  The electronic properties of cuprate high temperature superconductors in
their normal state are very two-dimensional: while transport in the ab plane is
perfectly metallic, it is insulating along the c-axis, with ratios between the
two exceeding 10^4. This anisotropy has been identified as one of the mysteries
of the cuprates early on, and while widely different proposals exist for its
microscopic origin, little is known empirically on the microscopic scale. Here,
we elucidate the properties of the insulating layers with a newly developed
scanning noise spectroscopy technique that can spatially map not only the
current but also the current fluctuations in time. We discover atomic-scale
noise centers that exhibit MHz current fluctuations 40 times the expectation
from Poissonian noise, more than what has been observed in mesoscopic systems.
Such behaviour can only happen in highly polarizable insulators and represents
strong evidence for trapping of charge in the charge reservoir layers. Our
measurements suggest a picture of metallic layers separated by polarizable
insulators within a three-dimensional superconducting state.
