Operative approach to quantum electrodynamics in dispersive dielectric objects based on a polarization-mode expansion

Abstract

In this paper, we deal with the macroscopic electromagnetic response of a finite size dispersive dielectric object, in unbounded space, in the framework of quantum electrodynamics, using the Heisenberg picture. We keep the polarization and the electromagnetic field distinct to enable the treatment of the polarization and electromagnetic fluctuations on equal footing in a self-consistent QED Hamiltonian. We apply a Hopfield type scheme to account for the dispersion and dissipation of the matter. We provide a general expression of the time evolution of the polarization density field observable as function of the initial conditions of the matter field observables and of the electromagnetic field observables. It is an integral operator whose kernel is a linear combination of the impulse responses of the dielectric object that we obtain within the framework of classical electrodynamics. The electric field observable is expressed in terms of the polarization density field observable by means of the full wave dyadic Green's function for the free space. The statistical functions of the observables of the problem can be expressed through integral operators of the statistics of the initial conditions of the matter field observables and of the electromagnetic field observables, whose kernels are linear or multilinear expressions of the impulse responses of the dielectric object. We expand the polarization density field observable in terms of the static longitudinal and transverse modes of the object to diagonalize the Coulomb and Ampere interaction energy terms of the Hamiltonian in the Coulomb gauge. Few static longitudinal and transverse modes are needed to calculate each element of the impulse response matrix for dielectric objects with sizes of the order up to $\underset{\ensuremath{\omega}}{min}{{c}_{0}/[\ensuremath{\omega}\sqrt{|\ensuremath{\chi}(\ensuremath{\omega})|}]}$, where $\ensuremath{\chi}(\ensuremath{\omega})$ is the susceptibility of the dielectric. We apply the proposed approach to different scenarios describing the dielectric susceptibility by the Drude-Lorentz model.