  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.
