  Based on the matrix product states method, we investigate numerically the
ground state properties of one-dimensional mixtures of repulsive bosons and
spin-imbalanced attractive fermions, the latter being in the
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, where Cooper pairs condense at a
finite momentum $k=k_{FFLO}$. We find that the visibility of such a state is
dramatically enhanced as the repulsive Bose-Fermi mixture is brought close to
the phase-separation point. In particular, large amplitude self-induced
oscillations with wave-vector $2k_{FFLO}$ appear in both the fermion total
density and the boson density profiles, leaving sharp fingerprints in the
corresponding static structure factors. We show that these features remain well
visible in cold atoms systems trapped longitudinally by a smooth flat-bottom
potential. Hence bosons can be used to directly reveal the modulated Fermi
superfluid in experiments.
