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Physical Review A

Analytic representations in the unit disk and applications to phase states and squeezing

Apostol Vourdas

Published Feb 1, 1992• 62 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.45.1943

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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.
The inner - outer part factorization of analytic representations in the unit disc is used for an effective characterization of the number-phase statistical properties of a quantum harmonic oscillator. It is shown that the factorization is intimately connected with the number-phase Weyl semigroup and its properties. In the Barut - Girardello analytic representation the factorization is implemented as a convolution. Several examples are given which demonstrate the physical significance of the factorization and its role for quantum statistics. In particular, we study the effect of phase-space interference on the factorization properties of a superposition state.
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.
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.