Fig. 1: Quasiparticle fluctuation measurement and the inferred effect of disorder on quasiparticle recombination.

a Micrograph of the microwave resonator consisting of a NbTiN capacitor and β-Ta inductor on a membrane, which is highlighted in red. Inside the white circle, the figure is two times enlarged. We measure fluctuations in resonance frequency (δf), which are proportional to complex conductivity fluctuations (δσ₂). b Sketch of the β-Ta inductor on the 110-nm-thick SiN membrane. When two quasiparticles (green) recombine into a Cooper-pair (blue), a phonon is emitted (yellow curvy arrows). A change in the number of quasiparticles or Cooper-pairs changes σ₂, which we measure. The emitted phonon is trapped by the membrane, as indicated. We measure two resonators: one with the inductor on a SiN membrane (as sketched) and one with the SiN patch on solid Si substrate. c Sketch of traditional quasiparticle recombination with emission of a phonon with energy ≥2Δ₀, which can subsequently break a Cooper-pair. The BCS density of state (DOS) is sketched on the right, which has the same energy y-axis. d Sketch of quasiparticle recombination in disordered superconductors. Disorder can suppress the gap locally, inducing quasiparticle localization at a typical length scale of r_c ~ ξ¹⁶. Quasiparticles can delocalize via absorption of a phonon, after which they relax and rapidly recombine on-site, emitting a phonon with less than 2Δ₀ energy. Since phonon absorption is slow in disordered superconductors^27,40 and subsequent on-site recombination is fast^41,57, delocalization limits this process. On the right, the position-averaged DOS is sketched with a broadened coherence peak, parameterized by η, and a subgap tail consisting of localized states, parameterized by Γ_tail^42,44.