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Physical Review Letters

Neural Counting and Photon Counting in the Presence of Dead Time

Malvin Carl Teich, William J. McGill

Published Mar 29, 1976• 34 citations
Publisher

Abstract

The usual stimulus-based neural counting model for audition is found to be mathematically identical to the well-know semiclassical formalism for photon counting. In particular, we explicitly demonstrate the equivalence of McGill's noncentral negative binomial distribution and Pe\ifmmode \check{r}\else \v{r}\fi{}ina's multimode confluent hypergeometric distribution for a coherent signal imbedded in chaotic noise. Dead-time corrections, important both in neural counting and in photon counting, are incorporated in a generalized form of this distribution. Some specific implications of these results are discussed.

DOI
10.1103/physrevlett.36.754

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The usual stimulus-based neural counting model for audition is found to be mathematically identical to the well-know semiclassical formalism for photon counting. In particular, we explicitly demonstrate the equivalence of McGill's noncentral negative binomial distribution and Pe\ifmmode \check{r}\else \v{r}\fi{}ina's multimode confluent hypergeometric distribution for a coherent signal imbedded in chaotic noise. Dead-time corrections, important both in neural counting and in photon counting, are incorporated in a generalized form of this distribution. Some specific implications of these results are discussed.
We report a series of optical experiments that verify the theoretical photocounting distributions obtained by Diament and Teich for triangularly and sinusoidally modulated laser radiation. Another series of experiments validates the (nonparalyzable) dead-time-modified versions of these formulas obtained by Cantor and Teich. A new expression is obtained for the counting distribution in the presence of modulation and paralyzable dead time. The results have application in photon counting, nuclear counting, and neural counting.
Abstract Photon counting provides a valuable tool for measuring quantum properties of light. Unavoidable imperfections of photon detectors lead to discrepancies between the statistics of photons and statistics of photon counts. Here we consider dead-time corrections and influence of intensity fluctuations for simulated Poissonian statistics of photo counts for the pulsed or continuous wave coherent light sources. We analyse dependence between the statistical moments of corrected distribution and the mean photon number, dead-time, and level of intensity fluctuations.
Abstract Measurement of photon statistics is an important tool for the verification of quantum properties of light. Due to the various imperfections of real single photon detectors, the observed statistics of photon counts deviates from the underlying statistics of photons. Here we analyze statistical properties of coherent states, and investigate a connection between Poissonian distribution of photons and sub-Poissonian distribution of photon counts due to the detector dead-time corrections. We derive a functional dependence between the mean number of photons and the mean number of photon counts, as well as connection between higher-order statistical moments, for the pulsed or continuous wave coherent light sources, and confirm the results by numerical simulations.
A computer simulation method is employed to investigate the dead-time modified photocount detection distribution where the ratio of coherence time of the incident radiation to the counting time τc/T ≦ 1. The principal result is to demonstrate that a dead-time correction formula that was derived for τc/T ≫ 1 may be modified to be applicable to arbitrary coherence times. The simulation technique was verified by comparing simulated results with known results for dead-time modified photocount distribution for radiation with τc/T ≫ 1 and for unmodified photocount distribution for radiation with τc/T ≦ 1.
We reconsider various approaches to the quantum theory of photodetection from the point of view of the quantum theory of measurement, and show that important differences between them depend upon the manner in which they take into account the modification of the field distribution produced by the presence of the detector. We show that the Mandel photon counting formula may lead to unphysical results, such as negative probabilities, in some situations just because this modification is not incorporated into the model. We also show that the recently developed quantum theory of continuous measurements provides a completely satisfactory framework for discussing photon counting experiments. As an illustration, an exact derivation of the photon statistics of a single-mode free field is presented, and some of the results are shown to be identical to those derived earlier by Mollow and others. We also indicate how our approach can be extended to discuss the photon statistics of fields in interaction with sources and reservoirs.
Exact forward photocounting distributions for a source of optical radiation with arbitrary statistics are obtained in the presence of photodetector dead time. In particular, we examine the counting and pulse-interval distributions that arise from amplitude-stabilized radiation, from chaotic radiation, and from a Van der Pol laser with arbitrary excitation. The exact dead-time-corrected Poisson distribution is graphically compared with a previous approximate result and with the uncorrected Poisson. Plots of the Bose–Einstein distribution clearly indicate the dramatic anti-bunching effects of the dead time in overcoming the inherent bunching of this distribution. A simplified approximate solution is also found. for the Van der Pol laser above threshold; this result is similar to light from an amplitude-stabilized source incident on a photodetector with a gaussian-distributed dead time. Information about photodetector dead-time variation can therefore be obtained either by using an amplitude-stabilized source or by properly choosing system parameters such that irradiance fluctuations are averaged out.