Coulomb bound states of strongly interacting photons

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.