Measurement-induced decoherence in electronic interferometry at nanoscale

We introduce a theoretical formalism describing a wide class of 'Which Path' experiments in mesoscopic/nanoscopic transport. The physical system involves a mesoscopic interferometer (e.g. an Aharonov-Bohm ring with embedded dots or a side-coupled quantum dot) which is electrostatically coupled to a nearby quantum point constriction. Due to the charge sensing effect the latter acts as a charge detector. Therefore the interference pattern can be monitored indirectly by looking at the current characteristics of the detector as shown in the experimental work of Buks et al. [E. Buks, R. Schuster, M. Heiblum, D. Mahalu, V. Umansky, Nature (London) 391 (1998) 871]. We use the non-equilibrium Green-Keldysh formalism and a second order perturbative treatment of the Coulomb interaction in order to compute the relevant transport properties. It is shown that in the presence of the Coulomb interaction the current through the detector exhibits oscillations as a function of the magnetic field applied on a single-dot AB interferometer. We also discuss the dependence of the visibility of the Aharonov-Bohm oscillations on the gate potential applied to the dot.