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Phase relaxation of electrons transferring through an electromechanicaltransistor is studied using the Aharonov-Bohm interferometer. With the approachof quantum master equation, the phase properties of an electron are numericallyanalyzed based on the interference fringes. Coherence of electron is partiallydestroyed by its scattering on excited levels of the local nanomechanicaloscillator. Transmission amplitudes with respect to two adjacent mechanicalvibrational levels have a phase difference of $\pi$. The character of phaseshift by $\pi$ depends on the oscillator frequency only and is robust for thewide range variance of the applied voltage, tunneling length and damping rateof the mechanical oscillator.

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