  We demonstrate that nonlinear response functions in many-body systems carry a
sharp signature of interactions between gapped low-energy quasiparticles. Such
interactions are challenging to deduce from linear response measurements. The
signature takes the form of a divergent-in-time contribution to the response --
linear in time in the case when quasiparticles propagate ballistically -- that
is absent for free bosonic excitations. We give an intuitive semiclassical
picture of this singular behaviour, validated against exact results from a
form-factor expansion of the Ising chain and tDMRG simulations in a
non-integrable model -- the spin-1 AKLT chain. We comment on extensions of
these results to more general settings, finite temperature, and higher
dimensions.
