Theoretical treatment of the interaction between two-level atoms and periodic waveguides

Light transport in periodic waveguides coupled to a two-level atom is investigated. By using optical Bloch equations and a photonic modal formalism, we derive semi-analytical expressions for the scattering matrix of one atom trapped in a periodic waveguide. The derivation is general, as the expressions hold for any periodic photonic or plasmonic waveguides. It provides a basic building block to study collective effects arising from photon-mediated multi-atom interactions in periodic waveguides.