  We consider a nonlinearly coupled electromechanical system, and develop a
quantitative theory for two-phonon cooling. In the presence of two-phonon
cooling, the mechanical Hilbert space is effectively reduced to its ground and
first excited states, allowing for quantum operations at the level of
individual phonons and preparing nonclassical mechanical states with negative
Wigner functions. We propose a scheme for performing arbitrary Bloch sphere
rotations, and derive the fidelity in the specific case of a $\pi$-pulse. We
characterise detrimental processes that reduce the coherence in the system, and
demonstrate that our scheme can be implemented in state-of-the-art
electromechanical devices.
