  We theoretically predict a new topological matter - topological inhomogeneous
Fulde-Ferrell superfluid - in one-dimensional atomic Fermi gases with equal
Rashba and Dresselhaus spin-orbit coupling near s-wave Feshbach resonances. The
realization of such a spin-orbit coupled Fermi system has already been
demonstrated recently by using a two-photon Raman process and the extra
one-dimensional confinement is easy to achieve using a tight two-dimensional
optical lattice. The topological Fulde-Ferrell superfluid phase is
characterized by a nonzero center-of-mass momentum and a non-trivial Berry
phase. By tuning the Rabi frequency and the detuning of Raman laser beams, we
show that such an exotic topological phase occupies a significant part of
parameter space and therefore it could be easily observed experimentally, by
using, for example, momentum-resolved and spatially resolved radio-frequency
spectroscopy.
