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Optica Acta International Journal of Optics

Photoelectron Counting Probabilities for Non-classical Fields

Daniel F. Walls, Gerard J Milburn, H. J. Carmichael

Published Sep 1, 1982• 7 citations
Publisher

Abstract

The single-mode photoelectron counting formula derived using the Glauber-Sudarshan diagonal coherent state representation is generalised to fields for which no well behaved Glauber-Sudarshan P-distribution exists using the generalized P-representation. From this formula the photocount distributions for a number-state and a squeezed-state are calculated.

DOI
10.1080/713820970

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The single-mode photoelectron counting formula derived using the Glauber-Sudarshan diagonal coherent state representation is generalised to fields for which no well behaved Glauber-Sudarshan P-distribution exists using the generalized P-representation. From this formula the photocount distributions for a number-state and a squeezed-state are calculated.
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.
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.
Abstract The Glauber-Sudarshan P-representation for bosons is well-known within quantum optics, and is widely applied to problems involving photon statistics. Less familiar, perhaps, is its fermionic counterpart, introduced by Cahill and Glauber. We present a derivation of both the bosonic and fermionic distributions and, in doing so, demonstrate the reason for the existence of two distinct fermionic forms and the relationship between these. We consider both single mode systems and also multiparticle systems with many modes. Expressions for the moments involving products of mode annihilation and creation operators are obtained. For simplicity only one type of boson or fermion will be considered, but generalising to more types is straightforward.
The Glauber-Sudarshan P-representation is well-known within quantum optics, and is widely applied to problems involving photon statistics. Less familiar, perhaps, is its fermionic counterpart. We present a derivation of both the bosonic and fermionic distributions and, in doing so, demonstrate the reason for the existence of two distinct fermionic forms and the relationship between these. We consider both single mode systems and also multiparticle systems with many modes. For simplicity only one type of boson or fermion will be considered.
Photon number probabilities for a single squeezed mode can exhibit sub-Poissonian or super-Poissonian statistics and can display sharp oscillations in appropriate conditions. We consider photocount probabilities for cw squeezed light occupying a single transverse mode. The squeezing spectrum is assumed to be flat over a bandwidth B, centered at optical frequency Ω/2π. We compute single-fold photoelectron probabilities for a counting interval of duration T. We proceed by decomposing the single transverse mode into a complete set of longitudinal wave-packet modes, defined in terms of prolate spheroidal wave functions. Each of these wave-packet modes is in a separate squeezed state. The Kelley-Kleiner photocount probability can then be written in terms of single-mode photon-number probabilities for these wave packet modes, which are detected with varying efficiencies. We need consider only a finite number of modes, because only about BT of the modes have high enough efficiency to contribute significantly to the photocount probability.