  We investigate the system of a heavy impurity embedded in a paired
two-component Fermi gas at the crossover from a Bose-Einstein condensate (BEC)
to a Bardeen--Cooper--Schrieffer (BCS) superfluid via an extension of the
functional determinant approach (FDA). FDA is an exact numerical approach
applied to study manifestations of Anderson\textquoteright s orthogonality
catastrophe (OC) in the system of a static impurity immersed in an ideal Fermi
gas. Here, we extend the FDA to a strongly correlated superfluid background
described by a BCS mean-field wavefunction. In contrast to the ideal Fermi gas
case, the pairing gap in the BCS superfluid prevents the OC and leads to
genuine polaron signals in the spectrum. Thus, our exactly solvable model can
provide a deeper understanding of polaron physics. In addition, we find that
the polaron spectrum can be used to measure the superfluid pairing gap, and in
the case of a magnetic impurity, the energy of the sub-gap Yu-Shiba-Rusinov
(YSR) bound state. Our theoretical predictions can be examined with
state-of-art cold-atom experiments.
