  We propose an implementation of a universal quantum gate between pairs of
spatially separated atoms in a microwave cavity at finite temperature. The gate
results from reversible laser excitation of Rydberg states of atoms interacting
with each other via exchange of virtual photons through a common cavity mode.
Quantum interference of different transition paths between the two-atom ground
and double-excited Rydberg states makes both the transition amplitude and
resonance largely insensitive to the excitations in the microwave cavity
"quantum bus" which can therefore be in any superposition or mixture of photon
number states. Our scheme for attaining ultralong-range interactions and
entanglement also applies to mesoscopic atomic ensembles in the Rydberg
blockade regime and is scalable to many ensembles trapped within a centimeter
sized microwave resonator.
