Broken Unitarity and Phase Measurements in Aharonov-Bohm Interferometers

Aharonov-Bohm mesoscopic solid-state interferometers yield a conductance which contains a term $\cos (\phi +\beta )$, where $\phi$ relates to the magnetic flux. Experiments with a quantum dot on one of the interfering paths aim to relate $\beta$ to the dot’s intrinsic Friedel transmission phase ${\alpha }_1$. For closed systems, which conserve the electron current (unitarity), the Onsager relation requires that $\beta =0\mathrm{}$ or $\pi$. For open systems, we show that in general $\beta$ depends on the details of the broken unitarity. Although it gives information on the resonances of the dot, $\beta$ is generally not equal to ${\alpha }_1$. A direct relation between $\beta$ and ${\alpha }_1$ requires specific ways of opening the system, which are discussed.