Quadratic pair-breaking absorption edge from Anderson localization in a superconducting wire

Abstract

We show that Anderson localization fundamentally reshapes dissipative superconducting electrodynamics, replacing the linear Mattis--Bardeen pair-breaking onset with a parametrically weaker quadratic Mott edge while leaving the quasiparticle spectrum unchanged. We derive this behavior for a weakly disordered single-channel wire with a spatially uniform $s$-wave pair potential using a Nambu-space extension of the Berezinskii diagram technique that resums elastic impurity scattering to all orders. Conservation of the Bogoliubov branch collapses the Nambu diagram hierarchy onto an exactly solvable localization problem. The localization length remains equal to its normal-state value, whereas the localization time diverges at the gap edge. The new absorption edge results from the product of the superconducting pair-creation coherence factor and the Mott-suppressed current matrix elements between localized orbitals, which generates the characteristic double logarithm. Localization also reduces the Hebel--Slichter coherence peak. The resulting theory provides an all-orders benchmark for microwave and spin-relaxation experiments on localized superconducting nanowires.