Proximity effect and spatial Kibble-Zurek mechanism in atomic Fermi gases with inhomogeneous pairing interactions

Introducing spatially tunable interactions to atomic Fermi gases makes it feasible to study two phenomena, the proximity effect and spatial Kibble-Zurek mechanism (KZM), in a unified platform. While the proximity effect of a superconductor adjacent to a normal metal corresponds to a step-function quench of the pairing interaction in real space, the spatial KZM is based on a linear drop of the interaction that can be modeled as a spatial quench. After formulating the Fermi gases with spatially varying pairing interactions by the Bogoliubov-de Gennes equation, we obtain the profiles of the pair wave function and its correlation function to study their penetration into the noninteracting region. For the step-function quench, both correlation lengths from the pair wave function and its correlation function follow the BCS coherence length and exhibit the same scaling behavior. In contrast, the scaling behavior of the two correlation lengths are different in the spatial quench, which then allows more refined analyses of the correlation lengths from different physical quantities. Moreover, adding a weakly interacting bosonic background does not change the scaling behavior. We also discuss relevant experimental techniques that may realize and verify the inhomogeneous phenomena.