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Journal of Physics Conference Series

Photon counting statistics with imperfect detectors

Aleksei Reutov, Сыч Денис Васильевич

Published Dec 1, 2021• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1742-6596/2086/1/012096

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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.
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.
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.
If a beam of light is coherent, then the photons constituting the beam arrive at random moments in time. Consequently, the photon counting statistics in a given time interval below the coherence time, should in theory follow a Poissonian distribution. When the measurement interval exceeds the coherence time, the distribution should broaden and become a super-Poissonian distribution. The aim of this study is to experimentally demonstrate the Poissonian nature of light, as well as quantifying the transition to a super-Poissonian distribution. In the experiment, a low-intensity laser beam was measured with single-photon detectors, able to capture individual photons. The arrival time of the photons was detected with an oscilloscope, and the data further analysed with our own Python code. The probability distribution of the arrival time of photons was constructed and fitted with a Poissonian distribution for different time intervals. The results showed a clear Poissonian distribution for all intervals from 0.2 μs up to 2000 μs. We did however, observe tendencies of randomness for the longest measured intervals. A Poissonian distribution is defined by the variance being equal to the mean, whilst for a super-Poissonian distribution the variance is higher. The biggest measured ratio between the variance and the mean in our experiment was 1.06 for interval time 2000 μs, which is too small to infer super-Poissonian. Due to limited data memory, our dataset was restricted in length and resolution. Therefore, longer intervals could not be studied without statistical uncertainty. To cross-check the result we also calculated the probability distribution for the arrival time itself. The result perfectly fitted an exponential distribution which was in accordance with the theory of coherent light. Hence, within the given measurement accuracy and statistical uncertainty, it is safe to conclude that the laser beam is coherent for all tested time intervals. However, this does not allow us to conclude that it is a single-mode laser.
Statistics of fluorescence photons emitted by a single two-level atom interacting with a continuous wave laser field are analyzed. The photon-counting distribution is calculated for the so-called intermediate fluorescence photons, i.e., those counted during the time interval between instants at which photons are emitted. The result is a sub-Poissonian (narrower than Poisson) distribution, which agrees with experimental observations. This intermediate-photon distribution is used to calculate the average number of fluorescence photons, the second factorial moment of the photon-counting distribution, and Mandel’s Q parameter commonly used to evaluate the deviation of photon statistics from the Poisson distribution. The theoretical expressions obtained for moments of the intermediate-photon distribution are different from well-known Mandel’s formulas.
Photon number resolving detectors can be highly useful for studying the statistics of multiphoton quantum states of light. In this work we study the counts statistics of different states of light measured on multiplexed on–off detectors. We put special emphasis on artificial nonclassical features of the statistics obtained. We show new ways to derive analytical formulas for counts statistics and their moments. Using our approach we are the first, to the best of our knowledge, to derive statistics moments for multimode thermal states measured on multiplexed on–off detectors. We use them to determine empirical Mandel parameters and recently proposed sub-binomial parameters suitable for tests of nonclassicality of the measured states. Additionally, we investigate sub-Poissonian and superbunching properties of the two-mode squeezed state measured on a pair of multiplexed detectors, and we present results of the Fano factor and second-order correlation function for these states.
Shot noise sets a fundamental limit on the sensitivity of classical optical measurements, with coherent emitters achieving the lowest possible shot-noise level. Emission from sub-Poissonian light provides a pathway to surpass this limit, and single-photon sources provide a natural platform for generating such light. However, it is commonly assumed that continuously excited single-photon sources exhibit Poissonian statistics. In this work, a theoretical model of a continuously driven two-level single-photon source is developed, treating both excitation and radiative decay as stochastic processes. The analysis demonstrates that photon emission can display sub-Poissonian statistics when excitation and decay rates are comparable, showing that continuous excitation does not inherently preclude nonclassical emission. The model is further extended to include finite detection efficiency and detector dead time, illustrating how these practical non-idealities can affect the experimental observation of sub-Poissonian statistics.