Compound atom-ion Josephson junction: Effects of finite temperature and ion motion

We consider a degenerate Bose gas confined in a double-well potential in interaction with a trapped ion in one dimension and investigate the impact of two relevant sources of imperfections in experiments on the system dynamics: ion motion and thermal excitations of the bosonic ensemble. Particularly, their influence on the entanglement generation between the spin state of the moving ion and the atomic ensemble is analyzed. We find that the detrimental effects of the ion motion on the entanglement protocol can be mitigated by properly choosing the double-well parameters as well as timings of the protocol. Furthermore, thermal excitations of the bosons affect significantly the system's tunneling and self-trapping dynamics at moderate temperatures; i.e., thermal occupation of a few double-well quanta reduces the protocol performance by about 10%. Hence, we conclude that finite temperature is the main source of decoherence in such junctions and we demonstrate the possibility to entangle the condensate motion with the ion vibrational state.