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Inhomogeneities play a crucial role in determining the properties of quantummaterials. Yet methods that can measure these inhomogeneities are few, andapply to only a fraction of the relevant microscopic phenomena. For example,the electronic properties of cuprate materials are known to be inhomogeneousover nanometer length scales, although questions remain about how such disorderinfluences supercurrents and their dynamics. Here, two-dimensional terahertzspectroscopy is used to study interlayer superconducting tunneling innear-optimally-doped La1.83Sr0.17CuO4. We isolate a 2 THz Josephson echo signalwith which we disentangle intrinsic lifetime broadening from extrinsicinhomogeneous broadening. We find that the Josephson plasmons are only weaklyinhomogeneously broadened, with an inhomogeneous linewidth that is three timessmaller than their intrinsic lifetime broadening. This extrinsic broadeningremains constant up to 0.7Tc, above which it is overcome by thethermally-increased lifetime broadening. Crucially, the effects of disorder onthe Josephson plasma resonance are nearly two orders of magnitude smaller thanthe in-plane variations in the superconducting gap in this compound, which havebeen previously documented using Scanning Tunnelling Microscopy (STM)measurements. Hence, even in the presence of significant disorder in thesuperfluid density, the finite frequency interlayer charge fluctuations exhibitdramatically reduced inhomogeneous broadening. We present a model that relatesdisorder in the superfluid density to the observed lifetimes.

Probing Inhomogeneous Cuprate Superconductivity by Terahertz Josephson Echo Spectroscopy