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Wigner function description of a.c. transport through a two-dimensional quantum point contact

We have calculated the admittance of a two-dimensional quantum point contact (QPC) using a novel variant of the Wigner distribution function (WDF) formalism. In the semiclassical approximation, a Boltzmann-like equation is derived for the partial WDF describing both propagating and non-propagating electron modes in an effective potential generated by the adiabatic QPC. We show that this quantum kinetic approach leads to the well known stepwise behaviour of the real part of the admittance (the conductance), and of the imaginary part of the admittance (the emittance), in agreement with the latest results derived by Christen and Büttiker, which is determined by the number of propagating electron modes. It is shown that the emittance is sensitive to the geometry of the QPC, and can be controlled by the gate voltage. We have established that the emittance has contributions corresponding to both quantum inductance and quantum capacitance. Stepwise oscillations in the quantum inductance are determined by the harmonic mean of the velocities for the propagating modes, whereas the quantum capacitance is a significant mesoscopic manifestation of the non-propagating (reflecting) modes.