Nonlinear Optics of Ultracold Atoms in an Optical Cavity.

We report observations of non-linear optical phenomena in an ultracold atomic gas in a Fabry-Perot cavity in the single atom strong coupling regime. Up to 5 x 10$^{4} \quad ^{87}$Rb atoms are trapped at the antinodes of an in-cavity far-off resonance optical standing wave. We have observed significant Kerr non-linearity and dispersive optical bistability in the transmission of a probe beam through the cavity at our lowest detectable intensities corresponding to 10$^{-2}$ photons in the cavity. The non-linear index of refraction responsible for these effects arises from the collective motion of atoms in the combined potential of the trap and probe.