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

TL;DR

This paper numerically solves the Bogoliubov-de Gennes equations at zero temperature to study Josephson effects across the BCS-BEC crossover, revealing enhanced currents near unitarity and detailed gap and density profiles.

Contribution

It provides a comprehensive, self-consistent numerical analysis of Josephson effects throughout the BCS-BEC crossover, including detailed profiles and stability analysis.

Findings

Josephson current is enhanced near unitarity across barriers

Friedel oscillations observed in the BCS limit

Critical current approaches Landau limit for vanishing barriers

Abstract

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.