Distribution of the magnetic environment in charge ordered La$_{1.885}$Sr$_{0.115}$CuO$_{4}$

We report the inverse Laplace transform (ILT) analysis of the $^{139}$La nuclear spin-lattice relaxation rate $1/T_1$ in charge ordered La$_{1.885}$Sr$_{0.115}$CuO$_4$ ($T_{charge} sim 80$K, $T_{c} simeq T_{spin}^{neutron}=30$K), and shed new light on its magnetic inhomogeneity. We deduce the probability density function $P(1/T_{1})$ of the distributed $1/T_1$ (i.e. the histogram of distributed $1/T_1$) by taking the inverse Laplace transform of the experimentally observed nuclear magnetization recovery curve $M(t)$. We demonstrate that spin freezing sets in in some domains precisely below the onset of charge order at $T_{charge}$, but their volume fraction grows only gradually toward $T_{c}$. Nearly a half of the sample volume exhibits properties expected for canonical high $T_c$ cuprates without charge order even near $T_c$. Our findings explain why charge order does not suppress $T_c$ of La$_{1.885}$Sr$_{0.115}$CuO$_4$ as significantly as in La$_{1.875}$Ba$_{0.125}$CuO$_4$.