We propose a scheme to prepare nanomechanical oscillators in nonclassicalsteady states, characterized by a pronounced negative Wigner function. In ouroptomechanical approach, the mechanical oscillator couples to multiple laserdriven resonances of an optical cavity. By lowering the resonance frequency ofthe oscillator via an inhomogeneous electrostatic field, we significantlyenhance its intrinsic geometric nonlinearity per phonon. This causes themotional sidebands to split into separate spectral lines for each phonon numberand transitions between individual phonon Fock states can be selectivelyaddressed. We show that this enables the preparation of the nanomechanicaloscillator in a single phonon Fock state. Our scheme can for example beimplemented with a carbon nanotube dispersively coupled to the evanescent fieldof a state of the art whispering gallery mode microcavity.
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