Dissipation by surface states in superconducting radio-frequency cavities

Recent experiments on superconducting cavities have found that under large rf electromagnetic fields the quality factor can improve with increasing field amplitude, a so-called “anti-$Q$ slope.” Linear theories of dissipation break down under these extreme conditions and are unable to explain this behavior. We numerically solve the Bogoliubov-de Gennes equations at the surface of a superconductor in a parallel AC magnetic field, finding that at large fields there are quasiparticle surface states with energies below the bulk value of the superconducting gap. As the field oscillates, such states emerge and disappear with every cycle. We consider the dissipation resulting from inelastic quasiparticle-phonon scattering into these states and investigate the ability of this mechanism to explain features of the experimental observations, including the field dependence of the quality factor. We find that this mechanism is likely not the dominant source of dissipation and does not produce an anti-$Q$ slope by itself; however, we demonstrate in a modified two-fluid model how these bound states can play a role in producing an anti-$Q$ slope.

Figure 1

The BdG quasiparticle bound states for $k_x=k_F, k_y=0$, at a field of 40 MV/m. For each state the wave functions $u$ and $v$ are plotted with a vertical shift according to their energy. The situation is somewhat analogous to a particle in a potential $\mathrm{\Delta }\left(z\right)$ with a hard wall on the left.

Figure 2

Top: Quasiparticle scattering from a bulk state into a bound state, releasing a phonon in the process. As the bound state energy rises, the field does work on the quasiparticle, resulting in net dissipation. Bottom: The macroscopic picture tracking the average occupation of an ensemble of such states for several values of $\omega \tau$. Dissipation is maximized when $\omega \tau \approx 1$.

Figure 3

Quality factor vs. field amplitude for a niobium cavity at frequency 1.3 GHz. The dashed lines show the quality factor computed from our inelastic scattering disequilibrium mechanism, using either a constant ${\tau }_h$ or the time-dependent ${\tau }_h\left(t\right)$ in Eq. (7). The dash-dot lines show the quality factor from this mechanism plus the three-fluid model as described in the text with ${\tau }_c=1$ ps. Markers show experimental data from Grassellino et al. [32] for a selection of electropolished (EP) or nitrogen-doped cavities.