We consider the electronic current through a one-dimensional conductor in the ballistic transport regime and show that the quantum oscillations of a weakly pinned single-scattering target results in a temperature- and bias-voltage independent excess current at large bias voltages. This is a genuine quantum effect on transport that derives from an exponential reduction of electron backscattering in the elastic channel due to quantum delocalisation of the scatterer and from a suppression of low-energy electron backscattering in the inelastic channels caused by the Pauli exclusion principle. We show that both the mass of the target and the frequency of its quantum vibrations can be measured by studying the differential conductance and the excess current. We apply our analysis to the particular case of a weakly pinned C60 molecule encapsulated by a single-wall carbon nanotube and find that the discussed phenomena are experimentally observable. Copyright © 2008 EPLA.