Josephson Critical Currents and Related Effects in Ultracold Atomic Superfluid Sytems

The Josephson and Proximity effects play a pivotal role in the design of superconducting devices for the implementation of quantum technology, ranging from the standard Al based to the more exotic twisted high-Tc junctions. Josephson critical currents have been recently investigated also in ultracold atomic systems where a potential barrier acts as a weak link. The unifying feature of the above systems, apart from being superconducting/superfluid, is the presence of spatial inhomogeneity, a feature that has to be properly taken into account in any theoretical approach employed to investigate them. In this work, we review the novel (dubbed LPDA for Local Phase Density Approximation) approach based on a coarse graining of the Bogoliubov-de Gennes (BdG) equations. Non-local and local forms of this coarse graining were utilized when investigating Proximity and Josephson effects. Moreover, the LPDA approach was further developed to include pairing fluctuations at the level of the non-self-consistent t-matrix approximation. The resulting approach, dubbed mLPDA (modified LPDA), can be used whenever inhomegeneity and fluctuations effects simultaneously play an important role.