  Motivated by recent experimental breakthroughs in generating spin-orbit
coupling in ultracold Fermi gases using Raman laser beams, we present a
systematic study of spin-orbit-coupled Fermi gases confined in a
quasi-one-dimensional trap in the presence of an in-plane Zeeman field (which
can be realized using a finite two-photon Raman detuning). We find that a
topological Fulde-Ferrell state will emerge, featuring finite-momentum Cooper
pairing and zero-energy Majorana excitations localized near the edge of the
trap based on the self-consistent Bogoliubov-de Genes (BdG) equations. We find
analytically the wavefunctions of the Majorana modes. Finally using the
time-dependent BdG we show how the finite-momentum pairing field manifests
itself in the expansion dynamics of the atomic cloud.
