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The density fluctuation spectrum captures many fundamental properties ofstrange metals. Using momentum-resolved electron energy-loss spectroscopy(M-EELS), we recently showed that the density response of the strange metalBi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ (Bi-2212) at large momentum, $q$, exhibits aconstant-in-frequency continuum [Mitrano, PNAS $\textbf{115}$, 5392 (2018);Husain, PRX $\textbf{9}$, 041062 (2019)] reminiscent of the marginal Fermiliquid (MFL) hypothesis of the late 1980s [Varma, PRL $\textbf{63}$, 1996(1989)]. However, reconciling this observation with infrared (IR) opticsexperiments, which show a well-defined plasmon excitation at $q \sim 0$, hasbeen challenging. Here we report M-EELS measurements of Bi-2212 using 4$\times$improved momentum resolution, allowing us to reach the optical limit. Formomenta $q<0.04$ r.l.u., the M-EELS data show a plasmon feature that isquantitatively consistent with IR optics. For $q>0.04$ r.l.u., the spectrabecome incoherent with an MFL-like, constant-in-frequency form. We speculatethat, at finite frequency, $\omega$, and nonzero $q$, some attribute of thisPlanckian metal randomizes the probe electron, causing it to lose informationabout its own momentum.

Consistency between reflection M-EELS and optical spectroscopy measurements of the long-wavelength density response of Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$