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Chinese Physics B

Estimation of photon counting statistics with imperfect detectors

Xiaochuan Han, Dongwei Zhuang, Yuxuan Li, Jun-Feng Song, Yong-Sheng Zhang

Published Jul 1, 2018• 2 citations
Publisher

Abstract

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.

DOI
10.1088/1674-1056/27/7/074208

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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.
We derive a closed photocounting formula, including noise counts and a finite quantum efficiency, for photon-number-resolving detectors based on on-off detectors. It applies to detection schemes such as array detectors and multiplexing setups. The result renders it possible to compare the corresponding measured counting statistics with the true photon number statistics of arbitrary quantum states. The photocounting formula is applied to the discrimination of photon numbers of Fock states, squeezed states, and odd coherent states. It is illustrated for coherent states that our formula is indispensable for the correct interpretation of quantum effects observed with such devices.
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.
We derive a computationally efficient expression of the photon-counting distribution for a uniformly illuminated array of single-photon detectors. The expression takes the number of single detectors, their quantum efficiency, and their dark count rate into account. Using this distribution we compute the error of the array detector by comparing the output to that of an ideal detector. We conclude from the error analysis that the quantum efficiency must be very high in order for the detector to resolve a handful of photons with high probability. Furthermore, we conclude that in the worst-case scenario the required array size scales quadratically with the number of photons that should be resolved. We also simulate a temporal array and investigate how large the error is for different parameters, and we compute the optimal size of the array that yields the smallest error.
The photon number distribution is reconstructed from the measured statistics of vacuum states in single-photon detection. Numerical results are presented for the coherent, thermal, number and squeezed-vacuum states. The quantum efficiency of single-photon detection should be high for the successful evaluation of the number state. The quantum efficiency provides the marginal influence for the evaluation of the coherent, thermal and squeezed-vacuum states. However, the accuracy of the reconstructed photon number distribution of the squeezed-vacuum state is low even though the quantum efficiency is unity. For a more accurate reconstruction, we must obtain a priori information on the photon number distribution of the squeezed-vacuum state.
We investigate fundamental bounds on the ability to determine photon number distribution and other related quantities from tomographically incomplete measurements with an array of M detectors that can only distinguish the absence or presence of photons. We show that the lower and upper bounds on photon number probabilities can be determined by solving a linear program. We present and discuss numerical results for various input states, including thermal states, coherent states, squeezed states, and highly nonclassical single-photon subtracted squeezed vacuum states. Besides photon number probabilities, we also investigate bounds on the parity of photon number distribution that determines the value of Wigner function of the state at the origin of phase space. Moreover, we also discuss estimation of mean photon number as an example of a quantity described by an unbounded operator. Our approach and results can provide quantitative guidance on the number of detection channels required to determine the photon number distribution with a given precision.
We analyse how imperfections in single-photon detectors impact the characterization of photon-pair sources. We perform exact calculations to reveal the effects of multi-pair emissions and of noisy, non-unit efficiency, nonphoton-number resolving detections on the Cauchy–Schwarz parameter, on the second-order auto-correlation and cross-correlation functions, and on the visibilities of both Hong–Ou–Mandel and Bell-like interferences. We consider sources producing either two-mode squeezed states or states with a Poissonian photon distribution. The proposed formulas are useful in practice to determine the impacts of multi-pair emissions and dark counts in standard tests used in quantum optics.