  We study a system of three photons in an atomic medium coupled to Rydberg
states near the conditions of electromagnetically induced transparency. Based
on the analytical analysis of the microscopic set of equations in the
far-detuned regime, the effective three-body interaction for these Rydberg
polaritons is derived. For slow light polaritons, we find a strong three-body
repulsion with the remarkable property that three polaritons can become
essentially non-interacting at short distances. This analysis allows us to
derive the influence of the three-body repulsion on bound states and
correlation functions of photons propagating through a one-dimensional atomic
cloud.
