  We present a scheme for tuning and controlling nano mechanical resonators by
subjecting them to electrostatic gradient fields, provided by nearby tip
electrodes. We show that this approach enables access to a novel regime of
optomechanics, where the intrinsic nonlinearity of the nanoresonator can be
explored. In this regime, one or several laser driven cavity modes coupled to
the nanoresonator and suitably adjusted gradient fields allow to control the
motional state of the nanoresonator at the single phonon level. Some
applications of this platform have been presented previously [New J. Phys. 14,
023042 (2012), Phys. Rev. Lett. 110, 120503 (2013)]. Here, we provide a
detailed description of the corresponding setup and its optomechanical coupling
mechanisms, together with an in-depth analysis of possible sources of damping
or decoherence and a discussion of the readout of the nanoresonator state.
