  Diffusion noise is a major source of spectral line broadening in liquid state
nano-scale nuclear magnetic resonance with shallow nitrogen-vacancy centres,
whose main consequence is a limited spectral resolution. This limitation arises
by virtue of the widely accepted assumption that nuclear spin signal
correlations decay exponentially in nano-NMR. However, a more accurate analysis
of diffusion shows that correlations survive for a longer time due to a
power-law scaling, yielding the possibility for improved resolution and
altering our understanding of diffusion at the nano-scale. Nevertheless, such
behaviour remains to be demonstrated in experiments. Using three different
experimental setups and disparate measurement techniques, we present
overwhelming evidence of power-law decay of correlations. These result in
sharp-peaked spectral lines, for which diffusion broadening need not be a
limitation to resolution.
