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Probing electronic order via coupling to low energy phonons in superconducting Bi$_2$Sr$_{2-x}$La$_x$CuO$_{6+δ}$

We report high-resolution inelastic x-ray scattering measurements of the acoustic phonons in the single-layer cuprate $Bi_{2}Sr_{2-x)La_{x}CuO_{6+δ}$. These measurements reveal anomalous broadening of the longitudinal acoustic phonon near the (1/4,1/4,0) wavevector. The observed wavevector and its doping dependence indicate the coupling of the phonons to an underlying electronic density wave state. In addition, a comparison of the scattered intensities for x-ray energy-gain and x-ray energy-loss indicates that both time-reversal and inversion symmetries are broken in the material. Upon cooling, the effects of symmetry breaking are enhanced in the pseudogap state.