QUANTUM ELECTRODYNAMICS IN PHOTONIC CRYSTALS

A macroscopic quantum electrodynamic formalism has been developed for a three-dimensional periodic dielectric lattice made of a linear, non-magnetic, isotropic, locally homogeneous, and lossless medium. The usual choice of Coulomb gauge can be shown to be inappropriate for the study of QED processes in photonic crystals. In the absence of charged particles and in a modified Coulomb gauge, the scalar potential can be forced to be zero and a vector potential is obtained which is consistent with Maxwell's equation and satisfies Bloch's theorem and the dispersion relation first derived by Ho et al. In the presence of sources, using a standard formalism from the appropriate lagrangian, the hamiltonian can be developed in a parallel way with the minimal coupling hamiltonian. However, several differences are present and the atomic hamiltonian is not the same as that of an isolated atomic system due to the fact that the scalar potential is not a solution of the Poisson equation, and the ordering of particle and radiation operators in the interaction hamiltonian is important. In the long wavelength approximation, the effective interaction hamiltonian for an infinitely heavy atom resembles that of the Coulomb gauge. In addition to the complete suppression of spontaneous emission within the forbidden band, we predict significant modifications outside the forbidden bands. The emission rate is found to be strongly position dependent and the atoms can emit light even in the direction of the oscillation, an effect which is totally absent in the free space case.