Effective nonlinear Hamiltonians in dielectric media

TL;DR

This paper derives an effective Hamiltonian for nonlinear parametric down conversion in absorbing dielectric media, incorporating local-field corrections and noise processes, advancing understanding of nonlinear quantum optics in realistic materials.

Contribution

It introduces a local-field corrected Hamiltonian that accounts for absorption and noise, extending the standard nonlinear Hamiltonian to realistic dielectric environments.

Findings

Derived the effective Hamiltonian including absorption effects.

Showed the Hamiltonian reduces to known forms without noise.

Demonstrated the Hamiltonian's capability to describe nonlinear noise processes.

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.