Charge Order Lock-in by Electron-Phonon Coupling in La$_{1.675}$Eu$_{0.2}$Sr$_{0.125}$CuO$_4$

We report an ultrahigh resolution resonant inelastic x-ray scattering (RIXS) study of the in-plane bond-stretching phonon mode in stripe-ordered cuprate La$_{1.675}$Eu$_{0.2}$Sr$_{0.125}$CuO$_4$. Phonon softening and lifetime shortening are found around the charge ordering wave vector. In addition to these self-energy effects, the electron-phonon coupling is probed by its proportionality to the RIXS cross section. We find an enhancement of the electron-phonon coupling around the charge-stripe ordering wave vector upon cooling into the low-temperature tetragonal structure phase. These results suggest that in addition to electronic correlations, electron-phonon coupling contributes significantly to the emergence of long-range charge-stripe order in cuprates.