Dephasing by electron-phonon interaction in an AB ring with an embedded quantum dot

We study the dephasing effect on the transport properties in an Aharonov-Bohm ring with an embedded quantum dot, due to the electron-phonon interaction. The differential conductance is calculated under finite bias using Keldysh Green function. The electron-phonon interaction is treated (i) by the canonical transformation, or (ii) by the second-order perturbation. By the latter method, we find that the Fano resonance is suppressed by the electron-phonon interaction and that the suppression increases with increasing bias voltage. The calculations by the former method do not yield the bias-voltage dependence of the dephasing effect.