  Ultracold atoms confined to periodic potentials have proven to be a powerful
tool for quantum simulation of complex many-body systems. We confine fermions
to one-dimension to realize the Tomonaga-Luttinger liquid model describing the
highly collective nature of their low-energy excitations. We use Bragg
spectroscopy to directly excite either the spin or charge wave for various
strength of repulsive interaction. We observe that the velocity of the spin and
charge excitations shift in opposite directions with increasing interaction, a
hallmark of spin-charge separation. The excitation spectra are in quantitative
agreement with the Tomonaga-Luttinger liquid theory, and furthermore, we find
that the spin excitations become dispersive at large interaction, signaling the
onset of the nonlinear Luttinger liquid regime.
