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Physical Review A

Expressions of various joint probability distributions of photoelectrons in terms of a photocount distribution P(n, t1 , t2 )

Masahito Ueda

Published Sep 1, 1988• 9 citations
Publisher

Abstract

Expressions of various joint probability distributions of photoelectrons in terms of the photocount distribution P(n,${t}_{1}$,${t}_{2}$) in which n photoelectrons are registered between ${t}_{1}$ and ${t}_{2}$ are systematically obtained with extensive use of an introduced simple notation. The expressions obtained are used to find two pairs of expressions of the second-order and associated intensity correlation functions in terms of P(n,${t}_{1}$,${t}_{2}$). Some applications to quantum optics are also made to obtain results in agreement with previous work.

DOI
10.1103/physreva.38.2937

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Expressions of various joint probability distributions of photoelectrons in terms of the photocount distribution P(n,t1{t}_{1}t1​,t2{t}_{2}t2​) in which n photoelectrons are registered between t1{t}_{1}t1​ and t2{t}_{2}t2​ are systematically obtained with extensive use of an introduced simple notation. The expressions obtained are used to find two pairs of expressions of the second-order and associated intensity correlation functions in terms of P(n,t1{t}_{1}t1​,t2{t}_{2}t2​). Some applications to quantum optics are also made to obtain results in agreement with previous work.
We express the exact probability density distribution function as the product of a gamma distribution and a series of associated Laguerre polynomials, with the expansion coefficients determined by moments of the integrated intensity. Orthogonal polynomials with respect to the exact probability distribution function are then expanded in similar fashion. These polynomials are then used to construct an expansion of the joint probability distribution function in the second-order photoelectron statistics. Since the polynomials are identical with the corresponding Laguerre polynomials when the exact probability distribution function is the gamma distribution, the new polynomials are generalized versions of the associated Laguerre polynomials. The joint photoelectron statistics may be studied with these new polynomials.
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.
The corrections to the photocount distributions, due to dead-time effects, are evaluated by use of the time-interval probability density of the counts. The normalized cumulative distribution function J k ( T , τ), associated with the detection of fluctuating light beams during a counting time interval T , with a counter having a constant dead-time τ, appear naturally as a convenient quantity to be used in comparing experimental results with theoretical predictions. The quantities J k ( T , τ) obey a simple recurrence relation and are readily derived from the photocount distribution p ( k, T , τ). A general expression is derived for J k ( T , τ) and p ( k, T , τ) associated with the detection of fluctuating light beams. Typical cumulative distribution functions, relating to the detection of thermal light, are evaluated as functions of the degree of polarization of the light, of the average number of photocounts < n 0 > in the absence of dead-time effects and of the ratio τ/ T . The distributions J k ( T , τ) and p ( k, T , τ) relating to detection of laser light are also calculated in a typical case.
In recent papers by Cohen-Tannoudji and Dalibard1 and Zoller et al.2 the statistics of quantum jumps in a three-level atom are analyzed in terms of the joint probability density W(t + τ;t) for a photon emission at time t followed by the next photon emission at time t+ τ. W(t+ τ;t) differs from the second-order correlation function G(2) (t + τ;t) often used to characterize photon statistics; the latter gives the joint emission probability without the condition that there be no emissions in the interval t to t + τ. Alter dividing by the density for single-photon emission, W(t + τ;t) gives the probability density for a delay τ between photon emissions.
The probability distribution of photoelectrons detected simultaneously at L different points of a photocathode is studied under the assumptions that the incident light is Gaussian with arbitrary state of coherence and polarization. The generating function is expressed as an infinite product of eigenvalues of a matrix integral equation, the kernel being the correlation matrix. The case of spectrally pure light beams is discussed in detail and the relation to previous work is given. The notion of polarizationally pure beams is introduced, for which the polarization state of light is independent of the spatial and temporal coherence. The generating function is then expressed in terms of eigenvalues of a simpler integral equation. An approximate distribution valid up to the second moment is derived.
Onefold photoelectron counting distributions are evaluated for coherent light scattered by an ensemble of particles. The ensemble contains, in general, a random number of nonidentical particles. Expressions are obtained for the probability density function and the moments of the scattering intensity. These results reduce to those of Pusey, Schaefer, and Koppel when the particles are identical. Photocount probabilities are also evaluated. The variance, coefficient of skewness, and coefficient of excess are evaluated when the probability density of the intensity contributed by each scatterer is specified by a narrow Gaussian distribution centered about the deterministic mean.