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Journal of the Optical Society of Korea

The Relation between Statistics of Input Photons and Photocounts

Yang Ha, Heonoh Kim, Jong-Tae Shin, Goo-Dong Park, Ill-Won Kim, Taesoo Kim, Hee-Dong Kang

Published Sep 1, 2000• 5 citations
Publisher

Abstract

The relation between the statistics of photons in input light on a detector and the measured photocounts by the detector is discussed. The averages and variances of the photocounts are compared with the averages and variances of input photons on practical detectors having quantum efficiencies of $\mu$ . This comparison was made for three kinds of inputs which include Fock state light, coherent light, and thermal light. The calculations were carried out based on the combined operator model for a detector having less-than-unit quantum efficiency.

DOI
10.3807/josk.2000.4.2.071

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The relation between the statistics of photons in input light on a detector and the measured photocounts by the detector is discussed. The averages and variances of the photocounts are compared with the averages and variances of input photons on practical detectors having quantum efficiencies of μ\muμ . This comparison was made for three kinds of inputs which include Fock state light, coherent light, and thermal light. The calculations were carried out based on the combined operator model for a detector having less-than-unit quantum efficiency.
The relation between the statistics of photons in input light on a detector and the measured photocounts is discussed for two types of light measurement, i.e., single photocount with one detector and coincidence detection with two detectors. The averages and variances of the measured photocounts are compared with the averages and variances of input photons on practical detectors having quantum efficiencies of /spl mu//sub 1/ and /spl mu//sub 2/. The calculations are carried out based on the combined operator model for a detector having less-than-unity quantum efficiency.
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The study on photon counting statistics is of fundamental importance in quantum optics. We theoretically analyzed the imperfect detection of an arbitrary quantum state. We derived photon counting formulae for six typical quantum states (i.e., Fock, coherent, squeeze-vacuum, thermal, odd and even coherent states) with finite quantum efficiencies and dark counts based on multiple on/off detector arrays. We applied the formulae to the simulation of multiphoton number detections and obtained both the simulated and ideal photon number distributions of each state. A comparison between the results by using the fidelity and relative entropy was carried out to evaluate the detection scheme and help select detectors for different quantum states.
Harnessing entanglement as a resource is the main workhorse of many quantum protocols, and establishing the degree of quantum correlations of quantum states is an important certification process that has to take place prior to any implementations of these quantum protocols. The emergence of photodetectors known as photon-number-resolving detectors (PNRDs) that allow for accounting of photon numbers simultaneously arriving at the detectors has led to the need for modeling accurately and applying them for use in the certification process. Here we study the variance of difference of photocounts (VDP) of two PNRDs, which is one measure of quantum correlations, under the effects of loss and saturation. We found that it would be possible to distinguish between the classical correlation of a two-mode coherent state and the quantum correlation of a twin-beam state within some photo count regime of the detector. We compare the behavior of two such PNRDs. The first for which the photocount statistics follow a binomial distribution accounting for losses, and the second is that of Agarwal, Vogel, and Sperling for which the incident beam is first split and then separately measured by ON/OFF detectors. In our calculations, analytical expressions are derived for the variance of difference where possible. In these cases, Gauss' hypergeometric function appears regularly, giving an insight to the type of quantum statistics the photon counting gives in these PNRDs. The different mechanisms of the two types of PNRDs leads to quantitative differences in their VDP.
The variance and average value of photocounts (counts of electrons emitted by the cathode of a photodetector) are calculated allowing for the existence of the following sources of noise in a frequency converter: fluctuations of the pump intensity, conversion of background radiation, thermal fluctuations of fields in the converter crystal, spontaneous generation of the difference frequency, and illumination of the photodetector with the pump radiation transmitted by filters. A function representing the distribution of photocounts is obtained for short time intervals allowing for fluctuations of the pumping intensity in the case of conversion of a random Gaussian signal.
The statistics of the sum of photocounts in partially coherent Gaussian light is investigated. Both the case of detection at LLL discrete points and the case of an extended surface of detection are considered. The combinatorial-analysis approach is applied to derive the moments and cumulants of the distribution. They all can be expressed in terms of the second-order correlation function. The results are generally valid independent of the ratio of the observation time to the coherence time of light. A comparison between the theoretical moments and the moments computed from experimental distributions as a function of the detector surface is carried out with the help of a pseudothermal source.