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The European Physical Journal D

Derivation of generalized quantum jump operators and comparison of the microscopic single photon detector models

Teppo Häyrynen, Joel Oksanen, Jukka Tulkki

Published Nov 6, 2009• 11 citations
Publisher
DOI
10.1140/epjd/e2009-00268-4

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Time development of electromagnetic fields in closed cavities under continuous detection of photons continues to be a subject of confusing controversy. Recently Dodonov et al. [Phys. Rev. A, 75, 013806, 2007] argued that their model of quantum superoperators (E model) invalidates some of the predictions of the previously introduced photon counting model of Srinivas and Davies [J. Mod. Optic. 28, 981, 1981] (SD model). Both the SD and the E models are based on two postulated quantum jump superoperators: (1) the one-count operator corresponding to the absorption of a single photon and (2) the no-count operator. In this work we develop a stochastic difference equation that describes the dissipative coupling of the cavity field and the detector. The difference equation is based on non-perturbative treatment of the cavity-detector coupling. In spite of being non-integrable due to the coupling of the detector with an external reservoir it can be used to derive the exact forms of the quantum jump superoperators. When applied to a particular photon counting measurement our theory gives predictions identical with those of the SD model which should be considered a non-perturbative and ab-initio result. It is pointed out that available experimental results coincide with the results given by the ab-initio SD model. We summarize some of the key characteristics of cavity fields and photon counting processes to demonstrate that the results given by the SD model are consistent with the principles of quantum mechanics while those given by the E model are not.
We study the back action of a single-photon detector on the electromagnetic field upon a photodetection by considering a microscopic model in which the detector is constituted of a sensor and an amplification mechanism. Using the quantum trajectories approach we determine the quantum jump superoperator (QJS) that describes the action of the detector on the field state immediately after the photocount. The resulting QJS consists of two parts: the bright-count term, representing the real photoabsorptions, and the dark-count term, representing the amplification of intrinsic excitations inside the detector. First we compare our results for the counting rates to experimental data, showing good agreement. Then we point out that by modifying the field frequency one can engineer the form of the QJS, obtaining the QJS's proposed previously in an ad hoc manner.
We discuss the quantum-jump operation in an open system and show that jump superoperators related to a system under measurement can be derived from the interaction of that system with a quantum measurement apparatus. We give two examples for the interaction of a monochromatic electromagnetic field in a cavity (the system) with two-level atoms and with a harmonic oscillator (representing two different kinds of detectors). We show that the derived quantum-jump superoperators have a ``nonlinear'' form $J\ensuremath{\rho}=\ensuremath{\gamma}\phantom{\rule{0.2em}{0ex}}\mathrm{diag}[F(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n})a\ensuremath{\rho}{a}^{\ifmmode\dagger\else\textdagger\fi{}}F(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n})]$, where the concrete form of the function $F(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n})$ depends on assumptions made about the interaction between the system and detector. Under certain conditions the asymptotical power-law dependence $F(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n})={(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}+1)}^{\ensuremath{-}\ensuremath{\beta}}$ is obtained. A continuous transition to the standard Srinivas-Davies form of the quantum-jump superoperator (corresponding to $\ensuremath{\beta}=0$) is shown.