entanglement of atom-ion quantum system in josephson junction

We investigate one dimensional tunneling dynamics of a single atom and a degenerate bosonic gas in a double-well potential in which an ion is in a simple harmonic trap localized in the center of it. using a model potential we will show that how it is possible to produce an entangled state between the ion motional state and a single atom state and how much the quantum speed limit of producing such an entanglement is. furthermore, by choosing a suitable parameters of the process an entangled state of the motional state of the ion and the bosonic gas have been obtained.