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arXiv:2110.04918

Existence Criterion of Solutions to the Inverse Problem of Photocount Statistics Obtained by the Inverse Bernoulli Transform

P. P. Gostev, Sergey A Magnitskiy, A. S. Chirkin

Published Oct 10, 2021• 0 citations
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Abstract

It is shown that the applicability conditions for the inverse Bernoulli transform method for solving the inverse problem of photocount statistics are determined by the fulfillment of the associativity condition for multiplying the matrices included in this transformation. A general criterion for evaluating the photocount distributions $Q_{m}$ in the case of few-photon light, which makes it possible to establish whether the solution to the inverse problem of photocount statistics by inverse Bernoulli transform method is applicable for $η<0.5$, is found. As an example of application of the obtained criterion, the critical quantum efficiency $η_{cr}$ is found for compound Poisson distribution, below which the solution of the inverse problem of photocount statistics becomes incorrect. Additionally it is shown that the normalization of $Q_{m}$ is not sufficient to obtain a correct solution using the inverse Bernoulli transform.

DOI
10.48550/arxiv.2110.04918

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It is shown that the applicability conditions for the inverse Bernoulli transform method for solving the inverse problem of photocount statistics are determined by the fulfillment of the associativity condition for multiplying the matrices included in this transformation. A general criterion for evaluating the photocount distributions QmQ_{m}Qm​ in the case of few-photon light, which makes it possible to establish whether the solution to the inverse problem of photocount statistics by inverse Bernoulli transform method is applicable for η<0.5η<0.5η<0.5, is found. As an example of application of the obtained criterion, the critical quantum efficiency ηcrη_{cr}ηcr​ is found for compound Poisson distribution, below which the solution of the inverse problem of photocount statistics becomes incorrect. Additionally it is shown that the normalization of QmQ_{m}Qm​ is not sufficient to obtain a correct solution using the inverse Bernoulli transform.
It is shown that the applicability conditions for the inverse Bernoulli transform method to solve the inverse problem of photocount statistics are determined by the fulfillment of the associativity condition for multiplying the matrices included in this transformation. A general criterion for evaluating the photocount distributions Qm{Q}_{m}Qm​ in the case of few-photon light, which makes it possible to establish whether the solution to the inverse problem of photocount statistics by the inverse Bernoulli transform method is applicable for $\ensuremath{\eta}<0.5$, is found. As an example of the application of the obtained criterion, the critical quantum efficiency ${\ensuremath{\eta}}_{\text{cr}}$ is found for compound Poisson distribution, below which the solution of the inverse problem of photocount statistics becomes incorrect. Additionally it is shown that the normalization of Qm{Q}_{m}Qm​ is not sufficient to obtain a correct solution using the inverse Bernoulli transform.
A general criterion for evaluating the photocount distributions QmQ_{m}Qm​ in the case of few-photon light, which makes it possible to establish whether the solution to the inverse problem of photocount statistics by inverse Bernoulli transform method is stable for η<0.5\eta<0.5η<0.5, is found. As an example of application of the obtained criterion, the critical quantum efficiency ηcr\eta_{cr}ηcr​ is found for compound Poisson distribution, below which the solution of the inverse problem of photocount statistics becomes unstable. It is also shown that the normalization of QmQ_{m}Qm​ is not sufficient to obtain a correct solution using the inverse Bernoulli transform.
An analytical recursive method for solving the inverse problem of photocount statistics, that is, for reconstructing the photon-number distribution from the photocount distribution, is proposed for few-photon light. It is shown that if the inverse problem is represented as a system of linear equations, then for finite distributions one can obtain a recurrence formula relating the photocount distribution and the photon-number distribution without restrictions on the quantum efficiency of photodetection.
The multidimensional photon statistics has been determined for the first time from the measurement of the statistics of electrons generated by the detected radiation. The two-dimensional case is described in detail. The used method is based on the application of the characteristic function, which makes it possible to obtain the relation between the two-dimensional photon distribution function and the two-dimensional photocount distribution function. The domain of application of the obtained results are few-photon and nonclassical light fields.
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.