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Optica Acta International Journal of Optics

Comment on ‘Photon Counting Probabilities in Quantum Optics’

L. Mandeļ

Published Nov 1, 1981• 37 citations
Publisher

Abstract

It is pointed out that some statements in a recent paper by Srinivas and Davies [1], concerning the validity of the so-called Mandel photon counting formula and its relation to another formula derived by the authors, are not correct. The former formula applies to an open system and the latter to a closed system, but conditions for the former are much more commonly encountered in practice.

DOI
10.1080/713820474

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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.

Physical Review A•Jun 1, 1994•17 citations
It is pointed out that some statements in a recent paper by Srinivas and Davies [1], concerning the validity of the so-called Mandel photon counting formula and its relation to another formula derived by the authors, are not correct. The former formula applies to an open system and the latter to a closed system, but conditions for the former are much more commonly encountered in practice.
We reconsider various approaches to the quantum theory of photodetection from the point of view of the quantum theory of measurement, and show that important differences between them depend upon the manner in which they take into account the modification of the field distribution produced by the presence of the detector. We show that the Mandel photon counting formula may lead to unphysical results, such as negative probabilities, in some situations just because this modification is not incorporated into the model. We also show that the recently developed quantum theory of continuous measurements provides a completely satisfactory framework for discussing photon counting experiments. As an illustration, an exact derivation of the photon statistics of a single-mode free field is presented, and some of the results are shown to be identical to those derived earlier by Mollow and others. We also indicate how our approach can be extended to discuss the photon statistics of fields in interaction with sources and reservoirs.
In this thesis we study continuously observed open quantum systems from the point of view of quantum filtering theory. The conditioned state of the open system obeys a non-commutative or quantum analogue of the Kushner-Stratonovich equation in classical filtering theory. The thesis consists out of four chapters. The first chapter is a brief introduction to quantum probability theory and quantum stochastic calculus. The second chapter is a description of a photon counting experiment in continuous time within the framework of Davies and Srinivas. The third chapter focusses on the derivation of the quantum filtering or Belavkin equation from the quantum stochastic differential equation governing the interaction between the open system and its environment, the electromagnetic field. The fourth chapter shows how the quantum filtering equation can be used to control quantum systems.
The concept of photodetection as a continuous quantum measurement introduced by Srinivas and Davies is extended to the detection of resonance fluorescence. This is a new approach to an old subject. This is also a new development in continuous photodetection in the sense that a fermion rather than the traditional boson field is the subject of monitoring. The superoperators for the no-count and the one-count processes are postulated, following the examples of Srinivas and Davies. The probability for the m-count process is then derived on the basis of these postulates. The first two factorial moments of the photon-number distribution are calculated, which are then used to evaluate the non-classical effects in resonance fluorescence. A parameter slightly different from Mandel's is found to be more suitable indicator of photon antibunching in the transient state.
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.