  We propose a scheme to prepare nanomechanical oscillators in nonclassical
steady states, characterized by a pronounced negative Wigner function. In our
optomechanical approach, the mechanical oscillator couples to multiple laser
driven resonances of an optical cavity. By lowering the resonance frequency of
the oscillator via an inhomogeneous electrostatic field, we significantly
enhance its intrinsic geometric nonlinearity per phonon. This causes the
motional sidebands to split into separate spectral lines for each phonon number
and transitions between individual phonon Fock states can be selectively
addressed. We show that this enables the preparation of the nanomechanical
oscillator in a single phonon Fock state. Our scheme can for example be
implemented with a carbon nanotube dispersively coupled to the evanescent field
of a state of the art whispering gallery mode microcavity.
