Free space photons can be made to strongly interact by dressing the photons with
atomic Rydberg states under conditions of electromagnetically induced transparency. Recently a
two-photon bound state was experimentally observed in such systems . Here we develop the
full scattering theory for two dark state polaritons and explore the various classes of bound states and scattering states that emerge. We then focus on the regime where the photonic bound states overlap with the continuum of purely atomic two-body bound states. In this continuum, we find the photonic bound states survive as metastable states whose energy spectrum is governed by the Coulomb problem, thus obtaining a photonic analogue of the hydrogen atom. These states eventually decay into bound pairs of Rydberg atoms, but are long-lived, in analogy to the quantum tunneling problem of a particle trapped by a potential barrier. Furthermore, we find that these states propagate with a negative group velocity. We demonstrate the metastability and backward propagation of these Coulomb bound states with full numerical simulations and discuss the prospects for experimental preparation and observation of these states.
 O. Firstenberg, et. al. Nature 502, 71 (2013).