  Two dimensional coherent spectroscopy (2DCS) probes the nonlinear optical
response of correlated systems. An interesting application is the study of
fractionalized excitations, which are challenging to distinguish unambiguously
in linear response. Here we demonstrate how the sensitivity of optical matrix
elements to variations in the photon polarization allows one to probe different
aspects of low lying excitations in models of the candidate fractionalized
materials Ce$_2$Zr$_2$O$_7$ and Nd$_2$Zr$_2$O$_7$, which host effective
one-dimensional spin chains when subjected to a [110] magnetic field. We show
how both fractionalized spinon excitations or conventional magnons can be
picked out in the 2DCS response, and how the response from polarized spin
chains can be used to probe the dipolar-octupolar mixing angle $\theta$ through
the relative intensity of one- and two-magnon signals. Further, we find that a
$[001]$ polarization of the probe field is particularly sensitive to lower band
edge of the spinon continuum and can be used as a measure of the proximity of a
quantum critical point in Ce$_2$Zr$_2$O$_7$. 2DCS can thus be employed to
provide invaluable and detailed information both on the constituent degrees of
freedom of a quantum material and on their collective behaviour.
