Influence of scattering on optical response of superconductivity

By using the gauge-invariant kinetic equation, we analytically investigate the influence of the scattering on the optical properties of superconductors in the normal-skin-effect region. Both linear and second-order responses are studied under a multi-cycle terahertz pulse. In the linear regime, we reveal that the optical absorption $\sigma_{1s}(\omega)$, which origins from the scattering, exhibits a crossover point at $\omega=2|\Delta|$. Particularly, it is further shown that when $\omega<2|\Delta|$, $\sigma_{1s}(\omega)$ from the scattering always exhibits a finite value even at low temperature, in contrast to the vanishing $\sigma_{1s}(\omega)$ in the anomalous-skin-effect region as the Mattis-Bardeen theory revealed. In the second-order regime, responses of the Higgs mode during and after the optical pulse are studied. During the pulse, we show that the scattering causes a phase shift in the second-order response of the Higgs mode. Particularly, this phase shift exhibits a significant $\pi$-jump at $\omega=|\Delta|$, which provides a very clear feature for the experimental detection. After the pulse, by studying the damping of the Higgs-mode excitation, we reveal a relaxation mechanism from the elastic scattering, which shows a monotonic enhancement with the increase of the impurity density.