Polarization exchange in colliding photon beams in a medium of an atomic gas

Photon-photon interactions mediated by an atomic gas can effect efficient polarization exchanges between two beams, leaving the medium exactly in its initial state. In, e.g., hydrogen, the distance required for macroscopic exchange is of order one tenth the distance in which the ordinary non-linear index of refraction would induce a phase change of pi. Several examples are worked out that show the variety of behaviors that can result, depending on the initial respective polarizations stated and the angle between the beams. Of particular interest are initial conditions in which there is no exchange at a mean field level, conventionally believed to apply when the number of photons, N, is large. Then the full theory leads both to large exchange and to large entanglement between the beams. Our most solid results indicate that one would have to wait a time proportional to log[N] to see this effect, but there are some indications that this behavior can be circumvented.