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International Journal of Theoretical and Mathematical Physics

Singularities in Quasidistributions, Their Regularization, and Nonclassical Number and Wave Statistics

Jan Peřina, J. Křepelka

Published Jan 1, 2017• 0 citations
OpenAlex

Abstract

Compared with earlier investigations we follow a way of arising singularities in quasidistributions as related to nonclassical photocount and wave statistics for nonlinear optical processes described by Gaussian statistics, also from the point of view of testing functions. Further we illustrate a process of regularization of singular quasidistributions so that regularized quasidistributions can provide measurable quantities. Results obtained can be applied to optical down-conversion as well as to Raman scattering provided that classical strong coherent pumping fields are used.

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Compared with earlier investigations we follow a way of arising singularities in quasidistributions as related to nonclassical photocount and wave statistics for nonlinear optical processes described by Gaussian statistics, also from the point of view of testing functions. Further we illustrate a process of regularization of singular quasidistributions so that regularized quasidistributions can provide measurable quantities. Results obtained can be applied to optical down-conversion as well as to Raman scattering provided that classical strong coherent pumping fields are used.
Abstract We consider nonclassical wave and number quantum statistics, and perform a decomposition of quasidistributions for nonlinear optical down-conversion processes using Bessel functions. We show that negative values of the quasidistribution do not directly represent probabilities; however, they directly influence measurable number statistics. Negative terms in the decomposition related to the nonclassical behavior with negative amplitudes of probability can be interpreted as positive amplitudes of probability in the negative orthogonal Bessel basis, whereas positive amplitudes of probability in the positive basis describe classical cases. However, probabilities are positive in all cases, including negative values of quasidistributions. Negative and positive contributions of decompositions to quasidistributions are estimated. The approach can be adapted to quantum coherence functions.
Nonlinear interaction of light with matter is described from the quantum-statistical point of view. The cases of two-photon absorption, Raman transition, sum-frequency generation, parametric amplification, and incoherent scattering are discussed. It is shown that the nonlinear optical effects depend strongly on the statistical properties of the light fields. The rate of nonlinear absorption, generation, and amplification is higher for chaotic than for coherent, and also higher for multimode than for single-mode pump fields. Measurements of the statistics of the output fields may yield information about the statistics of the input fields and the properties of the medium.
Statistical properties of optical fields in nonlinear couplers composed of two waveguides in which Raman or Brillouin processes (with classical pumping) are in operation and which are mutually connected through the mutual Stokes and/or anti-Stokes linear interactions are investigated within the framework of a generalized superposition of coherent fields and quantum noise. Heisenberg equations describing the couplers are solved both analytically under special conditions and numerically in general cases. Regimes for nonclassical properties of optical fields, such as sub-Poissonian photon-number statistics, negative reduced moments of integrated intensity and squeezing of quadrature fluctuations are discussed for the cases of single and compound fields. General results are compared with those from the short-length approximation.
A unified treatment of the description of the statistical dynamical properties of two-mode quadratic optical parametric processes with intense coherent or stochastic pumping (parametric amplification, frequency conversion, second subharmonic generation, etc. as special cases are involved) including the lossy mechanism is developed. The Heisenberg-Langevin equations are derived and their solutions are used to obtain the quantum characteristic functions and the corresponding quasidistributions. The Fokker-Planck equation approach is adopted as well and the equivalence of both the approaches is explicitly found.
In quantum optics, measurement statistics---for example, photocounting statistics---are considered nonclassical if they cannot be reproduced with statistical mixtures of classical radiation fields. We have formulated a necessary and sufficient condition for such nonclassicality. This condition is given by a set of inequalities that tightly bound the convex set of probabilities associated with classical electromagnetic radiation. Analytical forms for full sets and subsets of these inequalities are obtained for important cases of realistic photocounting measurements and unbalanced homodyne detection. As an example, we consider photocounting statistics of phase-squeezed coherent states. Contrary to a common intuition, the analysis developed here reveals distinct nonclassical properties of these statistics that can be experimentally corroborated with minimal resources.