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International Journal of Quantum Information

NONTRIVIAL PHOTON STATISTICS WITH LOW RESOLUTION-THRESHOLD PHOTON COUNTERS

Guido Zambra, Alessia Allevi, Alessandra Andreoni, Maria Bondani, Matteo G. A. Paris

Published Feb 1, 2007• 13 citations
Publisher

Abstract

We address the reconstruction of photon statistics using realistic photodetectors having low quantum efficiency and finite resolution as photon counters. Using a maximum-likelihood method, based on measurements taken at different quantum efficiencies, we experimentally reconstruct the nontrivial photon statistics of the state obtained by mixing at a beam-splitter two phase-averaged coherent states. The effect of using different resolution thresholds is also discussed.

DOI
10.1142/s0219749907002797

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We address the reconstruction of photon statistics using realistic photodetectors having low quantum efficiency and finite resolution as photon counters. Using a maximum-likelihood method, based on measurements taken at different quantum efficiencies, we experimentally reconstruct the nontrivial photon statistics of the state obtained by mixing at a beam-splitter two phase-averaged coherent states. The effect of using different resolution thresholds is also discussed.
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.
Reconstruction of photon statistics of optical states provides fundamental information on the nature of any optical field and helps with various relevant applications. Nevertheless, no detector that can reliably discriminate the number of incident photons is available. On the other hand, the alternative of reconstructing the density matrix by quantum tomography leads to various technical difficulties that are particularly severe in the pulsed regime (where mode matching between a signal and local oscillator is very challenging). Even if on/off detectors, as usual avalanche photodiodes operating in Geiger mode, seem useless as photo-counters, it was recently shown how reconstruction of photon statistics is possible by considering a variable quantum efficiency. Here we present experimental reconstructions of photon number distributions of both continuous-wave and pulsed light beams in a scheme based on on/off avalanche photodetection assisted by maximum-likelihood estimation. Reconstructions of the distribution for both semiclassical and quantum states of light (as single photon, coherent, pseudothermal, and multithermal states) are reported for single-mode and multimode beams. The stability and good accuracy obtained in the reconstruction of these states clearly demonstrate the interesting potentialities of this simple technique.
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.
We present an experimental scheme for the reconstruction of the Wigner function of optical states. The method is based on direct intensity measurements by non-ideal photodetectors operated in the linear regime. We mix, at a beam-splitter, the signal state with a set of coherent probes of known complex amplitudes, and measure the probability distribution of the detected photons for each probe. The Wigner function is given by a suitable sum of those probability distributions. For comparison, the same data are analyzed to obtain the number distributions and the Wigner functions for photons.