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Journal of Optics B Quantum and Semiclassical Optics

Quantum photodetection distributions with ‘nonlinear’ quantum jump superoperators

Alexandre V. Dodonov, Salomon Sylvain Mizrahi, V. V. Dodonov

Published Mar 10, 2005• 18 citations
Publisher

Abstract

We address the issue of the detection and counting of photons. We assume an electromagnetic field enclosed in an ideal cavity, which together with the detector constitute a closed system, so no photon is lost to the environment. Basing ourselves on a microscopic model consisting of a set of two-level atoms (the detector) interacting with the field, we derive a 'nonlinear' jump superoperator. We compare the count statistics calculated within our model with those obtained from previous models, such as the coincidence probability density , two-count conditional probability density , waiting times and the second order correlation function , for several field states.

DOI
10.1088/1464-4266/7/4/003

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We examine some aspects of the continuous photodetection model for photocounting processes in cavities. We work out a microscopic model that describes the field‐detector interaction and deduce a general expression for the Quantum Jump Superoperator (QJS), that shapes the detector's post‐action on the field upon a detection. We show that in particular cases our model recovers the QJSs previously proposed ad hoc in the literature and point out that by adjusting the detector parameters one can engineer QJSs.

AIP conference proceedings•Jan 1, 2008•0 citations
We address the issue of the detection and counting of photons. We assume an electromagnetic field enclosed in an ideal cavity, which together with the detector constitute a closed system, so no photon is lost to the environment. Basing ourselves on a microscopic model consisting of a set of two-level atoms (the detector) interacting with the field, we derive a 'nonlinear' jump superoperator. We compare the count statistics calculated within our model with those obtained from previous models, such as the coincidence probability density , two-count conditional probability density , waiting times and the second order correlation function , for several field states.
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 derive a generalized quantum jump superoperator that can be used in the quantum trajectory description of single photon detectors, light-emitting diodes (LEDs), and lasers. Our model describes an optical single-mode cavity field coupled to a reservoir through a two-state quantum system and includes three physical parameters: the coupling of the field to the two-state system, the coupling of the two-state system to the reservoir, and the cavity loss rate. In this setup, the two-state system can act as a photodetector or as an energy-adding mechanism. In the first case, we assume that the reservoir acts as a damping mechanism for an ideal cavity and derive reduced field operators describing the photon detection events. Our model coincides with the commonly known quantum trajectory based photon counting models at the weak- and strong-coupling limits and is, furthermore, also applicable between their validity regimes. In the second case, we assume that the reservoir injects energy into a lossy cavity through the two-state system. Again we derive the reduced field operators describing photon creation events into the lossy cavity. We show that this setup can act as an LED or as a laser depending on the strength of the injection. We also investigate how the setup operates at the close proximity of the lasing threshold.
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 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.
We examine some aspects of the continuous photodetection model for photocounting processes in cavities. First, we work out a microscopic model that describes the field-detector interaction and deduce a general expression for the Quantum Jump Superoperator (QJS), that shapes the detector's post-action on the field upon a detection. We show that in particular cases our model recovers the QJSs previously proposed ad hoc in the literature and point out that by adjusting the detector parameters one can engineer QJSs. Then we set up schemes for experimental verification of the model. By taking into account the ubiquitous non-idealities, we show that by measuring the lower photocounts moments and the mean waiting time one can check which QJS better describes the photocounting phenomenon.
We examine some aspects of the continuous photodetection model for photocounting processes in cavities. We work out a microscopic model that describes the field‐detector interaction and deduce a general expression for the Quantum Jump Superoperator (QJS), that shapes the detector's post‐action on the field upon a detection. We show that in particular cases our model recovers the QJSs previously proposed ad hoc in the literature and point out that by adjusting the detector parameters one can engineer QJSs.