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Journal of Physics A Mathematical Nuclear and General

The statistics of electron counting

I M Every

Published Sep 1, 1973• 5 citations
Publisher

Abstract

The probability P(n,t 1 ,T 2 ) of counting n electrons in the time interval from t 1 to t 2 is derived from a quantum-electrodynamic treatment of an electron multiplier. Although the expressions obtained are similar to those obtained by Glauber for photons there remain fundamental differences. The interest of these results for existing and proposed experiments on electron beams is outlined.

DOI
10.1088/0305-4470/6/9/012

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We consider the probability P(n)(t0,t){P}^{(n)}({t}_{0}, t)P(n)(t0​,t) of counting nnn photoelectrons during any time interval [t0,t{t}_{0}, tt0​,t] in a photodetection model built in the general framework of quantum field theory. Owing to an equilibrium assumption on the detector state, P(n)(t0,t){P}^{(n)}({t}_{0}, t)P(n)(t0​,t) is evaluated from the incident field and some electron-correlation functions. The conditions for P(n)(t0,t){P}^{(n)}({t}_{0}, t)P(n)(t0​,t) to be a compound Poisson distribution as expected are discussed. This leads to the definition of coherence properties for the electrons of the detector in connection with the reliability of the detection operation in counting experiments.
A theory of electromagnetic field measurement by means of photoionization is developed and applied to photoelectron counting. A probability theory involving multitime joint probability functions for a sequence of photoionizations is formulated. A general quantum-theory definition is proposed for the nonexclusive probability function which occurs in the probability theory. Approximations are then introduced to derive expressions for this probability function which involve correlation functions of the photoionization detector and the electromagnetic field-plus-source. The general theory is used to derive quantities of interest in photo-ionization counting experiments. Expressions are derived for (1) the probability Pn(t,t+T){P}_{n}(t, t+T)Pn​(t,t+T) that nnn photo-ionizations are observed in the time interval ttt to t+Tt+Tt+T, and (2) quantities related to Pn(t,t+T){P}_{n}(t, t+T)Pn​(t,t+T), such as its generating function and various moments. Pn(t,t+T){P}_{n}(t, t+T)Pn​(t,t+T) is found to be a compound Poisson distribution determined by the density operator of the field when the latter is expressed in Glauber's PPP representation. Using this result, the character of Pn(t,t+T){P}_{n}(t, t+T)Pn​(t,t+T) is examined for several specific density operators. These correspond to a coherent state, various fields with the mode phases distributed independently of the mode amplitudes, and a "spread-out" coherent state.
A theory of electron-counting processes is formulated in terms of the Liouville space method. The developed model is based on the quantum Markov processes developed by Davies and Srinivas. The time evolution of an electron system interacting with an electron counter and the counting probability distribution are investigated. The average value, fluctuation, and correlation function of the electron number registered by the counter are calculated, and the sub-Poissonian distribution and antibunching correlation of the electrons are obtained for an arbitrary noncorrelated initial state of electrons. For a correlated initial state, depending on the initial correlation among the electrons and on the electron counters used in the measurement, the statistics for the electron number registered by the counter is characterized as a sub-Poissonian, Poissonian, or super-Poissonian distribution, and the intensity correlation function exhibits antibunching or bunching correlation. As an example, the state with correlation between the up-spin and down-spin electrons is considered. The electron counting processes under the influence of a chaotic electron source are also considered, and the effect of the chaotic electron source on the counting statistics is investigated.