  Quantum phase transitions play an important role in shaping the phase diagram
of high-temperature cuprate superconductors. These cuprates possess intertwined
orders which interact strongly with superconductivity. However, the evidence
for the quantum critical point associated with the charge order in the
superconducting phase remains elusive. Here we show the short-range charge
orders and the spectral signature of the quantum fluctuations in
La$_{2-x}$Sr$_x$CuO$_4$ (LSCO) near the optimal doping using high-resolution
resonant inelastic X-ray scattering. On performing calculations through a
diagrammatic framework, we discovered that the charge correlations
significantly soften several branches of phonons. These results elucidate the
role of charge order in the LSCO compound, providing evidence for quantum
critical scaling and discommensurations associated with charge order.
