Abstract: In the presence of spin-orbit coupling and a linear Zeeman field, an interacting Fermi gas exhibits a topological phase transition from a regular superfluid phase to a topological superfluid phase, where the latter phase supports Majorana zero modes. These modes are long sought objects by solid state experiments, since they can show a non-Abelian exchange  statistics with promising potential for applications. Such a motivation has also triggered the search for Majorana zero modes in ultra-cold Fermi gases. They can be found when the fermionic pairing vanishes locally, and thus they are associated either with the system boundary, as edge states, or with internal defects that locally destroy superfluidity, as pinned modes. 

A particularly interesting example of the latter phenomenon in quasi one dimensional gases is the dark soliton, which was theoretically shown to exhibit novel dynamics in the topological phase, qualitatively distinct from the regular behavior of solitons. We have studied this topological excitation and found that there is not only one, but two types of dark solitons in spin-orbit-coupled Fermi gases. The existence of two Fermi surfaces, with different characteristic energy and length scales that feature distinct condensation peaks of fermionic pairs, allows for the emergence of two different types of dark solitons in the regular superfluid phase, while only one type has continuation into the topological phase, where it hosts Majorana zero modes at the core. The detection and identification of the two types of solitons in ultracold-gas experiments requires probing both the fermionic density and the order parameter.