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Operative Approach to Quantum Electrodynamics in Dispersive Dielectric Objects Based on a Polarization Modal Expansion

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 apply a Hopfield type scheme to account for the dispersion and dissipation of the matter. We provide a general expression of the polarization density field operator as functions of the initial conditions of the matter field operators and of the electromagnetic field operators. It is a linear functional whose kernel is a linear expression of the impulse response of the dielectric object that we obtain within the framework of classical electrodynamics. The electric field operator is expressed as a function of the polarization density field operator by means of the dyadic Green's function for the free space. The statistical functions of these operators are classical functionals of the statistics of the initial conditions of the matter field operators and of the electromagnetic field operators, whose kernels are linear or multilinear expressions of the impulse response of the dielectric object. We keep the polarization and the electromagnetic field distinct to enable the treatment of the polarization and electromagnetic fluctuations on equal footing. We expand the polarization density field operator 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. We expand the radiation fields in terms of the transverse plane wave modes of free space. 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 $\min\limits_ω\{c_0/[ω\sqrt{|χ(ω)|}]\}$ where $χ(ω)$ is the susceptibility of the dielectric.