Mobile magnetic impurities in a Fermi superfluid: A route to designer molecules

A magnetic impurity in a fermionic superfluid hosts bound quasiparticle states known as Yu-Shiba-Rusinov states. We argue here that, if the impurity is mobile (i.e., has a finite mass), the impurity and its bound Yu-Shiba-Rusinov quasiparticle move together as a midgap molecule, which has an unusual "Mexican-hat" dispersion that is tunable via the fermion density. We map out the impurity dispersion, which consists of an "atomic" branch (in which the impurity is dressed by quasiparticle pairs) and a "molecular" branch (in which the impurity binds a quasiparticle). We discuss the experimental realization and detection of midgap Shiba molecules, focusing on Li-Cs mixtures, and comment on the prospects they offer for realizing exotic many-body states.