Phase measurements in Aharonov-Bohm interferometers

In this paper we address measurements of the resonant quantum transmission amplitude $t_{QD}=-i|t_{QD}|e^{i\alpha_{QD}}$ through a quantum dot (QD), as function of the plunger gate voltage $V$. Mesoscopic solid state Aharonov-Bohm interferometers (ABI) have been used to measure the "intrinsic" phase, $\alpha_{QD}$, when the QD is placed on one of the paths. In a "closed" interferometer, connected to two terminals, the electron current is conserved, and Onsager's relations require that the conductance ${\cal G}$ through the ABI is an even function of the magnetic flux $\Phi=\hbar c\phi/e$ threading the ABI ring. Therefore, if one fits ${\cal G}$ to $A+B\cos(\phi+\beta)$ then $\beta$ only "jumps" between 0 and $\pi$, with no relation to $\alpha_{QD}$. Additional terminals open the ABI, break the Onsager relations and yield a non-trivial variation of $\beta$ with $V$. After reviewing these topics, we use theoretical models to derive three results on this problem: (i) For the one-dimensional leads, the relation $|t_{QD}|^2 \propto \sin^2(\alpha_{QD})$ allows a direct measurement of $\alpha_{QD}$. (ii) In many cases, the measured ${\cal G}$ in the closed ABI can be used to extract {\it both} $|t_{QD}|$ and $\alpha_{QD}$. (iii) For open ABI's, $\beta$ depends on the details of the opening. We present quantitative criteria (which can be tested experimentally) for $\beta$ to be equal to the desired $\alpha_{QD}$: the "lossy" channels near the QD should have both a small transmission and a small reflection.