Collective modes in terahertz field response of disordered superconductors

We consider a problem of nonlinear response to an external electromagnetic radiation in conventional disordered superconductors which contain a small amount of weak magnetic impurities. We focus on the diffusive limit and use Usadel equation to analyze the excitation energy and dispersion relation of the collective modes. We determine the resonant frequency and dispersion of both amplitude (Schmidt-Higgs) and phase (Carlson-Goldman) modes for moderate strength of magnetic scattering. We find that the minimum energy required for the excitation of the both of these modes decreases with an increase in spin-flip scattering. Surprisingly we also find that as a result the Carlson-Goldman mode becomes gapless and as a consequence can only be excited at some finite value of the threshold momentum. We thus discover yet another physical realization of a state with gapped momentum dispersion of one of its collective modes. The value of the threshold momentum is determined by the distance between the two consecutive spin-flip scattering events which, in turn, is proportional to the scattering time between two consecutive scattering events. The amplitude mode is diffusive and becomes strongly suppressed with the increase in spin-flip scattering. Possible ways to experimentally verify our results are also discussed.