High-order nonlinear terahertz probing of the two-band superconductor MgB$_2$

We report on high-order harmonic generation in the two-band superconductor MgB$_2$ driven by intense terahertz electromagnetic pulses. Third- and fifth-order harmonics are resolved in time-domain and investigated as a function of temperature and in applied magnetic fields crossing the superconducting phase boundary. The high-order harmonics in the superconducting phase reflects nonequilibrium dynamics of the superconducting order parameter in MgB$_2$, which is probed via nonlinear coupling to the terahertz field. The observed temperature and field dependence of the nonlinear response allows to establish the superconducting phase diagram.