Nonlocal Partial Differential Equations and Quantum Optics: Band Structure of Periodic Atomic Systems

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Abstract
We continue our study, initiated in [E. O. Hiltunen et al., SIAM J. Math. Anal., 56 (2024), pp. 3802–3831], of the quantum optics of a single photon interacting with a system of two level atoms. In this work, we investigate the case of a periodic arrangement of atoms. We provide a general structure theorem characterizing the band functions of this problem, which comprise the spectrum of the associated Hamiltonian. Additionally, we study atomic densities arising as periodically arranged scaled inclusions. For this family of examples, we obtain explicit asymptotic formulas for the band functions.

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