  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.
