Second-order optical response of superconductors induced by supercurrent injection

We develop a theory of the nonlinear optical responses in superconducting systems in the presence of a dc supercurrent. The optical transitions between particle-hole pair bands across the superconducting gap are allowed in clean superconductors as the inversion symmetry breaking by supercurrent. Vertex correction is included in optical conductivity to maintain the U(1) gauge symmetry in the mean-field formalism, which contains the contributions from collective modes. We show two pronounced current-dependent peaks in the second-harmonic generation ${\ensuremath{\sigma}}⁽²⁾(2\ensuremath{\omega},\ensuremath{\omega},\ensuremath{\omega})$ at the gap edge $2\ensuremath{\hbar}\ensuremath{\omega}=2\mathrm{\ensuremath{\Delta}}$ and $\ensuremath{\hbar}\ensuremath{\omega}=2\mathrm{\ensuremath{\Delta}}$ and one in the photocurrent effect ${\ensuremath{\sigma}}⁽²⁾(0,\ensuremath{\omega},\ensuremath{-}\ensuremath{\omega})$ at $\ensuremath{\hbar}\ensuremath{\omega}=2\mathrm{\ensuremath{\Delta}}$, all of which diverge in the clean limit. We demonstrate this in the models of a single-band superconductor with $s$-wave and $d$-wave pairings, and Dirac fermion systems with $s$-wave pairing. Our theory predicts that the current-induced peak in $\text{Im}[{\ensuremath{\sigma}}⁽²⁾(\ensuremath{\omega})]$ is proportional to the square of the supercurrent density in the $s$-wave single-band model, with the same order of magnitude as the recent experimental observation of second-harmonic generation in NbN by Nakamura et al. [Phys. Rev. Lett. 125, 097004 (2020)]. Supercurrent-induced nonlinear optical spectroscopy provides a valuable toolbox to explore novel superconductors.