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Solution of the Bogoliubov-de Gennes equations at zero temperature throughout the BCS-BEC crossover: Josephson and related effects

We investigate the evolution of the Josephson and related effects throughout the BCS-BEC crossover, by performing a systematic numerical solution of the (time-independent) Bogoliubov-de Gennes equations at zero temperature in a fully self-consistent fashion. The numerical strategies and algorithms we have adopted are described in detail, with the aim of easing further independent studies. Several results are obtained by the present calculation. The profiles of the magnitude and phase of the gap parameter across the Josephson barrier are determined under a variety of conditions. We find that the Josephson current is considerably enhanced at about unitarity for all barriers we have considered. A related enhancement is also found in the contribution to the total current from the Andreev bound states, which stem from the depression of the gap profile about the barrier.For vanishing barrier height, we find that the critical Josephson current approaches the limiting value predicted by the Landau criterion, which is determined by either pair-breaking or sound-mode excitations depending on the coupling value. In the BCS limit, we reveal the presence of Friedel oscillations in the oscillatory modulations of the gap and density profiles. In this limit, we also emphasize the special role played by the Andreev bound state in determining the critical Josephson current in the presence of a barrier. Finally, the stability of the two branches, out of which the Josephson characteristics are composed, is analyzed by calculating the energy required to produce a given spatial profile of the gap parameter.