Polaritons of dispersive dielectric media

We quantize the electromagnetic field in a polar medium starting with the fundamental equations of motion. In our model the medium is described by a Lorenz-type dielectric function ε(r,ω) appropriate e.g. for ionic crystals, metals and inert dielectrics. There are no restrictions on the spatial behavior of the dielectric function, i.e. there can be many different polar media with arbitrary shapes. We assume no losses in our system so the dielectric function for the whole space is assumed as real. The quantization procedure is based on an expansion of the total field (transverse and longitudinal) in terms of the coupled (polariton) eigenmodes, so our theory gives the Hamiltonian of polaritons of dispersive dielectric media. We pay particular attention to the derivation of the fundamental (equal-time) commutation relations between the conjugate field operators. As an example, we apply our theory to the quasi-two-dimensional Wigner crystal, formed by electrons at very low temperature. We discuss the influence of the quantized electromagnetic field on the dynamics of Wigner electrons, i.e., on the dispersion relation of Wigner phonons. We expect a significant influence in the case when frequencies of (surface) polaritons and Wigner phonons coincide, and we discuss the energy spectrum of such a system.