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arXiv:quant-ph/0405139 · Physical Review A

Photon statistics without counting photons

Andrea Renato Rossi, Stefano Olivares, Matteo G. A. Paris

Published Nov 1, 2004• 66 citations
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Abstract

We give a detailed analysis of an indirect method used to obtain the photon distribution of a single-mode field using only on/off avalanche photodetectors. The method is based on measuring the field at different quantum efficiencies and then inferring the photon distribution by maximum-likelihood estimation. We address the case when only a limited range of quantum efficiency is available and when these values are not precisely known. The convergence of the method and its robustness against fluctuations are illustrated by means of numerically simulated experiments.

DOI
10.1103/physreva.70.055801

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A method based on Maximum-Entropy (ME) principle to infer photon distribution from on/off measurements performed with few and low values of quantum efficiency is addressed. The method consists of two steps: at first some moments of the photon distribution are retrieved from on/off statistics using Maximum-Likelihood estimation, then ME principle is applied to infer the quantum state and, in turn, the photon distribution. Results from simulated experiments on coherent and number states are presented.
The output of a photodetector consists of a current pulse whose charge has the statistical distribution of the actual photon numbers convolved with a Bernoulli distribution. Photodetectors are characterized by a nonunit quantum efficiency, i.e., not all the photons lead to a charge, and by a finite resolution, i.e., a different number of detected photons leads to a discriminable values of the charge only up to a maximum value. We present a detailed comparison, based on Monte Carlo simulated experiments and real data, among the performances of detectors with different upper limits of counting capability. In our scheme the inversion of Bernoulli convolution is performed by maximum-likelihood methods assisted by measurements taken at different quantum efficiencies. We show that detectors that are only able to discriminate between zero, one and more than one detected photons are generally enough to provide a reliable reconstruction of the photon number distribution for single-peaked distributions, while detectors with higher resolution limits do not lead to further improvements. In addition, we demonstrate that, for semiclassical states, even on/off detectors are enough to provide a good reconstruction. Finally, we show that a reliable reconstruction of multipeaked distributions requires either higher quantum efficiency or higher resolution.