  Recent experiments on superconducting cavities have found that under large
radio-frequency (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.
