  We explore the quantum many-body physics of a three-component Bose-Einstein
condensate (BEC) in an optical lattices driven by laser fields in $V$ and
$\Lambda$ configurations. We obtain exact analytical expressions for the energy
spectrum and amplitudes of elementary excitations, and discover symmetries
among them. We demonstrate that the applied laser fields induce a gap in the
otherwise gapless Bogoliubov spectrum. We find that Landau damping of the
collective modes above the energy of the gap is carried by laser-induced roton
modes and is considerably suppressed compared to the phonon-mediated damping
endemic to undriven scalar BECs.
