Amplitude mode in a multi-gap superconductor MgB$_2$ investigated by terahertz two-dimensional coherent spectroscopy

We have investigated terahertz (THz) nonlinear responses in a multi-gap superconductor, MgB$_2$, using THz two-dimensional coherent spectroscopy (THz 2DCS). With broad-band THz drives, we identified a well-defined nonlinear response near the lower superconducting gap energy $2\Delta_{\pi}$ only at the lowest temperatures. Using narrow-band THz driving pulses, we observed first (FH) and third harmonic responses, and the FH intensity shows a monotonic increase with decreasing temperature when properly normalized by the driving field strength. This is distinct from the single-gap superconductor NbN, where the FH signal exhibited a resonant enhancement at temperatures near the superconducting transition temperature $T_{\text{c}}$ when the superconducting gap energy was resonant with the driving photon energy and which had been interpreted to originate from the superconducting amplitude mode. Our results in MgB$_2$ are consistent with a well-defined amplitude mode only at the lowest temperatures and indicate strong damping as temperature increases. This likely indicates the importance of interband coupling in MgB$_2$ and its influence on the nature of the amplitude mode and its damping.