  Using exact Bethe ansatz solution, we rigorously study excitation spectra of
the spin-1/2 Fermi gas (called Yang-Gaudin model) with an attractive
interaction. Elementary excitations of this model involve particle-hole
excitations, hole excitations and adding particles in the Fermi seas of pairs
and unpaired fermions. The gapped magnon excitations in spin sector show a
ferromagnetic coupling to the Fermi sea of the single fermions. By numerically
and analytically solving the Bethe ansatz equations and the thermodynamic Bethe
ansatz equations of this model, we obtain excitation energies for various
polarizations in the phase of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-like
state. For a small momentum (long-wavelength limit) and in the strong
interaction regime, we analytically obtained their linear dispersions with
curvature corrections, effective masses as well as velocities in particle-hole
excitations of pairs and unpaired fermions. Such a type of particle-hole
excitations display a novel separation of collective motions of bosonic modes
within paired and unpaired fermions. Finally, we also discuss magnon
excitations in spin sector and the application of the Bragg spectroscopy for
testing such separated charge excitation modes of pairs and single fermions.
