We introduce a large class of holomorphic quantum states by choosing their normalization functions to be given by generalized hypergeometric functions. We call them generalized hypergeometric states in general, and generalized hypergeometric coherent states in particular, if they allow a resolution of unity. Depending on the domain of convergence of the generalized hypergeometric functions, we distinguish generalized hypergeometric states on the plane, the open unit disk, and the unit circle. All states are eigenstates of suitably defined lowering operators. We then study their photon number statistics and phase properties as revealed by the Husimi and Pegg-Barnett phase distributions. On the basis of the generalized hypergeometric coherent states we introduce generalized hypergeometric Husimi distributions and corresponding phase distributions as well as new analytic representations of arbitrary quantum states in Bargmann and Hardy spaces.We introduce a large class of holomorphic quantum states by choosing their normalization functions to be given by generalized hypergeometric functions. We call them generalized hypergeometric states in general, and generalized hypergeometric coherent states in particular, if they allow a resolution of unity. Depending on the domain of convergence of the generalized hypergeometric functions, we distinguish generalized hypergeometric states on the plane, the open unit disk, and the unit circle. All states are eigenstates of suitably defined lowering operators. We then study their photon number statistics and phase properties as revealed by the Husimi and Pegg-Barnett phase distributions. On the basis of the generalized hypergeometric coherent states we introduce generalized hypergeometric Husimi distributions and corresponding phase distributions as well as new analytic representations of arbitrary quantum states in Bargmann and Hardy spaces.

Aug 1, 200317 citations

(q, ν)-deformation of generalized basic hypergeometric states

Mahouton Norbert Hounkonnou, E B Ngompe Nkouankam

We provide with a ( q , ν)-deformation of the generalized hypergeometric coherent states defined such that the normalization function is given by generalized basic hypergeometric functions. These states are eigenstates of suitably defined deformed lowering operators. We study the domain of convergence of the corresponding normalization function. On the basis of these states, we investigate generalized basic hypergeometric Husimi distributions and corresponding phase distributions as well as new analytic basis representations of arbitrary quantum states in Bargmann and Hardy spaces. The quantum statistical properties of the states, such as photon-counting statistics and quadrature squeezing are analytically and numerically discussed in the framework of conventional quantum optics.We provide with a ( q , ν)-deformation of the generalized hypergeometric coherent states defined such that the normalization function is given by generalized basic hypergeometric functions. These states are eigenstates of suitably defined deformed lowering operators. We study the domain of convergence of the corresponding normalization function. On the basis of these states, we investigate generalized basic hypergeometric Husimi distributions and corresponding phase distributions as well as new analytic basis representations of arbitrary quantum states in Bargmann and Hardy spaces. The quantum statistical properties of the states, such as photon-counting statistics and quadrature squeezing are analytically and numerically discussed in the framework of conventional quantum optics.

Journal of Physics A Mathematical and TheoreticalJan 14, 20094 citations

Quantum statistical properties of multiphoton hypergeometric coherent states and the discrete circle representation

S. Arjika, M. Calixto, J. Guerrero

We review the definition of hypergeometric coherent states, discussing some representative examples. Then, we study mathematical and statistical properties of hypergeometric Schrödinger cat states, defined as orthonormalized eigenstates of kth powers of nonlinear f-oscillator annihilation operators, with f of the hypergeometric type. These “k-hypercats” can be written as an equally weighted superposition of hypergeometric coherent states ∣zl⟩, l = 0, 1, …, k − 1, with zl = ze2πil/k a kth root of zk, and they interpolate between number and coherent states. This fact motivates a continuous circle representation for high k. We also extend our study to truncated hypergeometric functions (finite dimensional Hilbert spaces), and a discrete exact circle representation is provided. We also show how to generate k-hypercats by amplitude dispersion in a Kerr medium and analyze their generalized Husimi Q-function in the super- and sub-Poissonian cases at different fractions of the revival time.We review the definition of hypergeometric coherent states, discussing some representative examples. Then, we study mathematical and statistical properties of hypergeometric Schrödinger cat states, defined as orthonormalized eigenstates of kth powers of nonlinear f-oscillator annihilation operators, with f of the hypergeometric type. These “k-hypercats” can be written as an equally weighted superposition of hypergeometric coherent states ∣zl⟩, l = 0, 1, …, k − 1, with zl = ze2πil/k a kth root of zk, and they interpolate between number and coherent states. This fact motivates a continuous circle representation for high k. We also extend our study to truncated hypergeometric functions (finite dimensional Hilbert spaces), and a discrete exact circle representation is provided. We also show how to generate k-hypercats by amplitude dispersion in a Kerr medium and analyze their generalized Husimi Q-function in the super- and sub-Poissonian cases at different fractions of the revival time.

arXiv:1910.08180Journal of Mathematical PhysicsOct 1, 20196 citations

Generalized hypergeometric coherent states for special functions: Mathematical and physical properties

Messan Médard Akouetegan, Isiaka Aremua, Komi Sodoga +2 more

We investigate a class of generalized coherent states for associated Jacobi polynomials and hypergeometric functions, satisfying the resolution of the identity with respect to a weight function expressed in terms of Meijer’s G-function. We extend the state Hilbert space of the constructed states and discuss the property of the reproducing kernel and its analytical expansion. Further, we provide the expectation values of observables relevant to this quantum model. We also perform the quantization of the complex plane, compute and analyze the probability density and the temporal stability in these states. Using the completeness relation provided by the coherent states, we achieve the thermodynamic analysis in the diagonal P-representation of the density operator.We investigate a class of generalized coherent states for associated Jacobi polynomials and hypergeometric functions, satisfying the resolution of the identity with respect to a weight function expressed in terms of Meijer’s G-function. We extend the state Hilbert space of the constructed states and discuss the property of the reproducing kernel and its analytical expansion. Further, we provide the expectation values of observables relevant to this quantum model. We also perform the quantization of the complex plane, compute and analyze the probability density and the temporal stability in these states. Using the completeness relation provided by the coherent states, we achieve the thermodynamic analysis in the diagonal P-representation of the density operator.

Journal of Mathematical PhysicsJan 1, 20253 citations

New nonlinear coherent states based on hypergeometric-type operators

Alireza Dehghani, B. Mojaveri

The main goal of this paper is to present an alternative method to construct new kinds of nonlinear coherent states. To do this, we first establish a class of hypergeometric type of generalized displacement operators, 1 F r ([0], [0, 1, ..., r − 1], za †), act on the vacuum state of the harmonic oscillator and generate normalized quantum states of the Fock space which admit a resolution of the identity through a positive definite measure on the complex plane. Furthermore, realization of the compact form of these states, as functions of the position coordinate x for r = 2, leads to a generating function of the Hermite polynomials in terms of the modified Bessel function. Finally, studying some statistical characters reveals that they have indeed non-classical features such as squeezing, an anti-bunching effect and sub-Poissonian statistics, too.The main goal of this paper is to present an alternative method to construct new kinds of nonlinear coherent states. To do this, we first establish a class of hypergeometric type of generalized displacement operators, 1 F r ([0], [0, 1, ..., r − 1], za †), act on the vacuum state of the harmonic oscillator and generate normalized quantum states of the Fock space which admit a resolution of the identity through a positive definite measure on the complex plane. Furthermore, realization of the compact form of these states, as functions of the position coordinate x for r = 2, leads to a generating function of the Hermite polynomials in terms of the modified Bessel function. Finally, studying some statistical characters reveals that they have indeed non-classical features such as squeezing, an anti-bunching effect and sub-Poissonian statistics, too.

Journal of Physics A Mathematical and TheoreticalFeb 20, 201218 citations

Generalized hypergeometric coherent states for special functions: mathematical and physical properties

Isiaka Aremua, Messan Médard Akouetegan, Komi Sodoga +2 more

In continuation of our previous works J. Phys. A: Math. Gen. 35, 9355-9365 (2002), J. Phys. A: Math. Gen. 38, 7851 (2005) and Eur. Phys. J. D 72, 172 (2018), we investigate a class of generalized coherent states for associated Jacobi polynomials and hypergeometric functions, satisfying the resolution of the identity with respect to a weight function expressed in terms of Meijer's G-function. We extend the state Hilbert space of the constructed states and discuss the property of the reproducing kernel and its analytical expansion. Further, we provide the expectation values of observables relevant to this quantum model. We also perform the quantization of the complex plane, compute and analyze the probability density and the temporal stability in these states. Using the completeness relation provided by the coherent states, we achieve the thermodynamic analysis in the diagonal $P$ -representation of the density operator.In continuation of our previous works J. Phys. A: Math. Gen. 35, 9355-9365 (2002), J. Phys. A: Math. Gen. 38, 7851 (2005) and Eur. Phys. J. D 72, 172 (2018), we investigate a class of generalized coherent states for associated Jacobi polynomials and hypergeometric functions, satisfying the resolution of the identity with respect to a weight function expressed in terms of Meijer's G-function. We extend the state Hilbert space of the constructed states and discuss the property of the reproducing kernel and its analytical expansion. Further, we provide the expectation values of observables relevant to this quantum model. We also perform the quantization of the complex plane, compute and analyze the probability density and the temporal stability in these states. Using the completeness relation provided by the coherent states, we achieve the thermodynamic analysis in the diagonal $P$-representation of the density operator.

arXiv:2305.02194Apr 2, 20231 citation

P

-representation of the density operator.