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We investigate the charge current that is optically injected by interference between one- and two-photon excitation in the presence of excitonic effects. We consider a realistic V-shaped quantum wire excited slightly below the band gap by two simultaneous femtosecond laser pulses of frequency 2 and that interact, respectively, with the lowest B1 and A1 excitons. Using effective multiband Bloch equations for two-photon transitions, including the Coulomb interaction within the Hartree-Fock approximation, we show that, because of the different symmetry properties of the involved excitons, the generated charge current displays oscillations due to the quantum interference between the excitonic coherences.