Gazeau Klauder Research Articles

On the generalized intelligent states and certain related nonclassical states of a quantum exactly solvable nonlinear oscillator

We construct nonlinear coherent states or f-deformed coherent states for a nonpolynomial nonlinear oscillator which can be considered as placed in the middle between the harmonic oscillator and the isotonic oscillator (Cariñena et al 2008 J. Phys. A: Math. Theor. 41 085301). The deformed annihilation and creation operators which are required to construct the nonlinear coherent states in the number basis are obtained from the solution of the Schrödinger equation. Using these operators, we construct generalized intelligent states, nonlinear coherent states, Gazeau–Klauder coherent states and the even and odd nonlinear coherent states for this newly solvable system. We also report certain nonclassical properties exhibited by these nonlinear coherent states. In addition to the above, we consider the position-dependent mass Schrödinger equation associated with this solvable nonlinear oscillator and construct nonlinear coherent states, Gazeau–Klauder coherent states and the even and odd nonlinear coherent states for it. We also give explicit expressions of all these nonlinear coherent states by considering a mass profile which is often used for studying transport properties in semiconductors.

Superpositions of the dual family of nonlinear coherent states and their non-classical properties

Nonlinear coherent states (CSs) and their dual families were introduced recently. In this paper, we want to obtain their superposition and investigate their non-classical properties such as antibunching effect, quadrature squeezing and amplitude squared squeezing. For this purpose two types of superposition are considered. In the first type, we neglect the normalization factors of the two components of the dual pair, superpose them and then we normalize the obtained states, while in the second type we superpose the two normalized components and then again normalize the resultant states. As a physical realization, the formalism will then be applied to a special physical system with known nonlinearity function, i.e., Hydrogen-like spectrum. We continue with the (first type of) superposition of the dual pair of Gazeau–Klauder coherent states (GKCSs) as temporally stable CSs. An application of the proposal will be given by employing the Pöschl–Teller potential system. The numerical results are presented and discussed in detail, showing the effects of this special quantum interference.

Number-phase entropic uncertainty relations and Wigner functions for solvable quantum systems with discrete spectra

In this Letter, the “number-phase entropic uncertainty relation” and the “number-phase Wigner function” of generalized coherent states associated to a few solvable quantum systems with non-degenerate spectra are studied. We also investigate time evolution of “number-phase entropic uncertainty” and “Wigner function” of the considered physical systems with the help of temporally stable Gazeau–Klauder coherent states.

Ladder operators and coherent states for continuous spectra

The notion of ladder operators is introduced for systems with continuous spectra. We identify two different kinds of annihilation operators allowing the definition of coherent states as modified ‘eigenvectors’ of these operators. Axioms of Gazeau–Klauder are maintained throughout the construction.

New classes of nonlinear vector coherent states of generalized spin–orbit Hamiltonians

This paper deals with an extension of our previous work (Ben Geloun and Hounkonnou 2007 J. Phys. A: Math. Theor. 40 F817) by considering an alternative construction of canonical and deformed vector coherent states (VCSs) of the Gazeau–Klauder type associated with generalized spin–orbit Hamiltonians. We define an annihilation operator which takes into account the finite-dimensional space of states induced by the k-photon transition processes of the two-level atom interacting with the single-mode radiation field. The class of nonlinear VCSs (NVCSs) corresponding to the action of the annihilation operator is deduced and expressed in terms of generalized displacement operators. Various NVCSs including their ‘dual’ counterparts are also discussed. Also, by using the Hilbert space structure, a new family of NVCSs parametrized by unit vectors of the S3 sphere has been identified without making use of the annihilation operator.

Quons, coherent states and intertwining operators

We propose a differential representation for the operators satisfying the q-mutation relation B B † − q B † B = 1 which generalizes a recent result by Eremin and Meldianov, and we discuss in detail this choice in the limit q → 1 . Further, we build up non-linear and Gazeau–Klauder coherent states associated to the free quonic Hamiltonian h 1 = B † B . Finally we construct almost isospectral quonic Hamiltonians adopting the results on intertwining operators recently proposed by the author.

Coherent state of a nonlinear oscillator and its revival dynamics

The coherent state of a nonlinear oscillator having a nonlinear spectrum is constructed using Gazeau–Klauder formalism. The weighting distribution and the Mandel parameter are studied. Details of the revival structure arising from different time scales underlying the quadratic energy spectrum are investigated by the phase analysis of the autocorrelation function.

Coherent states in noncommutative quantum mechanics

Gazeau–Klauder coherent states in noncommutative quantum mechanics are considered. We find that these states share similar properties to those of ordinary canonical coherent states in the sense that they saturate the related position uncertainty relation, obey a Poisson distribution, and possess a flat geometry. Using the natural isometry between the quantum Hilbert space of Hilbert-Schmidt operators and the tensor product of the classical configuration space and its dual, we reveal the inherent vector feature of these states.

Vector coherent states and intertwining operators

In this paper, we discuss a general strategy to construct vector coherent states of the Gazeau–Klauder type and we use them to build up examples of isospectral Hamiltonians. For that we use a general strategy recently proposed by the author, which extends well-known facts on intertwining operators. We also discuss the possibility of constructing non-isospectral Hamiltonians with related eigenstates.

Extended SUSY quantum mechanics, intertwining operators and coherent states

We propose an extension of supersymmetric quantum mechanics which produces a family of isospectral Hamiltonians. Our procedure slightly extends the idea of intertwining operators. Several examples of the construction are given. Further, we show how to build up vector coherent states of the Gazeau–Klauder type associated to our Hamiltonians.

Comments on “Gazeau–Klauder coherent states for trigonometric Rosen–Morse potential” [J. Math. Phys. 49, 022104 (2008)

In a recently published paper in this journal [A. Cheaghlou and O. Faizy, J. Math. Phys. 49, 022104 (2008)], the authors introduce the Gazeau–Klauder coherent states for the trigonometric Rosen–Morse potential as an infinite superposition of the wavefunctions. It is shown that their proposed measure to realize the resolution of the identity condition is not positive definite. Consequently, the claimed coherencies for the trigonometric Rosen–Morse wavefunctions cannot actually exist.

Pseudoharmonic oscillator and their associated Gazeau–Klauder coherent states

In the paper we have constructed and examined the properties of the Gazeau–Klauder coherent states (GK-CSs) for the pseudoharmonic oscillator (PHO), one of three possible kinds in order to define the coherent states for this oscillator potential. In the second part, we have examined some nonclassical properties of these states. Our attention has been concentrated on the mixed states (thermal states). The diagonal P -representation of the corresponding density operator and some thermal expectations for the quantum canonical ideal gas of pseudoharmonic oscillators have also been examined. Like the CSs for the harmonic oscillator (HO), the GK-CSs for the PHO can be useful in the quantum information theory (QIT).

Gazeau–Klauder coherent states for trigonometric Rosen–Morse potential

The Gazeau–Klauder coherent states for the trigonometric Rosen–Morse potential are constructed. It is shown that the resolution of unity, temporal stability, and action identity conditions are satisfied for the coherent states. The Mandel parameter is also calculated for the weighting distribution function corresponding to the coherent states.

Gazeau–Klauder squeezed states associated with solvable quantum systems

A formalism for the construction of some classes of Gazeau–Klauder squeezed states, corresponding to arbitrary solvable quantum systems with a known discrete spectrum, are introduced. As some physical applications, the proposed structure is applied to a few known quantum systems and then statistical properties as well as squeezing of the obtained squeezed states are studied. Finally, numerical results are presented.

REPRESENTATIONS OF GENERALIZED Ar STATISTICS AND EIGENSTATES OF JACOBSON GENERATORS

We investigate a generalization of Ar statistics discussed recently in the literature. The explicit complete set of state vectors for the Ar statistics system is given. We consider a Bargmann or an analytic function description of the Fock space corresponding to Ar statistics of bosonic kind. This brings, in a natural way, the so-called Gazeau–Klauder coherent states defined as eigenstates of the Jacobson annihilation operators. The minimization of Robertson uncertainty relation is also considered.

Coherent states of non-Hermitian quantum systems

We use the Gazeau–Klauder formalism to construct coherent states of non-Hermitian quantum systems. In particular we use this formalism to construct coherent state of a PT symmetric system. We also describe the construction of coherent states following Klauder's minimal prescription.

Some physical appearances of vector coherent states and coherent states related to degenerate Hamiltonians

In the spirit of some earlier work on the construction of vector coherent states (VCS) over matrix domains, we compute here such states associated to some physical Hamiltonians. In particular, we construct vector coherent states of the Gazeau–Klauder type. As a related problem, we also suggest a way to handle degeneracies in the Hamiltonian for building coherent states. Specific physical Hamiltonians studied include a single photon mode interacting with a pair of fermions, a Hamiltonian involving a single boson and a single fermion, a charged particle in a three-dimensional harmonic force field and the case of a two-dimensional electron placed in a constant magnetic field, orthogonal to the plane which contains the electron. In this last example, which is related to the fractional quantum Hall effect, an interesting modular structure emerges for two underlying von Neumann algebras, related to opposite directions of the magnetic field. This leads to the existence of coherent states built out of Kubo-Martin-Schwinger (KMS) states for the system.

Construction of the dual family of Gazeau–Klauder coherent states via temporally stable nonlinear coherent states

Using the analytic representation of the so-called Gazeau–Klauder coherent states (CSs), we shall demonstrate that how a new class of generalized CSs, namely the family of dual states associated with theses states, can be constructed through viewing these states as temporally stable nonlinear CSs. Also we find that the ladder operators, as well as the displacement type operator corresponding to these two pairs of generalized CSs, may be easily obtained using our formalism, without employing the supersymmetric quantum mechanics (SUSYQM) techniques. Then, we have applied this method to some physical systems with known spectrum, such as Pöschl–Teller, infinite well, Morse potential and hydrogenlike spectrum as some quantum mechanical systems. Finally, we propose the generalized form of the Gazeau–Klauder CS and the corresponding dual family.

Gazeau-Klauder and Klauder-Perelomov coherert states for the anharmonic oscillator

Using the Kinani-Daoud method we have constructed the Gazeau-Klauder(GK) and KlauderPerelomov(KP) coherent states for the anharmonic oscillator. It is shown that the two types of coherent states have totally different forms in their expressions. We have also discussed the resolution of unity of the two coherent states and the Hilbert space of each state. The study of Mandel Q pa rameter of the states shows that GK coherent state coincides with the sub-Poissonian and KP coherent state coincides with the super-Poissonian statistical distributions.

Gazeau–Klauder quasi-coherent states for the Morse oscillator

In the Letter, we have constructed and investigated some properties of the Gazeau–Klauder quasi-coherent states for the Morse potential, previously deduced by Roy and Roy. We have focused our attention on the thermal states and we have found the analytical form for the diagonal P-representation of the density operator.