Consistency between reflection momentum-resolved electron energy loss spectroscopy and optical spectroscopy measurements of the long-wavelength density response of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{O}_{8+x}$

The density fluctuation spectrum captures many fundamental properties of strange metals. Using momentum-resolved electron energy loss spectroscopy (M-EELS), we recently showed that the density response of the strange metal ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{O}_{8+x}$ (Bi-2212) at large momentum, $q$, exhibits a constant-in-frequency continuum [M. Mitrano et al., Proc. Natl. Acad. Sci. USA 115, 5392 (2018); A. A. Husain et al., Phys. Rev. X 9, 041062 (2019)] reminiscent of the marginal Fermi liquid (MFL) hypothesis of the late 1980s [C. M. Varma et al., Phys. Rev. Lett. 63, 1996 (1989)]. However, reconciling this observation with infrared (IR) optics experiments, which show a well-defined plasmon excitation at $q\sim 0$, has been challenging. Here we report M-EELS measurements of Bi-2212 using $4\times$ improved momentum resolution, allowing us to reach the optical limit. For momenta $q<0.04$ reciprocal lattice unites (r.l.u.), the M-EELS data show a plasmon feature that is quantitatively consistent with IR optics. For $q>0.04$ r.l.u., the spectra become incoherent with an MFL-like, constant-in-frequency form. We speculate that, at finite frequency, $\omega$, and nonzero $q$, some attribute of this Planckian metal randomizes the probe electron, causing it to lose information about its own momentum.