  Multiband superconductors can host collective excitations with marked
differences with respect to their single-band counterpart. We first study the
spectrum of collective amplitude fluctuations in a clean two-bands
superconductor, showing that the spectral weight of the Higgs mode rapidly
deviates from the naive extension of the single band case as the interband
coupling is turned on. These results are then used to critically analyze the
non-linear optical response in MgB$_2$, providing an explanation for the
apparently contradictory results of recent experiments, pointing towards a
selective relevance either of the Leggett mode or of the amplitude fluctuations
at twice the lower gap. By using exact numerical simulations and realistic
estimate of disorder we compute the relative contribution of the quasiparticle,
amplitude and phase fluctuations to the non-linear optical response. We show
that at low pumping frequency only the resonance at twice the smaller gap
emerges, as due to the BCS response, while the Leggett mode dominates only in a
narrow range of higher pumping frequencies matching its low-temperature value.
Our findings provide a fresh perspective on the potential of non-linear THz
spectroscopy to detect collective modes in other multiband systems, as e.g.
iron-based superconductors.
