  Twodimensional spin-orbit-coupled Fermi gases subject to s-wave pairing can
be driven into a topological phase by increasing the Zeeman spin splitting
beyond a critical value. In the topological phase, the system exhibits the
hallmarks of chiral p-wave superfluidity, including exotic Majorana
excitations. Previous theoretical studies of this realization of a
twodimensional topological Fermi superfluid have focused on the BCS regime
where the s-wave Cooper pairs are only weakly bound and, hence, the induced
chiral p-wave order parameter has a small magnitude. Motivated by the goal to
identify potential new ways for the experimental realization of robust
topological superfluids in ultra-cold atom gases, we study the BCS-to-BEC
crossover driven by increasing the Cooper-pair binding energy for this system.
In particular, we obtain phase diagrams in the parameter space of two-particle
bound-state energy and Zeeman spin-splitting energy. Ordinary characteristics
of the BCS-to-BEC crossover, in particular the shrinking and eventual
disappearance of the Fermi surface, are observed in the nontopological phase.
In contrast, the topological phase retains all features of chiral p-wave
superfluidity, including a well-defined underlying Fermi surface, even for
large s-wave pair-binding energies. Compared to the BCS limit, the topological
superfluid in the BEC regime turns out to be better realizable even for only
moderate magnitude of spin-orbit coupling because the chiral p-wave order
parameter is generally larger and remnants of s-wave pairing are suppressed. We
identify optimal parameter ranges that can aid further experimental
investigations and elucidate the underlying physical reason for the persistence
of the chiral p-wave superfluid.
