Scheme for Realizing Kerr Nonlinearity in Cavity QED

We propose a scheme for realizing the Kerr-type nonlinearity for a cavity mode. In the scheme the cavity mode interacts with a single three-level atom dispersively. Under certain conditions, the evolution of the cavity field, decoupled from the atomic degree of freedom, corresponds to the Kerr effect. The scheme can be generalized to implement cross-Kerr effect and two-qubit phase gates for two cavity modes.