Deterministic generation of phononic Fock states via weak nonlinearities

We propose an efficient method to generate single-phonon Fock states by exploiting a previously unidentified phonon blockade mechanism in a hybrid spin-mechanical system. Specifically, this mechanism makes use of arbitrarily weak Duffing nonlinearity to modify matrix elements of an effective phonon excitation process, assisted by both one- and two-phonon drives. The weak Duffing nonlinearity of the phonons is induced by a nitrogen-vacancy (NV) center magnetically coupled to a cantilever, while the mechanical parametric driving is realized by modulating the spring constant of the cantilever using a pump. By implementing a time-dependent protocol for the linear and parametric drives, there exists a strict cutoff in the phonon number distribution, resulting in a pure Fock state. Furthermore, the nonclassical nature of the generated state remains robust against dissipations.