Motion-light parametric amplifier and entanglement distributor

We propose a scheme for entangling the motional mode of a trapped atom with a propagating light field via a cavity-mediated parametric interaction. We then show that if this light field is subsequently coupled to a second distant atom via a cavity-mediated linear-mixing interaction, it is possible to transfer the entanglement from the light beam to the motional mode of the second atom to create an Einstein-Podolsky-Rosen-type entangled state of the positions and momenta of two distantly separated atoms