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arXiv:1202.5106 · Physical Review A

True photocounting statistics of multiple on-off detectors

Jan Sperling, W. Vogel, G. S. Agarwal

Published Feb 17, 2012• 147 citations
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Abstract

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.

DOI
10.1103/physreva.85.023820

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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.
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.
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.
Discriminating between Fock states with a high degree of accuracy is a desirable feature for modern applications of optical quantum information processing. A well-known alternative to sophisticated photon number discriminating detectors is to split the field among a number of simple on/off detectors and infer the desired quantity from the measurement results. In this work we find an explicit analytical expression of the detection probability for any number of input photons, any number of on/off detectors, and we include quantum efficiency and a false count probability. This allows us to explicitly invert the conditional probability using Bayes' theorem and express the number of photons that we had at the input in the most unbiased way possible with ready-to-use formulas. We conclude with some examples.
Any single on-off photocounter, which can only detect the presence or absence of photons without discriminating their number, is not capable of identifying nonclassical nature of light. This limitation arises because any photocounting statistics obtained with such a detector can be easily reproduced with coherent states of a light mode. We show that a simple modification of an on-off detector -- introducing controlled attenuation as a tunable setting -- enables such detectors to reveal nonclassical properties of radiation fields.
Although some non-trivial photon number resolving detectors exist, it may still be convenient to discriminate photon number states with the method of multiplexed detection. Multiplexing can be performed with paths in real space, with paths in time, and in principle with any degree of freedom that has a sufficient number of eigenstates and that can be coupled to the photon number. Previous works have addressed the probabilities involved in these measurements with Monte Carlo simulations, or by restricting the number of detectors to powers of 2, or without including quantum efficiency or noise. In this work we find an analytical expression of the detection probabilities for any number of input photons and any number of on/off photon detectors with a quantum efficiency 0%≤η≤100%0\%\leqη\leq100\%0%≤η≤100% and a false count probability ε≥0\varepsilon\geq0ε≥0. This allows us to retrodict the number of photons that we had at the input in the least unbiased way possible. We conclude our work with some examples.