  Coupling an electromechanical resonator with carbon-nanotube quantum dots is
a significant method to control both the electronic charge and the spin quantum
states. By exploiting a novel micro-transfer technique, we fabricate two
strongly-coupled and electrically-tunable mechanical resonators on a single
carbon nanotube for the first time. The frequency of the two resonators can be
individually tuned by the bottom gates, and strong coupling is observed between
the electron charge and phonon modes of each resonator. Furthermore, the
conductance of either resonator can be nonlocally modulated by the phonon modes
in the other resonator. Strong coupling is observed between the phonon modes of
the two resonators, which provides an effective long distance electron-electron
interaction. The generation of phonon-mediated-spin entanglement is also
theoretically analyzed for the two resonators. This strongly-coupled nanotube
electromechanical resonator array provides an experimental platform for future
studies of the coherent electron-phonon interaction, the phonon mediated
long-distance electron interaction, and entanglement state generation.
