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arXiv:0812.0717 · AIP conference proceedings

Generalized squeezed states for the Jacobi group

Stefan Berceanu, Piotr Kielanowski, Anatol Odzijewicz, Martin Schlichenmaier, Theodore Voronov

Published Jan 1, 2008• 4 citations
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Abstract

We analyze the relationship between the covering of the Jacobi group and the squeezed states. We attach some nonclassical states to the Jacobi group. The matrix elements of the Jacobi group are presented.
DOI
10.1063/1.3043874

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The Jacobi group the semidirect product of the Heisenberg group and the symplectic group [1–3] is an important object in quantum mechanics, geometric quantization, optics [4–6]. The Jacobi group is responsible for the squeezed states [7–12] in quantum optics [5], [6], [13], [14]. Some coherent state systems based on SiegelJacobi domains [15], [16] have been investigated in the framework of quantum mechanics, geometric quantization, dequantization, quantum optics, nuclear structure, and signal processing [17], [18], [19], [20][23]. We have attached in [20] Perelomov’s coherent states [24] to the Jacobi group G1∗= H1(R)nSU(1,1), based on the Siegel-Jacobi disk D1 =C×D1, where D1 = {z ∈ C||z| . ez ∈ H denotes the Perelomov coherent state vector indexed by the points z of the homogeneous manifold M = G/H , where H is compact subgroup of G. The generalized squeezed states of the type introduced in [22], [23], [25] are not new [8, 26, 27]. The paper is organized as follows. §2 collects the main notation concerning
Infinitesimal holomorphic realizations for the Schrödinger-Weil representation and the discrete series representations of the Jacobi group are constructed. Explicit expressions of the basic differential operators are obtained. The squeezed states for the unitary irreducible representation of the Jacobi group are introduced. Matrix elements of the squeezed operators, expectation values of polynomial operators in infinitesimal generators of the Jacobi group, the squeezing region and a description of Mandel's parameter are presented.