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arXiv:quant-ph/9905029 · Journal of Optics B Quantum and Semiclassical Optics

Phase properties of hypergeometric states and negative hypergeometric states

Xiao-Guang Wang

Published Jan 19, 2000• 11 citations
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Abstract

We show that the three quantum states (Pólya states, the generalized non-classical states related to Hahn polynomials and negative hypergeometric states) introduced recently as intermediate states which interpolate between the binomial states and negative binomial states are essentially identical. By using the Hermitian-phase-operator formalism, the phase properties of the hypergeometric states and negative hypergeometric states are studied in detail. We find that the phase probability distribution exhibits one peak for the hypergeometric states and multiple peaks for the negative hypergeometric states.

DOI
10.1088/1464-4266/2/1/305

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We show that the three quantum states (Pólya states, the generalized non-classical states related to Hahn polynomials and negative hypergeometric states) introduced recently as intermediate states which interpolate between the binomial states and negative binomial states are essentially identical. By using the Hermitian-phase-operator formalism, the phase properties of the hypergeometric states and negative hypergeometric states are studied in detail. We find that the phase probability distribution exhibits one peak for the hypergeometric states and multiple peaks for the negative hypergeometric states.
“Hypergeometric states,” which are a one-parameter generalization of binomial states of the single-mode quantized radiation field, are introduced and their nonclassical properties are investigated. Their limits to the binomial states and to the coherent and number states are studied. The ladder operator formulation of the hypergeometric states is found and the algebra involved turns out to be a one-parameter deformation of su(2) algebra. These states exhibit highly nonclassical properties, like sub-Poissonian character, antibunching, and squeezing effects. The quasiprobability distributions in phase space, namely the Q and the Wigner functions are studied in detail. These remarkable properties seem to suggest that the hypergeometric states deserve further attention from theoretical and applicational sides of quantum optics.
We study the non-classical properties and algebraic characteristics of the negative binomial states introduced by Barnett recently. The ladder operator formalism and displacement operator formalism of the negative binomial states are found and the algebra involved turns out to be the SU(1,1) Lie algebra via the generalized Holstein-Primarkoff realization. These states are essentially Peremolov's SU(1,1) coherent states. We reveal their connection with the geometric states and find that they are excited geometric states. As intermediate states, they interpolate between the number states and geometric states. We also point out that they can be recognized as the nonlinear coherent states. Their non-classical properties, such as sub-Poissonian distribution and squeezing effect are discussed. The quasiprobability distributions in phase space, namely the Q and Wigner functions, are studied in detail.
New states of electromagnetic field, generalized excited negative binomial states are introduced here. These states interpolate between the superposition of two excited coherent states and number states. The non-classical properties for these states are discussed, such as, second order correlation function, squeezing phenomena [normal squeezing and amplitude squared squeezing], phase properties in Pegg–Barnett formalism and the quasi-probability distribution functions (Q-function and Wigner function).
The properties of the even and odd negative binomial states are investigated. Mainly we concentrate on the statistical properties for such states and we consider the squeezing phenomenon by examining the variation in the quadrature variances. Using the Pegg–Barnett formalism of phase operator, the phase probability distributions of the states are discussed, and also the phase and number squeezing with different parameters are studied. The numerical computation results show that the phase probability distributions for the even and odd NBSs can clearly exhibit the different features of quantum effects.