$^{139}$La NMR investigation of the interplay between the lattice, charge, and spin dynamics in charge ordered high $T_c$ cuprate La$_{1.875}$Ba$_{0.125}$CuO$_{4}$
We investigate the interplay between the lattice, charge, and spin dynamics in charge ordered high $T_c$ cuprate La$_{1.875}$Ba$_{0.125}$CuO$_{4}$ ($T_{c} =4$~K) based on the inverse Laplace transform (ILT) analysis of the $^{139}$La nuclear spin-lattice relaxation rate $1/T_1$ (dubbed ILTT$_{1}$ analysis here after). A major thrust of the ILTT$_{1}$ analysis is that one can deduce the probability density function $P(1/T_1)$ of distributed $1/T_1$. We demonstrate that $1/T_{1}^{lm}$, defined as the log-mean (i.e. the center of gravity on a logarithmic scale) of $P(1/T_1)$, can be well approximated by $1/T_{1}^{str}$ deduced from the phenomenological stretched fit, however, $P(1/T_1)$ can provide much richer insight into how the lattice, charge, and spin fluctuations and their distribution develop near and below the long range charge order at $T_{charge} \sim 54$~K. Upon entering the charge ordered state, a divergent increase of $1/T_{1}^{lm}$ toward the spin ordering at $T_{spin}^{\mu SR} \simeq 35$~K is accompanied by an asymmetric broadening of $P(1/T_1)$. Even deep inside the charge ordered state, $1/T_{1}$ at a gradually diminishing fraction of $^{139}$La sites continues to slow down as temperature is lowered, as expected for canonical superconducting CuO$_2$ planes without enhanced spin fluctuations. The fraction of such canonical $^{139}$La sites almost disappears by $\simeq 40$~K. In contrast, nearly a half of the $^{139}$La sites in La$_{1.885}$Sr$_{0.115}$CuO$_{4}$ ($T_{charge} \simeq 80$~K) still exhibits the canonical behavior without enhanced spin fluctuations even near its $T_{c} = 31$~K. These contrasting behaviors explain why superconductivity in La$_{1.875}$Ba$_{0.125}$CuO$_{4}$ is more strongly suppressed than in La$_{1.885}$Sr$_{0.115}$CuO$_{4}$ despite the lower onset temperature of the charge order.