  Local moments with a spin $S>1/2$ can exhibit a rich variety of elementary
quasiparticle excitations, such as quadrupolar excitations, that go beyond the
dipolar magnons of conventional spin-$1/2$ systems. However, the experimental
observation of such quadrupolar excitations is often challenging due to the
dipolar selection rules of many linear response probes, rendering them
invisible. Here we show that non-linear spectroscopy, in the form of
two-dimensional coherent spectroscopy (2DCS), can be used to reveal quadrupolar
excitations. Considering a family of spin-1 Heisenberg ferromagnets with
single-ion easy-axis anisotropy as an example, we explicitly calculate their
2DCS signature by combining exact diagonalization and generalized spin wave
theory. We further demonstrate that 2DCS can provide access to the quadrupolar
weight of an excitation, analogous to how linear response provides access to
the dipolar weight. Our work highlights the potential of non-linear
spectroscopy as a powerful tool to diagnose multipolar excitations in quantum
magnets.
