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arXiv:1212.6888

SU(1,1) Nonlinear Coherent States

B. Mojaveri, Alireza Dehghani

Published Dec 31, 2012• 0 citations
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Abstract

The idea of construction of the nonlinear coherent states based on the hypergeometric- type operators associated to the Weyl-Heisenberg group [J:P hys:A 45(2012) 095304], are generalized to the similar states for the arbitrary Lie group SU(1, 1). By using of a discrete, unitary and irreducible representation of the Lie algebra su(1, 1) wide range of generalized nonlinear coherent states(GNCS) have been introduced, which admit a resolution of the identity through positive definite measures on the complex plane. We have shown that realization of these states for different values of the deformation pa- rameters r = 1 and 2 lead to the well-known Klauder-Perelomov and Barut-Girardello coherent states associated to the Irreps of the Lie algebra su(1, 1), respectively. It is worth to mention that, like the canonical coherent states, GNCS possess the temporal stability property. Finally, studying some statistical characters implies that they have indeed nonclassical features such as squeezing, anti-bunching effect and sub-Poissonian statistics, too.

DOI
10.48550/arxiv.1212.6888

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The idea of construction of the nonlinear coherent states based on the hypergeometric- type operators associated to the Weyl-Heisenberg group [J:P hys:A 45(2012) 095304], are generalized to the similar states for the arbitrary Lie group SU(1, 1). By using of a discrete, unitary and irreducible representation of the Lie algebra su(1, 1) wide range of generalized nonlinear coherent states(GNCS) have been introduced, which admit a resolution of the identity through positive definite measures on the complex plane. We have shown that realization of these states for different values of the deformation pa- rameters r = 1 and 2 lead to the well-known Klauder-Perelomov and Barut-Girardello coherent states associated to the Irreps of the Lie algebra su(1, 1), respectively. It is worth to mention that, like the canonical coherent states, GNCS possess the temporal stability property. Finally, studying some statistical characters implies that they have indeed nonclassical features such as squeezing, anti-bunching effect and sub-Poissonian statistics, too.
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A quick review of some Lie algebras related to well-known groups is given. We start with the Heisenberg-Weyl algebra and after the definitions of the Fock states we give the definition of the coherent state of this group. This is followed by the exposition of the SU(2) and SU(1,1) algebras and their coherent states. From there we go on to describe the binomial states and their extensions as the finite dimensional pair coherent states and their nonlinear versions as realizations of the SU(2) group. This is followed by considering the negative binomial states, the single mode and two-mode squeezed states and their variants as realizations of the SU(1,1) group. Generation schemes based on physical systems are considered for some of these states.