Phase of Aharonov-Bohm oscillations in conductance of mesoscopic systems

Motivated by a recent experiment we analyze in detail the phase of Aharonov-Bohm oscillations across a 1D ring with a stub coupled to one of its arms, in the presence of a magnetic flux. We demonstrate that there are two kinds of conductance extremas. One class of them are fixed at particular flux values and can only change abruptly from a maxima to a minima as incident energy is varied. We show a different mechanism for such abrupt phase change in conductance oscillation. We demonstrate that these extremas can exhibit $``$phase locking". However, the second kind of extremas can shift continuously as the incident energy is varied.