Universal scaling of microwave dissipation in superconducting circuits

TL;DR

This work identifies a universal scaling between microwave dissipation and bulk superfluid density across a wide range of superconducting materials and device geometries, linking intrinsic loss to nonequilibrium quasiparticles trapped in disorder-induced gap variations. By modeling localized and delocalized quasiparticle states and compiling a large experimental dataset, it derives scaling laws such as $Q_i^{\max} \approx \kappa\,\sigma_2$ with $\kappa \sim (0.1-1)\ \Omega\cdot\mathrm{m}$ and reveals distinct dirty ($Q_i \propto \sigma_2$) and clean ($Q_i \propto \sigma_2^{3/2}$) limits, while identifying a planar-substrate–limited bound around $Q_i \sim 10^7$ for 2D circuits. The framework connects microscopic quasiparticle kinetics to macroscopic dissipation, explaining most measured trends and suggesting routes to push coherence limits via quasiparticle filtering, gap engineering, and improved packaging. It shows that bulk substrate loss sets a practical ceiling on planar devices, guiding material choice (Nb often offering the best ultimate performance) and architectures toward overcoming intrinsic bulk limits for scalable quantum circuits.

Abstract

Improving the coherence of superconducting qubits is essential for advancing quantum technologies. While superconductors are theoretically perfect conductors, they consistently exhibit residual energy dissipation when driven by microwave currents, limiting coherence times. Here, we report a universal scaling between microwave dissipation and the superfluid density, a bulk property of superconductors related to charge carrier density and disorder. Our analysis spans a wide range of superconducting materials and device geometries, from highly disordered amorphous films to ultra-clean systems with record-high quality factors, including resonators, 3D cavities, and transmon qubits. This scaling reveals an intrinsic bulk dissipation channel, independent of surface dielectric losses, that originates from a universal density of nonequilibrium quasiparticles trapped within disorder-induced spatial variations of the superconducting gap. Our findings identify an empirical coherence limit associated with intrinsic material properties and provide a data-driven basis for materials selection in future superconducting quantum circuits.

Figures (4)

• Figure 1: Microwave dissipation from disordered to clean superconductors. Internal quality factor $Q_{\rm i}$ of superconducting resonators and transmon qubits measured at low temperature ($T \lesssim 0.1$ K) and low power, plotted as a function of the imaginary part of the complex conductivity, $\sigma_2 = 1/\mu_0 \omega \lambda^2$, for various superconducting materials. The values of the magnetic penetration depth $\lambda$, and hence $\sigma_2$, span several orders of magnitude across materials, reflecting differences in disorder levels and charge carrier densities in cleaner superconductors. Dots represent devices in 2D geometries (coplanar waveguides or microstrip resonators), stars denote 2D transmon qubits, crosses correspond to suspended striplines, and empty triangles represent 3D resonant cavities. The two diagonal red stripes highlight the upper bound on microwave quality factors due to localized quasiparticles given by Eq. (\ref{['eq:Qi']}). For $\sigma_2 \lesssim 10^{8}$ S.m, the resulting dissipation follows the scaling $Q_{\rm i}^{\rm max} = \kappa \, \sigma_2$ with a proportionality factor $\kappa \approx (0.1-1)$$\Omega$.m. For $\sigma_2 \gtrsim 10^{8}$ S.m, the diagonal strip reflects the scaling $Q_{\rm i}^{\rm max} \propto \sigma_2^{3/2}$, specific for small kinetic inductance fraction. The dashed gray line is a visual guide with slope of 3/2, fitting the best-performing cavities. A horizontal line at $Q_{\rm i} \approx 10^7$ marks an apparent additional limit between planar devices (blue area) and 3D cavities (orange area), most likely related to dielectric substrate losses. Data and references are provided in Supplementary Materials Table \ref{['Table1']}.
• Figure 2: Frequency histogram of the datasets. Histogram of the frequencies corresponding to the datasets shown in Fig. \ref{['fig1']}.
• Figure 3: Dissipation mechanisms induced by nonequilibrium quasiparticles.a, Sketch of the quasiparticle density of states $\nu(E)$ as a function of energy near the superconducting gap edge. $\nu_0$ denotes the normal-state density of states per spin projection at the Fermi level; $\Delta_0$ is the superconducting gap in the clean limit; and $E_g \approx \Delta_0 (1 - \eta^{2/3})^{3/2}$ marks the mobility edge separating the delocalized states (pink) from the localized tail. The difference between $E_g$ and $\Delta_0$ arises from disorder-induced smearing of the BCS coherence peak, characterized by the depairing parameter $\eta$Abrikosov1960Larkin72. $\nu_{\text{loc}}(E)$ represents the subgap tail of localized quasiparticle states, with a characteristic energy scale $\varepsilon_T \ll \Delta_0$, and $E_C$ denotes the energy threshold below which quasiparticle recombination becomes inefficient Bespalov16. b--f, Schematic representation of possible dissipation mechanisms involving nonequilibrium quasiparticles, approximately ordered by decreasing dissipation strength (see also Fig. \ref{['fig:nonequil-qp-dissip_types_qualitative-diag']}): b, Standard Mattis-Bardeen-like transitions in the continuum of delocalized states Mattis58Fominov2011, subsec. \ref{['subsec:Dissipation_deloc-qps']}; c, Ionization of a bound quasiparticle state in a local well of suppressed superconducting order parameter (blue) with characteristic size $r_{\text{loc}}(E_C)$, subsecs. \ref{['subsec:Localized-states_direct-ionization_dirty-limit']}, \ref{['subsec:Localized-states_direct-ionization_clean-limit']}; d, Coherent Rabi oscillations between two bound states within a single well in the clean limit, subsec. \ref{['subsec:Localized-qp_clean-limit_coherent-TLS']}; e,f, Semiclassical dynamics involving multiphoton absorption and photon-assisted variable-range hopping between different localized wells, potentially in nonequilibrium regimes, subsec. \ref{['subsec:Localize-qps_low-freq_possible-loss-channels']}.
• Figure 4: Classification of the main dissipation mechanisms due to nonequilibrium quasiparticles. Green shading indicates the dominant dissipation mechanism in dirty superconducting thin films (left side of Fig. 1, main text), as discussed in subsec. \ref{['subsec:Localized-states_direct-ionization_dirty-limit']}. Orange indicates the dominant mechanism in clean films and 3D cavities (right side of Fig. 1, main text; see subsec. \ref{['subsec:Dissipation_deloc-qps']}). A red star marks those mechanisms for which a power-dependence analysis of the quality factor is available.