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Journal of Mathematical Physics

Analytic and contour representations in the unit disk based on SU(1,1) coherent states

Chao Lei, Apostol Vourdas, Alfred Wünsche

Published Nov 1, 2005• 5 citations
Publisher

Abstract

A contour representation in the unit disk based on SU(1,1) coherent states is introduced. The scalar product is given by a contour integral. The regions of convergence of the functions representing ket and bra states are studied. An analytic representation in the unit disk is also considered, where the scalar product is represented by an integral over the unit disk, with the Lobachevsky measure. Various relations which connect these analytic functions with other phase-space quantities are derived.

DOI
10.1063/1.2098527

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A contour representation in the unit disk based on SU(1,1) coherent states is introduced. The scalar product is given by a contour integral. The regions of convergence of the functions representing ket and bra states are studied. An analytic representation in the unit disk is also considered, where the scalar product is represented by an integral over the unit disk, with the Lobachevsky measure. Various relations which connect these analytic functions with other phase-space quantities are derived.
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Various Euclidean, hyperbolic and elliptic analytic representations are introduced and relations among them are discussed. The Bargmann analytic representation in the complex plane is considered and its relation to other phase-space methods for the harmonic oscillator is reviewed. The general theory that relates the growth of analytic functions with the density of their zeros is applied to Bargmann functions and it leads to theorems on the completeness of sequences of Glauber coherent states. Two hyperbolic analytic representations in the unit disc, based on SU (1, 1) coherent states and also on phase states are introduced. A third analytic representation in the complex plane based on Barut–Girardello states is also considered and transformations which relate it to the other ones are studied. In the case of systems with finite-dimensional Hilbert space, an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced. The Berezin formalism in the Euclidean, hyperbolic and elliptic cases is discussed. Contour analytic representations in these three cases are also presented.
We consider two analytic representations of the SU(1,1) Lie group: the representation in the unit disc based on the SU(1,1) Perelomov coherent states and the Barut - Girardello representation based on the eigenstates of the SU(1,1) lowering generator. We show that these representations are related through a Laplace transform. A `weak' resolution of the identity in terms of the Perelomov SU(1,1) coherent states is presented which is valid even when the Bargmann index k is smaller than . Various applications of these results in the context of the two-photon realization of SU(1,1) in quantum optics are also discussed.
The phase states of a harmonic oscillator are considered and used to define an analytic representation in the unit disk. The Weyl semigroup in the number-angle phase space is studied. More general states called negative binomial states are introduced, and the corresponding analytic representations in the unit disk are defined. M\"obius conformal mappings are used for the study of SU(1,1) transformations. They define alternative types of one-mode squeezing based on the discrete series of SU(1,1) representations.
Resolutions of the identity in terms of contour integrals that involve coherent states and their `complementary states', are derived. The complementary states are auxiliary states that help in the formulation of these resolutions of the identity. Since the coherent states are normalizable inside the unit disc and the complementary states are normalizable outside the unit disc, enlargements of the Hilbert space are considered which allow the construction of resolutions of the identity in terms of contour integrals in rings . Several examples of our formalism are presented.