Generating spin squeezing states and Greenberger-Horne-Zeilinger entanglement using a hybrid phonon-spin ensemble in diamond

Quantum squeezing and entanglement of spins can be used to improve the sensitivity in quantum metrology. Here we propose a scheme to create collective coupling of an ensemble of spins to a mechanical vibrational mode actuated by an external magnetic field. We find an evolution time where the mechanical motion decouples from the spins, and the accumulated geometric phase yields a squeezing of 5.9dB for 20 spins. We also show the creation of a Greenberger-Horne-Zeilinger spin state for 20 spins with a fidelity of ∼0.62 at cryogenic temperature. The numerical simulations show that the geometric-phase-based scheme is mostly immune to thermal mechanical noise.