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Physical Review A

Characterization of nonclassical optical fields by photodetection statistics

B. Picinbono, C. Bendjaballah

Published Jan 18, 2005• 19 citations
Publisher

Abstract

Experiments on nonclassical optical fields have recently been performed. One of the main properties reported is the antibunching effect of photoelectrons, a property that cannot be explained in the framework of the classical theory of optical fields. By carefully studying the random point process of the detection of the optical field, we show that bunching and antibunching effects can be fully explained by a concidence function. In the classical theory, this function is a correlation function which introduces necessarily a bunching effect. But this coincidence function has no reason to be in any case a correlation function. Therefore antibunching effects can simply be derived from the properties of the coincidence function. After having given a precise definition of this coincidence function, some of its properties are discussed and especially its relationship with bunching and antibunching effects. Similarly, its relationship with statistical properties of lifetimes and intervals between points of the process is established. Various examples are presented and analyzed. Several calculations and computer simulations highlight the theoretical results.

DOI
10.1103/physreva.71.013812

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Experiments on nonclassical optical fields have recently been performed. One of the main properties reported is the antibunching effect of photoelectrons, a property that cannot be explained in the framework of the classical theory of optical fields. By carefully studying the random point process of the detection of the optical field, we show that bunching and antibunching effects can be fully explained by a concidence function. In the classical theory, this function is a correlation function which introduces necessarily a bunching effect. But this coincidence function has no reason to be in any case a correlation function. Therefore antibunching effects can simply be derived from the properties of the coincidence function. After having given a precise definition of this coincidence function, some of its properties are discussed and especially its relationship with bunching and antibunching effects. Similarly, its relationship with statistical properties of lifetimes and intervals between points of the process is established. Various examples are presented and analyzed. Several calculations and computer simulations highlight the theoretical results.
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.
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.
We consider the situation in which the field of quantum particles such as photons (bosons) or electrons (fermions) undergoes a significant attenuation during the detection process. We characterize the second-order statistical properties of the field by the bunching for bosons and the antibunching for fermions. Bunching and antibunching effects are derived from the time intervals distributions of the random point process of detection instead of the moments of the number of particles registered in a given time interval. Fields of both beams are supposed to be in chaotic states and have second-order time correlation functions of exponential profile and of arbitrary correlation time. A test is proposed for nonclassical states: the behavior of the antibunching function is a nondecreasing function near the origin of the time axis.
This paper is concerned with the study of statistical properties of various optical fields obtained by several types of laser intensity modulation from the stationary random point process point of view. We essentially consider intensity correlations and time intervals distributions (TID). A generalization of the already known TID is given: It is the probability distribution (MTID) of the time interval between one photoelectron and the n th following photoelectron. Some theoretical and experimental results are presented in the general case (the time intervals considered are not necessarily small compared to the coherence time of the field). The modulation effect can be characterized by the photoelectron bunching effect. The TID and the MTID are significantly dependent upon the spectral profile of the field, if the used time scale is of the order of the coherence time. On the other hand, the MTID can be interpreted as the probability of the sum over time intervals between two successive photoelectrons. We treat two cases: If the variables of this sum are uncorrelated, we verify the central-limit theorem; if the variables are correlated we study the influence of the correlations when the number of summed variables becomes large.
We show that time distributions of photoelectrons can best be described by a generalized probability-density function which we call the probability-density functional. The probability-density functional gives complete information about the statistical properties of any random-point process, and provides the most natural definitions of the ensemble average and generating functional of a quantity distributed in time. Using this functional, we develop a new approach to photoelectron statistics that systematically relates various joint probability distributions of photoelectrons to one another and to the corresponding generating functionals. We demonstrate that this method can be used to derive previously obtained results and we use it to obtain new results. In particular, the amount of information about the statistical properties of photoelectrons that is contained in a probability distribution can be obtained by comparing the corresponding generating functional with the probability-density functional. Further, we examine time derivatives of the probability distribution for the number of counts and develop new relationships for probability distributions of triggered counting. Finally, we apply our results to obtain an ``inverse'' relationship between photon bunching (antibunching) and super-Poisson (sub-Poisson) statistics.