  The collective excitations that minimize the Helmholtz free energy of a
population-imbalanced mixture of a $^{6}$Li gas loaded in a quasi
one-dimensional optical lattice are obtained. These excitations reveal a
rotonic branch after solving the Bethe-Salpeter equation under a generalized
random phase approximation based on a single-band Hubbard Hamiltonian. The
phase diagram describing stability regions of Fulde-Farrell-Larkin-Ovchinnikov
and Sarma phases is also analyzed.
