Balancing force sensing and spin-motion entanglement in levitated diamond

Levitated systems, such as small diamond crystals in magnetic traps, have remarkable mechanical properties. They also benefit from integrated quantum spin sensors in the form of nitrogen vacancy (NV) centers, which enable a variety of potential experiments exploring the landscape of mechanical-spin interaction. We study a setup where the NV center spin is coupled to the center-of-mass motion of its host diamond in a magnetic trap. The large gradients of the magnetic field in the trap lead to a strong coupling between spin and motion. We find this enables a new paradigm for back-action-evading measurement. Alternatively, by carefully maximizing the back action, we are able to explore the possibility of entangling the NV center spin with the motion of the diamond. We propose an entanglement witness (EW) for this setup, a mathematical quantity with a bound which, if violated, determines that entanglement exists between the spin and the diamond.