Effective nonlinear Hamiltonians in dielectric media

Abstract

We derive an effective Hamiltonian for the nonlinear process of parametric down conversion 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, i.e. a nonlinear crystal. Here we show that the resulting effective Hamiltonian is trilinear in the electric and noise polarization fields, and is thus capable of describing nonlinear noise processes. Furthermore, it reduces to the phenomonological nonlinear Hamiltonian for the cases where the noise polarization field, and hence absorption, vanishes.

Updated to journal version. 12 pages, 1 figure

References in corpus (6)

• Quantum key distribution with entangled photon sources
• Simple and efficient quantum key distribution with parametric down-conversion
• Local-field correction to the spontaneous decay rate of atoms embedded in bodies of finite size
• Local-field correction to one- and two-atom van der Waals interactions
• Properties of entangled photon pairs generated in one-dimensional nonlinear photonic-band-gap structures
• Quantum theory of light and noise polarization in nonlinear optics

Cited by in corpus (4)

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• Nonlinear Quantum Electrodynamics of Epsilon-Near-Zero Nanocavities