Nonlocal PDEs and Quantum Optics: Bound States and Resonances

We consider the quantum optics of a single photon interacting with a system of two level atoms. This leads to the study of a nonlinear eigenproblem for a system of nonlocal partial differential equations. Two classes of solutions to these equations are studied. Bound states correspond to negative eigenvalues and resonances to eigenvalues with positive real parts. We have found necessary and sufficient conditions for the existence of bound states, along with an upper bound on the number of such states. We have also considered the eigenproblem for atomic models with small high contrast inclusions. In this setting, we have derived asymptotic formulas for the eigenvalues. Our results are illustrated with numerical computations.