Multi-orbital charge density wave excitations and concomitant phonon anomalies in Bi$_2$Sr$_2$LaCuO$_{6+\delta}$

Charge density waves (CDWs) are ubiquitous in under-doped cuprate superconductors. As a modulation of the valence electron density, CDWs in hole-doped cuprates possess both Cu-3d and O-2p orbital character owing to the strong hybridization of these orbitals near the Fermi level. Here, we investigate under-doped Bi$_2$Sr$_{1.4}$La$_{0.6}$CuO$_{6+\delta}$ using resonant inelastic X-ray scattering (RIXS) and find that a short-range CDW exists at both Cu and O sublattices in the copper-oxide (CuO2) planes with a comparable periodicity and correlation length. Furthermore, we uncover bond-stretching and bond-buckling phonon anomalies concomitant to the CDWs. Comparing to slightly over-doped Bi$_2$Sr$_{1.8}$La$_{0.2}$CuO$_{6+\delta}$, where neither CDWs nor phonon anomalies appear, we highlight that a sharp intensity anomaly is induced in the proximity of the CDW wavevector (QCDW) for the bond-buckling phonon, in concert with the diffused intensity enhancement of the bond-stretching phonon at wavevectors much greater than QCDW. Our results provide a comprehensive picture of the quasi-static CDWs, their dispersive excitations, and associated electron-phonon anomalies, which are key for understanding the competing electronic instabilities in cuprates.