  The conductance and the transmittance phase shifts of a two-terminal
Aharonov-Bohm (AB) ring are analyzed in the presence of mechanical
displacements due to coupling to an external can- tilever. We show that phase
rigidity is broken, even in the linear response regime, by means of inelastic
scattering due to phonons. Our device provides a way of observing continuous
variation of the transmission phase through a two-terminal
nano-electro-mechanical system (NEMS). We also propose measurements of phase
shifts as a way to determine the strength of the electron-phonon coupling in
NEMS.
