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arXiv:1910.08180 · Journal of Mathematical Physics

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

S. Arjika, M. Calixto, J. Guerrero

Published Oct 1, 2019• 6 citations
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Abstract

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.

DOI
10.1063/1.5099683

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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.
Canonical coherent states of a quantum harmonic oscillator have been generalized by requiring the conditions of normalizability, continuity in the label and resolution of the identity operator with a positive weight function. Superpositions of these states are considered in the present scenario as a generalization of the optical Schrödinger cat states. The Fock space is assumed to be canonical or finite-dimensional. The photon number distribution of these generalized Schrödinger cat states departs from the Poisson statistics in various ways for high photon numbers. For small nonvanishing values of the label, the photon number distribution is sub-Poissonian (nonclassical) or super-Poissonian, according to the interference properties. In fact, the sub- or super-Poissonian statistics is determined by the interplay between the relative phase and a critical value of the phase. The photon number distribution is uniquely sub-Poissonian for large values of the label.
We have introduced the notion of ‘four-photon nonlinear coherent states’ (FPNCSs) as right eigenstates of the fourth power of the generalized annihilation operator A4. It has been shown that there are four possible sets of such states which all can be expressed as the deformed Schrödinger cat type states derived from the superposition of four nonlinear coherent states (NCSs) which are π/2 out of phase. The nonclassical properties of the FPNCSs would be studied for two well-known quantum systems, a Kerr-like medium and a trapped laser-driven ion far from the Lamb-Dicke regime. We have found that the depth or the domain of the nonclassical features of FPNCSs in terms of sub-Poissonian photon statistics, higher order squeezing as well as amplitude-squared squeezing are higher than the corresponding standard states. Specifically, the proposed states in this paper exhibit sub-Poissonian statistics over an extensive range of real amplitudes which is in contrast with the lower level of sub-Poissonian characteristics of four-photon standard coherent states. Also the depth or domain of the nonclassical features of the FPNCSs can be modified by changing the magnitude of the nonlinearity parameters. Analysis of the quasi-probability distributions like Glauber–Sudarshan P-function, Husimi Q-function and Wigner distribution function verifies the nonclassical nature of the presented states. Studying the cavity field evolution in the interaction of a single two-level atom with an even NCS in the framework of f-deformed Jaynes-Cummings model shows that such a system can potentially be considered as a possible source for generating the proposed states.
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.
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 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 PPP-representation of the density operator.