  Phonon-cavity electromechanics allows the manipulation of mechanical
oscillations similar to photon-cavity systems. Many advances on this subject
have been achieved in various materials. In addition, the coherent phonon
transfer (phonon Rabi oscillations) between the phonon cavity mode and another
oscillation mode has attracted many interest in nano-science. Here we
demonstrate coherent phonon transfer in a carbon nanotube phonon-cavity system
with two mechanical modes exhibiting strong dynamical coupling. The
gate-tunable phonon oscillation modes are manipulated and detected by extending
the red-detuned pump idea of photonic cavity electromechanics. The first- and
second-order coherent phonon transfers are observed with Rabi frequencies 591
kHz and 125 kHz, respectively. The frequency quality factor product
fQ_m~2=10^12 Hz achieved here is larger thank k_B T_base/h, which may enable
the future realization of Rabi oscillations in the quantum regime.
