Open Quantum System Theory of Muon Spin Relaxation in Materials

Abstract

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.

Figures (5)

• Figure 1: Schematic representation of the muon spin relaxation as an open quantum system. By linearizing the magnetic density at the muon site, the effective magnetic field $\bm{B}(\bm{R}_{\mu},t)$ that couples to muon spin $\bm{S}$ is composed of a background field averaged over ion density ($\langle \rho_{\mathrm{i}} \rangle$) and the ion density fluctuation ($\delta \rho_{\mathrm{i}}$)-modulated field.
• Figure 2: Muon polarization $G_{z}(t)$ versus time for: (a) ZF, ion-dynamics only ($\nu_{\mu}=\Delta_{\mu}=0$); (b) ZF, static muon ($\nu_{\mu}=0$) with $\Delta_{\mu}=\Delta_{\mathrm i}$; (c) LF scans at different $B_{\mathrm L}$ (Gauss) for a static muon ($\nu_{\mu}=0$, $\Delta_{\mu}=\Delta_{\mathrm i}$) with $\nu_{\mathrm i}=0.5~\mu\mathrm{s}^{-1}$; (d) ZF, motional-narrowing regime with $\Delta_{\mu}=\Delta_{\mathrm i}$ and $\nu_{\mathrm i}=3.0~\mu\mathrm{s}^{-1}$ (see panel for the $\nu_{\mu}$ values). In all panels, $\Delta_{\mathrm i}=0.1~\mu\mathrm{s}^{-1}$ and $\alpha=0.0~\mu\mathrm{s}^{-2}$.
• Figure 3: Effect of the backaction (retarded-torque) strength $\Lambda$ (in units of $\mu s^{-2}$) on the longitudinal polarization $G_z(t)$ computed from the spin SDE Monte Carlo (solid curves). Dashed curves show the analytical reduction. All panels use an ion-modulated width $\Delta_{\mathrm{i}}=0.1~\mu\mathrm{s}^{-1}$, and $\Lambda$ is varied as indicated in the legend. (a) Purely dynamical limit in ZF: $\Delta_\mu=0$, with an intermediate fluctuation rate $\nu_{\mathrm{Li}}=0.5~\mu\mathrm{s}^{-1}$. (b) Mixed static-dynamic case in ZF: $\Delta_\mu=0.1~\mu\mathrm{s}^{-1}$ and $\nu_{\mathrm{Li}}=0.5~\mu\mathrm{s}^{-1}$. (c) Longitudinal-field response in the dynamical limit: $B_L=5$ G, $\Delta_\mu=0$, and $\nu_{\mathrm{Li}}=0.5~\mu\mathrm{s}^{-1}$. (d) Motional-narrowing regime in ZF with a quenched component: $\Delta_\mu=\Delta_{\mathrm{Li}}=0.1~\mu\mathrm{s}^{-1}$ and $\nu_{\mathrm{Li}}=1.0~\mu\mathrm{s}^{-1}$.
• Figure 4: Sensitivity of the spin-SDE polarization to the retarded-torque strength $\Lambda$ (in units of $\mu s^{-2}$) across fluctuation regimes set by the ion correlation rate $\nu_{\mathrm i}$ (MC results only). Panels (a)--(c) compare $G_z(t)$ for $\Lambda=0$ and $\Lambda=0.1$ at fixed $\Delta_{\mathrm i}$ while varying $\nu_{\mathrm i}$: (a) fast-fluctuation (short-memory) regime, $\nu_{\mathrm i}=3.0~\mu\mathrm{s}^{-1}$; (b) crossover regime, $\nu_{\mathrm i}=0.5~\mu\mathrm{s}^{-1}$; (c) quasi-static (long-memory) regime, $\nu_{\mathrm i}=10^{-4}~\mu\mathrm{s}^{-1}$. The impact of $\Lambda$ is non-monotonic in $\nu_{\mathrm i}$: it is weak when the bath is fast (a), most visible in the intermediate crossover regime (b), and can produce qualitatively different behavior when the bath is effectively frozen over the $\mu$SR window (c).
• Figure 5: Spin SDE description of $\mu$SR spectra of $\mathrm{Li}_{0.73}\mathrm{CoO}_2$ and temperature dependence of the fitted parameters (global ZF/weak-LF analysis). (a) Representative simultaneous fit of ZF, 5 G, and 10 G asymmetry spectra at a fixed temperature using a single parameter set. The fit reproduces the KT-like early-time dephasing (quenched width $\Delta_\mu$, with $\nu_\mu=0$), the systematic LF decoupling, and the remaining dynamical relaxation at long times. (b) Extracted Li-driven fluctuation rate $\nu_{\mathrm{Li}}(T)$. (c) Extracted field-width parameters: dynamical width $\Delta_{\mathrm{Li}}(T)$, quenched (static) width $\Delta_\mu(T)$, and $\Delta_{\mathrm{eff}}(T)\equiv \sqrt{\Delta_{\mathrm{Li}}^2+\Delta_\mu^2}$. (d) Extracted backaction (memory) strength $\Lambda(T)$. The data indicate a crossover from a low-$T$ regime requiring a substantial quenched component to a higher-$T$ regime dominated by Li-driven dynamics, with backaction effects most pronounced in the intermediate-temperature window.