Amplitude Higgs mode in superconductors with magnetic impurities

We study the nonlinear response of conventional superconducting alloys with weak magnetic impurities to an external alternating electromagnetic field. In particular, we calculate a correction to the superconducting order parameter $|\delta {\Delta }_{\Omega }|exp\left(i\Omega t\right)$ up to second order in the external vector potential and show that the frequency dependence of the order parameter amplitude has a characteristic resonant shape with a maximum at the frequency which is smaller than twice the magnitude of the pairing amplitude in equilibrium, $\Omega <2\Delta$, and at the same time exceeds the single-particle threshold energy. Our results suggest that in the presence of magnetic impurities the dynamics of the pairing amplitude in the collisionless regime will remain robust with respect to dissipative processes. We also evaluate the third harmonic contribution to the current as a function of the probe frequency and for various concentrations of magnetic impurities.