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 $\Delta_{P}$ 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 $\omega$. We analyze the resonator quality factor $Q(\omega,T)$ in the practically relevant range $\hbar\omega,\,T\ll\Delta\leq\Delta_{P}$, where $\Delta$ is the typical superconducting order parameter, distinct from $\Delta_{P}$. We show that low-$\omega$ dissipation is dominated by a new type of bulk localized collective modes arising from spatial inhomogeneity of the superconducting state. Consequently, $Q(\omega)$ decreases strongly with $\omega$ 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.