Gap Inhomogeneity in Cuprates: a view from Two-Dimensional Josephson Echo Spectroscopy

Novel theoretical developments have allowed to connect microscopic disorder in bosonic collective excitations to the signatures in two-dimensional terahertz spectroscopy (G\'{o}mez Salvador et al. 2025). Here, we employ this framework to analyze the recently measured Josepshon echoes in optimally doped La$_{2-x}$Sr$_x$CuO$_4$ (Liu et al. 2024). We consider the spatial gap inhomogeneities -- observed in scanning tunneling microscopy -- as input for the disorder in the superfluid density, and compute the resulting echo peaks. The excellent agreement supports the interpretation that the gap inhomogeneity arises solely from pairing gap fluctuations, with no evidence for non-superconducting competing local orders. Finally, we study the microscopic origin of the inelastic processes, contributing to the damping of the Josephson plasmon at low temperatures, and conclude that it can be attributed to nodal quasiparticles.