  We study a gated quantum wire contacted to source and drain electrodes in the
Fabry-Perot regime. The wire is also coupled to a third terminal (tip), and we
allow for an asymmetry of the tip tunneling amplitudes of right and left moving
electrons. We analyze configurations where the tip acts as an electron injector
or as a voltage-probe, and show that the transport properties of this
three-terminal set-up exhibit very rich physical behavior. For a
non-interacting wire we find that a tip in the voltage-probe configuration
affects the source-drain transport in different ways, namely by suppressing the
conductance, by modulating the Fabry-Perot oscillations, and by reducing their
visibility. The combined effect of electron electron interaction and finite
length of the wire, accounted for by the inhomogeneous Luttinger liquid model,
leads to significantly modified predictions as compared to models based on
infinite wires. We show that when the tip injects electrons asymmetrically the
charge fractionalization induced by interaction cannot be inferred from the
asymmetry of the currents flowing in source and drain. Nevertheless interaction
effects are visible as oscillations in the non-linear tip-source and tip-drain
conductances. Important differences with respect to a two-terminal set-up
emerge, suggesting new strategies for the experimental investigation of
Luttinger liquid behavior.
