolution relation in the Cλ-extended oscillator Fock space (or in some subspace thereof). They give rise to Bargmann representations of the latter, wherein the generators of the Cλ-extended oscillator algebra are realized as differential-operatorvalued matrices (or differential operators). The statistical and squeezing properties of the new coherent states are investigated over a wide range of parameters and some interesting nonclassical features are exhibited. 1 1

•Jan 1, 2001•18 citations
We display the relationship of Perelomov and Barut–Girardello coherent states associated with the discrete series of SU(1,1) by establishing the interwining operators. Both Perelomov and Barut–Girardello coherent states are deformations of Glauber coherent states of Heisenberg–Weyl group, but they approximate the latter from different sides and exhibit a certain duality. We also develop a symbol calculus of SU(1,1) algebra in terms of Barut–Girardello coherent states (Perelomov coherent states are not suitable for this purpose) and rederive the disentanglement (Baker–Campbell–Hausdorff) formula for SU(1,1) generators by the symbol calculus and parameter differentiation technique.
The idea of construction of the nonlinear coherent states based on the hypergeometric- type operators associated to the Weyl-Heisenberg group [J:P hys:A 45(2012) 095304], are generalized to the similar states for the arbitrary Lie group SU(1, 1). By using of a discrete, unitary and irreducible representation of the Lie algebra su(1, 1) wide range of generalized nonlinear coherent states(GNCS) have been introduced, which admit a resolution of the identity through positive definite measures on the complex plane. We have shown that realization of these states for different values of the deformation pa- rameters r = 1 and 2 lead to the well-known Klauder-Perelomov and Barut-Girardello coherent states associated to the Irreps of the Lie algebra su(1, 1), respectively. It is worth to mention that, like the canonical coherent states, GNCS possess the temporal stability property. Finally, studying some statistical characters implies that they have indeed nonclassical features such as squeezing, anti-bunching effect and sub-Poissonian statistics, too.
We introduce and study the properties of a class of coherent states for the group SU(1,1) X SU(1,1) and derive explicit expressions for these using the Clebsch-Gordan algebra for the SU(1,1) group. We restrict ourselves to the discrete series representations of SU(1,1). These are the generalization of the `Barut Girardello' coherent states to the Kronecker Product of two non-compact groups.The resolution of the identity and the analytic phase space representation of these states is presented. This phase space representation is based on the basis of products of `pair coherent states' rather than the standard number state canonical basis. We discuss the utility of the resulting `bi-pair coherent states' in the context of four-mode interactions in quantum optics.
We explicitly construct a Hamiltonian whose exact eigenfunctions are the generalized Laguerre functions. Moreover, we present the related raising and lowering operators. We investigate the corresponding coherent states by adopting three different methods which are due to (i)(i)(i) Barut-Girardello, (ii)(ii)(ii) Gazeau-Klauder and (iii)(iii)(iii) Perelomov associated to Lie algebra su(1,1)su(1,1)su(1,1). Resolution of unity and overlapping properties are examined in each method.
Two new types of coherent states associated with the Cλ-extended oscillator, where Cλ is the cyclic group of order λ, are introduced. The first ones include as special cases both the Barut-Girardello and the Perelomov su(1,1) coherent states for λ = 2, as well as the annihilation-operator coherent states of the Cλ-extended oscillator spectrum generating algebra for higher λ values. The second ones, which are eigenstates of the Cλ-extended oscillator annihilation operator, extend to higher λ values the paraboson coherent states, to which they reduce for λ = 2. All these states satisfy a unity resolution relation in the Cλ-extended oscillator Fock space (or in some subspace thereof). They give rise to Bargmann representations of the latter, wherein the generators of the Cλ-extended oscillator algebra are realized as differential-operatorvalued matrices (or differential operators). The statistical and squeezing properties of the new coherent states are investigated over a wide range of parameters and some interesting nonclassical features are exhibited. 1 1