Real-space Hopfield diagonalization of inhomogeneous dispersive media

TL;DR

This paper presents a real-space method extending the Hopfield approach to inhomogeneous, lossless, and dispersive media, enabling detailed analysis of polaritonic modes and their nonlinear interactions.

Contribution

It introduces a novel real-space technique for diagonalizing the Hopfield Hamiltonian in inhomogeneous media, including surface modes and nonlinear effects.

Findings

Successfully applied to a planar vacuum-dielectric interface

Allows treatment of propagative and surface polaritonic modes

Can be extended to dissipative materials

Abstract

We introduce a real-space technique able to extend the standard Hopfield approach commonly used in quantum polaritonics to the case of inhomogeneous lossless materials interacting with the electromagnetic field. We derive the creation and annihilation polaritonic operators for the system normal modes as linear, space-dependent superpositions of the microscopic light and matter fields, and we invert the Hopfield transformation expressing the microscopic fields as functions of the polaritonic operators. As an example, we apply our approach to the case of a planar interface between vacuum and a polar dielectric, showing how we can consistently treat both propagative and surface modes, and express their nonlinear interactions, arising from phonon anharmonicity, as polaritonic scattering terms. We also show that our theory can be naturally extended to the case of dissipative materials.