  Short ballistic graphene Josephson junctions sustain superconducting current
with a non-sinusoidal current-phase relation up to a critical current
threshold. The current-phase relation, arising from proximitized
superconductivity, is gate-voltage tunable and exhibits peculiar skewness
observed in high quality graphene superconductors heterostructures with clean
interfaces. These properties make graphene Josephson junctions promising
sensitive quantum probes of microscopic fluctuations underlying transport in
two-dimensions. We show that the power spectrum of the critical current
fluctuations has a characteristic $1/f$ dependence on frequency, $f$, probing
two points and higher correlations of carrier density fluctuations of the
graphene channel induced by carrier traps in the nearby substrate. Tunability
with the Fermi level, close to and far from the charge neutrality point, and
temperature dependence of the noise amplitude are clear fingerprints of the
underlying material-inherent processes. Our results suggest a roadmap for the
analysis of decoherence sources in the implementation of coherent devices by
hybrid nanostructures.
