  Inhomogeneities play a crucial role in determining the properties of quantum
materials. Yet methods that can measure these inhomogeneities are few, and
apply to only a fraction of the relevant microscopic phenomena. For example,
the electronic properties of cuprate materials are known to be inhomogeneous
over nanometer length scales, although questions remain about how such disorder
influences supercurrents and their dynamics. Here, two-dimensional terahertz
spectroscopy is used to study interlayer superconducting tunneling in
near-optimally-doped La1.83Sr0.17CuO4. We isolate a 2 THz Josephson echo signal
with which we disentangle intrinsic lifetime broadening from extrinsic
inhomogeneous broadening. We find that the Josephson plasmons are only weakly
inhomogeneously broadened, with an inhomogeneous linewidth that is three times
smaller than their intrinsic lifetime broadening. This extrinsic broadening
remains constant up to 0.7Tc, above which it is overcome by the
thermally-increased lifetime broadening. Crucially, the effects of disorder on
the Josephson plasma resonance are nearly two orders of magnitude smaller than
the in-plane variations in the superconducting gap in this compound, which have
been previously documented using Scanning Tunnelling Microscopy (STM)
measurements. Hence, even in the presence of significant disorder in the
superfluid density, the finite frequency interlayer charge fluctuations exhibit
dramatically reduced inhomogeneous broadening. We present a model that relates
disorder in the superfluid density to the observed lifetimes.
