Response of atoms in photonic lattices

It is rigorously shown that the rate of spontaneous emission of atoms placed in a dielectric is determined by a part of the local density of eigenmodes of the Maxwell equations: the local radiative density of states. Spontaneous emission is inhibited if the atom is located at a position where this local radiative density is vanishing, even when the total density of states is not small. The radiative density of states can be obtained from a purely classical calculation. We demonstrate this principle by a calculation of the optical bandstructure and density of states of a three-dimensional lattice of resonant two-level atoms in the dipole approximation. The formation of photonic bandgaps is exhibited. The bandstructure can be characterized by two dimensionless parameters. We find a longitudinal polarization mode as well as a class of vacuum modes that are unaltered by the interaction with matter.