Revealing novel aspects of light-matter coupling in terahertz two-dimensional coherent spectroscopy: the case of the amplitude mode in superconductors

Recently developed terahertz (THz) two-dimensional coherent spectroscopy (2DCS) is a powerful technique to obtain materials information in a fashion qualitatively different from other spectroscopies. Here, we utilized THz 2DCS to investigate the THz nonlinear response of conventional superconductor NbN. Using broad-band THz pulses as light sources, we observed a third-order nonlinear signal whose spectral components are peaked at twice the superconducting gap energy $2\Delta$. With narrow-band THz pulses, a THz nonlinear signal was identified at the driving frequency $\Omega$ and exhibited a resonant enhancement at temperature when $\Omega = 2\Delta$. General theoretical considerations show that such a resonance can only arise from a disorder-activated paramagnetic coupling between the light and the electronic current. This proves that the nonlinear THz response can access processes distinct from the diamagnetic Raman-like density fluctuations, which are believed to dominate the nonlinear response at optical frequencies in metals. Our numerical simulations reveal that even for a small amount of disorder, the $\Omega=2\Delta$ resonance is dominated by the superconducting amplitude mode over the entire investigated disorder range. This is in contrast to other resonances, whose amplitude-mode contribution depends on disorder. Our findings demonstrate the unique ability of THz 2DCS to explore collective excitations inaccessible in other spectroscopies.