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Journal of the Optical Society of America B

Generalized coherent state for bosonic realization of SU(2)Lie algebra

Vladimír Bužek, Tran Quang

Published Dec 1, 1989• 112 citations
Publisher

Abstract

We have investigated the statistical properties of fields in the SU(2) generalized coherent state built on the bosonic (Schwinger) representation of the generators of SU(2) Lie algebra. We have shown that there are sub-Poissonian photon statistics as well as anticorrelations. Schemes for the generation of the states under consideration are discussed. We show that in the relevant processes either fields that are highly sub-Poissonian can be generated or a field in one mode with sub-Poissonian statistics can be transformed into another field that also has sub-Poissonian statistics.

DOI
10.1364/josab.6.002447

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We have investigated the statistical properties of fields in the SU(2) generalized coherent state built on the bosonic (Schwinger) representation of the generators of SU(2) Lie algebra. We have shown that there are sub-Poissonian photon statistics as well as anticorrelations. Schemes for the generation of the states under consideration are discussed. We show that in the relevant processes either fields that are highly sub-Poissonian can be generated or a field in one mode with sub-Poissonian statistics can be transformed into another field that also has sub-Poissonian statistics.
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The mode of a photon-added double-mode SU(2) coherent state｜M,ξ;m〉=A_(Mξm)a~(+m)｜M,ξ〉is introduced and its quantum statistics property is investigated with numeric method. The results exhibit that when the photons are added to the mode aof the double mode SU(2) coherent state, the two field modes can preserve anti-relation character. The violation of Cauchy-Schwartz inequality, the sub-Poissonian photon statistics property of mode aand anti-bunching effect of mode bare enhanced, but the sub-Poissonian photon statistics property of mode band anti-bunching effect of mode a are weakened.
We have constructed the multimode bosonic realization of the su(1,1) Lie algebra, on the basis of which the su(1,1) generalized coherent state in the multimode Fock space is derived. It is shown that the multimode coherent state is, in fact, the generalized multimode squeezed vacuum state discussed by Lo and Sollie [Phys. Rev. A 47, 733 (1993)], and that the unitary displacement operator can be identified as the generalized multimode squeezing operator. Hence, similar to the single- and two-mode cases, our multimode bosonic realization of the su(1,1) Lie algebra has immediate relevance to the squeezing properties of boson fields.
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.