  We describe a Fourier transform spectroscopy technique for directly measuring
band structures, and apply it to a spin-1 spin-orbit coupled Bose-Einstein
condensate. In our technique, we suddenly change the Hamiltonian of the system
by adding a spin-orbit coupling interaction and measure populations in
different spin states during the subsequent unitary evolution. We then
reconstruct the spin and momentum resolved spectrum from the peak frequencies
of the Fourier transformed populations. In addition, by periodically modulating
the Hamiltonian, we tune the spin-orbit coupling strength and use our
spectroscopy technique to probe the resulting dispersion relation. The
frequency resolution of our method is limited only by the coherent evolution
timescale of the Hamiltonian and can otherwise be applied to any system, for
example, to measure the band structure of atoms in optical lattice potentials.
