Sharp optical phonon softening close to optimal doping in La${}_{2-x}$Ba${}_x$CuO${}_{4+\delta }$

We report a direct observation of a sharp Kohn-like anomaly in the doubly degenerate copper-oxygen bond-stretching phonon mode occurring at $\mathbf{q}=(0.3,0,0)$ in La${}_{2-x}$Ba${}_x$CuO${}_{4+\delta }$ with $\mathrm{x}=0.14\pm 0.01$, thanks to the high $\mathbf{Q}$ resolution of inelastic x-ray scattering. This anomaly is clearly seen when the inelastic signal is analysed using a single mode but is also consistent with a two mode hypothesis possibly due to a splitting of the degenerate modes due to symmetry breaking stripes. Our observation shows that the effect persists at the stripe propagation vector in a superconducting system close to optimal doping.