  Interacting bosons with two "spin'' states in a lattice show novel
superfluid-insulator phase transitions in the presence of spin-orbit coupling.
Depending on the parameter regime, bosons in the superfluid phase can condense
to either a zero momentum state or to one or multiple states with finite
momentum, leading to an unconventional superfluid phase. We study the response
of such a system to modulation of the optical lattice potential. We show that
the change in momentum distribution after lattice modulation shows distinct
patterns in the Mott and the superfluid phase and these patterns can be used to
detect these phases and the quantum phase transition between them. Further, the
momentum resolved optical modulation spectroscopy can identify both the gapless
(Goldstone) gapped amplitude (Higgs) mode of the superfluid phase and clearly
distinguish between the superfluid phases with a zero momentum condensate and a
twisted superfluid phase by looking at the location of these modes in the
Brillouin zone. We discuss experiments which can test our theory.
