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Physical Review A

Generalized coherent state for multimode bosonic realization of the su(2) Lie algebra

Chi‐Fai Lo, David Kiang

Published Sep 1, 1994• 2 citations
Publisher

Abstract

We have constructed the multimode bosonic realization of the su(2) Lie algebra, on the basis of which the SU(2) generalized coherent state in the multimode Fock space is derived. It is shown that in the multimode coherent state each bosonic mode has the sub-Poissonian statistics, and that the unitary displacement operator can be identified as the generalized multimode rotation operator.

DOI
10.1103/physreva.50.2781

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We have constructed the multimode bosonic realization of the su(2) Lie algebra, on the basis of which the SU(2) generalized coherent state in the multimode Fock space is derived. It is shown that in the multimode coherent state each bosonic mode has the sub-Poissonian statistics, and that the unitary displacement operator can be identified as the generalized multimode rotation operator.
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.
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.
We have constructed a new kind of two-mode bosonic realization of SU (1,1) Lie algebra, on the basis of which the SU (1,1) generalized coherent states in the two-mode Fock space are derived. These two-mode SU (1,1) coherent states, which are called uncorrelated two-mode SU (1,1) coherent states, include three special cases. For these states, we study the mean photon number distribution and their non-classical properties, which are photon anti-bunching, violations of Cauchy-Schwarz inequality and two-mode squeezing.
In this paper we have constructed the multiparticle ( q, p )-realization of the su (1, 1) Lie algebra, and studied its relation to the multimode bosonic realization and squeezing properties. We believe that this multiparticle realization should be very useful in the study of a system of coupled general time-dependent quantum oscillators whose Hamiltonian can be written as a linear combination of the three su (1, 1) generators; in particular, it is more capable to give a vivid picture of the system in the configuration space.
Abstract From the definition of the standard Perelomov coherent states we introduce the Perelomov number coherent states for any su(2) Lie algebra. With the displacement operator we apply a similarity transformation to the su(2) generators and construct a new set of operators which also close the su(2) Lie algebra, being the Perelomov number coherent states the new basis for its unitary irreducible representation. We apply our results to obtain the energy spectrum, the eigenstates and the partition function of two coupled oscillators. We show that the eigenstates of two coupled oscillators are the SU(2) Perelomov number coherent states of the two-dimensional harmonic oscillator with an appropriate choice of the coherent state parameters.