Time-Resolved Optical Conductivity And Higgs Oscillations In Two-Band Dirty Superconductors

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 superconducting condensate arises, which drastically enhances excitation. So far, most work concentrated on the periodic driving with light, where the third-harmonic generation (THG) response of the Higgs mode was shown to be enhanced. In this paper, we extend this analysis by calculating full temperature and frequency dependence of THG to better compare the theory with current experimental setups. 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. In contrast to previous work we calculate the response not only within a time-dependent density-matrix formalism but also in a diagrammatic picture derived from an effective action formalism, which gives a deeper insight into the microscopic processes.