  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.
