General Photon Counting Model for Beam Splitters and Optoelectronic Devices
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General Photon Counting Model for Beam Splitters and Optoelectronic Devices Teppo Häyrynen, Jani Oksanen and Jukka Tulkki Department of Biomedical Engineering and Computational Science, Helsinki University of Technology, P.O. Box 2200, FI-02015, TKK Finland, email: [email protected] ABSTRACT Creating a single photon to a light field or removing a single photon from a light field reveals interesting quantum phenomena. For theoretical modeling and understanding of these experiments it is essential to be able to model the coupling of the optical field to the measurement apparatus. We investigate a cascaded field–quantum system–reservoir setup to obtain a theoretical model that can, in contrast to previous models, simultaneously model both the weak and the strong coupling regimes of the field–detector system. Furthermore, our theory can be applied to model optical fields coupled to dissipative systems or to active amplifying systems including for example operation of various optical instruments and devices, such as detectors, LEDs and lasers. INTRODUCTION The complicated single photon cavity quantum electrodynamics experiments have become feasible only recently (see e.g. reference [1]). These experiments have given new insight into the validity of the SD photon counting model introduced by Srinivas and Davies in the 80’s and the E-model introduced by de Oliveira et al. in 2003 (see reference [2] and the references therein). Using the quantum trajectory approach [3] and comparing the numerical results to the experimental results, we have shown that in general the E model is valid for saturated detectors that absorb photons with constant rate independent of the photon number while the SD model is valid for nonsaturated detectors in which the absorption rate is proportional to the photon number [2]. We have also derived the operators for absorption of a single photon and at least one photon [4] corresponding to measurements by a single photon resolving detector and a nonresolving detector and shown the equivalence of the quantum trajectory based models and the beam splitter based models [5]. This equivalence allows performing simple experiments using beam splitters instead of the more complicated cavity photon counting experiments. In this work we generalize our model to cover also the regimes where the previous models are not valid. We also show how the generalized model can be used to calculate the photon statistics of LEDs and lasers. THEORY We consider a setup where an optical field (f) is coupled to a two state quantum system (s) with ground state g and excited state e which are, furthermore, coupled to a reservoir (R). The reservoir can act as a dissipative energy drain or as an energy source. Evolution of the combined f-s density operator interacting with the reservoir is governed by the Lindblad equation
dρˆ fs (t )
(
)
i ˆ H ρˆ fs − ρˆ fs Hˆ † + 2λσˆ a ρˆ fsσˆ b , (1) = dt where λ describes the strength of the system-reservoir coupling and the non-Hermitian operator Hˆ is a Jaynes-Cummings typ
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