Dephasing of electrons in the Aharonov-Bohm interferometer with a single-molecular vibrational junction

Phase relaxation of electrons transferring through an electromechanical transistor is studied using the Aharonov-Bohm interferometer. With the approach of quantum master equation, the phase properties of an electron are numerically analyzed based on the interference fringes. Coherence of electron is partially destroyed by its scattering on excited levels of the local nanomechanical oscillator. Transmission amplitudes with respect to two adjacent mechanical vibrational levels have a phase difference of $\pi$. The character of phase shift by $\pi$ depends on the oscillator frequency only and is robust for the wide range variance of the applied voltage, tunneling length and damping rate of the mechanical oscillator.