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.
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