Phase evolution of the transmission coefficient in an Aharonov-Bohm ring with Fano resonance

Phase of the transmission coefficient for a mesoscopic loop with Fano resonance has been investigated by using the scattering matrix method, for a model of an Aharonov-Bohm ring with a double barrier coupled to a $t$-stub resonator in one arm. It is found that the phase varies rapidly by π on the shoulder of a resonance peak. In addition, it is found that there is a sudden phase jump by $\pi$ near the tail of a resonance peak. Each resonance peak is found to return to the same phase via two processes; in one process, the phase shift by $\pi$ occurs rather smoothly through the excitation of the $h/2e$ oscillation near the resonance peak, and in the other process, the same amount of phase shift occurs very abruptly across the zero transmission of the resonator. These phase behaviors show some resemblance to the ones found in the recent phase measurement of a quantum dot via double-slit interference experiment.