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arXiv:quant-ph/0601175 · Laser Physics

Measuring the photon distribution with ON/OFF photodetectors

Marco Genovese, Marco Gramegna, G. Brida, Maria Bondani, Guido Zambra, Alessandra Andreoni, Andrea Renato Rossi, Matteo G. A. Paris

Published Feb 1, 2006• 32 citations
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Abstract

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.

DOI
10.1134/s1054660x06020320

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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.
Reconstruction of photon statistics of optical states provides fundamental information on the nature of any optical field and finds various relevant applications. Nevertheless, no detector that can accurately discriminate the number of incident photons is available, whereas reconstruction by quantum tomography is not an easily implementable technique suited for a diffuse use. Even if on/off detectors, as usual avalanche photodiodes operating in Geiger mode, seem useless as photocounters, recently it was shown that reconstruction of photon statistics is possible by considering variable quantum efficiency. We have experimentally demonstrated the potential of this technique by reconstructing photon statistics for various optical fields both in the cw and pulsed regime ranging from heralded photon states to multi-thermal and coherent ones. In this paper, after having introduced the method, a review of the results obtained for the mono-partite case is given and the extension to bi-partite case is presented in detail.
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.
The photon statistics of a strongly attenuated continuous-wave laser (1550 nm), characterized by an Avalanche Photodiode operating in Geiger mode, was reconstructed with the aid of an iterative Maximum Likelihood Estimation model and without photon counting. Statistical analysis of the reconstructed distribution was performed to determine the algorithm convergence criteria. Parameters such as reconstruction error, Fischer Dispersion Index, chi-square, and p-value were evaluated. The photon number distribution for attenuated coherent sources was successfully reconstructed according to the defined convergence criteria.
Single-photon avalanche detectors (SPADs) are crucial sensors of light for many fields and applications. However, they are not able to resolve photon number, so typically more complex and more expensive experimental setups or devices must be used to measure the number of photons in a pulse. Here, we present a methodology for performing photon number-state reconstruction with only one SPAD. The methodology, which is cost-effective and easy to implement, uses maximum-likelihood techniques with a detector model whose parameters are measurable. We achieve excellent agreement between known input pulses and their reconstructions for coherent states with up to ≈10 photons and peak input photon rates up to several Mcounts/s. When detector imperfections are small, we maintain good agreement for coherent pulses with peak input photon rates of over 40 Mcounts/s, greater than one photon per detector dead time. For anti-bunched light, the reconstructed and independently measured pulse-averaged values of g (2) (0) are also consistent with one another. Our algorithm is applicable to light pulses whose pulse width and correlation time scales are both at least a few detector dead times. These results, achieved with single commercially available SPADs, provide an inexpensive number-state reconstruction method and expand the capabilities of single-photon detectors.
The knowledge of the density matrix of a quantum state plays a fundamental role in several fields ranging from quantum information processing to experiments on foundations of quantum mechanics and quantum optics. Recently, a method has been suggested and implemented in order to obtain the reconstruction of the diagonal elements of the density matrix exploiting the information achievable with realistic on/off detectors, e.g. silicon avalanche photo-diodes, only able to discriminate the presence or the absence of light. The purpose of this paper is to provide an overview of the theoretical and experimental developments of the on/off method, including its extension to the reconstruction of the whole density matrix.