  We investigate the collective excitations of a Raman-induced spin-orbit
coupled Bose-Einstein condensate confined in a quasi one-dimension harmonic
trap using the Bogoliubov method. By tuning the Raman coupling strength, three
phases of the system can be identified. By calculating the transition strength,
we are able to classify various excitation modes that are experimentally
relevant. We show that the three quantum phases possess distinct features in
their collective excitation properties. In particular, the spin dipole and the
spin breathing modes can be used to clearly map out the phase boundaries. We
confirm these predictions by direct numerical simulations of the quench
dynamics that excites the relevant collective modes.
