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

Creating arbitrary quantum vibrational states in a carbon nanotube