Decoupling momentum and energy relaxation rates in cuprate strange metals via giant THz nonlinearities

Understanding the $T$-linear normal-state resistivity of cuprates remains a central physics challenge. The associated momentum relaxation rate, $\Gamma_M$, saturates near the conjectured ``Planckian" bound $\Gamma_M\sim kT/\hbar$, but the mechanism underlying the anomalous scattering remains unresolved. Here we employ nonlinear terahertz spectroscopy to systematically study La$_{2-x}$Sr$_x$CuO$_4$ across a broad temperature and doping range. We measure the normal-state third-order susceptibility, $|\chi^{(3)}|\approx 6\times10^{-9}$ m$^2$/V$^2$, among the largest in the THz regime, enabling direct access to the rarely measured electronic energy relaxation rate, $\Gamma_E$. Strikingly, $\Gamma_E$ is 10-40 times smaller than $\Gamma_M$, revealing that the scatterings responsible for momentum loss and $T$-linear resistivity do not remove appreciable energy from the electrons. While $\Gamma_M (T)$ is consistent with quasi-elastic scattering from bosonic modes above their characteristic energy scale, this is incompatible with the increasing temperature dependence of $\Gamma_E(T)$. Our results exclude phonons as the source of $T$-linear resistivity and impose strong constraints on possible mechanisms.