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Physical Review

Quantum Theory of an Optical Maser. III. Theory of Photoelectron Counting Statistics

Marlan O. Scully, Willis E. Lamb

Published Mar 10, 1969• 168 citations
Publisher

Abstract

In this paper we determine the photoelectron counting statistics produced by the fully quantum-mechanical laser considered in the first paper of this series. The problem of obtaining the photocount distribution from the now known photon statistics is solved in a completely quantum-mechanical fashion. The time evolution of the combined photodetector-laser system is derived. The techniques developed for the solution of this problem are of general interest in the area of nonequilibrium quantum statistical dynamics.

DOI
10.1103/physrev.179.368

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In this paper we determine the photoelectron counting statistics produced by the fully quantum-mechanical laser considered in the first paper of this series. The problem of obtaining the photocount distribution from the now known photon statistics is solved in a completely quantum-mechanical fashion. The time evolution of the combined photodetector-laser system is derived. The techniques developed for the solution of this problem are of general interest in the area of nonequilibrium quantum statistical dynamics.
It is well known that photoelectric measurements yield information about the statistical behavior of fluctuating light beams. The probability of n photoelectrons being ejected, in a fixed time interval, from the photosensitive surface of the photoelectric detector, upon which the light beam is normally incident, is a linear (Poisson) transform of the probability density for the intensity of the beam. With the help of some plausible assumptions, the present analysis provides solutions to the problem of inverting the Poisson transform, thus determining the probability density for the intensity from experimentally obtained photocounting distributions. The effectiveness of the method is demonstrated by an actual inversion of a typical experimental counting distribution. The technique is of particular interest in connection with efforts to understand the statistical behavior of optical fields, especially of laser fields.
In this paper we formulate the general problem of determining the photoelectron "counting" distribution resulting from an electromagnetic field impinging on a quantum detector. Although the detector model used was derived quantum mechanically, our treatment is wholly classical and includes all results known to date. This combination is commonly referred to as the semiclassical approach. The emphasis, however, lies in directing the problem towards optical communication. The electromagnetic field is assumed to be the sum of a deterministic signal and a zero-mean narrow-band Gaussian random process, and is expanded in a Karhunen-Loève series of orthogonal functions. Several examples are given. It is shown that all the results obtainable can be written explicitly in terms of the noise covariance function. Particular attention is given to the case of a signal plus white Gaussian noise, both of which are band-limited to\pm BHz. Since the result is a fundamental one, to add some physical insight, we show four methods by which it can be obtained. Various limiting forms of this distribution are derived, including the necessary conditions for those commonly accepted. The likelihood functional is established and is shown to be the product of Laguerre polynomials. For the problem of continuous estimation, the Fisher information kernel is derived and an important limiting form is obtained. The maximum a posteriori (MAP) and maximum-likelihood (ML) estimation equations are also derived. In the latter case the results are also functions of Laguerre polynomials.
A quantum statistical analysis of an optical maser is presented in generalization of the recent semiclassical theory of Lamb. Equations of motion for the density matrix of the quantized electromagnetic field are derived. These equations describe the irreversible dynamics of the laser radiation in all regions of operation (above, below, and at threshold). Nonlinearities play an essential role in this problem. The diagonal equations of motion for the radiation are found to have an apparent physical interpretation. At steady state, these equations may be solved via detailed-balance considerations to yield the photon statistical distribution ${\ensuremath{\rho}}_{n,n}$. The resulting distribution has a variance which is significantly larger than that for coherent light. The off-diagonal elements of the radiation density matrix describe the effects of phase diffusion in general and provide the spectral profile ${| E(\ensuremath{\omega}) |}^{2}$ as a special case. A detailed discussion of the physics involved in this paper is given in the concluding sections. The theory of the laser adds another example to the short list of solved problems in irreversible quantum statistical mechanics.
The photon counting statistics of light is studied, in which the various types of photon counting distribution can be represented by Nakagami'sm-distribution. To investigate this theoretical utility we have measured the photon counting distribution of laser light scattered by a very dilute solution in which non-Gaussian statistics appear. The comparison is also made by the factrial moment between the experimental value and theoretical one. The experimental distribution is in agreement with the theoretical distribution.
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.
A simple semiclassical derivation is given of the statistics of the photoelectrons emitted from a photosurface illuminated by light with statistically time dependent intensity. Comparison is made with the quantum mechanical expressions. Experiments are described that yield the probability of emission ofnphotoelectrons within a fixed time interval from a photocathode illuminated by a laser below and above threshold. The experimental results are compared with the theory developed in this paper and expressions derived by Glauber.