  We theoretically study the creation of single- and multi-phonon Fock states
and arbitrary superpositions of quantum phonon states in a nanomechanical
carbon nanotube (CNT) resonator. In our model, a doubly clamped CNT resonator
is initialized in the ground state and a single electron is trapped in a
quantum dot which is formed by a electric gate potential and brought into the
magnetic field of a micro-magnet. The preparation of arbitrary quantum phonon
states is based on the coupling between the mechanical motion of the CNT and
the electron spin which acts as a non-linearity. We assume that electrical
driving pulses with different frequencies are applied on the system. The
quantum information is transferred from the spin qubit to the mechanical motion
by the spin-phonon coupling and the electron spin qubit can be reset by the
single-electron spin resonance. We describe Wigner tomography which can be
applied at the end to obtain the phase information of the prepared phonon
states.
