Dissipation due to Bulk Localized Low-Energy Modes in Strongly Disordered Superconductors

Abstract

Strongly disordered superconductors (SDSCs) are widely used in qubits, microwave resonators, photon detectors, and other superconducting quantum devices. In SDSC-based devices, coherence times are limited by low-temperature microwave dissipation in the material. However, the standard Mattis-Bardeen theory fails in SDSCs because their single-particle spectrum exhibits a hard pseudogap $\hat{I}{”}_P$ both below and above the transition temperature $T_c$. We develop a novel microscopic theory of the dependence of \emph{ac }dissipation in such systems on temperature $T$ and frequency $\ddot{I}‰$. We analyze the resonator quality factor $Q(\ddot{I}‰,T)$ in the practically relevant range $\hslash \ddot{I}‰,T\ll \hat{I}”\le \hat{I}{”}_P$, where $\hat{I}”$ is the typical superconducting order parameter, distinct from $\hat{I}{”}_P$. We show that low-$\ddot{I}‰$ dissipation is dominated by a new type of bulk localized collective modes arising from spatial inhomogeneity of the superconducting state. Consequently, $Q(\ddot{I}‰)$ decreases strongly with $\ddot{I}‰$ and exhibits two-level-system-like growth with $T$ for $T\ll T_c$. Our theory provides a microscopic understanding of existing and future experiments on thin films of ${\mathrm{InO}}_x$, TiN, NbN, and similar SDSCs, and is phenomenologically relevant to granular aluminum films. The results suggest strategies to mitigate intrinsic microwave losses in SDSC-based quantum devices.

8 pages, 3 figures. Typos and bibliography formatting have been corrected to match the version published in Physical Review Letters