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Communications in Theoretical Physics

The Two-Variable Coherent States of the Quantum Algebra su q (1,1)

Gong Ren-shan

Published Sep 15, 1997• 0 citations
Publisher

Abstract

By means of the k -order q -boson realizations of su q (1,1), a new kind of q-coherent states, the two-variable coherent states, are introduced for the quantum algebra su q (1,1).They are relative to a definite irreducible representation of su q (1,1) and shown to satisfy a completeness relation in a subspace of q -boson's Fock space. With the help of these coherent states the z -representation, representation in functional space, for su q (1,1) is also discussed.

DOI
10.1088/0253-6102/28/2/231

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By means of the k -order q -boson realizations of su q (1,1), a new kind of q-coherent states, the two-variable coherent states, are introduced for the quantum algebra su q (1,1).They are relative to a definite irreducible representation of su q (1,1) and shown to satisfy a completeness relation in a subspace of q -boson's Fock space. With the help of these coherent states the z -representation, representation in functional space, for su q (1,1) is also discussed.
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.
A new method for describing the two-level pairing model in a many-fermion system is presented. The basic idea comes from the Schwinger boson representation for the su(2) ⊗ su(2)-algebra, in which four kinds of boson operators govern the dynamics. It is well known that in the original fermion space, it is almost impossible to introduce any trial state for the variation except the BCS state in the framework of the mean field approximation. However, in the Schwinger boson representation, various trial states can be defined on the condition that the mean field approximation is useful. In these states, the su(2) ⊗ su(1,1)-coherent state is mainly discussed in this paper.
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.