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arXiv:1210.4685

The Photodetection Process under the Conditions of an Imperfect Measurement Device

Alexander Trifanov, George P. Miroshnichenko

Published Oct 17, 2012• 0 citations
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Abstract

The process of cavity mode quantum state photodetection subject to a nonideal measurement device is under consideration. A set of nonorthogonal probabilistic operator valued measures (POVMs) describing the photodetection process is suggested. The superoperators of conditional system evolution and their Kraus representations are obtained. Developed formalism is used to the quantum state reconstruction problem. We recalculate the probabilities of hypotheses about the initial state of cavity quantum mode in the case of unknown superposition of vacuum and single photon states.

DOI
10.48550/arxiv.1210.4685

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In this report, we study the reduced conditional dynamics of a quantum system in the case of indirect quantum measurement. The detectors microscopic part (pointer) interacts with the measured system (target) and the environment, which results in a nonunitary interaction between target and pointer. The quantum state evolution conditioned by the measurement result is under investigation. Particularly, we are interested in explicit analytical expressions for the conditional evolution superoperators and basic information characteristics of this measurement process, which is applied to the cavity mode photodetection problem.
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.
The external photodetection of cavity radiation is considered as a continuous quantum measurement in the sense of Srinivas and Davies [Opt. Acta 28, 981 (1981)]. It is analyzed in terms of three elementary processes: the no-leak process, the one-leak-undetected process, and the one-leak-detected process. The first two form a no-count process and the last one, a one-count process. Superoperators for these elementary processes are postulated, following the examples of Srinivas and Davies. The superoperators for the one-count process, the no-count process, the m-count process, and the nonreferring process are then derived based on the postulates. The following implications of these superoperators are found: (1) the change of average photon number due to the one-count process is identical to the well-known Q parameter introduced by Mandel [Opt. Lett. 4, 205 (1979)]; (2) the no-count process is governed by a pseudomaster equation; (3) The probability for the m-count process is the Poisson transform of the initial P function of the cavity radiation; and (4) the nonreferring measurement process is governed by a master equation.
We develop a theory of the discrete photodetection model where individual unexcited atoms from the atomic beam passing through a cavity are used as pointers. Atomic energy state selective detectors (ionization chambers) measure the escape statistics of ground-or excited-state atoms. Mandelstam’s method of successive indirect quantum measurements underlies the theory being developed. In contrast to the available works, the suggested approach takes into account the atomic beam statistics, the operation nonideality (quantum efficiencies) of ionization chambers, the detector recovery time, the back action of the probe on the quantized mode, and the relaxation process in the time interval when the cavity is empty (without atoms). We derive formulas that relate the a posteriori probabilities of photodetector clicks to the elements of the main diagonal of the initial density matrix for the cavity mode. Our numerical simulations confirm that the inverse problem of reconstructing the initial photon number distribution in a cavity can be solved using the statistics of detector clicks measured during the transient process.