Disorder-driven coexistence of distinct dynamical states in frustrated Sr$_3$CuNb$_2$O$_9$: a microscopic $μ$SR and $^{93}$Nb NMR study

Despite recent progress in identifying the exotic random singlet (RS) state in disordered frustrated magnets as a distinct correlated phase, three-dimensional (3D) realizations remain scarce. Sr$_3$CuNb$_2$O$_9$ was proposed to be one of such 3D frustrated systems with magnetic site disorder hosting an RS ground state. Here, we report a detailed microscopic investigation of Sr$_3$CuNb$_2$O$_9$ employing muon spin relaxation ($μ$SR) and $^{93}$Nb nuclear magnetic resonance (NMR) techniques. The $μ$SR zero-field relaxation rate reveals a power-law divergence of the relaxation rate as a function of temperature. Also, a power-law divergence is present in the relaxation rate as a function of applied longitudinal field, consistent with the formation of an RS phase. The $^{93}$Nb NMR spectra unambiguously resolve two components with distinct local magnetic environments, whose nature is further elucidated through spin-lattice relaxation measurements analyzed via an inverse Laplace transform (ILT) of the nuclear magnetization recovery. The relaxation-rate distribution obtained from ILT reveals two well-separated channels: a fast component, $(1/T_1)_{\mathrm{fast}}$, and a slow component, $(1/T_1)_{\mathrm{slow}}$. Both components follow distinct power-law temperature dependences ($T^α$), with $α= 0.6$ and $1.1$ for the fast and slow channels, respectively. The combined spectral and relaxation data demonstrate that the fast channel qualitatively represents an RS-like state, whereas the slow channel exhibits quantum spin liquid (QSL) like behavior, thereby establishing the microscopic coexistence of RS and QSL-like phases in Sr$_3$CuNb$_2$O$_9$.