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.
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.
We define a quasiprobability distribution
SNP(n,\ensuremath{\theta}) which describes the quantum statistics of the photon number and phase observables of a single-mode field (or, equivalently, a harmonic oscillator). The properties of
SNP(n,\ensuremath{\theta}) are the photon number and phase analogies of the properties of Wigner's original function, which describes the position and momentum observables. For example, the marginals of
SNP(n,\ensuremath{\theta}) are the continuous phase and the discrete photon-number probability distributions. We give examples of the
SNP(n,\ensuremath{\theta}) representation of various states and show, in particular, that
SNP(n,\ensuremath{\theta}) displays the quantum interference associated with Schr\"odinger cat states. We also describe how
SNP(n,\ensuremath{\theta}) can be determined from quantities that are, in principle, measurable.
Abstract Given its well known spectral decomposition profile, the 1-dim harmonic oscillator potential modified by an inverse square (1-dim angular momentum-like) contribution works as an efficient platform for probing classical and quantum information quantifiers in the context of the phase-space Weyl-Wigner formalism. In particular, the phase-space informational content related to the canonical ensemble driven by such a singular oscillator can be quantified in terms of well established analytical structures. Considering that, on one hand, the singular oscillator produce a spectral decomposition profile equivalent to that one of the unmodified harmonic system – in the sense that they result into identical thermodynamic statistics, even for different statistical mixtures – on the other hand, a more complete scrutinization of their phase-space information content can capture some different aspects of the encoded information for the related quantum ensembles. Besides the identification of decoherence effects, the Wigner flow analysis is presumedly useful in identifying stable quantum configurations, according to finite temperature and interaction parameter values. Unexpectedly, our results show that the equivalence between the statistical (quantum) mechanics of the anharmonic singular oscillator and an ordinary harmonic oscillator can also be extended to the phase-space quantum purity quantifier, which is analytically computed and reproduces exactly the same quantum ensemble statistical mixture profile, which does not depend on interaction parameter values.