We show theoretically that it is possible to coherently transfer vibrational excitation between trapped neutral atoms over a micrometer apart. To this end we consider three atoms, where two are in the electronic ground state and one is excited to a Rydberg state whose electronic orbital overlaps with the positional wave functions of the two ground-state atoms. The resulting scattering of the Rydberg electron with the ground-state atoms provides the interaction required to transfer vibrational excitation from one trapped atom to the other. By numerically investigating the
dependence of the transfer dynamics on the distance between traps and their relative frequencies we find that there is a ‘sweet spot’ where the transfer of a vibrational excitation is nearly perfect and fast compared to the Rydberg lifetime. We investigate the robustness of this scenario with respect to changes of the parameters. In addition, we derive a intuitive effective Hamiltonian which explains the observed dynamics.