  We study frequency dependent noise of a suspended carbon nanotube quantum dot
nanoelectromechanical resonator induced by electron-vibration coupling. By
using rigorous Keldysh diagrammatic technique, we build a formal framework to
connect the vibration properties and the electrical measurement. We find that
the noise power spectrum has a narrow resonant peak at the frequency of
vibrational modes. This fine structure feature disappears due to a coherent
cancellation effect when tuning tunneling barriers to a symmetric point. We
note that measuring the electrical current noise spectra provides an
alternative and ultra-sensitive detection method for determining the damping
and dephasing of the quantum vibration modes.
