The doping evolution of the charge density wave and charge density fluctuations in La$_{2-x}$Sr$_x$CuO$_4$

Cuprate superconductors show various collective charge correlations that are intimately connected with their electronic properties. In particular, charge order in the form of an incommensurate charge density wave (CDW) order with an in-plane wavevector $\delta_{\text{CDW}} \approx$ 0.23--0.35~r.l.u. appears to be universally present. In addition to CDW, dynamic charge density fluctuations (CDF) are also present with wavevectors comparable to $\delta_{\text{CDW}}$. CDFs are present up to $\sim300\;$K and have relatively short correlation lengths of $\xi \sim 20$\;\AA. Here we use Cu-$L_3$ and O-$K$ resonant inelastic X-ray scattering (RIXS) to study the doping dependence of CDW and CDFs in La$_{2-x}$Sr$_x$CuO$_4$. We fit our data with (quasi)elastic peaks resulting from the CDW and up to four inelastic modes associated with oxygen phonons that can be strongly coupled to the CDFs. Our analysis allows us to separate the charge correlations into three components: the CDW with wavevector $\delta_{4a-\text{CDW}} \approx 0.24$ and two CDF components with $\delta_{4a-\text{CDF}} \approx 0.24$ and $\delta_{3a-\text{CDF}} \approx 0.30$. We find that for $T \approx T_c$ the CDW coexists with the CDFs for dopings near $x=p \sim 1/8$. The $4a$-CDW disappears beyond $x=0.16$ and the $4a$-CDF beyond $x=0.19$, leaving only a weak $3a$-CDF at the highest doping studied, $x=0.22$. Our data suggest that low-energy charge fluctuations exist up to doping $x=0.19=p^{\star}$, where the pseudogap disappears, however, we find no evidence that they are associated with a quantum critical point.