  Hybrid quantum systems involving cold atoms and microwave resonators can
enable cavity-mediated infinite-range interactions between atomic spin systems
and realize atomic quantum memories and transducers for microwave to optical
conversion. To achieve strong coupling of atoms to on-chip microwave
resonators, it was suggested to use atomic Rydberg states with strong electric
dipole transitions. Here we report on the realization of coherent coupling of a
Rydberg transition of ultracold atoms trapped on an integrated superconducting
atom chip to the microwave field of an on-chip coplanar waveguide resonator. We
observe and characterize the cavity driven Rabi oscillations between a pair of
Rydberg states of atoms in an inhomogeneous electric field near the chip
surface. Our studies demonstrate the feasibility, but also reveal the
challenges, of coherent state manipulation of Rydberg atoms interacting with
superconducting circuits.
