  We study the ground-state properties of a two-component one-dimensional
system of a few ultra-cold fermions with attractive interactions. We show that,
by ramping up an external potential barrier felt by one of the components, it
is possible to induce regions of exotic superfluid phases, characterized by a
tunable finite net momentum of the Cooper pair, without changing the overall
spin populations. We show that these phases, which are the few-body analogs of
the celebrated Fulde-Ferrell-Larkin-Ovchinnikov state, can be distinguished by
analyzing a specific two-particle correlation encoded in the noise correlation
function. Our theoretical results can be addressed in current experiments with
cold atoms confined in spin-selective optical traps.
