  The introduction of crystalline defects or dopants can give rise to so-called
"dirty superconductors", characterized by reduced coherence length and
quasiparticle mean free path. In particular, granular superconductors such as
Granular Aluminum (GrAl), consisting of remarkably uniform grains connected by
Josephson contacts have attracted interest since the sixties thanks to their
rich phase diagram and practical advantages, like increased critical
temperature, critical field, and kinetic inductance. Here we report the
measurement and modeling of circuit quantum electrodynamics properties of GrAl
microwave resonators in a wide frequency range, up to the spectral
superconducting gap. Interestingly, we observe self-Kerr coefficients ranging
from $10^{-2}$ Hz to $10^5$ Hz, within an order of magnitude from analytic
calculations based on GrAl microstructure. This amenable nonlinearity, combined
with the relatively high quality factors in the $10^5$ range, open new avenues
for applications in quantum information processing and kinetic inductance
detectors.
