Quantum phase gates with trapped atoms coupling to a superconducting transmission line resonator

We propose a hybrid approach with atoms trapped above the surface of a microwave superconducting transmission line resonator to implement a universal two-qubit phase gate based on geometric or dynamical phase shifts. These proposals exploit the Rydberg excitation blockade mechanism that results from the effective long-range dipoledipole interactions mediated by the stripline cavity mode. The geometric one relies on the stimulated Raman adiabatic passage techniques involving only the ground states of the trapped atoms, while the dynamical phase gate employs the pulse area techniques. These proposals may provide interesting applications in the field of on-chip quantum computations.