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The nomenclature quantum transport has been coined for the phenomenon of electron motion through constrictions of transverse dimensions smaller than the electron wavelength, e.g., quantum-point contacts, quantum wires, molecules, etc. To describe transport properties on such a small scale, a quantum theory of transport is required. In this Chapter we focus on quantum transport problems whose experimental setup is schematically displayed in Fig. 32.1a. A central region of meso- or nanoscopic size is coupled to two metallic electrodes which play the role of charge reservoirs. The whole system is initially in a well defined equilibrium configuration, described by a unique temperature and chemical potential (thermodynamic consistency). No current flows through the junction, the charge density of the electrodes being perfectly balanced. In the previous Chapter, Gebauer et al. proposed to join the left and right remote parts of the system so to obtain a ring geometry, see Fig. 30.1. In their approach the electromotive force is generated by piercing the ring with a magnetic field that increases linearly in time. Here, we consider the longitudinal geometry of Fig. 32.1a and describe an alternative approach. As originally proposed by Cini [Cini 1980], we may drive the system out of equilibrium by exposing the electrons to an external time-dependent potential which is local in time and space. For instance, we may switch on an electric field by putting the system between two capacitor plates far away from the system boundaries, see Fig. 32.1b. The dynamical formation of dipole layers screens the potential drop along the electrodes and the total potential turns out to be uniform in the left and right bulks. Accordingly, the potential drop is entirely limited to the central region. As the system size increases, the remote parts are less disturbed by the junction, and the density inside the electrodes approaches the equilibrium bulk density.This work was supported by the European Community 6th framework Network of Excellence NANOQUANTA (NMP4-CT-2004-500198) and by the Research and Training Network EXCITING. AR acknowledges support from\udthe EC project M-DNA (IST-2001-38051), Spanish MCyT and the University\udof the Basque Country.Peer reviewe

Time-Dependent Transport Through Single Molecules: Nonequilibrium Green’s Functions