Title: Investigation of Quasi-particle Relaxation in Strongly Disordered Superconductor Resonators

Abstract: In this paper, we investigate the quasi-particle (QP) relaxation of strongly disordered superconducting resonators under optical illumination at different bath temperatures with the Rothwarf and Taylor equations and the gap-broadening theory described by the Usadal equation. The analysis is validated with various single-photon responses of Titanium Nitride (TiN) microwave kinetic inductance detectors (MKIDs) under pulsed 405~nm laser illumination. The QP relaxation in TiN is dominated by QPs with energy below the energy gap smeared by the disorder, and its duration is still inversely proportional to the QP density. The QP lifetime versus temperature can be fitted. The relaxation of the resonator can be further modeled with QP diffusion. The fitted QP diffusion coefficient of TiN is significantly smaller than expected. Our result also shows a significant increase in QP generation efficiency as the bath temperature increases.