  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 wavefunction
and its correlation function to study their penetration into the noninteracting
region. For the step-function quench, both correlation lengths from the pair
wavefunction 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.
