Optical conductivity of a dirty current-carrying superconductor

Physical Review B: Condensed Matter and Materials Physics. 2026. Vol. 113. No. 17. Article 174516.

Polkin A., Skvortsov M.

We develop a full microscopic theory for the optical conductivity 𝜎⁡(𝜔) of a dirty current-carrying superconductor. Within the Keldysh sigma model formalism, we obtain the general analytical expression for 𝜎⁡(𝜔), applicable for arbitrary frequency 𝜔, temperature 𝑇, and dc supercurrent 𝐼. In addition to altering the usual Mattis-Bardeen conductivity 𝜎1⁡(𝜔), a finite supercurrent introduces two new contributions: 𝜎qp2⁡(𝜔) from quasiparticle redistribution and 𝜎SH2⁡(𝜔) 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 Re⁢𝜎⁡(𝜔) above the optical gap and a sign change of Im⁢𝜎⁡(𝜔), 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.

Keywords: критический ток, superconductivity, сверхпроводимость, critical current, optical conductivity, комплексная проводимость сверхпроводников, Usadel equations