We develop a full microscopic theory for the optical conductivity $\sigma \left(\omega \right)$ of a dirty current-carrying superconductor. Within the Keldysh sigma model formalism, we obtain the general analytical expression for $\sigma \left(\omega \right)$, applicable for arbitrary frequency $\omega$, temperature $T$, and dc supercurrent $I$. In addition to altering the usual Mattis-Bardeen conductivity ${\sigma }_1\left(\omega \right)$, a finite supercurrent introduces two new contributions: ${\sigma }_2^{\text{qp}}\left(\omega \right)$ from quasiparticle redistribution and ${\sigma }_2^{\text{SH}}\left(\omega \right)$ 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 $\mathrm{Re}\sigma \left(\omega \right)$ above the optical gap and a sign change of $\mathrm{Im}\sigma \left(\omega \right)$, 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.

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• Critical current
• Optical conductivity
• Superconductivity
• Superfluid density
• Superconductors
• Methods in superconductivity
• Usadel equation