Negative Binomial and Multinomial States: probability distributions and coherent states

Abstract

Following the relationship between probability distribution and coherent states, for example the well known Poisson distribution and the ordinary coherent states and relatively less known one of the binomial distribution and the $su(2)$ coherent states, we propose ``interpretation'' of $su(1,1)$ and $su(r,1)$ coherent states ``in terms of probability theory''. They will be called the ``negative binomial'' (``multinomial'') ``states'' which correspond to the ``negative'' binomial (multinomial) distribution, the non-compact counterpart of the well known binomial (multinomial) distribution. Explicit forms of the negative binomial (multinomial) states are given in terms of various boson representations which are naturally related to the probability theory interpretation. Here we show fruitful interplay of probability theory, group theory and quantum theory.

24 pages, latex, no figures

References in corpus (3)

• Generalized Binomial States: Ladder Operator Approach
• Exponential and Laguerre Squeezed States for su(1,1) Algebra and Calogero-Sutherland Model
• Generally Deformed Oscillator, Isospectral Oscillator System and Hermitian Phase Operator

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• Coherent control of quantum topological states of light in Fock-state lattices
• Quantum localized states in photonic flat-band lattices
• Analysis of generalized negative binomial distributions attached to hyperbolic Landau levels
• Excited Binomial States and Excited Negative Binomial States of the Radiation Field and Some of their Statistical Properties
• Negative Binomial States of the Radiation Field and their Excitations are Nonlinear Coherent States
• Probability Distributions and Coherent States of $B_r$, $C_r$ and $D_r$ Algebras
• Quantifying athermality and quantum induced deviations from classical fluctuation relations
• Intermediate coherent-phase(PB) states of radiation fields and their nonclassical properties
• su(2) and su(1,1) displaced number states and their nonclassical properties