Open Quantum System Theory of Muon Spin Relaxation in Materials

This paper presents a non-Markovian open quantum system theory for muon spin relaxation using a Schwinger-Keldysh influence-functional formulation to derive a spin stochastic equation with colored fluctuations and memory effects, enabling a quantitative global analysis of $\mu$SR spectra in materials like $\mathrm{Li}_{0.73}\mathrm{CoO}_2$ that goes beyond standard strong-collision approximations.

The Gist

The Big Picture: The Muon as a "Spy Drone"

Imagine you want to know how people are moving inside a crowded, chaotic concert hall. You can't see everyone, but you can send in a tiny, invisible spy drone (the Muon) that has a built-in compass (its Spin).

Once the drone lands in the crowd, it starts spinning. The way it spins tells you everything about the people around it. If the people are standing still, the drone spins in a predictable pattern. If the people are running around, the drone gets bumped and wobbles. By watching how the drone's spin changes over time, scientists can figure out how fast the people (ions) are moving.

The Problem:
For decades, scientists used a simple rulebook (called Kubo-Toyabe) to interpret the drone's wobbling. This rulebook assumed that the people bumping the drone were like a random crowd of strangers: they bump the drone, and then immediately forget about it. It's like a game of "pin the tail on the donkey" where everyone spins you once and walks away instantly.

But in real materials (like the batteries in your phone), the "people" (Lithium ions) are more complex. They don't just bump and forget. They might bump the drone, and then remember they bumped it a moment later, or they might move in a coordinated dance. The old rulebook couldn't handle this "memory." It was like trying to predict the weather using only yesterday's temperature, ignoring the fact that the wind is still blowing from a storm that happened three hours ago.

The New Theory: Giving the Drone a "Memory"

The authors of this paper, Elvis Arguelles and Osamu Sugino, built a new, much smarter rulebook. They treat the muon not just as a passive observer, but as an Open Quantum System.

Think of it this way:

• The Old Way: The drone is hit by a random gust of wind. Whoosh! It spins. The wind is gone.
• The New Way: The drone is hit by a gust of wind, but the air itself is "sticky." When the wind hits the drone, it leaves a little trail of turbulence behind it. A second later, that turbulence swirls back and hits the drone again. This is called Non-Markovian physics (or "memory effects").

The authors used a complex mathematical tool called the Schwinger-Keldysh path integral.

• Analogy: Imagine trying to record a conversation in a noisy room. The old method just recorded the average noise level. The new method records the conversation and the echo, realizing that what you hear right now is a mix of what was said a second ago and what is being said now.

The "Back-Action" (The Echo)

The most exciting part of their discovery is something they call Retarded Back-Action or Memory Torque.

Imagine you are trying to walk through a crowd.

1. Standard View: You bump into someone, they move out of the way, and you keep walking.
2. This Paper's View: You bump into someone, and for a split second, they push back because you bumped them. Or, the crowd shifts in a way that anticipates your movement.

In the material Li₀.₇₃CoO₂ (a common battery material), the Lithium ions are hopping around. The new theory shows that the muon feels a "drag" or a "twist" from the ions that isn't just random noise; it's a structured response. The ions remember the muon's presence for a tiny fraction of a second, creating a "memory kernel."

Testing the Theory: The Battery Lab

To prove their theory works, they looked at data from Li₀.₇₃CoO₂ (a lithium-cobalt oxide battery material).

• The Setup: They watched the muon drone at different temperatures.
  • Cold: The Lithium ions are frozen (like people sitting in chairs). The drone spins in a specific pattern (the "Kubo-Toyabe" shape).
  • Hot: The Lithium ions are running wild (like people dancing). The drone spins very fast and smooths out.
  • The "Goldilocks" Zone (Intermediate Temperature): This is where the magic happens. The ions are moving, but not fast enough to be random. They are in a "crossover" zone.

The Result:
The old rulebook failed in this "Goldilocks" zone. It couldn't explain why the drone's spin was stabilizing in a weird way.
The new theory, with its Memory Torque, fit the data perfectly. It showed that:

1. The Lithium ions move in a thermally activated way (they need heat to jump, like a frog needing a push to hop).
2. There is a distinct "memory" effect where the ions' movement creates a feedback loop that the muon feels.

Why Does This Matter?

This isn't just about math; it's about better batteries.

1. Seeing the Invisible: This new method allows scientists to separate the "static" noise (frozen ions) from the "dynamic" noise (moving ions) much more clearly than before.
2. The "Memory" Signature: They found a clear signature of "non-Markovian" behavior. This means that in these battery materials, the movement of ions is correlated. They aren't just random; they are influencing each other over time.
3. Future Batteries: By understanding exactly how ions move and interact with their environment (even on a quantum level), engineers can design better battery materials that charge faster and last longer.

Summary in One Sentence

The authors created a new, "memory-aware" mathematical model for how muons spin in materials, allowing them to finally see the subtle, coordinated dance of ions in battery materials that older, simpler models missed.

Technical Summary

1. Problem Statement

Muon Spin Relaxation ($\mu$SR) is a powerful probe for local magnetic dynamics and ion transport in condensed matter. However, standard analysis relies on the Kubo-Toyabe (KT) framework and its dynamic extensions. These models suffer from fundamental limitations:

• Markovian Assumption: They assume field fluctuations are uncorrelated in time (white noise), neglecting memory effects.
• Strong-Collision Approximation: They assume instantaneous, randomizing collisions, which fails to capture systems with slow dynamics, correlated hopping, or glassy disorder.
• Phenomenological Nature: They often cannot distinguish between static field broadening and slow dynamical fluctuations, leading to misinterpretation of ion diffusion rates, particularly in complex battery materials like $Li_xCoO_2$.

There is a need for a non-Markovian, microscopic theory that treats the implanted muon as an open quantum system coupled to a temporally correlated magnetic environment, capable of separating static (quenched) and dynamic (ion-driven) components without relying on the strong-collision approximation.

2. Methodology

The authors develop a unified framework combining stochastic processes and open quantum system theory using the Schwinger-Keldysh influence functional formalism.

• Model Hamiltonian: The muon spin ($S$) is treated as an open quantum system coupled to a composite magnetic bath. The local field $B(t)$ is decomposed into:

  1. Background Field ($B^{(\mu)}$): Quasi-static nuclear dipolar fields (quenched disorder).
  2. Ion-Modulated Field ($B^{(i)}$): Fluctuating fields driven by ion hopping (e.g., Li ions).
     The authors derive the autocorrelation tensor $C_{\alpha\beta}(t)$ for these fields, showing they follow Drude-Lorentz spectral densities characterized by correlation rates ($\nu_\mu, \nu_i$) and widths ($\Delta_\mu, \Delta_i$).

• Spin-Boson Mapping: The muon is mapped to an $SU(2)$ coherent state coupled to two independent bosonic baths (background and ion-modulated). The bath degrees of freedom are integrated out to yield an influence functional.

• Stochastic Equation of Motion (SDE): The influence functional generates a non-Markovian stochastic differential equation for the muon spin direction $\mathbf{n}(t)$:
  $$\dot{\mathbf{n}}(t) = \mathbf{n}(t) \times \left[ \gamma_\mu \mathbf{B}_L + \sum_{\zeta} \left( \frac{1}{S} \int dt' \Gamma^{(\zeta)}(t-t') \cdot \mathbf{n}(t') - \frac{\xi^{(\zeta)}(t)}{S} \right) \right]$$

  • $\xi(t)$: Colored noise (Ornstein-Uhlenbeck process) representing field fluctuations.
  • $\Gamma(t)$: A retarded (memory) kernel representing a causal backaction torque. This term accounts for the history-dependent response of the bath to the spin, a key non-Markovian feature.

• Numerical Implementation: The authors implement a Monte Carlo (MC) simulation of the SDE. To handle the memory integral efficiently, they introduce auxiliary "memory" variables, converting the integro-differential equation into a local system of differential equations (Markovian embedding).

• Analytical Reduction: Under specific limits (static muon, small-angle approximation, rotating frame), they derive a closed-form analytical expression for the polarization $G_z(t)$. This involves a factorization into static and dynamic components and a generalized Abragam function with a mixture of correlation rates.

3. Key Contributions

1. Non-Markovian Theory: The derivation of a spin SDE where colored fluctuations and retarded memory torque appear on equal footing, moving beyond the Markovian strong-collision approximation.
2. Unified Framework: A single theoretical model that seamlessly interpolates between the static KT limit, the dynamic KT limit, and the motional-narrowing regime, while explicitly handling the interplay between static disorder and dynamic fluctuations.
3. Backaction Mechanism: Identification of a retarded backaction (memory) torque parameter ($\Lambda$). The theory demonstrates that this backaction can suppress depolarization and induce "motional-narrowing-like" effects even in regimes where standard KT theory predicts strong relaxation.
4. Analytical vs. Numerical Benchmarking: The authors rigorously compare their full numerical SDE solution with their analytical reduction, defining the precise regime of validity for the analytical approach (valid for intermediate fluctuation rates and weak backaction; breaks down in quasi-static limits with strong memory).

4. Results

• Benchmarking:

  • In the purely dynamic limit, the SDE reproduces standard dynamic KT results quantitatively.
  • In the quasi-static limit, the analytical reduction fails (over-depolarization), while the SDE correctly recovers the static KT recovery tail.
  • Longitudinal Field (LF) Effects: The model successfully reproduces the decoupling of muon polarization from transverse fields as $B_L$ increases.
  • Muon Hopping: The theory predicts that muon diffusion itself acts as a fluctuating field, suppressing the KT tail and driving the system toward complete depolarization at high hopping rates.

• Non-Markovian Signatures:

  • The strength of the backaction parameter $\Lambda$ has a non-monotonic effect depending on the fluctuation rate $\nu_i$.
  • The effect is most pronounced in the intermediate crossover regime ($\nu_i \sim \Delta_i$), where the memory kernel reshapes the line shape significantly. In fast-fluctuation or quasi-static limits, the effect is weaker or qualitatively different.

• Application to $Li_{0.73}CoO_2$:

  • The framework was applied to global fits of Zero-Field (ZF) and Weak Longitudinal-Field (5 G, 10 G) spectra.
  • Separation of Components: The model successfully separated the static nuclear width ($\Delta_\mu$) from the Li-driven dynamic width ($\Delta_{Li}$) and fluctuation rate ($\nu_{Li}$).
  • Thermally Activated Dynamics: The extracted $\nu_{Li}(T)$ follows an Arrhenius law with an activation energy $E_a \approx 90$ meV, consistent with previous studies.
  • Memory Effects: A clear non-Markovian signature was identified. The backaction parameter $\Lambda(T)$ was found to be most significant in the intermediate-temperature window, indicating that Li-driven fluctuations induce a history-dependent feedback on the muon spin that standard KT models miss.

5. Significance

This work provides a microscopically motivated, fit-ready tool for quantitative $\mu$SR analysis in complex materials.

• Beyond Phenomenology: It replaces the phenomenological strong-collision approximation with a rigorous open quantum system derivation.
• Ion Transport: It offers a robust method to extract ion diffusion rates in battery cathodes (like $LiCoO_2$) even when ion motion is correlated or slow, where standard NMR or QENS might struggle due to paramagnetic interference or thermal instability.
• New Physics: It highlights the importance of non-Markovian backaction in spin relaxation. The discovery that memory effects can stabilize polarization (motional narrowing) in the crossover regime suggests that previous analyses of $\mu$SR data in functional materials may have underestimated or misinterpreted ion dynamics.
• Extensibility: The framework is readily extendable to anisotropic kernels, multiple bath channels, and correlated ionic motion, setting a new standard for studying ion-driven magnetic fluctuations.