  We propose that the long-sought Fulde-Ferrell superfluidity with nonzero
momentum pairing can be realized in ultracold two-component Fermi gases of
$^{40}$K or $^{6}$Li atoms by optically tuning their magnetic Feshbach
resonances via the creation of a closed-channel dark state with a
Doppler-shifted Stark effect. In this scheme, two counterpropagating optical
fields are applied to couple two molecular states in the closed channel to an
excited molecular state, leading to a significant violation of Galilean
invariance in the dark-state regime and hence to the possibility of
Fulde-Ferrell superfluidity. We develop a field theoretical formulation for
both two-body and many-body problems and predict that the Fulde-Ferrell state
has remarkable properties, such as anisotropic single-particle dispersion
relation, suppressed superfluid density at zero temperature, anisotropic sound
velocity and rotonic collective mode. The latter two features can be
experimentally probed using Bragg spectroscopy, providing a smoking-gun proof
of Fulde-Ferrell superfluidity.
