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We present theoretical calculations of collective modes of the one-bandattractive Hubbard model which is widely used to study the s-wave superfluidphases of atomic Fermi gases of two-hyperfine states loaded in a deep opticallattice. To make our theory applicable for both superconductivity andsuperfluidity, we assume the more general $t-U-J$ Hamiltonian.  Using the functional differentiation we derive Schwinger-Dyson equations forthe single-particle Green's functions.  The method of Legendre transform is used to give a systematic derivation ofthe Bethe-Salpeter (BS) equation for the two-particle Green's function and theassociated collective modes. The numerical solution of the BS equation in thelimit $J\rightarrow 0$ shows the existence of two rotonlike collective modeswith different low-energy Goldstone dispersions and different positions of therotonlike minima. The two rotonlike modes lie outside of the region determinedby the lower boundary of the particle-hole continuum, and therefore, the twomodes are not damped and they should be experimentally observable.  In the presence of superfluid flow at a certain critical flow momentum, theminimum of the first rotonlike mode reaches zero energy, but this occurs beforethe minimum of the second mode and the lower boundary of the particle-holecontinuum do, i.e. there are two critical flow momenta related to the existenceof two rotonlike excitations.

Superfluidity of fermion atoms loaded in a deep optical lattice: the existence of two rotonlike modes