Resonant Interaction Energy between Two Identical Atoms in a Photonic Crystal

We consider the resonant interaction energy between two identical atoms, one in an excited state and the other in the ground state, placed inside a photonic crystal. We consider two different models of a photonic crystal: a one-dimensional crystal and an isotropic three-dimensional crystal. The two atoms, having the same orientation of their transition dipole moment, are supposed prepared in their entangled symmetrical state and interacting with the quantum electromagnetic field in the multipolar coupling scheme. We consider both the case of an atomic transition frequency outside the photonic band gap and the case of a transition frequency inside the gap. When the transition frequency is outside the photonic band gap and close to its edge, so that the effective mass approximation can be used, we show that the resonant interatomic energy and force can be strongly enhanced by the presence of the photonic crystal, in both cases of 1D crystal and isotropic 3D crystal, as a consequence of the modified dispersion relation and density of states. The distance dependence of the force is however the same as for atoms in the vacuum space. Differences among the two models considered of photonic crystal are discussed in detail, as well as comparison with the analogous system of two impurity atoms in a quantum semiconductor wire. A numerical estimate of the effect in a realistic situation is also discussed. When the atomic transition frequency is inside the band gap, we find that strength of the resonant interaction force is reduced and that it decreases more rapidly with the distance compared to the case of atoms in the vacuum space. However, in this case, the spontaneous decay of the atoms is strongly inhibited, so that the state considered has a much longer lifetime. This could make easier the experimental measurement of the interatomic resonant force, which requires maintaining for a sufficiently long time the coherence of the correlated state of the two atoms.