Optical conductivity of a dirty current-carrying superconductor

We develop a full microscopic theory for the optical conductivity, $\sigma(\omega)$, of a dirty current-carrying superconductor. Within the Keldysh sigma model formalism, we obtain the general analytical expression for $\sigma(\omega)$, applicable for arbitrary frequency $\omega$, temperature $T$, and dc supercurrent $I$. In addition to altering the usual Mattis-Bardeen conductivity, $\sigma_1(\omega)$, a finite supercurrent introduces two new contributions: $\sigma_2^\text{qp}(\omega)$ from quasiparticle redistribution and $\sigma_2^\text{SH}(\omega)$ from the amplitude (Schmid-Higgs) mode excitation by the ac field. We investigate, both analytically and numerically, the main features of the optical conductivity in the presence of a dc supercurrent. They include a peak in $\text{Re}\,\sigma(\omega)$ above the optical gap and a sign change of $\text{Im}\,\sigma(\omega)$, with both effects becoming more pronounced at higher $I$ and lower $T$. We also elucidate the role of inelastic relaxation, which governs the low-frequency response, leading to a giant microwave absorption and a suppression of the apparent superfluid density at the critical current. The optical conductivity measurement of a superconductor biased by a finite dc supercurrent enables the direct observation of the Schmid-Higgs mode via transport measurements.