Magnon-Scattering Reservoir Computing

• Magnon-scattering reservoirs are high-dimensional systems that employ nonlinear magnon interactions through three-magnon splitting in confined ferromagnetic structures.
• They transform temporal input sequences into complex spectral and time-multiplexed outputs, demonstrating strong short-term memory and nonlinear transformation capabilities.
• The reservoir's performance is robust across both modal and temporal output schemes, highlighting its intrinsic, physics-driven computational efficiency.

A magnon-scattering reservoir is a physical system in which nonlinear magnon–magnon interactions, typically realized in confined or engineered ferromagnetic structures, are exploited as a high-dimensional dynamical platform for computational functions, notably in the framework of reservoir computing. This system leverages the inherent nonlinearity and transient, multimodal scattering behavior of geometrically quantized magnons to perform complex tasks such as short-term memory and nonlinear transformations, with its utility determined by the physical dynamics rather than by the choice of output sampling or read-out scheme (Heins et al., 4 Feb 2025).

1. Fundamental Mechanisms and Reservoir Principle

In the magnon-scattering reservoir, the internal degrees of freedom are discrete magnon modes established in microstructured ferromagnets, typically disks patterned in the magnetic vortex state. Magnons—collective, quantized spin-wave excitations—are excited in specific eigenmodes via pulsed microwave protocols. When driven with powers above a threshold, these primary magnon modes undergo nonlinear three-magnon splitting, a process governed by conservation of energy and angular momentum. The splitting produces secondary magnons (with opposite azimuthal mode indices) according to the selection rules such as $f_\mathrm{A} = f_{\mathrm{A}+} + f_{\mathrm{A}-}$. Importantly, temporal overlap and cross-stimulation between multiple drive frequencies or input channels amplify the nonlinearity: the presence of an already-split mode from one input can enhance or redirect the splitting pathway of another, leading to a dynamic, time-dependent high-dimensional mapping of inputs to outputs.

Crucially, the nonlinear dynamics arise from intrinsic magnon–magnon interactions without the need for engineered or digital nonlinearities in the output. The reservoir functions by mapping temporal input sequences into complex mode populations and spectral signatures, which can then be linearly processed to accomplish computational tasks. The continuity and complexity of the response arise from the reservoir’s physical processes rather than any property of the read-out.

2. Experimental Realization and Read-Out Schemes

The experimental implementation features a single NiFe (Permalloy) disk (diameter ≈ 5.1 μm) in the vortex magnetization state, excited via an ω-shaped microwave antenna. Two input channels (binary “1” and “0”) are encoded as microwave pulses at $f_\mathrm{A}$ ≈ 8.9 GHz and $f_\mathrm{B}$ ≈ 7.4 GHz, corresponding to the third and second radial modes, respectively. Input sequences of 14 ns pulses are chosen to probe the transient nonlinear regime, prior to steady-state regime saturation.

The reservoir’s response is probed via time-resolved micro-focused Brillouin light scattering (TR-µBLS), yielding both high temporal (≈1 ns) and fine spectral resolution. The dynamic scattering spectrum, which includes both the driven (primary) and nonlinear (secondary, split) modes, is sampled in two complementary output spaces:

• Modal (Spectral) Output Space: The full time-averaged spectrum is discretized into a number of frequency bins, with the vector of binned spectral intensities serving as the reservoir output feature set.
• Temporal (Time-Multiplexed) Output Space: The intensities of several of the strongest modes are tracked as time traces over each pulse interval, divided into sub-time windows to build a high-dimensional output vector.

Notably, the experimental findings establish that, provided the output dimensionality is sufficient to resolve the nonlinear features, both output spaces yield equivalent benchmarking performance. The capacity of the reservoir is fundamentally set by the underlying magnon dynamics, not by output sampling strategy or complexity of the read-out scheme (Heins et al., 4 Feb 2025).

3. Memory and Nonlinear Transformation Performance

The efficacy of the magnon-scattering reservoir is quantified using canonical benchmarks from physical reservoir computing:

• Short-Term Memory (STM): The ability to recover a previous input, quantified via the correlation coefficient between the system output and a delayed version of the input sequence.
• Parity-Check (PC): The ability to compute a nonlinear transformation (XOR over a window of τ+1 bits), which evaluates the system’s nonlinear transformation capacity.

Both tasks leverage the fact that the magnon-scattering reservoir’s nonlinearities, particularly three-magnon splitting and cross-stimulation between overlapping pulses, generate a high-dimensional mapping of input histories. For $N$ pulses,

$$\hat{y}_\mathrm{STM}(T_j, \tau) = u(T_{j-\tau}), \qquad \hat{y}_\mathrm{PC}(T_j, \tau) = u(T_j) \oplus u(T_{j-1}) \oplus \dots \oplus u(T_{j-\tau}),$$

where $u(T_j)$ is the binary input at time $T_j$. The output is processed via a linear matrix ($\mathbb{W}$) fit to maximize the squared correlation $r^2(\tau)$. The summed capacities over a range of delays, $C = \sum_{\tau=1}^{10} r^2(\tau)$, reach $C_\mathrm{STM} \approx 2.59$ and $C_\mathrm{PC} \approx 2.15$ for optimal output dimension, values which are competitive with the best reported in other physical systems (Heins et al., 4 Feb 2025).

4. Nonlinear Three-Magnon Dynamics and Cross-Stimulation

The nonlinear transformation capability of the reservoir is rooted in the three-magnon splitting process. When a mode (e.g., at frequency $f_A$) crosses its nonlinearity threshold, it splits into two secondary modes with frequencies $f_{A+}$ and $f_{A-}$ such that $f_A = f_{A+} + f_{A-}$. If two input pulses are temporally proximate (overlapping or closely spaced), cross-stimulation occurs, and the pre-existence of one split mode can determine, enhance, or redirect the splitting pathway of the subsequent pulse. This phenomenon results in:

• Increased feature diversity in the output, as the spectral signatures are sensitive to both the input symbol and its timing relative to previously applied pulses.
• Nontrivial temporal correlations in the magnon population evolution, essential for the encoding of memory.

The dynamical complexity introduced by cross-stimulation accounts for the system’s ability to map input history into its internal (observable) nonlinear degrees of freedom, thus underpinning both memory and nonlinear transformation benchmarks.

5. Independence from Output Measurement Scheme

A key finding of the reservoir benchmarking is that the memory and nonlinear transformation capacities are not reliant on the choice of output measurement scheme, as long as the output space is sufficiently high-dimensional to capture all nonlinear features generated by magnon–magnon interactions. Both the modal (frequency-binned) and temporal (time-binned) output spaces provide equivalent capacities when properly optimized for dimensionality. This substantiates the statement that the computational properties—and ultimately the reservoir’s capacity—are intrinsic to the magnonic device’s physics, not to the post-processing or read-out protocol (Heins et al., 4 Feb 2025).

The following table summarizes this independence:

Output Scheme	Features Captured	Capacity Scaling
Modal (Spectral)	Full nonlinear spectrum (bins)	Increases with number of bins; saturates when nonlinearity is fully sampled
Temporal	Time evolution of mode intensities	Increases with time slices; saturates at sufficient sampling

The plateau in task performance with increasing output dimension occurs when the nonlinear spectral content is fully sampled, underscoring that no additional digital processing or measurement complexity is required to harness the physical reservoir’s computational power.

6. Broader Implications and Outlook

The magnon-scattering reservoir demonstrates that nonlinear magnon–magnon dynamics in patterned ferromagnetic microstructures enable reservoir computing with high memory and nonlinear transformation capacities. The physical origin of the computational power—three-magnon splitting and temporal cross-stimulation—guarantees that the system functions as an intrinsic analog reservoir, independent of read-out complexity. This paradigm points toward the feasibility of ultrafast, energy-efficient, all-magnonic computation and neuromorphic devices, with potential for integration into hybrid systems and extension to a variety of geometries or materials to further optimize nonlinearity and capacity.

Possible extensions include combining magnonic reservoirs with electronics or photonics for heterogenous computing, tuning geometry or material composition to enhance desired nonlinearities, and scaling such reservoirs for system-level neuromorphic architectures (Heins et al., 4 Feb 2025).

• Benchmarking a magnon-scattering reservoir with modal and temporal multiplexing (Heins et al., 4 Feb 2025)