Absence of thermal smearing for the transmission phase through a quantum dot

We derive the phase θ of the Aharonov-Bohm oscillations of a multi-terminal electron ring with a quantum dot in one of its arms. We show i) that θ varies smoothly by π at each transmission peak of the system. We also find ii) that θ drops sharply by π between the peaks if consecutive peaks have similar resonance parameters. The characteristic energy scale for ii) is of the order of the intrinsic resonance width ${\mit\Gamma}$. This fact can be employed to measure ${\mit\Gamma}$ in a novel way.