  Inhomogeneous superfluidity lies at the heart of many intriguing phenomena in
quantum physics. It is believed to play a central role in unconventional
organic or heavy-fermion superconductors, chiral quark matter, and neutron star
glitches. However, so far even the simplest form of inhomogeneous
superfluidity, the Fulde-Ferrell (FF) pairing state with a single
centre-of-mass momentum, is not conclusively observed due to the intrinsic
complexibility of any realistic Fermi systems in nature. Here we theoretically
predict that the controlled setting of ultracold fermionic atoms with synthetic
spin-orbit coupling induced by a two-photon Raman process, demonstrated
recently in cold-atom laboratories, provides a promising route to realize the
long-sought FF superfluidity. At experimentally accessible low temperatures
(i.e., $0.05T_{F}$, where $T_{F}$ is the Fermi temperature), the FF superfluid
state dominates the phase diagram, in sharp contrast to the conventional case
without spin-orbit coupling. We show that the finite centre-of-mass momentum
carried by Cooper pairs is directly measurable via momentum-resolved
radio-frequency spectroscopy. Our work opens the way to direct observation and
characterization of inhomogeneous superfluidity.
