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Mathematische Nachrichten

Characterization of Classical and Quantum Poisson Systems by Thinnings and Splittings

Karl‐Heinz Fichtner, Wolfgang Freudenberg, Volkmar Liebscher

Published Oct 1, 2000• 4 citations
Publisher

Abstract

There exist several well–known characterizations of Poisson and mixed Poisson point processes (Cox processes) by thinning and splitting procedures. So a point process is necessarily a Cox process if for arbitrary small thinning parameter it can be obtained by a thinning of some other point process [30]. Poisson processes are characterized by the independence of the two random subconfigurations obtained by an independent splitting of the configuration into two parts [11]. For quantum mechanical particle systems beam splittings which are well–known in quantum optics provide analogous procedures. It is shown that coherent states respectively mixtures of them can be characterized in the same way as Poisson processes and Cox processes. Moreover, for the position distributions of these states which are “classical” point processes just the above mentioned characterizations are obtained. As example of mixed coherent states we consider Gaussian states which arise as equilibrium states of ideal Bose gases.

DOI
10.1002/1522-2616(200010)218:1<25::aid-mana25>3.0.co;2-y

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There exist several well–known characterizations of Poisson and mixed Poisson point processes (Cox processes) by thinning and splitting procedures. So a point process is necessarily a Cox process if for arbitrary small thinning parameter it can be obtained by a thinning of some other point process [30]. Poisson processes are characterized by the independence of the two random subconfigurations obtained by an independent splitting of the configuration into two parts [11]. For quantum mechanical particle systems beam splittings which are well–known in quantum optics provide analogous procedures. It is shown that coherent states respectively mixtures of them can be characterized in the same way as Poisson processes and Cox processes. Moreover, for the position distributions of these states which are “classical” point processes just the above mentioned characterizations are obtained. As example of mixed coherent states we consider Gaussian states which arise as equilibrium states of ideal Bose gases.
Based on a model for general beam splittings we search for states of boson systems which are invariant under the combination of the evolution given by the splitting procedure and some inherent evolution. It turns out that for finite systems only trivial invariant normal states may appear. However, for locally normal states on a related quasilocal algebra representing states of infinite boson systems, one can find examples of nontrivial invariant states. We consider as example a beam splitting combined with a contraction compensating the loss of intensity caused by the splitting process. In general, we observe interesting connections between the splitting procedure and certain thinning operations in classical probability theory. Several applications to physics seem to be natural since these beam splitting models are used to describe measuring procedures on electromagentic fields.
The statistical distribution for the number of bosons in a specified subsystem of a finite-dimensional multilevel system is obtained and investigated under conditions for which the total number of particles is a random variable characterized by the Poisson distribution. The resulting hyper-Poisson distribution is determined by the confluent hypergeometric function and contains as its particular limiting cases the Poisson distribution, as well as the single-mode and multimode thermal distributions. The developed model can be efficiently used to describe the thermal characteristics of bounded quantum systems, as well as in the problems of statistical reconstruction of optical quantum states, processes, and detectors.
We examine the coherence properties of stationary light obtained by the superposition of nonstationary independent emissions occurring at random space–time points. The positions of the points are independent and uniformly distributed over the volume of the source. The emission times fluctuate in accordance with a stationary renewal point process. This process admits of super-Poisson, sub-Poisson, or Poisson behavior. The individual emissions are assumed to be in a coherent, chaotic, or n state. The statistical nature of the random-emission space–time points plays an important role in determining the coherence properties and photon statistics of the total field. This is manifested in the normalized second-order correlation function, which turns out to have the usual form for chaotic light with two additional terms. The first of these is determined by the statistical nature of the individual emissions (it is positive for coherent or chaotic, but zero for single-photon, emissions). The second term is governed by the statistics of the primary excitations (it is positive for super-Poisson, zero for Poisson, and negative for sub-Poisson excitations). Both additional terms become small for light with a high degeneracy parameter (many total photons per emission lifetime). The behavior of the light is then asymptotically chaotic. In the opposite limit, when the degeneracy parameter is small (or the emissions are instantaneous), the correlation properties of the primary excitations are directly transferred to the correlation properties of the photons. The first-order spatial-coherence properties of the field turn out to be identical with those of chaotic light (i.e., the van Cittert–Zernike theorem is obeyed), although the second-order properties differ. The photon-counting distribution reflects the character of the correlation function. Thus sub-Poisson primary excitations, together with n-state emissions, can lead to sub-Poisson photon counts under certain conditions. Such nonclassical light may be made arbitrarily intense if interference effects are eliminated by detecting many spatial modes. As an example of our theory, we cite a Franck–Hertz experiment excited by a space-charge-limited electron beam. If the electron excitations are represented as a sub-Poisson renewal point process, and the photon emissions as one-photon states, the light generated should be antibunched and sub-Poisson.
The analysis of the statistical properties of the point process (PP) of photon detection times can be used to determine whether or not an optical field is classical, in the sense that its statistical description does not require the methods of quantum optics. This determination is, however, more difficult than ordinarily admitted and the first aim of this paper is to illustrate this point by using some results of the PP theory. For example, it is well known that the analysis of the photodetection of classical fields exhibits the so-called bunching effect. But this property alone cannot be used to decide the nature of a given optical field. Indeed, we have presented examples of point processes for which a bunching effect appears and yet they cannot be obtained from a classical field. These examples are illustrated by computer simulations. Similarly, it is often admitted that for fields with very low light intensity the bunching or antibunching can be described by using the statistical properties of the distance between successive events of the point process, which simplifies the experimental procedure. We have shown that, while this property is valid for classical PPs, it has no reason to be true for nonclassical PPs, and we have presented some examples of this situation also illustrated by computer simulations.
We study the universal spatial statistics of point-like topological defects formed during the nonequilibrium condensation of a coherently coupled Bose gas using the stochastic projected Gross-Pitaevskii equation. The symmetry-breaking transition is driven by a linear quench of the chemical potential, leading to stochastic vortex nucleation in the individual condensate components. When the two components are considered together, these elementary defects may combine across components to emerge as composite topological defects known as full quantum vortices. Beyond the mean defect density predicted by the Kibble-Zurek mechanism (KZM), we investigate the spatial organization of both the elementary and composite defects and show that their positions are well described by a Poisson point process, revealing a universal stochastic geometry. This universality is further described through Voronoi tessellation, whose cell-area statistics follow Poisson-Voronoi predictions. We also introduce the spatial form factor for characterizing the vortex configurations and demonstrate the emergence of a characteristic dip-ramp-plateau structure. Our results establish universal stochastic geometry of topological defects beyond conventional Kibble-Zurek scaling and identify it as a fundamental feature of nonequilibrium condensation in coherently coupled Bose gases.