  We show that two photons coupled to Rydberg states via electromagnetically
induced transparency can interact via an effective Coulomb potential. This
interaction gives rise to a continuum of two-body bound states. Within the
continuum, metastable bound states are distinguished in analogy with
quasi-bound states tunneling through a potential barrier. We find multiple
branches of metastable bound states whose energy spectrum is governed by the
Coulomb potential, thus obtaining a photonic analogue of the hydrogen atom.
Under certain conditions, the wavefunction resembles that of a diatomic
molecule in which the two polaritons are separated by a finite "bond length."
These states propagate with a negative group velocity in the medium, allowing
for a simple preparation and detection scheme, before they slowly decay to
pairs of bound Rydberg atoms.
