Entanglement witness for combined atom interferometer-mechanical oscillator setup

We investigate how to entangle an atom interferometer and a macroscopic mechanical oscillator in order to create non-classical states of the oscillator. We propose an entanglement witness, from whose violation, the generation of entanglement can be determined. We do this for both the noiseless case and when including thermal noise. Thermal noise can arise from two sources: the first being when the oscillator starts in a thermal state, and the second when a continuous thermal bath is in contact with the oscillator. We find that for the appropriate oscillator quality factor $Q$, violation always exists for any value of magnetic coupling $\lambda$ and thermal occupancy $\bar n$. Cooling the oscillator to its ground state provides an $O(\bar n)$ improvement in the EW violation than starting in a thermal state. However, this still requires at least 10$^{5}$ measurements to be determined. We then consider how to use magnetic interactions to realize this idea.