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

About a new family of coherent states for some SU(1,1) central field potentials

Dušan Popov, Vjekoslav D. Sajfert, Nicolina Pop, Viorel Chiritoiu

Published Mar 1, 2013• 0 citations
Publisher

Abstract

In this paper, we shall define a new family of coherent states which we shall call the “mother coherent states,” bearing in mind the fact that these states are independent from any parameter (the Bargmann index, the rotational quantum number J, and so on). So, these coherent states are defined on the whole Hilbert space of the Fock basis vectors. The defined coherent states are of the Barut-Girardello kind, i.e., they are the eigenstates of the lowering operator. For these coherent states we shall calculate the expectation values of different quantum observables, the corresponding Mandel parameter, the Husimi's distribution function and also the P- function. Finally, we shall particularize the obtained results for the three-dimensional harmonic and pseudoharmonic oscillators.

DOI
10.1063/1.4795137

Related papers

A new set of coherent states for the isotropic harmonic oscillator: Coherent angular momentum states

Anthony J. Bracken, Howard I. Leemon

The Hamiltonian for the oscillator has earlier been written in the form where ν† and ν are raising and lowering operators for ν†ν, which has eigenvalues k (the ’’radial’’ quantum number), and λ† and λ are raising and lowering 3-vector operators for λ†⋅λ, which has eigenvalues l (the total angular momentum quantum number). A new set of coherent states for the oscillator is now defined by diagonalizing ν and λ. These states bear a similar relation to the commuting operators H, L2, and L3 (where L is the angular momentum of the system) as the usual coherent states do to the commuting number operators N1, N2, and N3. It is proposed to call them coherent angular momentum states. They are shown to be minimum-uncertainty states for the variables ν, ν† , λ, and λ†, and to provide a new quasiclassical description of the oscillator. This description coincides with that provided by the usual coherent states only in the special case that the corresponding classical motion is circular, rather than elliptical; and, in general, the uncertainty in the angular momentum of the system is smaller in the new description. The probabilities of obtaining particular values for k and l in one of the new states follow independent Poisson distributions. The new states are overcomplete, and lead to a new representation of the Hilbert space for the oscillator, in terms of analytic functions on C×K3, where K3 is the three-dimensional complex cone. This space is related to one introduced recently by Bargmann and Todorov, and carries a very simple realization of all the representations of the rotation group.

Journal of Mathematical Physics•Apr 1, 1981•7 citations

A set of nonlinear coherent states for the pseudoharmonic oscillator

Khalid Ahbli, Hamidou Kassogué, Patrick K. Kayupe +1 more

We construct two-parameters family of nonlinear coherent states by replacing the factorial in coefficients $z^n/\sqrt{n!}$ of the canonical coherent states by a specific generalized factorial $x_n^{γ,σ}!$ where parameters $γ$ and $σ$ satisfy some conditions for which the normalization condition and the resolution of identity are verified. The obtained family is a generalization of the Barut-Girardello coherent states and those of the philophase states. In the particular case of parameters $γ$ and $σ$, we attache these states to the pseudo-harmonic oscillator depending on two parameters $α,β> 0$. The obtained nonlinear coherent states are superposition of eigenstates of this oscillator. The associated Bargmann-type transform is defined and we derive some results.

arXiv:1907.08881•Jul 20, 2019•0 citations

Barut-Girardello coherent states of the pseudoharmonic oscillator

Dušan Popov

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.

Journal of Physics A Mathematical and Theoretical•Jun 15, 2001•48 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

PERELOMOV AND BARUT–GIRARDELLO SU(1, 1) COHERENT STATES FOR HARMONIC OSCILLATOR IN ONE-DIMENSIONAL HALF SPACE

Quanhui Liu, H. ZHUO

The Perelomov and the Barut–Girardello SU(1, 1) coherent states for harmonic oscillator in one-dimensional half space are constructed. Results show that the uncertainty products ΔxΔp for these two coherent states are bound from below [Formula: see text] that is the uncertainty for the ground state, and the mean values for position x and momentum p in classical limit go over to their classical quantities respectively. In classical limit, the uncertainty given by Perelomov coherent does not vanish, and the Barut–Girardello coherent state reveals a node structure when positioning closest to the boundary x = 0 which has not been observed in coherent states for other systems.

International Journal of Modern Physics A•May 10, 2006•1 citation

Pair-coherent states of the pseudoharmonic oscillator

Dušan Popov, Vjekoslav D. Sajfert

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.

Physica Scripta•Jul 1, 2009•4 citations
In this paper, we shall define a new family of coherent states which we shall call the “mother coherent states,” bearing in mind the fact that these states are independent from any parameter (the Bargmann index, the rotational quantum number J, and so on). So, these coherent states are defined on the whole Hilbert space of the Fock basis vectors. The defined coherent states are of the Barut-Girardello kind, i.e., they are the eigenstates of the lowering operator. For these coherent states we shall calculate the expectation values of different quantum observables, the corresponding Mandel parameter, the Husimi's distribution function and also the P- function. Finally, we shall particularize the obtained results for the three-dimensional harmonic and pseudoharmonic oscillators.
The Hamiltonian for the oscillator has earlier been written in the form where ν† and ν are raising and lowering operators for ν†ν, which has eigenvalues k (the ’’radial’’ quantum number), and λ† and λ are raising and lowering 3-vector operators for λ†⋅λ, which has eigenvalues l (the total angular momentum quantum number). A new set of coherent states for the oscillator is now defined by diagonalizing ν and λ. These states bear a similar relation to the commuting operators H, L2, and L3 (where L is the angular momentum of the system) as the usual coherent states do to the commuting number operators N1, N2, and N3. It is proposed to call them coherent angular momentum states. They are shown to be minimum-uncertainty states for the variables ν, ν† , λ, and λ†, and to provide a new quasiclassical description of the oscillator. This description coincides with that provided by the usual coherent states only in the special case that the corresponding classical motion is circular, rather than elliptical; and, in general, the uncertainty in the angular momentum of the system is smaller in the new description. The probabilities of obtaining particular values for k and l in one of the new states follow independent Poisson distributions. The new states are overcomplete, and lead to a new representation of the Hilbert space for the oscillator, in terms of analytic functions on C×K3, where K3 is the three-dimensional complex cone. This space is related to one introduced recently by Bargmann and Todorov, and carries a very simple realization of all the representations of the rotation group.
We construct two-parameters family of nonlinear coherent states by replacing the factorial in coefficients zn/n!z^n/\sqrt{n!}zn/n!​ of the canonical coherent states by a specific generalized factorial xnγ,σ!x_n^{γ,σ}!xnγ,σ​! where parameters γγγ and σσσ satisfy some conditions for which the normalization condition and the resolution of identity are verified. The obtained family is a generalization of the Barut-Girardello coherent states and those of the philophase states. In the particular case of parameters γγγ and σσσ, we attache these states to the pseudo-harmonic oscillator depending on two parameters α,β>0α,β> 0α,β>0. The obtained nonlinear coherent states are superposition of eigenstates of this oscillator. The associated Bargmann-type transform is defined and we derive some results.
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.
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
The Perelomov and the Barut–Girardello SU(1, 1) coherent states for harmonic oscillator in one-dimensional half space are constructed. Results show that the uncertainty products ΔxΔp for these two coherent states are bound from below [Formula: see text] that is the uncertainty for the ground state, and the mean values for position x and momentum p in classical limit go over to their classical quantities respectively. In classical limit, the uncertainty given by Perelomov coherent does not vanish, and the Barut–Girardello coherent state reveals a node structure when positioning closest to the boundary x = 0 which has not been observed in coherent states for other systems.