# Benchmarking a magnon-scattering reservoir with modal and temporal multiplexing

Christopher Heins, Joo-Von Kim, Lukas Körber, Jürgen Fassbender, Helmut Schultheiss, Katrin Schultheiss

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Christopher Heins, Joo-Von Kim, Lukas Körber, Jürgen Fassbender, Helmut Schultheiss, et al., Benchmarking a magnon-scattering reservoir with modal and temporal multiplexing. Physical Review Applied, 2025, 23 (5), pp.054087. 10.1103/PhysRevApplied.23.054087. hal-05103079

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Distributed under a Creative Commons Attribution 4.0 International License
Benchmarking a magnon-scattering reservoir with modal and temporal multiplexing

Christopher Heins,¹,² Joo-Von Kim,³ Lukas Körber,¹,²,⁴ Jürgen Fassbender,¹,²
Helmut Schultheiss,¹,*, and Katrin Schultheiss¹,†

¹Institut für Ionenstrahlphysik und Materialforschung, Helmholtz-Zentrum Dresden–Rossendorf,
D-01328 Dresden, Germany

²Fakultät Physik, Technische Universität Dresden, D-01062 Dresden, Germany

³Centre de Nanosciences et de Nanotechnologies, CNRS, Université Paris-Saclay, 91120 Palaiseau, France

⁴Institute of Molecules and Materials, Radboud University, Heyendaalseweg 135, 6525 AJ Nijmegen,
The Netherlands

<figure><img src="image_2.png" /></figure>

(Received 5 February 2025; revised 25 April 2025; accepted 1 May 2025; published 29 May 2025)

Physical reservoir computing has emerged as a powerful framework for exploiting the inherent nonlinear dynamics of physical systems to perform computational tasks. Recently, we presented the magnon-scattering reservoir, whose internal nodes are given by the fundamental wavelike excitations of ferromagnets called magnons. These excitations can be geometrically quantized and, in response to an external stimulus, show transient nonlinear scattering dynamics that can be harnessed to perform memory and nonlinear transformation tasks. Here, we test a magnon-scattering reservoir in a single magnetic disk in the vortex state against two key performance indicators for physical reservoir computing: the short-term memory and parity-check tasks. Using time-resolved Brillouin light scattering microscopy, we measure the evolution of the reservoir’s spectral response to an input sequence consisting of random binary inputs encoded in microwave pulses with two distinct frequencies. Two different output spaces of the reservoir are defined: one based on the time-averaged frequency spectra and another based on temporal multiplexing. Our results demonstrate that the memory and nonlinear transformation capability do not depend on the chosen readout scheme, provided that the dimension of the output space is large enough to capture all nonlinear features provided by the magnon-magnon interactions. This further shows that it is solely the nonlinear magnons in the physical system—not the readout—that determine the reservoir’s capacity.

DOI: 10.1103/PhysRevApplied.23.054087

I. INTRODUCTION

Physical reservoir computing (PRC) [1–3] is a computa-
tional paradigm that exploits the natural dynamics of com-
plex, nonlinear physical systems to process information.
Originally inspired by recurrent neural networks, reser-
voir computing (RC) [4–7] simplifies training by keeping
a fixed set of randomly connected neurons—the reser-
voir—and only optimizing the output layer, rather than
the entire network. This architecture has proven particu-
larly effective in temporal tasks such as pattern recognition,
signal processing, and chaotic time-series prediction. In
PRC, the reservoir is not an artificial neural network but an

actual physical system with intrinsic nonlinear dynamics
that map a complex input into a high-dimensional, linearly
separable output space. Limiting the problem to a simple
linear classification, without the need for complex control
over internal states, makes PRC a promising alternative
for tasks requiring fast and energy-efficient information
processing.

In recent years, various substrates have been sug-
gested for physical reservoirs, including optical fibers [8–
11], memristive circuits [12–15], spintronic devices [16–
21], arrays of nanomagnetic elements [22–24], skyrmions
[25–27], and magnons [28–34]. In our work, we study the
computational capabilities of a magnon-scattering reser-
voir [35]. This realization relies on the nonlinear interac-
tions of quantized magnons, the collective excitations in
a magnetically ordered system. In particular, we evalu-
ate the short-term-memory (STM) and parity-check (PC)
tasks, which are both fundamental benchmarks for assess-
ing the memory and nonlinear transformation capabilities
of a computational system. The STM task evaluates a
system’s ability to maintain temporal information over

Published by the American Physical Society under the terms of
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ther distribution of this work must maintain attribution to the
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DOI.

<footnote>*Contact author: h.schultheiss@hzdr.de</footnote>

<footnote>†Contact author: k.schultheiss@hzdr.de</footnote>
short durations. This is critical for any temporal processing model, as many real-world signals involve retaining information over sequential time steps. The PC task tests the system's ability to perform nonlinear transformations, as determining parity involves identifying and processing specific nonlinear relationships among sequential inputs. Both tasks can be applied to the same input sequence, differing only in separately trained output layers that map the response of the magnon-scattering reservoir to the respective transformed target.

In our experiments, the temporal evolution of the magnon reservoir's spectral response is measured by means of time-resolved Brillouin light scattering microscopy. To characterize and read out the response of the magnon-scattering reservoir, we compare two distinct output spaces of the reservoir: one based on time-averaged frequency spectra of the magnon system, and another defined by the temporal evolution of the individual magnon modes.

## II. METHODS

### A. Sample preparation

The magnon-scattering reservoir is based on a single magnetic disk with a diameter of 5.1 µm, which is patterned from a Ti(2 nm)/Ni₈₁Fe₁₉(50 nm)/Ti(5 nm) film deposited on a SiO₂ substrate using electron-beam lithography, electron-beam evaporation, and subsequent lift off. To excite magnetization dynamics, we pattern an Ω-shaped antenna around the disk from a Ti(2 nm)/Au(200 nm) film, also using electron-beam lithography, electron-beam evaporation, and lift off. The inner and outer diameters of the antenna are 8.3 and 11.1 µm, respectively. An illustration of the sample, which is based on a scanning electron microscopy (SEM) image, is shown in Fig. 1(a).

As depicted in the SEM image, the ground state in such a disk is a magnetic vortex with the magnetic moments (black arrows) curling in-plane around the vortex core, a narrow region in the disk's center where the magnetic moments align perpendicularly to the film plane (white arrow). The magnon eigenmodes in this system have well-defined frequencies and are described by their mode numbers (n, m), with n (m) counting the number of nodes in the radial (azimuthal) direction across half the disk. A few examples of mode profiles are depicted in Fig. 1(a). By applying a microwave current to the Ω-shaped antenna, the symmetry of the homogeneous out-of-plane Oersted field solely allows for the excitation of pure radial modes with m = 0.

### B. Nonlinear three-magnon splitting

If the excitation power crosses a certain threshold, non-linear three-magnon splitting sets in [36,37], as sketched in Fig. 1(b). The excited magnon mode (n<sub>A</sub>, 0) at frequency

<figure><img src="image_3.png" /><figcaption>FIG. 1. (a) Magnon-scattering reservoir consisting of a 50-nm-thick, 5.1-µm-diameter NiFe disk in the vortex state placed inside an Ω-shaped antenna, with schematic intensity profiles of different magnon modes (n, m). (b) Two radial magnon modes f<sub>A</sub> and f<sub>B</sub> spontaneously split into two magnons with m ≠ 0 and frequencies equally spaced around half their respective excitation frequencies. (c), (d) The pulsed excitation of f<sub>A</sub> and f<sub>B</sub> with a temporal overlap results in cross-stimulation of the two channels, where the split modes of the later pulse depend on the temporal order of the two inputs.</figcaption></figure>

f<sub>A</sub> spontaneously splits into two secondary magnon modes under conservation of energy f<sub>A</sub> = f<sub>A+</sub> + f<sub>A-</sub> and angular momentum m<sub>A+</sub> = -m<sub>A-</sub>. Inside a single disk, many different channels for this splitting exist, as indicated in Fig. 1(b) for excitation frequencies f<sub>A</sub> and f<sub>B</sub>.

These different splitting channels can mutually influence each other via cross-stimulated three-magnon scattering [35]. During transient dynamics, this influence between different channels can be nonreciprocal. Figures 1(c) and 1(d) depict the simplified case with two magnon modes excited by nanosecond-long microwave pulses. The spontaneous splitting of frequency f<sub>A</sub> forces f<sub>B</sub> into a specific pair of split magnons, which deviate from its spontaneous split modes [Fig. 1(c)]. If the temporal order of the two pulses is reversed [Fig. 1(d)], f<sub>B</sub> splits spontaneously and forces f<sub>A</sub> into different split modes during the transient dynamics. Therefore, it is possible to distinguish the temporal order of the input signals by only measuring the nonlinearly excited split modes.
<figure><img src="image_4.png" /><figcaption>FIG. 2. (a) Excerpt of a binary input sequence “0010111010” which is encoded by alternating microwave pulses, each 14 ns long and resembling one time step $T_j$. Here, “1” (“0”) is represented by the frequency $f_A$ ($f_B$). (b) Excerpt from the time-resolved µBLS data measured on the magnon-scattering reservoir with the recorded magnon intensity color coded. The colored boxes highlight areas of the spectrum that are used to evaluate the system’s performance based on two distinct output schemes. (c) On the one hand, the spectral response is integrated for each time step $T_j$, and the binned intensity is used to construct output space $\mathcal{F}$. (d) On the other hand, the time evolution of all involved magnon modes is tracked individually and then concatenated to generate output space $\mathcal{T}$.</figcaption></figure>

This nonlinear interaction of quantized magnon modes in reciprocal space underpins the magnon-scattering reservoir.

### C. Signal generation

To perform the STM and PC task experimentally, we generate a sequence of 2000 random binary inputs ($u \in [0, 1]$), an excerpt of which is shown in Fig. 2(a). This input is then transformed into 2000 radio-frequency (rf) pulses, each with a nominal length of 14 ns and representing an individual time step $T_j$ with $0 \le j \le 2000$. The pulse duration is chosen to be as short as possible to place the pulse transitions within a time window where the dynamics

are not yet in the steady state. For the microwave equipment used, this corresponds to 14 ns. The binary input “1” corresponds to the third radial magnon mode (2, 0) excited at frequency $f_A = 8.9$ GHz, while the input “0” corresponds to the second radial mode (1, 0) at frequency $f_B = 7.4$ GHz. The excitation powers $P_A = 20$ dBm and $P_B = 24$ dBm are strong enough to reach the nonlinear regime spontaneously for each of the targeted modes A and B. Due to the higher threshold for $f_B$, the power is larger for that frequency to achieve similar intensities of the split modes.

The rf pulses are generated by two separate microwave sources set to pulse mode at fixed frequencies $f_A$ and $f_B$. To synchronize the two sources, a pattern generator (Pulse Streamer by Swabian Instruments) is used to create the
series of pulses gating the rf sources. The two generated
microwave signals are combined and fed into the Ω-shaped
antenna using picoprobes.

D. Time-resolved Brillouin light scattering microscopy

The response of the magnon-scattering reservoir to the
input sequence is probed as a function of frequency and
time by time-resolved microfocused Brillouin light scat-
tering spectroscopy (TR-μBLS) [38]. A monochromatic
(532 nm) continuous-wave laser is focused onto the sample
surface using a 100× microscope lens with a high numeri-
cal aperture (0.75), allowing us to reach a spatial resolution
of about 300 nm. The backscattered light is sent into a
tandem Fabry-Pérot interferometer [39] to measure the fre-
quency shift caused by inelastic scattering of photons and
magnons. Control signals that encode the current state of
the interferometer, signals from the photon counter, and a
clock signal from the pattern generator are acquired contin-
uously by a time-to-digital converter (Time Tagger 20 by
Swabian Instruments) with a temporal resolution of 1 ns.
From these data, the temporal evolution of the magnon
spectra with respect to the stroboscopic rf excitation is
reconstructed.

During the experiments, the investigated structure is
imaged using a red light-emitting diode and a charge-
coupled device camera. The displacements and drifts of
the sample are tracked by an image-recognition algorithm
and compensated by the positioning system (XMS linear
stages by Newport). To capture magnon modes with dif-
ferent spatial distributions, we integrate the signal over
12 positions covering one-quarter of the disk, four in the
azimuthal direction and three in the radial direction. All
measurements are conducted at room temperature.

E. Micromagnetics simulations

To support the experimental work, we also per-
formed micromagnetic simulations of the response of
the magnon-scattering reservoir to the same sequence
of rf pulses. We used the MUMAX3 code, which per-
forms numerical time integration of the Landau-Lifshitz-
Gilbert equation using the finite-difference method [40].
In the simulation setup, we model the 5-µm diameter,
50-nm-thick permalloy disk using 512 × 512 × 1 cells,
with an exchange constant of A = 13 pJ/m, a satura-
tion magnetization of M<sub>S</sub> = 810 kA/m, a Gilbert damp-
ing constant of α = 0.008, and a gyromagnetic constant
of γ/2π = 29.6 GHz/T. Finite-temperature effects are
accounted for by adding a stochastic term to the effective
field, where we assume T = 300 K. The rf field excitation
is modeled with a spatially uniform perpendicular field,
b<sub>rf</sub> = b<sub>i</sub> sin(2πf<sub>i</sub>t)z̃, where b<sub>A</sub> = 3.0 mT for f<sub>A</sub> = 8.9 GHz
and b<sub>B</sub> = 3.5 mT for f<sub>B</sub> = 7.4 GHz.

Following the approach outlined in Ref. [35], we esti-
mate the power spectra of magnon excitations using a

coarse-grained approach. We subdivide the simulation
geometry into a triangular mesh and record the spatial
average of the magnetization over each mesh region as a
function of time, with a time step of 20 ps. The total power
spectrum is then computed by summing the power spec-
tra of these mesh regions, obtained using a discrete Fourier
transform of the fluctuating magnetization over an inter-
val corresponding to the pulse length, 14 ns. This results
in a sequence of 2000 spectra, corresponding to a random
input sequence of 2000 rf pulses with frequencies fA and
fB. To mimic the experiment as closely as possible, which
involves multiple repetitions of the same pulse sequence,
we repeat this simulation 100 times, but with different real-
izations of the stochastic thermal field. The average power
spectrum over these 100 runs is subsequently used for
computing the STM and PC metrics.

### III. RESULTS AND DISCUSSION

An excerpt from the magnon response is shown in
Fig. 2(b), with the recorded magnon intensity color coded
on a logarithmic scale. Due to the long measurement dura-
tion of 28.1 µs, the full input sequence and the complete
time-resolved BLS spectrum are not shown, but they are
available in Ref. [41]. The gray dashed lines separate the
14-ns-long time steps T<sub>j</sub>. The directly excited modes at f<sub>A</sub>
and f<sub>B</sub> yield the most intense response. Their duration is
slightly longer than the supplied input pulses due to the
finite magnon lifetime.

In addition to the directly excited modes, one can see
a multitude of secondary modes in the frequency range
from 2 to 7 GHz, i.e., around half the frequencies of
the initially excited modes. As explained in the previous
section, the secondary modes are subject to continuous
cross-stimulation from one another; however, in addition
to the simple cross-stimulation of two consecutive pulses
depicted in Figs. 1(c) and 1(d), the split modes in a longer
input sequence also depend on excitation pulses further in
the past. An example of this long-term manipulation can be
seen in the TR-μBLS spectrum [Fig. 2(b)] when compar-
ing the transitions T<sub>31</sub> → T<sub>32</sub> and T<sub>33</sub> → T<sub>34</sub>. Both cases
show the transition f<sub>B</sub> → f<sub>A</sub>, but the intensities of the split
modes are clearly different due to the long-term history.

At the beginning of time step T<sub>33</sub>, f<sub>B</sub> = 7.4 GHz is
forced to split into modes at 2.5 and 4.9 GHz [horizon-
tal white lines for T<sub>33</sub> in Fig. 2(b)]. For the transition
T<sub>33</sub> → T<sub>34</sub>, the already existing population of the mode
at 4.9 GHz stimulates the splitting of f<sub>A</sub> = 8.9 GHz into
modes at 3.8 and 5.1 GHz directly at the beginning of T<sub>34</sub>,
despite being slightly off-resonant [black lines in T<sub>34</sub> of
Fig. 2(b)]; however, the initial mode populations are differ-
ent for the transition T<sub>31</sub> → T<sub>32</sub>. At the beginning of T<sub>32</sub>,
the mode at 4.9 GHz is no longer populated from the pre-
vious time step due to the limited lifetime of the mode at
2.5 GHz. Therefore, it takes almost 7 ns for the splitting of
$f_A$ to build up in $T_{32}$ [horizontal black lines in Fig. 2(b)]. This gives one example of how the splitting of the directly excited modes is influenced by time steps further in the past.

To analyze the memory and nonlinear capabilities of the magnon-scattering reservoir more quantitatively, we use two different output schemes for our analysis. The first output space $\mathcal{F}$ is constructed following modal multiplexing, as introduced in Ref. [35]. As shown in Fig. 2(c) for one time step $T_{32}$, the spectral response is integrated over time within each 14-ns-long pulse. The integrated spectra are then separated into $N_f$ different frequency bins, in which the detected intensities are averaged. This is conducted for each time step $T_j$. The resulting vectors $x_f(T_j)$ form a matrix $X_f$ representing the output space $\mathcal{F}$.

Alternatively, a more common approach for reservoir computing in the time domain is used, as illustrated in Fig. 2(d). Here, the intensities of the ten most intense magnon modes are tracked by integrating a 300-MHz window around the center frequency of the respective mode. Then, these time traces for the different modes are appended to form one long timestream, which is divided into a number of time steps $N_t$. The resulting vectors $x_t(T_j)$ for different time steps are concatenated to create the output matrix $X_t$. This defines the temporal output space $\mathcal{T}$.

Both output spaces, $\mathcal{F}$ and $\mathcal{T}$, are extracted for different frequency bin sizes and for varying resolutions of the time traces, respectively. This allows for a direct comparison between output spaces with similar dimensions $N_f \sim N_t$, which will be explained in more detail below.

Following calculations of the memory capacity in Refs. [42,43], the target vectors $\hat{y}_{\text{STM}}(\tau)$ for the STM task are constructed by delaying the original binary input sequence by $\tau$ time steps into the future, as illustrated for $\tau = 1, 2$ in Fig. 3(a):

$$ \hat{y}_{\text{STM}}(T_j, \tau) = u(T_{j-\tau}). \quad (1) $$

This results in the current output $x(T_j)$ at time step $T_j$ being used to recall the preceding input $u(T_{j-\tau})$ at $\tau$ steps in the past.

For the PC task, the target vectors $\hat{y}_{\text{PC}}(\tau)$ are expressed as the parity of the sum of the last ($\tau + 1$) inputs of the binary sequence $u$, as shown for $\tau = 1, 2$ in Fig. 3(b). This is equivalent to $\tau$ consecutive XOR ($\oplus$) operations on the input sequence:

$$ \hat{y}_{\text{PC}}(T_j, \tau) = u(T_j) \oplus u(T_{j-1}) \oplus \cdots \oplus u(T_{j-\tau}). \quad (2) $$

The reservoir's output layer is then trained to map the experimentally acquired output matrix $X$ to the previously described target functions, with $W$ being the matrix of the individual output weights. The weight matrix is trained

<figure><img src="image_5.png" /><figcaption>FIG. 3. Principle of (a) the short-term memory (STM) task and (b) the parity check (PC) task, plotting the target outputs for the first two delays $\tau = 1, 2$. For the STM task, the input is shifted forward in time by $\tau$. The PC task determines the parity of the sum over the previous ($\tau + 1$) time steps. (c), (d) With the spectral output space $\mathcal{F}$, the output is predicted for different delays $\tau$ (continuous lines) and is compared to the target values (dashed lines). (e), (f) Correlation between the target and predicted output as a function of delay $\tau$ for (e) the STM task and (f) the PC task. The colored areas yield the respective capacities $C$ for each task.</figcaption></figure>

separately for each task:

$$ W_{\text{STM}}X = \hat{Y}_{\text{STM}}, \quad (3) $$

$$ W_{\text{PC}}X = \hat{Y}_{\text{PC}}. \quad (4) $$

To find the matrices $W$, the output is split into a training dataset containing 1550 pulses and a set of 400 testing pulses. The first ten pulses are discarded to let the system settle in the beginning. With the pseudoinverse of the training pulse output, one can construct the weight matrix for
<figure><img src="image_6.png" /><figcaption>FIG. 4. (a) Distinction between the full spectrum, directly excited modes, and nonlinear split modes, which are used to determine reservoir metrics based on the spectral output space $\mathcal{F}$. Colored bars with different sizes indicate varying sizes of the frequency bins. (b) Distinction between the full time trace, directly excited modes, and nonlinear split modes, which are used to determine reservoir metrics based on the temporal output space $\mathcal{T}$. Colored bars with different sizes indicate varying sizes of the time bins. (c)–(f) Comparison of STM capacity (upper row) and PC capacity (bottom row) as a function of the bin size and based on the different output spaces. (g), (h) Capacities plotted as a function of output space dimension $N$, independent of its type.</figcaption></figure>

each delay $\tau$ of the STM and the PC task:

$$W_{\text{STM}} = \hat{Y}_{\text{STM}}X^{-1}, \quad (5)$$

$$W_{\text{PC}} = \hat{Y}_{\text{PC}}X^{-1}. \quad (6)$$

In Figs. 3(c) and 3(d), the calculated weight matrices are used to predict the output for the STM and PC tasks based on the test data of the spectral output space $\mathcal{F}$. Note that we plot the results as a function of absolute time $t$, with 14-ns-long rf pulses corresponding to one time step $T_j$ of the binary input sequence, with the axis starting at the first pulse after the training data. It can be seen that the reconstruction works reasonably well for the first few delays $\tau$, which confirms that the magnon-scattering reservoir exhibits short-term memory and nonlinear transformation capabilities.

This is further quantified by calculating the square of the correlation coefficient between the predicted output $y_{\text{out}}$

and the expected target output $\hat{y}_{\text{out}}$:

$$r^2(\tau) = \frac{\text{cov}(y_{\text{out}}(\tau), \hat{y}_{\text{out}}(\tau))^2}{\text{cov}(y_{\text{out}}(\tau), y_{\text{out}}(\tau)) \times \text{cov}(\hat{y}_{\text{out}}(\tau), \hat{y}_{\text{out}}(\tau))}, \quad (7)$$

where $\text{cov}(x, y)$ is the covariance between vectors $x$ and $y$. A value of 1 for the correlation coefficient corresponds to a perfect prediction, while values close to 0 mean that the prediction is random. Figures 3(e) and 3(f) show the resulting squared correlations up to $\tau = 10$ for the STM and PC tasks, respectively. Note that at zero delay ($\tau = 0$), the direct input is simply reconstructed and is therefore neglected in the further evaluation. Therefore, the capacity of the reservoir is defined by the area below the remaining squared correlation function shaded in purple:

$$C = \sum_{\tau=1}^{10} r^2(\tau). \quad (8)$$
To systematically assess the impact of different types of output spaces, we repeat our analysis for the spectral output space $\mathcal{F}$ as well as for the temporal output space $\mathcal{T}$. In addition, we further distinguish different data ranges for our analysis to better understand the origin of the reservoir's memory and computing capabilities, as illustrated in Figs. 4(a) and 4(b).

For the spectral output space $\mathcal{F}$, we consider three different frequency ranges [Fig. 4(a)]. First, we begin with the full spectrum from 2 to 10 GHz to evaluate the reservoir's full potential. Second, we limit our analysis to the frequency range from 7 to 10 GHz. This range is closest to a simple linear transformation of the input because it only contains the directly excited modes and no nonlinear split modes. Nonetheless, these excited modes are also influenced by nonlinear intensity changes and frequency shifts. Third, we choose a frequency window from 2 to 7 GHz, focusing only on the nonlinear split modes. Magnons in this range are excited purely through nonlinear three-magnon splitting and thus nicely capture the reservoir's nonlinear properties. For the temporal output space $\mathcal{T}$, we make a similar distinction [Fig. 4(b)], first looking at the full time trace and then separating the directly excited modes from the nonlinear split modes. Additionally, we vary the bin sizes for the spectral and temporal binning to study the influence of the output space dimension and allow for a more direct comparison between the different output spaces with similar dimensions of the weight matrices.

Figures 4(c) and 4(d) summarize the capacities for the STM and PC tasks, which are analyzed based on the spectral output space $\mathcal{F}$. The memory capacity of the magnon-scattering reservoir is largest when combining all modes, and it reaches its maximum of $C_{\text{STM}} = 2.59$ for a bin size of 112.5 MHz [Fig. 4(c)]. Comparing the nonlinear and excited modes reveals that the largest contribution to this memory capacity originates from the nonlinear split modes; however, even the directly excited modes contribute, as evidenced by the fact that their spectra also contain nonlinear contributions. Up to a frequency bin size of 200 MHz, the memory capacity only changes marginally. Further increasing the frequency bin size, however, results in a significant drop in the memory capacity, which is even more pronounced in the directly excited modes than in the nonlinear split modes. For larger bin sizes, above 1 GHz, the memory is completely lost for the excited modes.

The PC task shows similar behavior for smaller bin sizes [Fig. 4(d)]. When analyzing the full spectrum, the nonlinear transformation capability reaches a maximum of $C_{\text{PC}} = 2.0$ for a bin size of 112.5 MHz and is rather constant up to 300 MHz. The nonlinear modes contribute most to the nonlinear transformation capabilities of the magnon-scattering reservoir. Nonetheless, the performance is improved by including the directly excited modes as well. Interestingly, for bin sizes larger than 300 MHz, the nonlinear capacity

is similar when analyzing the excited modes and the nonlinear modes separately.

For the temporal output space $\mathcal{T}$, the directly excited modes only contain the time evolution of a 300-MHz integrated window around 7.4 and 8.9 GHz. The chosen scattered modes are shown in Fig. 1(d) and correspond to the nine most intense split modes. For the smallest time bin size, the memory capacity of the nonlinear modes [Fig. 4(e)] shows very similar results to the spectral output space $\mathcal{T}$. It can also be seen that for the memory, the nonlinear modes contain all relevant information, and adding the excited modes does not further increase the maximum value of $C_{\text{STM}} = 2.54$ at 2-ns bin size. Interestingly, this is not the case for the PC task. Here, the capacity improves when analyzing the full-time trace compared to only evaluating the nonlinear response. This is surprising, since the system's nonlinearity in the form of three-magnon splitting should be more relevant for the PC task than for the STM task. Overall, the capacity for the parity check reaches a slightly higher maximum capacity $C_{\text{PC}} = 2.15$ at a bin size of 2 ns than for the spectral output space $\mathcal{F}$.

In general, comparing the spectral and temporal output spaces reveals that the scattered modes are more important for the metrics of the magnon-scattering reservoir but that the directly excited frequencies also contain valuable information, whereby their inclusion in the output space improves the overall capacities.

Changing the bin sizes has a direct impact on the number of output features. Therefore, the different output spaces

<figure><img src="image_7.png" /><figcaption>FIG. 5. Reservoir computing metrics obtained from micro-magnetic simulations. (a),(c) STM and (b),(d) PC capacities as a function of (a),(b) bin size and (c),(d) output space dimension $N$.</figcaption></figure>
are compared regarding their dimension *N* in Figs. 4(g) and 4(h). This shows that the directly excited modes perform worse when considered alone for the same number of features; however, combining both directly excited and nonlinear modes yields better results, even with a similar number of output variables. Overall, the performance depends not on which output space was chosen but on the number of features when considering a combination of excited and nonlinear modes. In addition, one can see that the presence of too many features will degrade the capacity due to an excess of training weights, leading to overfitting.

Similar results are obtained in micromagnetic simulations for the STM and PC capacities using the output space $\mathcal{F}$, as shown in Fig. 5. As in the experiments, we observe the best performance when the full spectrum is used to construct the output space vector, which slightly outperforms the case in which only the nonlinear excitations are used. Overall, the capacities for both the STM and PC metrics are very close to the experimental results, with maximum values of $C_{STM} = 2.57$ and $C_{PC} = 2.37$. They follow a similar decrease as a function of the bin size and increase with the output space dimension. This also indicates that the experimental values stem from the magnon-scattering reservoir itself and are unrelated to possible nonlinearities of the microwave circuitry.

## IV. CONCLUSION

To benchmark the memory and nonlinear transformation potential of a magnon-scattering reservoir comprising a single micrometer-sized magnetic disk in the vortex state, we performed the short-term memory and parity-check tasks, which are two established metrics for physical reservoir computing devices. For this, a random binary sequence was encoded into two oscillation frequencies in the GHz regime, and these were used to excite dynamics in the magnon reservoir using a microwave antenna. The resulting highly complex nonlinear scattering of the quantized magnon modes, which gives rise to the reservoir computing capabilities, was subsequently measured using time-resolved Brillouin light scattering microscopy and used to define two distinct types of reservoir output spaces. In addition to the modal output space $\mathcal{F}$, given by the frequency spectra averaged over the time intervals of the input sequence, a temporal output space $\mathcal{T}$ was used, which was constructed by directly tracking the temporal evolution of the ten most dominant modes, averaged along a small frequency window around the central frequency of the respective mode. The capacity reached maximum values of $C_{STM} = 2.59$ for the short-term memory task and $C_{PC} = 2.15$ for the parity-check task, which is comparable to other physical substrates [17,19,33].

Importantly, for the optimal output dimension, the two output spaces $\mathcal{F}$ and $\mathcal{T}$ yield comparable capacities. On

the one hand, this highlights that it is not necessary to track the full temporal evolution of the scattering reservoir to achieve the best performance. Instead, provided that the nonlinear magnon dynamics are sufficiently well resolved, it is equivalent to tracing the time-averaged spectra of the active modes in the system. On the other hand, the equivalence between the two output spaces clearly shows that neither readout scheme changes the nonlinearity of the system in a relevant way. In light of the fact that in physical reservoir computing, it can be difficult to disentangle how much capacity is carried by the (possibly nonlinear) readout scheme, this clearly shows that the obtained capacities in this work are dominated solely by the nonlinear magnon interactions.

## ACKNOWLEDGMENTS

This work received funding from the EU Research and Innovation Programme Horizon Europe under Grant Agreement No. 101070290 (NIMFEIA). Support by the Nanofabrication Facilities Rossendorf (NanoFaRo) at the IBC is gratefully acknowledged.

H.S., C.H., and K.S. conceptualized the presented work. J.V.K., H.S., and K.S. acquired funding. K.S. fabricated the sample. C.H. conducted the experiments and analyzed the data. J.V.K. performed and analyzed the micromagnetic simulations. All authors discussed the results. C.H., H.S., L.K., and K.S. visualized the results. C.H., L.K., and K.S. wrote the original draft of the paper. All authors reviewed and edited the paper.

The authors have no conflicts of interest to disclose.

## DATA AVAILABILITY

The data that support the findings of this article are
openly available [41].

[1] G. Tanaka, T. Yamane, J. B. Héroux, R. Nakane, N. Kanazawa, S. Takeda, H. Numata, D. Nakano, and A. Hirose, Recent advances in physical reservoir computing: A review, Neural Netw. 115, 100 (2019).

[2] K. Nakajima, Physical reservoir computing—an introduc-
tory perspective, Jpn. J. Appl. Phys. 59, 060501 (2020).

[3] X. Liang, J. Tang, Y. Zhong, B. Gao, H. Qian, and H. Wu, Physical reservoir computing with emerging electronics, Nat. Electron. 7, 193 (2024).

[4] H. Jaeger and H. Haas, Harnessing nonlinearity: Predicting chaotic systems and saving energy in wireless communication, Science 304, 78 (2004).

[5] W. Maass, T. Natschläger, and H. Markram, Real-time computing without stable states: A new framework for neural computation based on perturbations, Neural Comput. 14, 2531 (2002).
[6] D. Verstraeten, B. Schrauwen, M. D'Haene, and D. Stroobandt, An experimental unification of reservoir computing methods, *Neural Netw.* **20**, 391 (2007).

[7] K. Nakajima and I. Fischer, Eds., *Reservoir Computing: Theory, Physical Implementations, and Applications*, Natural Computing Series (Springer Singapore, Singapore, 2021).

[8] F. Duport, B. Schneider, A. Smerieri, M. Haelterman, and S. Massar, All-optical reservoir computing, *Opt. Express* **20**, 22783 (2012).

[9] G. Van Der Sande, D. Brunner, and M. C. Soriano, Advances in photonic reservoir computing, *Nanophotonics* **6**, 561 (2017).

[10] Y. K. Chembo, Machine learning based on reservoir computing with time-delayed optoelectronic and photonic systems, *Chaos: Interdiscip. J. Nonlinear Sci.* **30**, 013111 (2020).

[11] M. Yan, C. Huang, P. Bienstman, P. Tino, W. Lin, and J. Sun, Emerging opportunities and challenges for the future of reservoir computing, *Nat. Commun.* **15**, 2056 (2024).

[12] C. Du, F. Cai, M. A. Zidan, W. Ma, S. H. Lee, and W. D. Lu, Reservoir computing using dynamic memristors for temporal information processing, *Nat. Commun.* **8**, 2204 (2017).

[13] J. Moon, W. Ma, J. H. Shin, F. Cai, C. Du, S. H. Lee, and W. D. Lu, Temporal data classification and forecasting using a memristor-based reservoir computing system, *Nat. Electron.* **2**, 480 (2019).

[14] Y. Zhong, J. Tang, X. Li, B. Gao, H. Qian, and H. Wu, Dynamic memristor-based reservoir computing for high-efficiency temporal signal processing, *Nat. Commun.* **12**, 408 (2021).

[15] Y. Zhong, J. Tang, X. Li, X. Liang, Z. Liu, Y. Li, Y. Xi, P. Yao, Z. Hao, B. Gao, H. Qian, and H. Wu, A memristor-based analogue reservoir computing system for real-time and power-efficient signal processing, *Nat. Electron.* **5**, 672 (2022).

[16] T. Kanao, H. Suto, K. Mizushima, H. Goto, T. Tanamoto, and T. Nagasawa, Reservoir computing on spin-torque oscillator array, *Phys. Rev. Appl.* **12**, 024052 (2019).

[17] S. Tsunegi, T. Taniguchi, K. Nakajima, S. Miwa, K. Yakushiji, A. Fukushima, S. Yuasa, and H. Kubota, Physical reservoir computing based on spin torque oscillator with forced synchronization, *Appl. Phys. Lett.* **114**, 164101 (2019).

[18] D. Marković, N. Leroux, M. Riou, F. Abreu Araujo, J. Torrejon, D. Querlioz, A. Fukushima, S. Yuasa, J. Trastoy, P. Bortolotti, and J. Grollier, Reservoir computing with the frequency, phase, and amplitude of spin-torque nanoscillators, *Appl. Phys. Lett.* **114**, 012409 (2019).

[19] T. Yamaguchi, N. Akashi, K. Nakajima, H. Kubota, S. Tsunegi, and T. Taniguchi, Step-like dependence of memory function on pulse width in spintronics reservoir computing, *Sci. Rep.* **10**, 19536 (2020).

[20] T. Taniguchi, A. Ogihara, Y. Utsumi, and S. Tsunegi, Spintronic reservoir computing without driving current or magnetic field, *Sci. Rep.* **12**, 10627 (2022).

[21] N. Akashi, Y. Kuniyoshi, S. Tsunegi, T. Taniguchi, M. Nishida, R. Sakurai, Y. Wakao, K. Kawashima, and K. Nakajima, A coupled spintronics neuromorphic approach for high-performance reservoir computing, *Adv. Intell. Syst.* **4**, 2200123 (2022).

[22] J. C. Gartside, K. D. Stenning, A. Vanstone, H. H. Holder, D. M. Arroo, T. Dion, F. Caravelli, H. Kurebayashi, and W. R. Branford, Reconfigurable training and reservoir computing in an artificial spin-vortex ice via spin-wave fingerprinting, *Nat. Nanotechnol.* **17**, 460 (2022).

[23] D. A. Allwood, M. O. A. Ellis, D. Griffin, T. J. Hayward, L. Manneschi, M. F. K. Musameh, S. O'Keefe, S. Stepney, C. Swindells, M. A. Trefzer, E. Vasilaki, G. Venkat, I. Vidamour, and C. Wringe, A perspective on physical reservoir computing with nanomagnetic devices, *Appl. Phys. Lett.* **122**, 040501 (2023).

[24] J. H. Jensen, A. Strømberg, I. Breivik, A. Penty, M. A. Niño, M. W. Khaliq, M. Foerster, G. Tufte, and E. Folven, Clocked dynamics in artificial spin ice, *Nat. Commun.* **15**, 964 (2024).

[25] D. Prychynenko, M. Sitte, K. Litzius, B. Krüger, G. Bourianoff, M. Kläui, J. Sinova, and K. Everschor-Sitte, Magnetic skyrmion as a nonlinear resistive element: A potential building block for reservoir computing, *Phys. Rev. Appl.* **9**, 014034 (2018).

[26] D. Pinna, G. Bourianoff, and K. Everschor-Sitte, Reservoir computing with random skyrmion textures, *Phys. Rev. Appl.* **14**, 054020 (2020).

[27] Y. Sun et al., Experimental demonstration of a skyrmion-enhanced strain-mediated physical reservoir computing system, *Nat. Commun.* **14**, 3434 (2023).

[28] R. Nakane, G. Tanaka, and A. Hirose, Reservoir computing with spin waves excited in a garnet film, *IEEE Access* **6**, 4462 (2018).

[29] A. Papp, G. Csaba, and W. Porod, Characterization of nonlinear spin-wave interference by reservoir-computing metrics, *Appl. Phys. Lett.* **119**, 112403 (2021).

[30] M.-K. Lee and M. Mochizuki, Reservoir computing with spin waves in a skyrmion crystal, *Phys. Rev. Appl.* **18**, 014074 (2022).

[31] R. Nakane, A. Hirose, and G. Tanaka, Performance enhancement of a spin-wave-based reservoir computing system utilizing different physical conditions, *Phys. Rev. Appl.* **19**, 034047 (2023).

[32] W. Namiki, D. Nishioka, Y. Yamaguchi, T. Tsuchiya, T. Higuchi, and K. Terabe, Experimental demonstration of high-performance physical reservoir computing with nonlinear interfered spin wave multidetection, *Adv. Intell. Syst.* **5**, 2300228 (2023).

[33] S. Nagase, S. Nezu, and K. Sekiguchi, Spin-wave reservoir chips with short-term memory for high-speed estimation of external magnetic fields, *Phys. Rev. Appl.* **22**, 024072 (2024).

[34] W. Namiki, D. Nishioka, T. Tsuchiya, and K. Terabe, Fast physical reservoir computing, achieved with nonlinear interfered spin waves, *Neuromorph. Comput. Eng.* **4**, 024015 (2024).

[35] L. Körber, C. Heins, T. Hula, J.-V. Kim, S. Thlang, H. Schultheiss, J. Fassbender, and K. Schultheiss, Pattern
recognition in reciprocal space with a magnon-scattering reservoir, Nat. Commun. **14**, 3954 (2023).

[36] K. Schultheiss, R. Verba, F. Wehrmann, K. Wagner, L. Körber, T. Hula, T. Hache, A. Kákay, A. A. Awad, V. Tiberkevich, A. N. Slavin, J. Fassbender, and H. Schultheiss, Excitation of whispering gallery magnons in a magnetic vortex, Phys. Rev. Lett. **122**, 097202 (2019).

[37] L. Körber, K. Schultheiss, T. Hula, R. Verba, J. Fassbender, A. Kákay, and H. Schultheiss, Nonlocal stimulation of three-magnon splitting in a magnetic vortex, Phys. Rev. Lett. **125**, 207203 (2020).

[38] T. Sebastian, K. Schultheiss, B. Obry, B. Hillebrands, and H. Schultheiss, Micro-focused Brillouin light scattering: Imaging spin waves at the nanoscale, Front. Phys. **3**, 35 (2015).

[39] R. Mock, B. Hillebrands, and R. Sandercock, Construction and performance of a Brillouin scattering set-up using a

triple-pass tandem Fabry-Perot interferometer, J. Phys. E: Sci. Instr. **20**, 656 (1987).

[40] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyenberge, The design and verification of MuMax3, AIP Adv. **4**, 107133 (2014).

[41] C. Heins, J.-V. Kim, L. Körber, J. Faßbender, H. Schultheiß, and K. Schultheiß, Data publication: Benchmarking a magnon-scattering reservoir with modal and temporal multiplexing (2025), https://doi.org/10.14278/rodare.3558.

[42] H. Jaeger, Short Term Memory in Echo State Networks, GMD-Forschungszentrum Informationstechnik (2001), https://doi.org/10.24406/publica-fhg-291107.

[43] M. Dale, J. F. Miller, S. Stepney, and M. A. Trefzer, A substrate-independent framework to characterize reservoir computers, Proc. R. Soc. A: Math. Phys. Eng. Sci. **475**, 20180723 (2019).