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arXiv:0906.3668 · New Journal of Physics

Statistical inference from imperfect photon detection

Koenraad M. R. Audenaert, Stefan Scheel

Published Nov 30, 2009• 4 citations
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Abstract

We consider the statistical properties of photon detection with imperfect detectors that exhibit dark counts and less than unit efficiency, in the context of tomographic reconstruction. In this context, the detectors are used to implement certain POVMs that would allow to reconstruct the quantum state or quantum process under consideration. Here we look at the intermediate step of inferring outcome probabilities from measured outcome frequencies, and show how this inference can be performed in a statistically sound way in the presence of detector imperfections. Merging outcome probabilities for different sets of POVMs into a consistent quantum state picture has been treated elsewhere [K.M.R. Audenaert and S. Scheel, New J. Phys. 11, 023028 (2009)]. Single-photon pulsed measurements as well as continuous wave measurements are covered.

DOI
10.1088/1367-2630/11/11/113052

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We present a novel quantum tomographic reconstruction method based on Bayesian inference via the Kalman filter update equations. The method not only yields the maximum likelihood/optimal Bayesian reconstruction but also a covariance matrix expressing the measurement uncertainties in a complete way. From this covariance matrix the error bars on any derived quantity can be easily calculated. This is a first step towards the broader goal of devising an omnibus reconstruction method that could be adapted to any tomographic setup with little effort and that treats measurement uncertainties in a statistically well-founded way. In this first part, we restrict ourselves to the important subclass of tomography based on measurements with discrete outcomes (as opposed to continuous ones), and we also ignore any measurement imperfections (dark counts, less than unit detector efficiency, etc), which will be treated in a follow-up paper. We illustrate our general theory on real tomography experiments of quantum optical information processing elements.
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.
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.
Quantum optics is central to emerging quantum technologies such as quantum metrology and quantum information. The continuous variable (CV) approach to encoding information in a physical system, for example the electromagnetic field quadratures has significant advantages over discrete variables (DV) encoding. CV offers higher density encoding in a single optical state and on-demand state preparation for example. However, techniques for CV preparation, manipulation and measurement are not as well developed as the corresponding DV approach. In this thesis we present methods to characterize quantum detectors that operate in the CV regime, and discuss their application to CV state preparation. A quantum detector is described by its positive operator valued measure (POVM) set, with one operator for each measurement outcome. We present an experimental scheme for detector tomography, that is reconstruction of the POVM set of an unknown detector by means of measurement outcomes for a set of tomographically complete input states. We use this scheme to experimentally determine the POVM set of an avalanche photodiode (APD) and a time-multiplexed-detector (TMD) and compare the results to models of the detectors. We introduce a weak-homodyne photon-number-resolving detector (PNRD) , which uses a weak local oscillator beam to set a phase reference for the detection process. We develop a theoretical description of the sensitivity regimes for this detector using the Wigner phase space picture. We also present experimental detector tomography for a weak-homodyne detector based on two APDs. In many cases it is sufficient to calibrate the efficiencies of previously characterized PNRDs. We experimentally demonstrate a scheme using correlated photons produced by spontaneous parametric downconversion (SPDC) to determine the efficiencies of PNRDs, based on the photon number statistics of the source. The same SPDC source is also used to demonstrate preparation of photonic states in the photon number basis and for a first proof-of-principle experiment with a weak-homodyne PNRD based on a TMD.
In quantum illumination, various detection schemes have been proposed for harnessing the remaining quantum correlations of the entanglement-based resource state. To date, the only successful implementation in the microwave domain [R. Assouly, R. Dassonneville, T. Peronnin, A. Bienfait, and B. Huard, Nat. Phys. 19, 1418 (2023)] has relied on a specific mixing operation of the respective return and idler modes, followed by single-photon counting in one of the two mixer outputs. We investigate the performance of this scheme for realistic detection parameters in terms of the detection efficiency, dark-count probability, and photon-number resolution. Furthermore, we take the second mixer output into account and investigate the advantage of correlated photon counting (CPC) for a varying thermal background and optimum postprocessing weighting in CPC. We find that the requirements for photon-number resolution in the two mixer outputs are highly asymmetric due to different associated photon-number expectation values.