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arXiv:1012.0771 · Physical Review A

Coincident count rates in absorbing dielectric media

J. A. Crosse, Stefan Scheel

Published Feb 22, 2011• 7 citations
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Abstract

A study of the effects of absorption on the nonlinear process of parametric down-conversion is presented. Absorption within the nonlinear medium is accounted for by employing the framework of macroscopic QED and the Green tensor quantization of the electromagnetic field. An effective interaction Hamiltonian, which describes the nonlinear interaction of the electric field and the linear noise polarization field, is used to derive the quantum state of the light leaving a nonlinear crystal. The signal and idler modes of this quantum state are found to be a superposition of the electric and noise polarization fields. Using this state, the expression for the coincident count rates for both type-I and type-II down-conversion are found. The nonlinear interaction with the noise polarization field was shown to cause an increase in the rate on the order of ${10}^{\ensuremath{-}12}$ for absorption of 10% per centimeter. This astonishingly small effect is found to be negligible compared to the decay caused by linear absorption of the propagating modes. From the expressions for the biphoton amplitude, it can be seen that the maximally entangled states can still be produced even in the presence of strong absorption.

DOI
10.1103/physreva.83.023815

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A study of the effects of absorption on the nonlinear process of parametric down-conversion is presented. Absorption within the nonlinear medium is accounted for by employing the framework of macroscopic QED and the Green tensor quantization of the electromagnetic field. An effective interaction Hamiltonian, which describes the nonlinear interaction of the electric field and the linear noise polarization field, is used to derive the quantum state of the light leaving a nonlinear crystal. The signal and idler modes of this quantum state are found to be a superposition of the electric and noise polarization fields. Using this state, the expression for the coincident count rates for both type-I and type-II down-conversion are found. The nonlinear interaction with the noise polarization field was shown to cause an increase in the rate on the order of 10\ensuremath−12{10}^{\ensuremath{-}12}10\ensuremath−12 for absorption of 10% per centimeter. This astonishingly small effect is found to be negligible compared to the decay caused by linear absorption of the propagating modes. From the expressions for the biphoton amplitude, it can be seen that the maximally entangled states can still be produced even in the presence of strong absorption.
We derive an effective Hamiltonian for the nonlinear process of parametric downconversion in the presence of absorption. Based upon the Green-function method for quantizing the electromagnetic field, we first set up Heisenberg's equations of motion for a single atom driven by an external electric field and in the presence of an absorbing dielectric material. The equations of motion are then solved to second order in perturbation theory which, in rotating-wave approximation, yields the standard effective interaction Hamiltonian known from free-space nonlinear optics. In a second step, we derive the local-field-corrected Hamiltonian for an atom embedded in a dielectric host medium, that is, a nonlinear crystal. Here we show that the resulting effective Hamiltonian is found to be trilinear in the electric and noise-polarization fields and is thus capable of describing nonlinear noise processes. Furthermore, it reduces to the phenomenological nonlinear Hamiltonian for the cases where absorption, and hence the noise-polarization field, vanishes.
An effective Hamiltonian for the second-order nonlinear process of parametric downconversion in the presence of absorption is derived. The electromagnetic field is quantized using the Green function method. Heisenberg's equations of motion for the interaction of a single atom with a driving electric field are then solved to second order in perturbation theory in the rotating-wave approximation. The real-cavity model is used to embed the interacting atom in a bulk medium and correct for local field effects of the material. The resulting Hamiltonian is found to be trilinear in the electric and noise polarization fields and reduces to the phenomenological nonlinear Hamiltonian for the cases where absorption vanishes.
We present a quantization scheme for the electromagnetic field in locally responding nonlinear dielectric materials with dispersion and absorption. Based on the Green tensor expansion of the macroscopic electromagnetic field, we derive an expression for the nonlinear polarization that contains, besides the reactive part which is responsible for the nonlinear interaction, a noise contribution that has hitherto been disregarded. For the process of third-harmonic generation, we find that the magnitude of the nonlinear noise polarization is proportional to the intensity of the pump fields.
The effect of enhancing nonlinear generation of entangled photons in the process of interaction of the external coherent electromagnetic field with nonlinear dispersive medium is studied in this paper. Taking into account the second and third order of the susceptibility tensor of the crystal, it is demonstrated that in good cavity approximation the bistable behavior of the two photon generation coefficient as a function of intensity of the pump laser field is possible. This effect is stimulated by decreasing the detuning between the frequency of the cavity mode and pump frequency as a function of an harmonicity terms in polarization.
The effect of enhancing nonlinear generation of entangled photons in the process of interaction of the external coherent electromagnetic field with a nonlinear dispersive medium is studied in this paper. Taking into account the second- and third-order susceptibility tensors of the crystal, it is demonstrated that in the good cavity approximation the bistable behaviour of the two-photon generation coefficient as a function of intensity of the pump laser field is possible. This effect is stimulated by decreasing the detuning between the frequency of the cavity mode and pump frequency as a function of anharmonicity terms in polarization.
We study conditions for the generation of strongly correlated photons of two extremely different frequency ranges, optical and terahertz, via the spontaneous parametric down-conversion effect in a nonlinear medium. Analytical expressions for the correlation function are derived using equations of the generalized Kirchhoff law for the second-order field moments with account of both classical thermal and quantum vacuum field fluctuations, possible inherent absorption of the medium at terahertz frequencies, and angular divergences of signal and idler radiation caused by a transverse limited spatial profile of the Gaussian pump beam. Associated with the current absence of terahertz photon counting detectors and coincidence schemes, the possible experimental limitations in measuring the second-order optical-terahertz field correlations are studied. Numerical calculations of noncollinear eee-geometry parametric down-conversion in a Mg:LiNbO3\mathrm{Mg}:\mathrm{LiNb}{\mathrm{O}}_{3}Mg:LiNbO3​ crystal demonstrate angular characteristics, particular features of the temperature and spectral behavior of the second-order quantum correlation function of optical-terahertz biphotons.