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Journal of Experimental and Theoretical Physics Letters

Inverse Quantum Problem of Multidimensional Photocount Statistics

A. S. Chirkin

Published Apr 1, 2025• 2 citations
Publisher

Abstract

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.

DOI
10.1134/s0021364025605937

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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.
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.
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.
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.
An original approach has been developed to extract information on the photon distribution function recorded by the photocount method in the presence of interference. Photon statistics reconstruction has been studied in the presence of two types of interference: dark counts and background radiation. The calculation has been performed using a characteristic function. Correctly reconstructing photon statistics and appropriate experimental conditions have been discussed. The obtained results can be used in measuring photon statistics of low-photon radiation sources and non-classical light fields.
This paper is concerned with the determination of the quantum statistics of the photoelectric detection, and the E-field detection of multiple-mode superposed chaotic and coherent radiation. The Glauber P-representation is utilized extensively, and results are obtained by a fully quantum-mechanical treatment. The generating function, the mean, and the variance of the photocount statistics are derived for illumination times of arbitrary duration and for modes of arbitrary statistical interdependence. For short illumination times, the complete photocount distribution is obtained. The distribution has the same form as that for a single mode of superposed chaotic and coherent radiation. The multiple-mode distribution, however, is nonstationary. The difference between experimental determinations of the mean and variance which utilize periodic sampling and those which utilize random sampling is derived. A method for determining the P-representation for modulated fields is described, and the detection statistics are derived. The statistics of the E field are found to be Gaussian, and it is found that for multiple-mode classical statistics and quantum statistics differ.