  We propose a scheme for implementing cross Kerr nonlinearity between two
superconducting transmission line resonators (TLR) via their interaction with a
coupler which is constructed by two superconducting charge qubits connected to
each other via a superconducting quantum interference device. When suitably
driven, the coupler can induce very strong cross phase modulation (XPM) between
the two TLRs due to its N-type level structure and the consequent
electromagnetically induced transparency in its lowest states. The flexibility
of our design can lead to various inter-TLR coupling configurations. The
obtained cross Kerr coefficient is large enough to allow many important quantum
operations in which only few photons are involved. We further show that this
scheme is very robust against the fluctuations in solid state circuits. Our
numerical calculations imply that the absorption and dispersion resulted from
the decoherence of the coupler are very small compared with the strength of the
proposed XPM.
