  We theoretically analyze the dynamics of an atomic double-well system with a
single ion trapped in its center. We find that the atomic tunnelling rate
between the wells depends both on the spin of the ion via the short-range
spin-dependent atom-ion scattering length and on its motional state with
tunnelling rates reaching hundreds of Hz. A protocol is presented that could
transport an atom from one well to the other depending on the motional (Fock)
state of the ion within a few ms. This phonon-atom coupling is of interest for
creating atom-ion entangled states and may form a building block in
constructing a hybrid atom-ion quantum simulator. We also analyze the effect of
imperfect ground state cooling of the ion and the role of micromotion when the
ion is trapped in a Paul trap. Due to the strong non-linearities in the
atom-ion interaction, the micromotion can cause couplings to high energy
atom-ion scattering states, preventing accurate state preparation and
complicating the double-well dynamics. We conclude that the effects of
micromotion can be reduced by choosing ion/atom combinations with a large mass
ratio and by choosing large inter-well distances. The proposed double-well
system may be realised in an experiment by combining either optical traps or
magnetic microtraps for atoms with ion trapping technology.
