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Multiband superconductors can host collective excitations with markeddifferences with respect to their single-band counterpart. We first study thespectrum of collective amplitude fluctuations in a clean two-bandssuperconductor, showing that the spectral weight of the Higgs mode rapidlydeviates from the naive extension of the single band case as the interbandcoupling is turned on. These results are then used to critically analyze thenon-linear optical response in MgB$_2$, providing an explanation for theapparently contradictory results of recent experiments, pointing towards aselective relevance either of the Leggett mode or of the amplitude fluctuationsat twice the lower gap. By using exact numerical simulations and realisticestimate of disorder we compute the relative contribution of the quasiparticle,amplitude and phase fluctuations to the non-linear optical response. We showthat at low pumping frequency only the resonance at twice the smaller gapemerges, as due to the BCS response, while the Leggett mode dominates only in anarrow range of higher pumping frequencies matching its low-temperature value.Our findings provide a fresh perspective on the potential of non-linear THzspectroscopy to detect collective modes in other multiband systems, as e.g.iron-based superconductors.

Contribution of Collective Excitations to Third Harmonic Generation in Multiband Superconductors: the case of MgB$_2$