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Applied Categorical Structures

Analytic Representations Based on SU(1,1) Lie Algebra Coherent States for Squeezed Displaced Fock States

G. M. Abd Al-Kader, A.‐S. F. Obada

Published Nov 6, 2006• 0 citations
Publisher
DOI
10.1007/s10485-006-9032-9

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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 introduce the concept of algebra eigenstates which are defined for an arbitrary Lie group as eigenstates of elements of the corresponding complex Lie algebra. We show that this concept unifies different definitions of coherent states associated with a dynamical symmetry group. On the one hand, algebra eigenstates include different sets of Perelomov's generalized coherent states. On the other hand, intelligent states (which are squeezed states for a system of general symmetry) also form a subset of algebra eigenstates. We develop the general formalism and apply it to the SU(2) and SU(1,1) simple Lie groups. Complete solutions to the general eigenvalue problem are found in the both cases, by a method that employs analytic representations of the algebra eigenstates. This analytic method also enables us to obtain exact closed expressions for quantum statistical properties of an arbitrary algebra eigenstate. Important special cases such as standard coherent states and intelligent states are examined and relations between them are studied by using their analytic representations.