Dispersive properties and giant Kerr non-linearities in Dipole Induced Transparency

We calculate the dispersive properties of the reflected field from a cavity coupled to a single dipole. We show that when a field is resonant with the dipole it experiences a 90 degree phase shift relative to reflection from a bare cavity if the Purcell factor exceeds the bare cavity reflectivity. We then show that optically Stark shifting the dipole with a second field can be used to achieve giant Kerr non-linearites. It is shown that currently achievable cavity lifetimes and cavity quality factors can allow a single emitter in the cavity to impose a nonlinear $\pi$ phase shift at the single photon level.