Abstract: We will start by introducing a real space model of a scalar electromagnetic field coupled to a continuum collection of two level atoms. From this we will obtain a pair of nonlocal partial differential equations describing the energy eigenstates that have at most one photon present in the field.
Next, we will consider the case that the density of atoms is periodic with respect to a general lattice and describe a structure theorem for the spectral bands of the one photon Hamiltonian.
Finally we will discuss the existence of linear crossings of the spectral bands (Dirac Points) in two cases:
1. Atomic densities which are weak perturbations of a constant. (Low contrast)
2. Atomic densities which are sums of shifted scaled atomic inclusions. (High contrast).
This work is joint with Erik Hiltunen, John Schotland, and Michael Weinstein.