Transmission phase shift of phonon-assisted tunneling through a quantum dot

The influence of electron-phonon interaction on the transmission phase shift of an electron passing through a quantum dot is investigated by using the scattering theory. The transmission phase versus the intra-dot level shows a serial of phonon-induced dips. These dips are highly sensitive to electron-phonon interaction strength $\lambda$, and they are much more pronounced than phonon-assisted sub-peaks appeared in the conductance. Phonon-induce dephasing is also studied, and the results show that the dephasing probability $T_d$ monotonically increases with the electron-phonon interaction strength $\lambda$. The dephasing probability $T_d \propto\lambda^2$ for small $\lambda$ but $T_d\propto \lambda$ at large $\lambda$.