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Journal of Mathematical Physics

Generalized coherent states and their statistical characteristics in power-law potentials

Shahid Iqbal, Farhan Saif

Published Aug 1, 2011• 21 citations
Publisher

Abstract

Generalized coherent states based on Gazeau-Klauder formalism are developed for one-dimensional power-law potentials and their quantum statistical characteristics, together with generalized Heisenberg algebra coherent states, are reported. We show that these states exhibit super-Poissonian, Poissonian, or sub-Poissonian distributions as a function of the power-law exponent. The analytical results are supported by numerical calculations. In addition, we explain possible sources of errors in numerical analysis.

DOI
10.1063/1.3626936

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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.
(q;l,λ)(q;l,λ)(q;l,λ)-deformed Heisenberg algebra: coherent states, their statistics and geometry
The Heisenberg algebra is deformed with the set of parameters q,l,λ{q, l,λ}q,l,λ to generate a new family of generalized coherent states respecting the Klauder criteria. In this framework, the matrix elements of relevant operators are exactly computed. Then, a proof on the sub-Poissonian character of the statistics of the main deformed states is provided. This property is used to determine the induced generalized metric.
In the present scenario, coherent states of a quantum harmonic oscillator are generalized with positive Fox H auxiliary functions. The novel generalized coherent states provide canonical coherent states and Mittag-Leffler or Wright generalized coherent states, as particular cases, and resolve the identity operator, over the Fock space, with a weight function that is the product of a Fox H function and a Wright generalized hypergeometric function. The novel generalized coherent states, or the corresponding truncated generalized coherent states, are characterized by anomalous statistics for large values of the number of excitations: the corresponding decay laws exhibit, for determined values of the involved parameters, various behaviors that depart from exponential and inverse-power-law decays, or their product. The analysis of the Mandel Q factor shows that, for small values of the label, the statistics of the number of excitations becomes super-Poissonian, or sub-Poissonian, by simply choosing sufficiently large values of one of the involved parameters. The time evolution of a generalized coherent state interacting with a thermal reservoir and the purity are analyzed.