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Journal of Physics A Mathematical and General

Population statistics and the counting process

Eric Jakeman, Terence J. Shepherd

Published Oct 1, 1984• 30 citations
Publisher

Abstract

The authors investigate the outcome of different monitoring schemes applied to small classical populations. Explicit calculation for a solvable model establishes the relationship between the statistical properties of the population and those of the counting processes. Parallels are drawn with operator ordering in the second quantised formulation of photodetection.

DOI
10.1088/0305-4470/17/14/004

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The authors investigate the outcome of different monitoring schemes applied to small classical populations. Explicit calculation for a solvable model establishes the relationship between the statistical properties of the population and those of the counting processes. Parallels are drawn with operator ordering in the second quantised formulation of photodetection.
The semiclassical formalism for integrated statistics in photoelectron counting is compared with a method of population monitoring for a single-mode cavity field evolving according to an arbitrary Markov process. Equations governing the exact integrated statistics in the two approaches are shown to be identical up to a single-photon loss process associated with the photodetection. Specific examples are provided, illustrating the relation of these methods to explicitly second-quantized formulations.
A model for photodetection is presented which permits evolution of the detected field during the detection process. The integrated univariate and bivariate statistics of the photocount are obtained exactly for detection of single-mode thermal, coherent and free-field cavity radiation. The scheme encompasses the results of three hitherto disparate photocounting models, and demonstrates explicitly the relation between the univariate photon counting statistics and the photon statistics of the field. First-order photocount correlation functions are derived and compared with the corresponding cavity photon population correlations. Photocount antibunching is examined and found to be possible in this model.
One can attempt to derive some information of the field from the measured random properties of the photoelectron process: this is the object of photodetection theory developed first for classical fields. This chapter presents a review of photodetection theory, quantum theory and population monitoring model. It discusses two interesting applications to photostatistics. The photodetection equation is used to classify quantum and classical fields. The chapter presents an example of collective antibunching of photons. The double random character of the photodetection process prevents a simple correspondence between the statistical properties of the intensity and the statistics of the photocounts. However it is more instructive to connect the photoelectron process with the quantum nature of the electromagnetic field. Although the problem of photodetection appears to be satisfactorily solved for free fields interacting only with the detector, a heuristic model permits evolution of the detected field during the detection process.