Spatially-resolved dynamics of the amplitude Schmid-Higgs mode in disordered superconductors

We investigate the spatially-resolved dynamics of the collective amplitude Schmid-Higgs (SH) mode in disordered $s$-wave superconductors and fermionic superfluids. By analyzing the analytic structure of the zero-temperature SH susceptibility in the complex frequency plane, we find that when the coherence length greatly exceeds the mean free path: (i) the SH response at fixed wave vectors exhibits late-time oscillations decaying as $1/t^2$ with frequency $2\Delta$, where $\Delta$ is the superconducting gap; (ii) sub-diffusive oscillations with a dynamical exponent $z{=}4$ emerge at late times and large distances; and (iii) spatial oscillations at fixed frequency decay exponentially, with a period that diverges as the frequency approaches $2\Delta$ from above. When the coherence length is comparable to the mean free path, additional exponentially-decaying oscillations at fixed wave vectors appear with frequency above $2\Delta$. Furthermore, we show that the SH mode induces an extra peak in the third-harmonic generation current at finite wave-vectors. The frequency of this peak is shifted from the conventional resonance at $\Delta$, thereby providing an unambiguous signature of order parameter amplitude dynamics.