  Recent studies have emphasized the importance of impurity scattering for the
optical Higgs response of superconductors. In the dirty limit, an additional
paramagnetic coupling of light to the superconducting condensate arises which
drastically enhances excitation. So far, most work concentrated on the periodic
driving with light, where the third-harmonic generation response of the Higgs
mode was shown to be enhanced. In this work, we additionally calculate the
time-resolved optical conductivity of single- and two-band superconductors in a
two-pulse quench-probe setup, where we find good agreement with existing
experimental results. We use the Mattis-Bardeen approach to incorporate
impurity scattering and calculate explicitly the time-evolution of the system.
Calculations are performed both in a diagrammatic picture derived from an
effective action formalism and within a time-dependent density matrix
formalism.
