Preparation and light-mediated distribution of motional state entanglement

We describe and analyze numerically schemes (i) for entangling orthogonal motional modes of one or a few harmonically trapped atoms or ions and (ii) for transferring the entanglement from one of these local modes to a distant trapped atom (or atoms) via a light-mediated quantum-state transfer procedure proposed in previous work [A.S. Parkins and H.J. Kimble, J. Opt. B: Quantum Semiclass. Opt. $1,$ 496 (1999)]. Possibilities arising from these schemes include the generation of an Einstein-Podolsky-Rosen state in the positions and momenta of distantly separated trapped atoms and the preparation of delocalized mesoscopic vibrational states.