Gazeau-Klauder coherent states in position-deformed Heisenberg algebra

In this paper, we present coherent states à la Gazeau-Klauder for a free particle in square well potential within position-deformed Heisenberg algebra . These states satisfy the Klauder’s mathematical requirement to build coherent states. Some statistical properties such as the probability distribution, the intensity correlation function and the Mandel parameter are calculated and analyzed. We find that these states are sub-Poissonian in nature. We also construct for these coherent states, the even cat states and we evaluate its Wigner function which analyses the quasiprobability distribution of these states. We graphically demonstrate that these states exhibit nonclassical behavior.


Introduction
The study of coherent states has remained, over the past four decades a constant source of application in different branches of physics. They were first discovered in connection with the quantum harmonic oscillator by Schrödinger in 1926, who referred to them as states of minimum uncertainty product. In fact, 35 years after Schrödingerʼs pioneering idea, the importance of coherent states was put forward by Glauber [1,2] and Sudarshan [3] in the framework of quantum optics. The construction of these states has motivated the introduction of different sorts of coherent states [4][5][6][7][8] and has found considerable applications in different fields of theoretical and experimental physics [9][10][11][12][13][14][15].
The same states have also been reintroduced by Klauder, who investigated their mathematical properties [16,17]. He has noted that these states must satisfy the following minimum conditions: normalizability, continuity in the label, and the existence of a resolution of unity with a positive definite weight function. In 1999, Gazeau and Klauder (GK) [18] proposed new coherent states for semibounded Hamiltonian operators having either a discrete or continuous spectrum. These states, which have been constructed for a large variety of quantum systems [19][20][21][22][23][24][25][26][27], also satisfy the Klauderʼs minimum requirements.
Recently, we have studied the dynamics of a free particle in square-well potential within position-deformed Heisenberg algebra with maximal length uncertainty [28]. It has been shown that, this maximal length induces strong deformation in the quantum energy levels allowing particles to jump from one point to another with high probability densities. The obtained deformed-spectrum of this system generalised the ordinary one of quantum mechanics. In this study, we construct the GK coherent states for this system's deformed-spectrum. We show that these states satisfy the Klauder's mathematical requirement to build coherent states. We also explore the statistical properties [29][30][31][32][33][34][35][36] of these states, such as the photon distribution, the photon mean number, the intensity correlation and the Mandel parameter. We find that these states are sub-Poissonian in nature. With these coherent states at hand, we construct the corresponding even cat states [22,37]. We demonstrate that the quasidistribution function namely the Wigner function of these new states exhibit nonclassical behaviour. This paper is organised as follows: In the next section, we review in one dimension (1D) the representation of the position-deformed Heisenberg algebra that was recently introduced in [38]. In section 3, we construct GK coherent states and GK even cat states for the deformed-spectrum of a free particle in a square well potential recently determined [28]. We discuss the quantum statistical properties of the constructed coherent states. Finally, we conclude this work in section 4.

Position-deformed Heisenberg algebra
be the Hilbert space of square integrable functions. The operatorsX andP that act in this space are defined by [28,38] where the Hermitian operatorsx andp satisfy the ordinary Heisenberg algebra . The operatorsX andP satisfy the following relation [28,38] [ˆˆ] (ˆˆ) ( ) where τ ä (0, 1) is the GUP deformed parameter [39][40][41][42][43]. Let f(x) and f(p) be respectively the position and momentum representations defined on . The action of the operators (1) on these square integrable functions reads as follows For both representations, the corresponding completeness relations are given by [44] Consequently, the scalar product between two states |Ψ〉 and |Φ〉 and the orthogonality of eigenstates become . 8 For an operatorˆ{ˆˆ} = A X P , , its expectation value for both representations are given by For any representation, an interesting feature can be observed from the commutation relation (2) through the following uncertainty relation : Using the relationˆ( )á ñ = D + á ñ X X X 2 2 2 , the equation (13) can be rewritten as a second order equation for ΔXˆ( By setting the equation (14) intoˆˆ( the solutions ΔX are given by This equation leads to the absolute minimal uncertainty DP min in P-direction and the absolute maximal It is well known that [11], the existence of minimal uncertainty raises the question of the loss of representation i.e., the space is inevitably bounded by minimal quantity beyond which any further localization of particles is not possible. In the presente situation, the minimal momentum DP min leads to a loss of f(p)-representation and a maximal f(x)-representation. Thus, the corresponding representation of operators are given bŷ Using this equation (18), one can recover the algebra (2).
As one can see from the representation of operators in equation (1) or in equation (18), the position operatorX is Hermitian while the momentum operatorP is notˆˆˆ(ˆ) To restore the Hermicity of this operator, we have to restrict the action ofP in a physical dense subset,   Ì , which is defined by The restriction to dense subset guarantees the existence of the adjoint operatorˆ † P , a necessary condition for one to obtain the Hermicity of this operator. The adjoint domain is defined by x 2 Thus, we may write (ˆ) (ˆ) †   Ì P P , which means that the domain ofP is a proper subset of the domain of its adjointˆ † P . To show the Hermicity of the oparatorP, we consider the functional F(ξ, j) defined by Using the relation (5) and by a straightforward computation of this functional, we have , and ξ(x) can reach any arbitrary value at the boundaries. This lead to the vanishing of F(ξ, j) i.e., F(ξ, j) = 0. Consequently, the operatorP is a Hermitian in (ˆ)  P such that Despite the fact that the momentum is Hermitian, it is not always a self-adjoint operator because its domain includes the domain ofˆ † P . It could have none, or it could have an infinite number of self-adjoint extensions. Note that, as rule in quantum mechanics, the operators that act on square integrable functions are essentially self-adjoint. There are exceptions to the rule. This is because the basic quantization requirement that operators whose expectation values are real do not strictly require these operators be self-adjoint. Indeed, the Hermicity result (24) is a sufficient condition to ensure that all expectation values of the momentum operator are real. Let us considerĤ , the operator Hamiltonian of a system defined within this space bŷ where V is the potential energy of the system. The time-dependent deformed Schrödinger equation is x t

2
The probability density η(x, t) = |f(x, t)| 2 obeys the continuity equation , 0, 27 x where the current density is given by 3. GK coherent states for a particle in square well potential Let us consider the Hamiltonian of the above quantum system (25) confined in an infinite square-well potential, defined as For standing waves in a null potential, the corresponding time-independent Schrödinger equation reads as Then, the energy spectrum of the particle is written as The generalized wave function and the probability density corresponding to the energies (31) are given by [28] ⎜ where the normalized constant A is given by At the limit τ → 0, we have The GK-coherent states [18] for a Hermitian HamiltonianĤ with discrete, bounded below and nondegenerate eigenspectrum are defined as a two parameter set 4. Temporal stability: e − iHt |J, γ, 〉 = |J, γ + ωt〉, with ω = const.

Construction of GK coherent states
For the system under consideration, the dimensionless form of the energy eigenvalues defined in equation (31), can be obtained as: The parameter ρ(n) is defined as As a result, the coherent states given in equation (38) may be expressed as 1 46 n i e n n 0 n n 2 By multiplying the above equation by the vector 〈x| we express the coherent states (46), in term of the discrete wave function (32)

Mathematical properties
In this subsection, we will discuss the above properties of these states (46) by analysing the non-orthogonality, the conditions of continuity in the label, normalizability, the resolution of identity by finding the weight function ( )  J and the temporal stability.

The non-orthogonality
In order to characterize these states, we can see that the scalar product of two coherent states does not vanish

The Label continuity
The label continuity condition of the |J, γ〉 can then be stated as,

Resolution of unity
The overcompleteness relation reads as follows  Comparing equations (55) and (56), we obtain that figure 1 illustrates the weight function (58) versus J for various values of the deformed parameter τ. One can observe that, the weight function globally decreases when the parameter τ increases.

The temporal stability
The time evolution of coherent states |J, γ〉 can be obtained by unitary transformation |ˆ( ) , where the time evolution operator is given asˆ( )= -U t e iHt . In the present case, the time evolution of the GK coherent states (46) is given by n ie t n n 0 n n 2 By multiplying the above equation by the vector 〈x|, we have where N is the number operator which is defined as the operator which diagonalizes the basis for the number statesˆ| The Mandel Q-parameter is related to the intensity correlation function by [46,47]  In figure 3, the intensity correlation function g (2) (0) and the Mandel Q-parameter have been plotted in terms of the parameter J for different values of the deformed parameter τ. One can see that the Mandel Q-parameter is negative and the intensity correlation function g (2) (0) < 1 which indicates that, the GKCS (46) have sub-Poissonian statistics.

GK even cat states (GKECs)
The even cat states (ECs) |Ψ ecs 〉 are defined as the coherent superposition of the two ordinary coherent states (42) |α〉 and | − α〉 which can be given in the form [48]  where N ecs is the normalization constant. These states are well known in the literature [49,50] and are useful in the field of quantum information [51,52], quantum metrology [52], in teleportation protocols [53] and quantum spectroscopy [54]. The GK coherent states |J, γ〉 can be exploited for a generalization of the states |Ψ ecs 〉. An example of such generalization is given by the relation [22,37]  With the above results at hand, we evaluate the quasiprobability function of GKECs, namely, the Wigner function ( )  a function of GKECs. It is also worth recalling that the Wigner function can be measured experimentally [55], including the measurements of its negative values [56]. Thus, negative values of the Wigner function indicates nonclassical states which are usually used in the relevant literatures such as the Gibbs entropy, the quadrature squeezing and the Husimi distribution etc [22].