We report a fully self-consistent calculation of the complex renormalization by exchange interactions and hence the complete phase diagram of conventional $s$-wave superconductors with magnetic impurities as well as the related physical properties including the optical response. We show that a small amount of magnetic disorder can drive the system into a gapless superconducting state, where the single-particle excitation gap vanishes whereas the superconducting order parameter ${\mathrm{\Delta }}_0$ remains finite. In this phase, the linear optical conductivity exhibits a finite absorption over the low-frequency regime, particularly for photon energies below the conventional threshold $2|{\mathrm{\Delta }}_0|$, even at low temperatures, in sharp contrast to the gapped state. The nonlinear response, however, remains coherent and is dominated by the Higgs-mode dynamics rather than the gapless quasiparticle background. These findings reveal a fundamental distinction between dissipative single-particle excitations and coherent collective dynamics of the condensate, a feature likely general to other gapless superconductors, and introduce a fundamentally different detection scheme, using THz spectroscopy to probe the signatures of Shiba states.

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• Impurities in superconductors
• Light-matter interaction
• Superconducting phase transition
• Superconductors