  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, thus forming a mechanical qubit. This allows for
performing quantum operations at the level of individual mechanical 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.
