Open Quantum System Theory of Muon Spin Relaxation in Materials

We present a non-Markovian theory of muon spin relaxation that treats the implanted muon as an open quantum spin coupled to a temporally correlated local magnetic environment. Using a Schwinger-Keldysh influence-functional formulation, we derive a spin stochastic equation of motion in which colored fluctuations and retarded memory torque appear on equal footing. In the appropriate limits, the theory reduces to standard Kubo-Toyabe descriptions. This enables quantitative, global analysis of zero-field (ZF) and weak longitudinal-field (LF) $μ$SR spectra beyond the strong-collision approximation. Applied to $\mathrm{Li}_{0.73}\mathrm{CoO}_2$, the approach separates quenched broadening from Li-driven fluctuations and extracts a thermally activated fluctuation rate over the intermediate-temperature window. It also reveals a distinct non-Markovian line-shape signature captured by a retarded backaction (memory) kernel that is most evident in the crossover between quasi-static and fast-fluctuation limits.