  We consider the resonance interaction energy between two identical entangled
atoms, where one is in the excited state and the other in the ground state.
They interact with the quantum electromagnetic field in the vacuum state and
are placed in a photonic-bandgap environment with a dispersion relation
quadratic near the gap edge and linear for low frequencies, while the atomic
transition frequency is assumed to be inside the photonic gap and near its
lower edge. This problem is strictly related to the coherent resonant energy
transfer between atoms in external environments. The analysis involves both an
isotropic three-dimensional model and the one-dimensional case. The resonance
interaction asymptotically decays faster with distance compared to the
free-space case, specifically as $1/r^2$ compared to the $1/r$ free-space
dependence in the three-dimensional case, and as $1/r$ compared to the
oscillatory dependence in free space for the one-dimensional case. Nonetheless,
the interaction energy remains significant and much stronger than dispersion
interactions between atoms. On the other hand, spontaneous emission is strongly
suppressed by the environment and the correlated state is thus preserved by the
spontaneous-decay decoherence effects. We conclude that our configuration is
suitable for observing the elusive quantum resonance interaction between
entangled atoms.
