  Motivated by the recent experimental realization of a candidate to the
Fulde-Ferrell (FF) and the Larkin-Ovchinnikov (LO) states in one dimensional
(1D) atomic Fermi gases, we study the quantum phase transitions in these
enigmatic, finite momentum-paired superfluids. We focus on the FF state and
investigate the effects of the induced interaction on the stability of the FFLO
phase in homogeneous spin-imbalanced quasi-1D Fermi gases at zero temperature.
When this is taken into account we find a direct transition from the fully
polarized to the FFLO state. Also, we consider the effect of a finite lifetime
of the quasi-particles states in the normal-superfluid instability. In the
limit of long lifetimes, the lifetime effect is irrelevant and the transition
is directly from the fully polarized to the FFLO state. We show, however, that
for sufficiently short lifetimes there is a quantum critical point (QCP), at a
finite value of the mismatch of the Fermi wave-vectors of the different
quasi-particles, that we fully characterize. In this case the transition is
from the FFLO phase to a normal partially polarized state with increasing
mismatch.
