  We study the phase diagram of a one-dimensional, two-component attractive
fermions on optical lattices in an off-diagonal confinement. We identify in
this system a pure Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state with spatially
modulated superfluidity and polarization in a large parameter window, which
provides an easier playground to detect the FFLO state experimentally. The
clear signature of FFLO state is analyzed in the ground state properties, the
pair correlation functions, and the magnetic structure factor, using the
density-matrix renormalization group method.
