  The realization of cross-Kerr nonlinearity is an important task for many
applications in quantum information processing. In this work, we propose a
method for realizing cross-Kerr nonlinearity interaction between two
superconducting coplanar waveguide resonators coupled by a three-level
superconducting flux qutrit (coupler). By employing the qutrit-resonator
dispersive interaction, we derive an effective Hamiltonian involving two-photon
number operators and a coupler operator. This Hamiltonian can be used to
describe a cross-Kerr nonlinearity interaction between two resonators when the
coupler is in the ground state. Because the coupler is unexcited during the
entire process, the effect of coupler decoherence can be greatly minimized.
More importantly, compared with the previous proposals, our proposal does not
require classical pulses. Furthermore, due to use of only a three-level qutrit,
the experimental setup is much simplified when compared with previous proposals
requiring a four-level artificial atomic systems. Based on our Hamiltonian, we
implement a two-resonator qubits controlled-phase gate and generate a
two-resonator entangled coherent state. Numerical simulation shows that the
high-fidelity implementation of the phase gate and creation of the entangled
coherent state are feasible with current circuit QED technology.
