  Reservoir engineering enables the robust preparation of pure quantum states
in noisy environments. We show how a new family of quantum states of a
mechanical oscillator can be stabilized in a cavity that is parametrically
coupled to both the mechanical displacement and the displacement squared. The
cavity is driven with three tones, on the red sideband, on the cavity resonance
and on the second blue sideband. The states so stabilized are (squeezed and
displaced) superpositions of a finite number of phonons. They show the unique
feature of encompassing two prototypes of nonclassicality for bosonic systems:
by adjusting the strength of the drives, one can in fact move from a
single-phonon- to a Schrodinger-cat-like state. The scheme is deterministic,
supersedes the need for measurement-and-feedback loops and does not require
initialization of the oscillator to the ground state. As such, it enables the
unconditional preparation of nonclassical states of a macroscopic object.
