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Physical Review A

Unified quantum jump superoperator for optical fields from the weak- to the strong-coupling limit

Teppo Häyrynen, Jani Oksanen, Jukka Tulkki

Published Jun 3, 2010• 6 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.81.063804

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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.
We consider a single-mode optical cavity field with simultaneous dissipative and amplifying couplings to a reservoir via a two-state system. In previous works, the reservoir has either solely absorbed energy (detector setup) or added energy (amplifying setup) to the cavity. In this work, we allow both interactions simultaneously and derive a reduced dynamic model for the optical field. Our model includes four physical parameters: the field-two-state system coupling γ, the excitation and de-excitation couplings of the two-state system to the reservoir λA and λD, respectively, and the mirror losses of the cavity C. We show that, depending on the parameters, our setup can operate as a detector, a light-emitting diode or a laser. We also consider single-photon subtraction and addition schemes based on beam splitters and calculate the output fields and the probabilities of the subtraction and addition events. Furthermore, we discuss which are the parameter regimes in which our model is equivalent to beam splitter-based schemes.
Analysis of quantum optical experiments and the simulation of optical devices require detailed quantum mechanical models, especially in the case of weak optical fields. In this thesis the quantum dynamics of cavity fields are investigated and new tools for modeling cavity fields interacting with an energy reservoir are developed. Using the quantum trajectory approach the field dynamics during photon detection processes are investigated. Two experimentally feasible detector models, the resolving and the non-resolving detector scheme, are derived and applied to single photon detection and coincidence photon detection experiments. Furthermore, equivalence of the cavity field model to the beam splitter based single photon subtraction and addition schemes is shown. In addition to the detection schemes described above, a reduced model for fields in a non-ideal cavity interacting with a dissipative and amplifying reservoir through multiple two state systems is derived. The reduced model can be used to describe e.g. light emitting diodes and lasers depending on the relative strengths of the losses and energy injection. In these cases the model reproduces fields that approach a thermal or a coherent field, respectively. The derived models can be applied to wide variety of cavity field experiments. The reduced field model can be applied to modeling the optical fields of semiconductor devices or to describe cavity field based quantum information processing experiments. Furthermore, fundamental quantum optics experiments of single photon addition, single photon subtraction, coincidence detection, and their combinations can be analyzed using the derived models.
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.
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.
Complete density-matrix equations and a one-photon counting probability are used to examine the photon statistics of fluorescence light of a single three-level atom, weakly coupled to an external cavity of low QQQ value. The internal structure of the atom is supposed to be characterized by a weak and a strong transition. The strong transition is coherently driven by an external laser field, whereas no external field is acting on the weak transition. In free space the system exhibits quantum jumps. By placing the atom within the mirrors of an external cavity, tuned close to the transition frequency of the two upper levels, modifications in the quantum jump behavior of the atom are shown due to a modified spontaneous decay rate between the two states. In particular, it is demonstrated that the effect of the cavity is to modify the repetition rate of quantum jumps into the metastable dark state and the average duration of bright periods of fluorescence, whereas the average duration of dark periods is not changed in first order of the spontaneous decay rates involved. In view of current experiments in our laboratory a quantitative analysis is presented for the special case of a single indium ion located between the mirrors of a high-order symmetric confocal cavity.