  We consider a weakly-interacting fermionic gas of alkali-metal atoms
characterized by two hyper- fine states which are Rabi coupled. By using a
Hartree approximation for the repulsive interaction we determine the
zero-temperature equation of state of this Fermi gas in D spatial dimensions (D
= 1, 2, 3). Then, adopting the Landau-Vlasov equation and hydrodynamic
equations, we investi- gate the speed of first sound and zero sound. We show
that the two sounds, which occur respectively in collisional and collisionless
regimes, crucially depend on the interplay between interaction strength and
Rabi coupling. Finally, we discuss for some experimentally relevant cases the
effect of a trapping harmonic potential on the density profiles of the
fermionic system.
