  We present the collective excitation spectrum analysis of binary
Bose-Einstein condensates (BECs) with spin-orbit (SO) and Rabi couplings in a
quasi-two-dimensional system. In particular, we investigate the role of SO and
Rabi coupling strengths in determining the dynamical stability of the coupled
BECs using Bogoliubov-de Gennes (BdG) theory. Using the eigenergy of BdG
spectrum, we confirm the existence of phonon, roton, and maxon modes with weak
repulsive intra- and inter-species contact interactions. The depth of the
minimum corresponding to the roton mode depends strongly on the coupling
strength. We find that the increase of the SO coupling leads to instability,
while the increase in the Rabi coupling stabilizes the system. Also the
eigenvectors of BdG spectrum indicates the presence of density like mode in the
stable regime and spin like modes in unstable regimes. A phase diagram
demonstrating the stability regime in the plane of SO and Rabi coupling
strengths is obtained. Finally, we complement the observation of the excitation
spectrum with the direct numerical simulation results of coupled
Gross-Pitaevskii equations.
