Mode hopping via nonlinear magnon-magnon coupling in a synthetic antiferromagnet

Domain: biology
Published: 2026-03-12
Editorial outcome: major_revision

Abstract
Identifying nonlinear interactions among magnons is crucial for advancing the field of
magnonics and developing next-generation spintronic devices. In this work, we report the
experimental observation of nonlinear magnon mode hopping and hysteretic behavior in a synthetic
antiferromagnet (SAF). Under strong radio-frequency excitation, the magnon system exhibits abrupt
transitions between acoustic and optic modes, accompanied by GHz-scale frequency jumps—orders
of magnitude larger than previously reported in magnon-based hybrid systems. We further show
that this mode hopping is hysteretic, reflecting multistable nonlinear dynamics, and support these
findings through theory and micromagnetic simulations. These results establish a new regime of
nonlinear magnon dynamics, going beyond previously reported strong-coupling effects, and highlight
the potential of SAFs as platforms for magnon-based frequency converters and switches for
information processing.

Introduction
Nonlinear phenomena play a crucial role in a wide range of physical systems, offering deep insights
into complex interactions and emergent behaviors. From the turbulence of fluid dynamics1,2 to the formation
of solitons in optics2,3, nonlinear physics has advanced our understanding of many-body systems4 and
collective excitations5. In condensed matter physics, nonlinear interactions often give rise to unique
phenomena6–8 such as chaos, pattern formation9, and bifurcations10, all of which are essential for advancing
both fundamental theory11 and technological applications.

In magnonic systems, nonlinear physics has also been a focus of considerable attention due to its
potential for manipulating spin waves and enabling new functionalities in devices12,13. Nonlinear magnon
interactions14 have been predominantly explored through multi-magnon scattering, which can lead to
phenomena, such as magnon Bose-Einstein condensation (BEC)15–17, which is typically observed in
incoherent magnon interactions where the magnons have random phases. More recently, magnon
nonlinearity has been explored in hybrid systems18,19, where phase-defined coherent magnons couple with
other quasiparticles, such as phonons or photons20–25, giving rise to phenomena such as frequency bi-
stability26, gain-driven auto-oscillation27 and anticrossing gap closing28.

Direct nonlinear magnon-magnon interactions, however, remain far less understood. While several
recent reports have demonstrated nonlinear magnon-magnon coupling phenomena12,29–32, most of these
works have focused on spectral signatures such as frequency mixing or gap modulation33–37. What has not
yet been explored, and what we focus on here, is the emergence of mode hopping with hysteresis—an
abrupt transition between two magnon modes under strong drive.

In this work, we experimentally demonstrate such nonlinear mode hopping in a synthetic
antiferromagnet (SAF). The SAF supports two distinct magnon modes: optic (high-frequency) and acoustic
(low-frequency) magnons. By tuning an external magnetic field, we coupled these modes and observed
anticrossing behavior, consistent with established strong coupling of two magnon modes. More importantly,
under strong RF excitation, we uncovered a series of nonlinear effects that constitute the central findings
of this work: (1) abrupt mode hopping between the optic and acoustic modes, accompanied by frequency
jumps up to 5 GHz; (2) pronounced hysteresis of the mode hopping depending on the field sweep direction;
and (3) a frequency redshift and closure of the anticrossing gap.

To understand these effects, we developed a theoretical model that qualitatively matches our
experimental results. The model shows that the observed nonlinearities stem from interactions among
propagating and Kittel magnon modes, both optic and acoustic branches. As the RF driving power increases,
these magnons then participate in a three-magnon process between the optic and acoustic modes, resulting
in the observed mode hopping and its hysteretic character. These findings advance the understanding of
nonlinear magnon dynamics beyond previously reported strong-coupling effects, and highlight SAFs as a
promising platform for magnon-based switches and frequency converters where large GHz-scale jumps can
be exploited for information processing.

Results
Samples and measurement set-up
For this study, we fabricated a sample with a SAF structure consisting of Ru (3 nm)/CoFeB (15
nm)/Ru (0.6 nm)/CoFeB (15 nm)/Pt (1 nm)/Ru (3 nm)/Ta (3 nm). The two CoFeB layers with in-plane
magnetization are separated by a 0.6 nm Ru spacer to induce antiferromagnetic coupling via the Ruderman-
Kittel-Kasuya-Yoshida (RKKY) interaction. The sample was patterned into a strip with dimensions of 10
μm in width and 100 μm in length, and a 60 nm-thick SiO insulating layer was deposited above the strip.
Finally, two coplanar waveguide (CPW) antennas were patterned on top of the insulating layer to generate
the RF driving field. The distance between the centers of the two CPW antennas was set to 1 μm, as shown
in Fig. 1(a) [see Methods and Supplementary Note S1 for details].

Figure 1(b) illustrates the schematic of our experimental setup. We drive an RF field through CPW1
to generate spin waves, which propagate towards CPW2 within external in-plane magnetic field at an angle
𝜃 respect to propagating direction. The transmitted spin waves are then detected using the S parameter
k 21
of a vector network analyzer (VNA). Since our sample has a SAF structure, it supports two distinct magnon
modes: the optic mode and the acoustic mode38, as depicted in Figs. 1(c) and 1(d). In the optic mode (Fig.
1c), the magnetizations of the two CoFeB layers precess out-of-phase, resulting in a higher frequency,
whereas in the acoustic mode (Fig. 1d), they precess in-phase, leading to a lower frequency. These two
modes can be selectively excited by adjusting the angle of the applied magnetic field. When the field is
applied at an angle of 0°, tilting the magnetization along the x-axis, RF excitation predominantly drives the
optic mode. Conversely, when the magnetic field is applied at 90°, tilting the magnetization along the y-
axis, the in-phase acoustic mode is excited [see Supplementary Fig. S1]. The crossing of two magnon modes
can be achieved when the magnetic field is applied at a 45° angle, where both modes are excited, resulting
in their coupling. This coupling is manifested as an anticrossing with large cooperativity [see
Supplementary Note 1] 33–35,37. The observation of anticrossing indicates that the propagating spin wave is
dominant in our measurement, rather than the non-propagating Kittel mode, since the coupling between
optic and acoustic modes in SAF occurs only for propagating spin waves through dynamic dipolar
interaction37. We note that, unlike previous reports on van der Waals magnets36—where an anticrossing gap
was observed by applying out-of-plane (OOP) magnetic field—our SAF system shows no such OOP-
induced anticrossing [see Supplementary Note 2], which can be attributed to the relatively weak interlayer
exchange field in our system36,39.

Nonlinear magnon-magnon coupling
We next investigate the nonlinear behavior of the magnon-magnon coupling. To do this, we fixed
the magnetic field angle at 45° and gradually increased the RF driving power while measuring the spin
wave spectra. To avoid the influence of magnetization switching, the sample was first saturated at positive
field and then returned to zero before increasing the field again [the effect of RF driving power on
magnetization switching is discussed in Supplementary Note 3]. As shown in Figs. 2(a)-(c), from -20 dBm
to 0 dBm, the spectra show minimal change, with the anticrossing behavior clearly visible. However, as the
driving power increases further, the spectra begin to distort, and nonlinear effects become apparent, as seen
in Figs. 2(d)-(f).

Several distinct nonlinear phenomena are observed for high RF driving power. First, above the
threshold, the overall resonant frequency experiences a redshift. Figure 2(g) quantifies this, showing the
magnon frequency as a function of driving power, where a sudden frequency drop occurs once the threshold
is exceeded. Second, we observe the closure of the anticrossing gap beyond this threshold. For clearer
visualization, Fig. 2(h) presents the frequency spectrum at the anticrossing point as the driving power
increases, showing the gradual closing of the gap. These observations—frequency redshift and gap
closure—suggest that the propagating magnons, which are primarily responsible for the frequency
anticrossing, weaken in the nonlinear regime, and the lower-frequency Kittel magnons may become
dominant.

Most notably, mode hopping is observed once the power exceeds the threshold. As we increase the
magnetic field, we observed that at around 20 mT—where the frequency of the optic magnon, 𝜔 , is
op
nearly twice that of the acoustic magnon, 𝜔 , —the optic mode abruptly transitions into the acoustic mode
ac
(as highlighted by the red guide in Figs. 2(d)-2(f)). This suggests that power-induced nonlinearity in the
magnons may trigger an optic-to-acoustic mode transition via a three-magnon scattering process40,41, where
one optic magnon with 𝜔 splits into two acoustic magnons with 𝜔 . Importantly, additional
op ac
measurements under out-of-plane magnetic fields and at different in-plane field angles are presented in
Supplementary Note 4, which further support the robustness of these nonlinear behaviors.

It should be noted that although our measurements employ a single excitation frequency, nonlinear
interactions between magnons at different frequencies can still manifest. Recent studies have clearly
demonstrated this point: ref. 29 showed that single-frequency FMR excitation in a SAF can nevertheless
reveal acoustic–optic three-magnon scattering, while ref. 42 reported nonlinear frequency doubling under
high-power drive using heterodyne MOKE. These findings establish that single-frequency measurements
can capture signatures of multi-frequency magnon interactions, consistent with the observed mode hopping.

Hysteretic behavior in nonlinear magnon-magnon coupling
To further investigate the nonlinear-induced mode hopping, we explore the field sweep direction
dependence, as hysteresis is commonly observed in nonlinear systems. Under the same experimental
conditions as before, we measured the spin wave spectra while sweeping the magnetic field forward
(increase) and backward (decrease). As shown in Figs. 3(a) and 3(b), below the threshold power (0 dBm),
there is no noticeable difference in the spin wave spectra for forward (Fig. 3a) or backward (Fig. 3b) field
sweeps. However, at an RF power of 2.5 dBm, which exceeds the threshold, the spin wave spectra exhibit
clear hysteresis, with the mode hopping point depending on the direction of the field sweep, as seen in Figs.
3(c) (forward) and 3(d) (backward).

To provide a more quantitative analysis, we extract the resonant frequencies from each spectrum
and plotted them as a function of magnetic field for forward (red) and backward (blue) sweep, as shown in
Figs. 3(e)-(h). As the power increases, we observe that the hysteresis gap gradually narrows: at 2.5 dBm,
the gap is approximately 15 mT (Fig. 3f), while at 10 dBm, the gap reduces to less than 5 mT (Fig. 3h). The
presence of hysteresis as well as the narrowing of the hysteresis gap suggests that the mode hopping process
involves more than a simple three-magnon scattering mechanism based on 𝜔 ≈ 2𝜔 and may involve
op ac
a more complex process.

Theoretical explanation
To understand the physical mechanisms behind the observed nonlinear behaviors, we developed
a theoretical model for SAF magnons including nonlinear interactions. Given that the CPW antenna
typically excites both Kittel (𝑘 = 0) and propagating (𝑘 ≠ 0) magnons43, and considering the short distance
between the two CPW antennas, it is reasonable to consider the presence of four distinct magnon modes:
Kittel acoustic and optic magnons, as well as propagating acoustic and optic magnons [Fig. 4(a)]. Our
experimental results can be qualitatively reproduced with the following theory which we aim to keep
minimal. The Hamiltonian for the four SAF magnons that are of our interest can be written as27,44:

𝐻̂ = 𝐻̂ 0 +𝐻̂ P𝑚 +𝐻̂ int . (1)

Here, 𝐻̂ represents the Hamiltonian for Kittel magnons, 𝐻̂ denotes the Hamiltonian for propagating
0 𝑚
magnons, and 𝐻̂ describes the interaction between Kittel and propagating magnons. The Hamiltonian
int
𝐻̂ 0 for Kittel magnons is given by:

𝐻̂ = ℏω0 𝑎̂†𝑎̂+ℏω0 𝑏̂†𝑏̂ −ℏ𝐺 (𝑏̂𝑎̂†𝑎̂†+𝑎̂𝑎̂𝑏̂†), (1-1)
0 ac op 0

where 𝑎̂/ 𝑎̂† and 𝑏̂/ 𝑏̂† are the annihilation/creation operators for the Kittel acoustic and optic magnons,
respectively, ω0 and ω0 are the frequencies of the Kittel acoustic and optic magnons, and 𝐺 is the
ac op 0
nonlinear three-magnon scattering coefficient 45,46 between Kittel optic and acoustic magnons, which can
be originated from, e.g., the dipolar coupling47. Similarly, the Hamiltonian 𝐻̂ for propagating magnons
𝑚
can be written as:

𝐻̂ = ∫ℏ𝑑𝑘(ω𝑘 𝑚̂ † 𝑚̂ +ω𝑘 𝑚̂ † 𝑚̂ +𝑔 (𝑚̂ † 𝑚̂ +𝑚̂ † 𝑚̂ ))
𝑚 m,1 1,𝑘 1,𝑘 m,2 2,𝑘 2,𝑘 m 1,𝑘 2,𝑘 2,𝑘 1,𝑘
(1-2)

where 𝑚̂ /𝑚̂ † and 𝑚̂ /𝑚̂ † are the annihilation/creation operators for the propagating acoustic and
1,𝑘 1,𝑘 2,𝑘 2,𝑘
optic magnons with wavevector 𝑘, ω𝑘 and ω𝑘 are their respective frequencies, 𝑔 is the linear
m,1 m,2 m
hybridization (coupling between acoustic- and optic magnons) term for the propagating magnons36,37. The
interaction Hamiltonian 𝐻̂ is given by:
int

𝐻̂ = ∫ℏ𝑑𝑘[𝑖𝜆 (𝑎̂†𝑚̂ +𝑚̂ † 𝑎̂)+𝑖𝜆 (𝑏̂†𝑚̂ +𝑚̂ † 𝑏̂)+𝜇 (𝑎̂†2 𝑚̂ +𝑚̂ † 𝑎̂2)
int 𝑎 1,𝑘 1,𝑘 𝑏 2,𝑘 2,𝑘 𝑎 1,𝑘 1,𝑘
+𝜇 𝑏 (𝑏̂† 2 𝑚̂ 2,𝑘 +𝑚̂ 2 † ,𝑘 𝑏̂2)+𝜆 𝑚 𝑒𝑖𝜑0∫𝑑𝑘′(𝑚̂ 2 † ,𝑘 𝑚̂ 1,𝑘−𝑘′ 𝑚̂ 1,𝑘′ +𝑚̂ 1 † ,𝑘−𝑘′ 𝑚̂ 1 † ,𝑘′ 𝑚̂ 2,𝑘 )]
(1-3)

where 𝜆 and 𝜆 are the linear coupling coefficients between one Kittel magnon and one propagating
𝑎 𝑏
magnon, while 𝜇 and 𝜇 represent the nonlinear coupling coefficients between two Kittel magnons and
𝑎 𝑏
one propagating magnon, for the acoustic and optic modes, respectively. The 𝜆 is the nonlinear three-
𝑚
magnon scattering coefficient between the propagating optic and acoustic magnons, and 𝜑 is the phase
0
lag of the propagating magnons detected at the antenna. We note that the interaction between two
propagating magnons and one Kittel magnon is neglected due to energy conservation considerations, as the
energy of propagating magnons is higher than that of Kittel magnons.

The interaction Hamiltonian 𝐻̂ indicates that propagating magnons can be converted into
int
Kittel magnons through the coupling, leading to an accumulation of Kittel magnons. This is consistent with
our experimental observations in Figs. 2(g) and 2(h), as the transition from propagating to Kittel magnons
results in a redshift in frequency (Fig. 2(g)) and the closing of the anticrossing gap (Fig. 2(h)). These results
suggest that the Kittel mode becomes more dominant in the nonlinear regime. To further explore this, we
derive the equations of motion for the Kittel magnons based on the above Hamiltonian (see Supplementary
Note 5 for detailed calculations):

𝑑𝑎
𝑑𝑡
= −𝑖(𝜔
𝑎
0
𝑐
+𝑖(𝐺
1
−𝜅
𝑎𝑐
−𝜉|𝑎|2)) 𝑎+𝑖𝐺
0
𝑎†𝑏+𝑖𝜆
𝑎
𝜆
𝑚
𝑒𝑖𝜑0〈𝑚̂
1
†
,𝑘′
𝑚̂
2,𝑘+𝑘′
〉
𝑘𝑘′
(2-1)

𝑑𝑏
𝑑𝑡
= −𝑖(𝜔
𝑜
0
𝑝
+𝑖(𝐺
2
−𝜅
𝑜𝑝
−𝜂|𝑏|2))𝑏+𝑖𝐺
0
𝑎2+𝑖𝜆
𝑎
𝜆
𝑚
𝑒𝑖𝜑0〈𝑚̂
1,𝑘′
𝑚̂
1,𝑘−𝑘′
〉
𝑘𝑘′
(2-2)

where, 𝐺 = 𝜋𝜆2 and 𝐺 = 𝜋𝜆2 correspond to the linear magnonic gain for Kittel acoustic and optic
1 𝑎 2 𝑏
magnons, respectively, and 𝜉 = 2𝜋𝜇2 and 𝜂 = 2𝜋𝜇2 are the nonlinear damping for Kittel acoustic and
𝑎 𝑏
optic magnons. The last terms of each equation indicate the integral of the average values of nonlinear
interaction among propagating magnons, ∫𝑑𝑘𝑑𝑘′ 〈𝑚̂ † 𝑚̂ 〉 ≡ 〈𝑚̂ † 𝑚̂ 〉 and,
1,𝑘′ 2,𝑘+𝑘′ 1,𝑘′ 2,𝑘+𝑘′ 𝑘𝑘′
∫𝑑𝑘𝑑𝑘′ 〈𝑚̂
1,𝑘′
𝑚̂
1,𝑘−𝑘′
〉 ≡ 〈𝑚̂
1,𝑘′
𝑚̂
1,𝑘−𝑘′
〉
𝑘𝑘′
. The magnonic gain (𝐺
1
and 𝐺
2
) drives the conversion of
propagating magnons into Kittel magnons, causing their amplitudes to increase exponentially until they
reach steady-state values, where the nonlinear damping (ξ and η) counteracts the gain. In Eqs. (2-1) and (2-
2), the resonance frequency, which can be tuned by external magnetic field, can be described by38

𝜔0 = 𝜇 𝛾√(𝐻 cos𝜙−𝐻 cos2𝜙±𝐻 cos2𝜙)(𝐻 cos𝜙−𝐻 cos2𝜙+𝑀 ±𝐻 ) ,
ac(+),op(−) 0 ext I𝐸 𝐸 ext 𝐸 𝑠 𝐸

where 𝛾 is the gyromagnetic ratio, 𝐻 is external field, 𝐻 is the exchange field (=25 mT in our SAF
ext E
sample), 𝑀 is the saturation magnetization (=1521 mT in our sample), and 𝜙 = cos−1(𝐻 /𝐻 ), the
s ext 𝐸
angle between canted magnetizations. The (+) and (-) signs denote the acoustic and optic magnon
frequencies, respectively (the exact values of the parameters are specified at the Supplementary Note S2).

We numerically solve the above differential equations by varying 𝐺 , while keeping 𝐺 −𝜅 =
1 2 op
−0.01 GHz constant, based on the assumption that the magnonic gain for acoustic magnons is greater than
that for optic magnons due to their lower energy. The normalized steady-state amplitudes of each mode are
shown in Figs. 4(b) and 4(c), respectively. The red plot in Fig 4(b) represents the amplitude of the acoustic
magnons as the magnetic field increases, which shows a sudden jump above a critical magnetic field,
indicating that the initially negligible acoustic magnons in the low-field regime become dominant in the
high-field regime above the critical field. The grey plot in Fig. 4(b) corresponds to the backward magnetic
field sweep, showing a sudden decrease in amplitude below the critical field. This suggests that the initially
significant acoustic magnons in the high-field regime become less dominant in the low-field regime. It is
worth noting that the hysteretic behavior was observed, when comparing the red and grey plots in Fig. 4(b).
Similarly, the blue and grey plots in Fig. 4(c) represents the amplitude of the optic magnons, indicating that
the optic magnons dominate in the low-field regime and also exhibit hysteresis. This behavior suggests that
the system undergoes mode hopping depending on the magnetic field sweep, which causes hysteresis above
the threshold power27. We note that the presence of hysteresis implies a memory effect in the system. In
our numerical calculations, different initial conditions were imposed depending on the sweep direction, and
hysteresis naturally emerged from this procedure. In the experiment, however, the microscopic origin of
the memory effect is not entirely clear, since, the magnons are expected to be fully relaxed before the next
measurement. One plausible scenario is that the magnetization configuration at a given field slightly differs
depending on the sweep direction, thereby biasing the system toward either the optic or the acoustic mode.
This interpretation is further supported by the angle-dependent threshold measurements presented in
Supplementary Note 4, which demonstrate that the preferred mode strongly depends on the magnetization
state.

To further reproduce the experimental observations, we reconstructed the f-H relations for
different magnonic gain 𝐺 . We calculated the frequencies of the acoustic and optic magnons while varying
1
the magnetic field, and plotted the dominant mode based on the hysteretic mode hopping observed in Figs.
4(b) and 4(c). In the linear regime (𝐺 = 0,𝐺 −𝜅 < 0), we only detect linearly hybridized propagating
1 1 𝑎𝑐
magnon mode as shown in the Fig. 4 (d) which is calculated by only considering propagating magnonic
contribution. However, in the nonlinear regime, where 𝐺 > 0 with 𝐺 −𝜅 > 0, mode hopping begins
1 1 𝑎𝑐
to emerge: the optic mode suddenly transitions to the acoustic mode with increasing magnetic field, while
the acoustic mode transitions to the optic mode with decreasing magnetic field, as shown in Figs. 4(e)-(g).
Furthermore, we found that as the magnonic gain 𝐺 increases, the hysteresis gap narrows, closely
1
matching the experimental observations in Figs. 3(f)-3(h), which are obtained by increasing the RF driving
power.

Methods
Film preparation and device fabrication
The Ru (3 nm)/ CoFeB (15 nm)/ Ru (0.6 nm)/ CoFeB (15 nm)/ Pt (1 nm)/ Ru (3 nm)/ Ta (3 nm)
synthetic antiferromagnetic film (illustrative sample stack is described at the Supplementary Note S1) were
deposited on Si/SiO (200 nm) substrates using AJA compact UHV sputtering system under a base pressure
2
of ~10-9 Torr. The working pressure was maintained at 3 mTorr with an Ar gas flow of 30 sccm into the
chamber. All the metallic layers were deposited by DC magnetron sputtering.

The film was patterned into the strip-line structure with dimensions of 10 µm in width and 100 µm
in length using photolithography (MicroWriter ML3 Baby, Durham Magneto Optics) and Ar ion milling
(KVET-B2000, KOREA VACUUM TECH) under a low base pressure of 7×10-7 Torr. The Ar ion beam
was generated by flowing 10 sccm Ar gas into the chamber, maintaining a working pressure of 0.5 mTorr.
On top of the pattern, we deposited the with 65 µm-width 90 µm-wide and 60 nm-thick SiO insulating
2
layer (blue region in Fig. 1a) to prevent flowing of leakage current from metallic sample to coplanar
waveguide (CPW) antennae.

The CPW antennae for the GSG contact were fabricated using a Ti (5 nm)/Au (150 nm) through
photolithography, following the same process as the sample patterning, while the central parts were
designed by E-beam lithography (TESCAN essence MIRA, EBL kit). After the lift-off process, the CPW
geometry is given by 1.28 µm-width signal-line and 0.68 µm-width ground-line. The centre-to-centre
distance between signal lines is 4.94 µm, while the shortest distance between ground lines is just 1 µm, as
illustrated in Fig. 1(a), with an enlarged view provided in Supplementary Fig. S1(d).

RF measurements set-up
The spin-wave spectra were measured using a vector network analyser (VNA), specifically the
Keysight PNA E5080, while applying an in-plane magnetic field. The RF-signal processing was conducted
over a frequency range of 100 MHz to 15 GHz, with IF-bandwidth of 500 Hz. The magnetic field was
applied using a customized in-plane electromagnet from KR-TECH (Korea), with a resolution of 0.1 mT.
The measurements were performed using a home-built probe station equipped with GSG probes and RF
measurement toolkits.

Data availability
The datasets generated and/or analyzed during the current study are available from the corresponding
authors on reasonable request.

Code Availability
The relevant codes for numerical calculation/simulation of this work are available from corresponding
authors upon reasonable request.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Recently, magnon-magnon coupling in synthetic antiferromagnet has widely investigated experimentally, theoretically, and\nmicromagnetic simulations. In this work, the authors experimentally and theoretically the magnon-magnon coupling in SAF\nand the high RF power induced nonlinear phenomenon. The observed mode hoping and the hysteretic behavior\nphenomena is very interesting for the field of nonlinear magnonics. However, I cannot make a recommendation for the\ncurrent format of the manuscript. I suggest that the authors revise the manuscript with more focus on the mode hoping and\nhysteretic behavior of magnon modes since the strong and even ultrastrong coupling between optical and acoustic magnons\nin synthetic antiferromagnets have been already reported. Followings are my suggestions.\n1.The hysteretic behavior of magnon mode should relate to the magnetization hysteresis loop. Is it possible to add a part on\ndiscussing the effect of RF power on the static MH loops?\n2.Micromagnetic simulations have been widely used to understand the coupling mechanism since it can provide the\ndirection precession information of different magnon modes. The observed “mode hoping” should be further understood with\nthe help of micromagnetic simulations.\n3.Regarding the external magnetic field, here, the authors studied the directions of external magnetic field in the plane of\nsample. Generally, the magnon-magnon coupling can also be realized by tilting the field toward out-of-plane direction. The\nobserved mode hoping can also appear when tilting the magnetic field out-of-plane? It is appreciated if they can give a\ndiscussion.\n4.The results and schematic presented in Fig. 1 are nothing new, I think these can be shown in supplemental, especially\nFig. 1e-f.\n5.In Fig. 2, the angle of magnetic field should be indicated in the caption. In Fig. 2g, the label should be inserted.\n6.Fig. 2h is used to present the observed closing of anticrossing gap. Is there a better field than 27 mT. I suggest they chose\nthe field at which the intensity of the optical and acoustic is similar.\n7.In Fig. 3e-h, four typical RF power is provided. I am wondering that the RF power 2.5 dBm is the threshold for a specific\ndirection of external magnetic field or for all in-plane angles? The hysteretic behavior appears abruptly or slowly?\n8.Since the results in Fig. 4 is theoretical calculations, is it possible give a discussion on how the threshold RF power\nvarying with extremal field? It is still not quite clear why the mode hopping decreases with G1? Is it related to the variation of\nthe precession angle of magnetizations (m, m1, m2)? Micromagnetic simulations should be helpful.", "reviewer_2": "In this manuscript, Mujin You and co-authors report strong coupling and nonlinear interactions involving two GHz-frequency\nmagnons in a synthetic antiferromagnetic structure driven by RF excitations. They observe a range of nonlinear effects\nabove a certain RF power threshold, including frequency shifts, anticrossing gap closure, mode hopping with large\nfrequency jumps, and hysteresis. The authors further develop a phenomenological model that considers nonlinear\ninteractions between Kittel and propagating magnon modes to explain their experimental findings. The manuscript is well-\nwritten, and the study is scientifically sound. Therefore, I recommend that the paper be considered for publication in Nature\nCommunications, provided that the authors address the following questions:\n1. Since the anticrossing originates from dipolar interactions, the size of the anticrossing gap is expected to depend on the\nexcitation wavevector and, consequently, the width of the coplanar waveguide. Could the authors clarify the influence of the\ncoplanar waveguide structure on the coupling and the nonlinear interactions observed?\n2. What is the physical origin of the narrowing hysteresis gap with increasing RF power? While the numerical simulations\nreproduce this effect as the magnonic gain G_1 increases, a detailed discussion of the underlying mechanism is also\nneeded.\n3. The Hamiltonian in the theoretical model primarily considers three-magnon interactions to account for the observed\nnonlinearities. Could four-magnon processes, such as magnon self-Kerr and cross-Kerr interactions, also play a role in the\nobserved phenomena? If so, could the authors comment on their potential contributions?\nMinor point: The intensity scales should be defined in all the 2D color plots throughout Figures 1–3.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature\nCommunications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career\nResearchers who co-review manuscripts.", "reviewer_4": "The manuscript reports a study of magnon spectra in a synthetic antiferromagnet under strong microwave excitation. The\nauthors show that with increasing excitation power, the spectrum undergoes strong modifications, which are described in\nterms of nonlinear mode hoping and explained by magnon-magnon interaction. The observed modifications are hysteretic\nand depend on the direction, in which the magnetic field is changed. The authors also present a model, that reproduces the\nobserved hysteresis.\nUnfortunately, I cannot recommend publication of this work for the following reasons:\n1. First of all, the claims of novelty in the abstract and introduction are an exaggeration. In fact, there are many works\ndemonstrating nonlinear coupling between coherent magnons. One can mention Appl. Phys. Lett. 121, 112404 (2022) as\none of the recent examples. If the authors conduct a thorough literature search, they will find many more examples.\n2. The interpretation of the results in terms of nonlinear coupling of coherent magnons is highly controversial.\na) All the considered “interacting” magnons (acoustic and optic Kittel magnons and propagating magnons) have different\nfrequencies. In the experiment, they are excited by sweeping the frequency of the microwave source, i.e., at different\nmoments in time. These magnons never coexist in the sample unless the frequency is swept on a temporal scale shorter\nthan the magnon relaxation time, which I believe is impossible. Therefore, the idea of the interaction of coherent magnons is\nnot applicable here. For such an interaction to exist, interacting magnons must be excited simultaneously by two sources (as\nin Appl. Phys. Lett. 121, 112404 (2022) mentioned above).\nb) Let us now consider the hysteresis (e.g., Fig. 3f). It appears that the state of the system found at 20 mT depends on\nwhether previous measurements were made at larger or smaller field. However, all magnons excited in these previous\nmeasurements relax completely before the measurement at 20 mT is started. Therefore, the observed hysteretic behavior\ncannot be due to the magnon accumulation. Obviously, it is caused by a process characterized by a much longer relaxation\ntime. For example, it can be caused by heating the sample with intense microwaves."}, "author_response": "We sincerely thank all the reviewers for their insightful comments. Below, we provide detailed\nresponses to each of the comments. Revised parts in the manuscript are highlighted in blue. A summary\nof the major revisions made to the manuscript and supplementary information is provided below.\nList of major changes in the main text\n1) We have added new author (Jun Seok Seo) for his contribution to the micromagnetic simulations\nduring the revision process.\n2) We have also added new author (Daiju Hayashi) for provision of supporting optical measurement\nresults (via Heterodyne magneto-optical Kerr effect microscope).\n3) We have moved the Fig. 1 (e)-(g) to Supplementary Note as Fig. S1 (f)-(h), following the comment\nfrom Reviewer #1.\n4) We have revised the abstract and introduction, which previously emphasized strong coupling\namong magnon modes. Instead, we now focus on the nonlinear phenomena in magnon–magnon\nhybrid systems, as suggested by Reviewer #1.\n5) In the Samples and Measurement Set-up section of the main text, we have moved the calculation\nof the cooperativity for acoustic and optic magnons to Supplementary Note S1.\n6) In the Fig. 2 and Fig. 3, we have added the scale-bar for the every VNA-measured contour map as\nthe Reviewer #2 requested.\nList of changes in Supplementary Note\n1) We have newly added Supplementary Notes S2-to-S4 and S6-to-S7, which includes 5 notes, 8\nfigures.\n2) We revised the Supplementary Note S1, Fig. S1.\n3) We moved the Supplementary Note S2 (Microscopic description of the three-magnon induced\nmode hopping) to S5.\n4) Note S1, Fig. S1 (e)-(h): The VNA-measured contour maps with respect to in-plane (external) field\ndirection which were in the Fig. 1 in the main text.\n5) Note S2: Effect of out-of-plane magnetic field on the magnon-magnon coupling in the SAF.\n6) Note S3: Influence of RF Excitation on magnetization switching and MH loops.\n7) Note S4: Magnetic field angle-dependent mode hopping in the SAF.\n8) Note S6: Micromagnetic simulation of mode hopping in the SAF.\n9) Note S7: Discussions on four-magnon scattering.\nResponse to Reviewer #1\nRecently, magnon-magnon coupling in synthetic antiferromagnet has widely investigated\nexperimentally, theoretically, and micromagnetic simulations. In this work, the authors experimentally\nand theoretically the magnon-magnon coupling in SAF and the high RF power induced nonlinear\nphenomenon. The observed mode hoping and the hysteretic behavior phenomena is very interesting for\nthe field of nonlinear magnonics. However, I cannot make a recommendation for the current format of\nthe manuscript. I suggest that the authors revise the manuscript with more focus on the mode hoping and\nhysteretic behavior of magnon modes since the strong and even ultrastrong coupling between optic\nand acoustic magnons in synthetic antiferromagnets have been already reported. Followings are my\nsuggestions.\nResponse:\nWe thank the reviewer for recognizing the importance of our work and for suggesting that we more clearly\nemphasize its distinction from earlier studies. Motivated by this valuable feedback, we have reformulated the\nrevised manuscript to focus on the mode hopping and hysteretic behavior of magnon modes, which represent\nthe central and novel aspects of our study. The main text and figures have been reorganized accordingly, and\nadditional discussion and data have been included to highlight these features. We hope that the reviewer finds\nour response satisfactory and the revised manuscript acceptable.\nComment 1. The hysteretic behavior of magnon mode should relate to the magnetization hysteresis loop. Is\nit possible to add a part on discussing the effect of RF power on the static MH loops?\nResponse:\nWe thank the reviewer for this insightful comment. As the reviewer pointed out, both the magnon mode\nand the magnetization switching exhibit hysteretic behaviors, suggesting that they may be related. To clarify\nthis point, we performed magneto-optical Kerr effect (MOKE) measurements of the magnetization hysteresis\nloops. The schematic of the measurement setup is shown in Fig. R1a. The MOKE sensitivity was adjusted to\ndetect the magnetization along the x-axis, while the external magnetic field was applied at a 45° angle,\nconsistent with the magnon experiment. The results show that magnetization switching occurs at\napproximately 5 mT. Considering that the magnon mode hopping appears above 10 mT, we conclude that\nmagnetization switching and magnon mode hopping are not directly correlated. In other words, when the\nmagnetic field is gradually increased, magnetization switching first occurs, and only after the magnetization\nis well defined with respect to the field direction does the mode hopping emerge.\nOne may expect that the magnetization switching around 5 mT could affect the mode-hopping\nmeasurements, as the field is swept from zero to positive values and thus abrupt changes might appear near 5\nmT. However, we note that no signal change is observed around 5 mT in the f–H map (Fig. 3 in the main text).\nThis is because the sample was first saturated at positive field and then returned to zero before increasing the\nfield again (we have specified this in the revised manuscript). As a result, the magnon signal does not capture\nany feature directly associated with the magnetization switching process.\nTo further address the reviewer’s suggestion regarding the effect of RF power on the static MH loops, we\nmeasured MH loops while varying the RF excitation power. Although the ideal measurement would be\nperformed in between the two antennas where magnons are most strongly excited, the laser spot size was too\nlarge to access this narrow region. Therefore, we performed the measurement just outside the right antenna.\nAccording to a recent report by D. Hayashi et al. [Appl. Phys. Lett. 126, 082401 (2025)], nonlinear magnons\nare also efficiently generated in the area just outside the antenna, ensuring that any influence of magnons on\nthe MH loops could still be detected.\nRF signals ranging from –10 dBm to +10 dBm were applied to the coplanar waveguide (CPW) antenna\nusing a signal generator, and the MOKE signal was recorded at frequencies corresponding to nonlinear mode\nhopping. As shown in Figs. R1b–d, the MH loops do not exhibit any noticeable change with increasing RF\npower. This indicates that the influence of RF excitation on the static magnetization is minimal. We attribute\nthis to the different nature of the two measurements: RF excitation drives a time-dependent oscillation of the\nmagnetization, whereas the MH loop reflects the time-averaged static magnetization. As a result, even under\nRF excitation, the static MH loop remains almost unchanged.\nWe have added a Supplementary Note 3 to provide detailed information on the negligible effect of RF\npower on the magnetization hysteresis loop in the mode-hopping field regime.\nFigure R1.1 | Longitudinal Magneto-optical Kerr Effect (L-MOKE) hysteresis loop measurement\nunder RF excitation. (a) Schematic illustration of the MOKE set-up for measuring magnetization\nhysteresis loops. (b)-(d) MOKE hysteresis loops measured under various RF excitation powers applied\nvia the signal generator. The RF frequency is set as (b) 5GHz, (c) 7GHz, (d) 9GHz.\nComment 2. Micromagnetic simulations have been widely used to understand the coupling mechanism since\nit can provide the direction precession information of different magnon modes. The observed “mode hoping”\nshould be further understood with the help of micromagnetic simulations.\nResponse:\nWe thank the reviewer for a thoughtful comment and agree that micromagnetic simulation will help\nunderstand the observed effects more precisely. To provide an intuitive interpretation of the experimental\nresults and explicitly demonstrate the magnetization dynamics, we performed micromagnetic simulations for\nSAF magnon mode hopping using the Mumax3 platform.\nThe simulation assumes a 1-dimensional array of cells consisting of two antiferromagnetically coupled\nferromagnetic layers. Each ferromagnetic layer is discretized into 32 unit cells along the propagation direction,\nyielding 64 cells in total for the two layers. Similar to the experimental situation, we create a spin wave at one\nedge of the sample using an external RF magnetic field 𝐵 and perform analysis at the other edge. The\nCPW\nin-plane static magnetic field 𝜇 𝐻 is applied at 𝜃 = 45o, consistent with the experimental situation.\n0 in k\nFurther details are discussed in Supplementary Note 6. Although the simulation is simplified and does\nnot fully capture all experimental complexities, it is sufficient to reproduce the essential features\nobserved in the experiment.\nFigure R1.2 summarizes the simulated magnon spectra as a function of the in-plane bias field 𝜇 𝐻 .\n0 in\nAs described above, the field was swept for each frequency while recording the transmitted signal\nintensity. Under weak RF drive (𝐵 = 10 mT), the spectra clearly reveal both acoustic and optic\nCPW\nmodes [Fig. R1.2a], demonstrating that the simulation successfully reproduces the characteristic\nfeatures of the SAF system. Note that no anticrossing gap is observed here, since the interlayer dipolar\ncontribution is suppressed by the ultrathin layer thickness (~ 0.28 nm in our simulation). When the drive\nfield is increased (𝐵 = 80 mT), the spectra exhibit discontinuities [Fig. R1.2b], consistent with the\nCPW\nexperimentally observed mode hopping. In particular, around 𝜇 𝐻 = 30 mT, the system abruptly\n0 in\nswitches from the optic to the acoustic mode, confirming that the simulation captures the nonlinear\nmode hoping.\nTo gain insight into the nonlinear mode hopping, we compared the spectra at low and high drive\npower. Figures R1.2a and R1.2b show that the resonance frequency exhibits a pronounced redshift at\nhigh power, in agreement with the experimental observations. In the main text, we interpret this redshift\nas a transition from propagating magnons to localized Kittel magnons. To verify this, we examined the\nk-resolved magnon distribution at representative regime: the acoustic-mode–dominant regime (blue\npoints in Figs. R1.2a,b) and the optic-mode–dominant regime (red points in Figs. R1.2a,b). At low\npower [Figs. R1.2c,e], the excited magnons extend into high- k regions, regardless of whether the mode\nis acoustic or optic. In contrast, under strong drive [Figs. R1.2d,f], the spectral weight collapses toward\nk = 0. This confirms that nonlinear excitation indeed promotes a transition from propagating magnons\nto Kittel magnons, which accounts for the mode hopping via three-magnon scattering (discussed in\nmore detail in Comment 8). We note that this transition has also been observed experimentally using k-\nresolved MOKE, as recently reported by D. Hayashi et al. [Appl. Phys. Lett. 126, 082401 (2025)].\nTherefore, the simulations not only reproduce the experimentally observed nonlinear mode hopping\nbut also provide independent evidence for the transition from propagating to Kittel magnons,\nconfirming the role of three-magnon scattering in the optic–acoustic mode conversion. These results\nsupport both the experimental findings and the proposed theoretical framework, establishing a coherent\npicture of nonlinear magnon dynamics in SAFs.\nFigure R1.2 | Mode hopping via micromagnetic simulation. (a) The f-H relation at the low CPW\npower. Conventional acoustic and optic magnon signals are detected. The red (blue) circular point\nindicates the analysis point for the (c)-(f). (b) The f-H relation at the high CPW power. The clear\ndiscontinuous mode hopping at the 30 mT is observed. (c) and (d) describe the magnon population with\nrespect to wavevector in the 𝜇 𝐻 = 40 mT, 𝑓 ~ 4 GHz acoustic mode condition for the low and\n0 in\nhigh CPW powers. (e) and (f) describe the magnon population with respect to wavevector in the\n𝜇 𝐻 = 20 mT, 𝑓 ~ 6 GHz optic mode condition for the low and high CPW powers.\n0 in\nComment 3. Regarding the external magnetic field, here, the authors studied the directions of external\nmagnetic field in the plane of sample. Generally, the magnon-magnon coupling can also be realized by tilting\nthe field toward out-of-plane direction. The observed mode hoping can also appear when tilting the magnetic\nfield out-of-plane? It is appreciated if they can give a discussion.\nResponse:\nWe thank the reviewer for raising this important point. As the reviewer pointed out, out-of-plane (OOP)\nmagnetic fields can in some cases induce magnon–magnon coupling by lifting the degeneracy between optic\nand acoustic modes [D. MacNeill et al., Phys. Rev. Lett. 123, 047204 (2019); A. H. Comstock et al., Nat.\nCommun. 14, 1834 (2023)]. To clarify the role of OOP fields in our system, we conducted additional\nmeasurements as follows.\nFirst, we investigated whether magnon–magnon coupling could be induced in our SAF by applying a tilted\nmagnetic field. We measured FMR spectra with tilt angles of 90°, 60°, 30°, and 10° relative to the OOP\ndirection (see Fig. R1.3). Unlike previous reports on van der Waals magnets [e.g., Fig. 4 of MacNeill et al.,\nPRL (2019); Fig. 3 of Comstock et al., Nat. Commun. (2023)], our measurements did not reveal any signature\nof an anticrossing gap. Instead, we only observed a systematic shift of the mode crossing point toward higher\napplied fields as the tilt angle increased, which we attribute to the change in the in-plane projection of the\nexternal field. These results indicate that in our CoFeB/Ru/CoFeB SAF, an OOP field does not induce\nsymmetry breaking sufficient to generate magnon–magnon coupling.\nNext, we examined whether an OOP field influences nonlinear mode hopping. We applied an OOP bias\nfield of 67 mT—comparable to the in-plane field used for mode-hopping measurements—and repeated the\nexperiments. As shown in Fig. R1.4, the spectra obtained with [Fig. R1.4b] and without [Fig. R1.4a] the OOP\nfield are nearly identical, confirming that the OOP field has negligible impact on mode hopping in our system.\nWe interpret these observations as follows. In our SAF, magnon–magnon coupling is mediated primarily\nby dynamic dipolar interactions. The resulting anticrossing emerges when the magnetic field is applied in-\nplane and is not collinear with the magnon propagation direction, providing the necessary symmetry breaking.\nIn contrast, prior studies of van der Waals antiferromagnets and low-symmetry layered magnets have shown\nthat OOP fields can break structural or parity symmetries, leading to gap opening. In those systems, intrinsic\nasymmetries (e.g., inequivalent layers, inversion-symmetry breaking, or Dzyaloshinskii–Moriya interaction)\nallow OOP fields to couple the optic and acoustic modes.\nOur CoFeB/Ru/CoFeB trilayer differs in key respects: the two ferromagnetic layers are nominally identical,\nthe Ru spacer only mediates interlayer exchange, and the system exhibits strong in-plane magnetic anisotropy.\nAs a result, the symmetry-breaking pathway for OOP-field–induced coupling is negligible. We therefore\nconclude that in our SAF, the OOP magnetic field has negligible influence on both magnon–magnon coupling\nand mode hopping.\nWe have added a Supplementary Note 2 to provide the discussion on the negligible effect of OOP field to\nmagnon-magnon coupling in our SAF.\nFigure. R1.3 | Ferromagnetic resonance (FMR) spectra under tield magnetic fields. (a) Schematic of the\nexternal magnetic field geometry, where the polar angle 𝜑 and azimuthal angle 𝜃 define the tilt relative\n0 k\nto the out-of-plane (OOP) axis. (b–e) Color maps of the measured FMR spectra as functions of drive\nfrequency and external field for tilt angles of 𝜑 = 90o,60o,30o and 10o, respectively. While the mode\n0\ncrossing point systematically shifts toward higher fields with decreasing tilt angle, no anticrossing gap is\nobserved. This indicates that in our CoFeB/Ru/CoFeB SAF, tilting the field out of plane does not induce\nmagnon–magnon coupling.\nFigure. R1.4 | Effect of out-of-plane (OOP) magnetic field on nonlinear magnon spectra. (a)\nExperimental geometry (left) and measured spectra as a function of in-plane field under different RF powers\n(P = −10, 0, 2.5 and 10 dBm) with no OOP bias field applied (μ 𝐻 = 0 mT). (b) Same measurements\n0 oop\nperformed under an additional OOP bias field of μ 𝐻 = 67 mT. The overall mode structure, including\n0 oop\nthe onset of nonlinear mode hopping at high drive powers, remains essentially unchanged compared to (a),\nindicating that the OOP field has negligible influence on both magnon–magnon coupling and mode hopping\nin the CoFeB/Ru/CoFeB SAF.\nComment 4. The results and schematic presented in Fig. 1 are nothing new, I think these can be shown in\nsupplemental, especially Fig. 1e-f.\nResponse: We thank the reviewer for this constructive comment. In the revised manuscript, we have moved\nFigs. 1e–g to Supplementary Note 1, while retaining the schematics in Fig. 1 to convey essential experimental\ninformation such as the field angle, propagation direction, and CPW geometry. We have also revised the main\ntext accordingly.\nComment 5. In Fig. 2, the angle of magnetic field should be indicated in the caption. In Fig. 2g, the label\nshould be inserted.\nResponse: We thank the reviewer for pointing this out. In the revised manuscript, we have added the magnetic\nfield angle (45°) to the caption of Fig. 2 and included labels indicating the field strength in Fig. 2g to avoid\nany ambiguity.\nComment 6. Fig. 2h is used to present the observed closing of anticrossing gap. Is there a better field than 27\nmT. I suggest they chose the field at which the intensity of the optic and acoustic is similar.\nResponse: We thank the reviewer for this helpful suggestion. As the reviewer pointed out, showing a line\nprofile where the optic and acoustic modes have comparable intensity would indeed provide a clearer\nillustration of the gap closing. We also considered this point and examined line profiles at various fields (see\nFig. R1.5). Among the fields tested, 27 mT provided the clearest signature of both peaks, which is why we\npresented this line profile in the main text.\nWhile it would be ideal to find a field where the optic and acoustic intensities are nearly equal, such a\ncondition does not occur in our system. Both modes are present across the entire field range, but the intensity\nof the upper branch is always larger. This asymmetry arises from the different origins of the two signals: in\nour SAF, the optic mode intensity corresponds to variations of the longitudinal component of magnetization\nalong the external field direction, whereas the acoustic mode corresponds to variations of the transverse\ncomponent. At low fields (nearly spin-flop state), the spins align almost perpendicular to the field, so the\nlongitudinal variation is large while the transverse variation is small—resulting in strong optic and weak\nacoustic signals. At high fields (nearly spin-flip state), the spins align nearly parallel to the field, so the\ntransverse variation becomes dominant while the longitudinal variation is suppressed—leading to strong\nacoustic and weaker optic signals. Consequently, the upper branch remains stronger throughout the entire field\nrange. We note that this behavior is consistent with previous reports, for example Y. Shiota et al., Phys. Rev.\nLett. 125, 017203 (2020).\nFigure R1.5 | Line profiles of magnon spectra at different bias fields. Left: f–H map at P = −10 dBm,\nwhere vertical dashed lines indicate the fields chosen for line-profile analysis (μ 𝐻 =27, 30, 35 mT).\n0 in\nRight: magnon spectra measured at these fields as a function of drive frequency for various RF powers. At 27 mT,\nboth optic (red arrows) and acoustic (blue arrows) peaks can be resolved, whereas at 30 mT and 35 mT one\nof the modes dominates in intensity. These results highlight that 27 mT provides the clearest observation of\nboth modes, although the optic branch remains stronger across all fields.\nComment 7. In Fig. 3e-h, four typical RF power is provided. I am wondering that the RF power 2.5 dBm is\nthe threshold for a specific direction of external magnetic field or for all in-plane angles? The hysteretic\nbehavior appears abruptly or slowly?\nResponse: We thank the reviewer for this important question. To address it, we performed additional\nexperiments by systematically varying the in-plane angle of the external magnetic field. The results are\nsummarized in Fig. R1.6. We find that the threshold power for mode hopping strongly depends on the in-\nplane field angle. Specifically, at 𝜃 = 30o, no clear threshold was observed; at 𝜃 = 45o, the threshold\nk k\npower is about 2.5 dBm; and at 𝜃 = 60o, the threshold is reduced to nearly 0 dBm. Similarly, the critical\nk\nfield for mode hopping at a fixed power also shifts with angle: for example, at P =10 dBm, mode hopping\noccurs around 20 mT at 𝜃 = 45o, but at a lower field of about 15 mT at 𝜃 = 60o. These results clearly\nk k\nshow that the threshold conditions depend sensitively on the in-plane field angle.\nWe interpret this angle dependence in terms of the magnon population. As shown in Fig. 1c of the main\ntext, the optic mode dominates at small field angles 𝜃 , while at larger angles approaching 90° the acoustic\nk\nmode becomes dominant (Fig. 1d of the main text). Since the nonlinear mode hopping corresponds to a\ntransition from the optic to the acoustic mode, this transition becomes progressively easier as the acoustic\npopulation increases toward larger in-plane angles. This explains why the threshold decreases with increasing\nfield angle.\nRegarding the reviewer’s final question, we note that the nonlinear transition occurs abruptly rather than\ngradually. As shown in Fig. 2g of the main text, the onset of nonlinear redshift takes place at a well-defined\nthreshold power, consistent with an abrupt transition.\nWe have added a Supplementary Note 4 to provide the discussion on the in-plane angle dependence of\nmode hopping threshold.\nFigure. R1.6 | Angle dependence of threshold power and field for nonlinear mode hopping. (a–c)\nMeasured magnon spectra as functions of in-plane magnetic field μ 𝐻 and frequency at different in-\n0 in\nplane field angles: (a) 𝜃 = 30o, (b) 𝜃 = 45o, and (c) 𝜃 = 60o. Data are shown for representative\nk k k\nRF drive powers of –10, 0, 2.5, and 10 dBm.\nComment 8. Since the results in Fig. 4 is theoretical calculations, is it possible to give a discussion on how\nthe threshold RF power varying with extremal field? It is still not quite clear why the mode hopping decreases\nwith G1? Is it related to the variation of the precession angle of magnetizations (m, m1, m2)? Micromagnetic\nsimulations should be helpful.\nResponse: We thank the reviewer for this thoughtful question. Strictly speaking, the observed behavior is less\nrelated to an increase of the precession angle itself and more closely associated with a transition from\npropagating magnons to non-propagating Kittel magnons under strong RF drive.\nAs the reviewer noted, increasing the RF power reduces the critical field for mode hopping, which in turn\nleads to narrowing of the hysteresis. In our theoretical framework, this effect is captured by the parameter G :\n1\nlarger G reproduces the experimentally observed narrowing of hysteresis, implying that G effectively\n1 1\nreflects the influence of RF power. In our model, G represents the linear magnonic gain associated with the\n1\ntransition from propagating magnons to Kittel magnons. Increasing G therefore corresponds to an increased\n1\npopulation of Kittel magnons.\nThis picture naturally explains the experimental results: as RF power increases, propagating magnons are\nprogressively converted into Kittel magnons, and the growing Kittel magnon population drives the hysteresis\ngap narrowing. The physical reason can be understood in terms of three-magnon scattering processes as\nfollows. Mode hopping arises when the resonance condition ω = 2ω is satisfied, enabling the\nop ac\nconversion between optic and acoustic magnons. If only Kittel magnons were present, mode hopping would\noccur only at this sharp resonance condition, and hysteresis would be minimal. However, in our system both\nKittel and propagating magnons coexist. For propagating magnons, broader scattering channels are allowed,\ne.g., ω (𝑘+𝑞) → ω (𝑘)+ω (𝑞) where two acoustic magnons with different frequencies and wave\nop ac ac\nvectors participate. These relaxed conditions extend the field range over which mode hopping can occur,\nthereby broadening the hysteresis. More specifically, as the magnetic field increases, the three-magnon\nprocess involving Kittel magnons is triggered first. However, if the population of Kittel magnons is\ninsufficient, the field must be further increased until the condition for the propagating three-magnon process\nis satisfied, at which point mode hopping can be observed. Therefore, as the relative population of Kittel\nmagnons increases with RF power, the system favors the sharper Kittel scattering channel, and the hysteresis\ngap narrows.\nThe key question is therefore whether RF power indeed converts propagating magnons into Kittel magnons.\nIn the original manuscript, we presented the frequency redshift and gap closing as indirect evidence of this\ntransition. In the revised version, we address this more directly by combining simulations with new\nexperiments. As described in our response to Comment 2, the micromagnetic simulations clearly show that\nincreasing RF power drives the transition from propagating to Kittel magnons (Fig. R1.2). In addition, during\nthe revision process, our co-author group performed k-resolved MOKE measurements and directly observed\nthat the population near k = 0 increases with RF power, consistent with enhanced Kittel magnons (Fig. R1.7,\npublished during the revision).\nTaken together, these additional simulation and experimental results strongly support our interpretation that\nthe RF-power–induced conversion from propagating magnons to Kittel magnons underlies both the onset of\nmode hopping and the narrowing of hysteresis in our SAF system.\nWe have added a detailed discussion of the underlying mechanism in our theoretical model in the revised\nmanuscript.\nFigure. R1.7 | k-resolved spectra measured by the heterodyne MOKE, showing the evolution from\npropagating magnons to Kittel magnons with increasing RF power. As the RF power increases, spectral\nweight progressively shifts toward k = 0, indicating a growing population of Kittel magnons. [D. Hayashi et\nal., Appl. Phys. Lett. 126, 082401 (2025)].\nReviewer #2\nIn this manuscript, Mujin You and co-authors report strong coupling and nonlinear interactions\ninvolving two GHz-frequency magnons in a synthetic antiferromagnetic structure driven by RF\nexcitations. They observe a range of nonlinear effects above a certain RF power threshold, including\nfrequency shifts, anticrossing gap closure, mode hopping with large frequency jumps, and hysteresis.\nThe authors further develop a phenomenological model that considers nonlinear interactions between\nKittel and propagating magnon modes to explain their experimental findings. The manuscript is well-\nwritten, and the study is scientifically sound. Therefore, I recommend that the paper be considered for\npublication in Nature Communications, provided that the authors address the following questions:\nComment 1. Since the anticrossing originates from dipolar interactions, the size of the anticrossing gap is\nexpected to depend on the excitation wavevector and, consequently, the width of the coplanar waveguide.\nCould the authors clarify the influence of the coplanar waveguide structure on the coupling and the nonlinear\ninteractions observed?\nResponse: We thank the reviewer for this insightful comment. The reviewer is correct that magnon–magnon\ncoupling in SAFs arises from dynamic dipolar interactions and therefore exhibits a clear dependence on the\nexcitation wavevector. This k-dependence has already been systematically investigated in the previous work\nby our co-author [Shiota et al. Phys. Rev. Lett. 125, 017203 (2020)], whose key results are reproduced in Fig.\nR2.1. As shown here, the anticrossing gap increases with increasing wavevector.\nIn the present study, the coplanar waveguide (CPW) has a 0.64–1.28–0.64 μm geometry, corresponding to\na characteristic excitation wavevector of approximately 1.6 μm−1. According to Fig. R2.1, this value lies in\nthe regime where the magnon–magnon coupling is strongest. This was precisely the motivation behind our\nchoice of antenna geometry: to maximize the coupling strength and thereby allow us to focus on the nonlinear\nphenomena—particularly mode hopping—arising from strong coupling.\nTherefore, while the 𝑘-dependence of magnon–magnon coupling is indeed important and has been\nestablished previously, the central novelty of our work lies in the observation of nonlinear mode hopping and\nhysteretic behavior under strong coupling conditions—features that were also highlighted as particularly\ninteresting by Reviewer #1.\nFigure. R2.1 | k-dependence of magnon-magnon coupling in a synthetic antiferromagnet (SAF).\nExperimental results reproduced from Shiota et al. [Phys. Rev. Lett. 125, 017203 (2020)]. The coupling\nstrength g increases systematically with wavevector k, demonstrating that stronger magnon–magnon coupling\nis achieved at larger wavevectors.\nComment 2. What is the physical origin of the narrowing hysteresis gap with increasing RF power? While\nthe numerical simulations reproduce this effect as the magnonic gain G increases, a detailed discussion of\n1\nthe underlying mechanism is also needed.\nResponse: We thank the reviewer for this thoughtful question. As the reviewer pointed out, increasing the RF\npower leads to narrowing of the hysteresis. In our theoretical framework, this effect is captured by the\nparameter G : larger G reproduces the experimentally observed narrowing, indicating that G reflects the\n1 1 1\ninfluence of RF power. In our model, G represents the linear magnonic gain associated with the conversion\n1\nof propagating magnons into Kittel magnons; increasing G therefore corresponds to an enhanced Kittel\n1\nmagnon population.\nThis interpretation naturally accounts for the observed narrowing. Mode hopping originates from three-\nmagnon scattering processes. For Kittel magnons, the condition is sharp, ω = 2ω , so mode hopping\nop ac\noccurs only at this resonance and hysteresis is minimal. For propagating magnons, however, additional\nchannels such as ω (𝑘+𝑞) → ω (𝑘)+ω (𝑞) are allowed, which relax the resonance condition and\nop ac ac\nbroaden the field range of mode hopping. More specifically, as the magnetic field increases, the Kittel three-\nmagnon process is triggered first. If the Kittel population is insufficient, the field must be further increased\nuntil the propagating-magnon process is satisfied, at which point mode hopping appears. Consequently, as RF\npower increases and propagating magnons are converted into Kittel magnons, the system favors the sharp\nKittel channel and the hysteresis gap narrows.\nThe crucial point is whether RF power truly converts propagating magnons into Kittel magnons. In the\noriginal manuscript, we presented the frequency redshift and gap closing as indirect evidence of this transition.\nIn the revised version, we address this more directly by combining simulations with new experiments.As\ndescribed in the revised Supplementary Note 6, micromagnetic simulations demonstrate the RF-power–\ninduced transition from propagating to Kittel magnons (Fig. R2.2). In addition, recent k-resolved MOKE\nmeasurements performed by our co-authors directly confirm that the population near k = 0 increases with RF\npower (Fig. R2.3, published during the revision).\nTaken together, these results provide consistent evidence that the narrowing of the hysteresis originates\nfrom the RF-power–driven conversion of propagating magnons into Kittel magnons, which governs the onset\nand character of nonlinear mode hopping in our SAF system.\nFigure. R2.2 | Micromagnetic simulation of the transition from propagating magnons to Kittel\nmagnons under strong RF excitation. (a,b) Simulated magnon spectra as functions of frequency and\nin-plane magnetic field for (a) low CPW power (𝐵 = 10 mT) and (b) high CPW power (𝐵 =\nCPW CPW\n80 mT). The colored circles mark representative points used for k-resolved analysis. (c–f) Fourier-\ntransformed wavevector spectra at the selected conditions. At low power (c,e), magnons with finite k\nare present, indicating propagating modes. At high power (d,f), the spectral weight collapses toward k\n= 0, demonstrating the growth of the Kittel magnon population. These results confirm that increasing\nRF power drives the conversion from propagating to Kittel magnons, consistent with the experimental\nobservations and the interpretation of nonlinear mode hopping in SAFs.\nFigure. R2.3 | k-resolved spectra measured by the heterodyne MOKE, showing the evolution from\npropagating magnons to Kittel magnons with increasing RF power. As the RF power increases, spectral\nweight progressively shifts toward k = 0, indicating a growing population of Kittel magnons. [D. Hayashi et\nal., Appl. Phys. Lett. 126, 082401 (2025)].\nComment 3. The Hamiltonian in the theoretical model primarily considers three-magnon interactions to\naccount for the observed nonlinearities. Could four-magnon processes, such as magnon self-Kerr and cross-\nKerr interactions, also play a role in the observed phenomena? If so, could the authors comment on their\npotential contributions?\nResponse: We thank the reviewer for raising this point regarding possible contributions from four-magnon\ninteractions. We also considered such processes in our analysis, including self-Kerr (𝐾 𝑎̂†𝑎̂†𝑎̂𝑎̂) and cross-\nself\nKerr (𝐾 𝑎̂†𝑏̂†𝑏̂𝑎̂ ) interactions. The expected influence of these interactions was calculated and is\ncross\npresented in Supplementary Note 7.\nBriefly summarizing the results, the self-Kerr interaction in the acoustic–optic hybrid system leads to fold-\nover effects and frequency shifts in the magnon spectra [Figs. R2.4(b,c) for field dependence and Figs.\nR2.4(e,f) for frequency dependence]. The cross-Kerr interaction produces multiple discontinuities in the\nspectra, as illustrated in Figs. R2.4(g,h). These behaviors are consistent with previous theoretical predictions\nfor hybrid magnonic systems including four-magnon interactions [Y. P. Wang et al., Phys. Rev. Lett. 120,\n057202 (2018)].\nIn our experiments, we indeed observed frequency shifts with increasing RF power, but we did not detect\nthe fold-over effect in the frequency–intensity plots (Fig. R2.5), which is considered the key signature of four-\nmagnon Kerr interactions. We therefore conclude that, under our experimental conditions, the contribution of\nfour-magnon interactions is negligible compared to the dominant three-magnon processes. Nevertheless, we\nexpect that Kerr-type nonlinearities would become significant at sufficiently high RF powers. Exploring these\neffects in SAFs would be an interesting avenue for future work, though it is beyond the scope of the present\nstudy.\nFigure. R2.4 | Theoretical calculation of four-magnon Kerr interactions in SAFs. (a) Schematic f-H\nrelation for the acoustic and optic magnons with the fixed frequency (black dashed line). The (b) acoustic and\n(c) optic magnon bistability revealed by the hysteresis at the fixed frequency 7 GHz. (d) Schematic f-H\nrelation with the fixed field (black dashed line). The (e) acoustic and (f) optic magnon bistability revealed by\nthe hysteresis at the fixed field point 35 mT. (g) The combined effect of self and cross Kerr effect on the\nhysteresis for the (g) acoustic and (h) optic mode magnons.\nFigure. R2.5 | Experimental frequency spectra under varying RF power. (a–c) Measured transmission\nspectra 𝑆 as a function of frequency at in-plane bias fields of (a) μ 𝐻 =10 mT, (b) μ 𝐻 =35 mT, and\n21 0 in 0 in\n(c) μ 𝐻 =60 mT. Curves correspond to different RF drive powers ranging from –15 dBm to +10 dBm, as\n0 in\nindicated in the legend. With increasing RF power, a clear frequency redshift of the resonance is observed.\nHowever, the characteristic fold-over distortion expected from strong four-magnon Kerr interactions (see Fig.\nR2.4) is absent, indicating that Kerr-type nonlinearities make only a minor contribution in our system\ncompared to three-magnon processes.\nMinor point: The intensity scales should be defined in all the 2D color plots throughout Figures 1–3.\nWe thank the reviewer for pointing this out. In the revised manuscript, we have defined the intensity scales in\nall 2D color plots throughout Figures 1–3 for clarity.\n----------------------------------------------------------------------------------------\nReviewer #3\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the\nNature Communications initiative to facilitate training in peer review and to provide appropriate recognition\nfor Early Career Researchers who co-review manuscripts.\nResponse:\nWe thank the reviewer for co-reviewing our manuscript.\n----------------------------------------------------------------------------------------\nReviewer #4\nThe manuscript reports a study of magnon spectra in a synthetic antiferromagnet under strong\nmicrowave excitation. The authors show that with increasing excitation power, the spectrum undergoes\nstrong modifications, which are described in terms of nonlinear mode hoping and explained by magnon-\nmagnon interaction. The observed modifications are hysteretic and depend on the direction, in which\nthe magnetic field is changed. The authors also present a model, that reproduces the observed hysteresis.\nUnfortunately, I cannot recommend publication of this work for the following reasons:\nComment 1. First of all, the claims of novelty in the abstract and introduction are an exaggeration. In fact,\nthere are many works demonstrating nonlinear coupling between coherent magnons. One can mention Appl.\nPhys. Lett. 121, 112404 (2022) as one of the recent examples. If the authors conduct a thorough literature\nsearch, they will find many more examples.\nResponse: We appreciate the reviewer’s comment and agree that nonlinear coupling between coherent\nmagnons has been reported in a number of previous works, including Appl. Phys. Lett. 121, 112404 (2022)\nand others. Our intention was not to overstate novelty in this regard. Rather, the novelty of the present study\nlies in the observation of nonlinear mode hopping and hysteretic behavior of magnon modes in SAFs, which,\nto our knowledge, has not been previously reported.\nSpecifically, we demonstrate that under high RF power, the system exhibits an abrupt transition from the\noptic to the acoustic mode, with a frequency drop as large as 4–5 GHz. We further show that this transition is\nhysteretic, and we establish its microscopic origin through both theoretical modeling and micromagnetic\nsimulations. These features go beyond earlier demonstrations of nonlinear coupling by directly linking strong\nnonlinearity to mode conversion and hysteresis.\nImportantly, such abrupt and hysteretic mode conversion is not only of fundamental interest in nonlinear\nmagnonics but also carries potential for applications such as magnon-based mode converters and frequency\nswitches. Reviewer #1 also highlighted this point, noting that “the observed mode hopping and the hysteretic\nbehavior phenomena is very interesting for the field of nonlinear magnonics,” and the Editor likewise\nrecognized the significance of these findings.\nWe therefore believe that the focus of our work—on nonlinear mode hopping and hysteresis in SAFs—\nprovides a distinct and meaningful contribution that justifies consideration in Nature Communications. To\naddress this concern, we have revised the Abstract and Introduction to more clearly articulate that the novelty\nof our work lies in the observation and explanation of nonlinear mode hopping and hysteresis, rather than in\nmagnon–magnon coupling itself.\nComment 2. The interpretation of the results in terms of nonlinear coupling of coherent magnons is highly\ncontroversial.\nResponse: We agree with the reviewer that the interpretation of nonlinear magnon–magnon coupling remains\ncontroversial. This is largely because the field is still at an early stage, and direct experimental evidence has\nbeen limited. In the case of synthetic antiferromagnets (SAFs), the situation is further complicated by the\ncoexistence of optic and acoustic magnons, as well as the distinction between k = 0 Kittel magnons and\npropagating magnons with k ≠ 0. Such complexity makes the manifestation of nonlinear effects diverse and\nnontrivial.\nOur study provides experimental evidence of nonlinear behaviors in SAFs, in particular magnon mode\nhopping and hysteretic responses. Although the theoretical framework is not yet fully established, we believe\nthat presenting these experimental results is crucial for advancing understanding in this field. In particular,\nthey may help guide the development of a comprehensive theory and clarify the respective roles of Kittel and\npropagating magnons in nonlinear coupling.\nBelow, we provide detailed responses to the reviewer’s specific comments. As the reviewer rightly\npoints out, some assumptions are unavoidable given the current state of the field, but we believe that our results will\ncontribute to the broader effort to build a comprehensive understanding of nonlinear magnon dynamics.\na) All the considered “interacting” magnons (acoustic and optic Kittel magnons and propagating magnons)\nhave different frequencies. In the experiment, they are excited by sweeping the frequency of the microwave\nsource, i.e., at different moments in time. These magnons never coexist in the sample unless the frequency is\nswept on a temporal scale shorter than the magnon relaxation time, which I believe is impossible. Therefore,\nthe idea of the interaction of coherent magnons is not applicable here. For such an interaction to exist,\ninteracting magnons must be excited simultaneously by two sources (as in Appl. Phys. Lett. 121, 112404\n(2022) mentioned above).\nResponse: We thank the reviewer for raising this important point. It is correct that in our measurements we\nexcite and detect magnons at a single frequency. However, this does not necessarily mean that only single-\nfrequency magnons are involved; even in single-frequency measurements, the spectra can include signatures\nof interactions among magnons at different frequencies. This has become increasingly clear from recent\nstudies, which demonstrate that nonlinear interactions can manifest under such measurement conditions.\nFor example, in a very recent work by A. Sud et al. [Phys. Rev. Lett. 134, 246704 (2025)], which was\npublished during the revision of our manuscript, three-magnon scattering between acoustic and optic modes\nin a SAF was directly observed using FMR. As shown in Figs. 2(c) and 2(d) of that paper, even though only\na single excitation frequency was applied, mode splitting occurred due to nonlinear three-magnon processes.\nThe authors demonstrated that strongly excited acoustic magnons can combine to generate optic magnons,\nthereby giving rise to effective acoustic–optic coupling.\nSimilarly, D. Hayashi et al. [Appl. Phys. Lett. 126, 082401 (2025)], which was also published during the\nrevision of our manuscript, used heterodyne MOKE to study nonlinear effects in the same SAF system. Their\nFig. 3 clearly shows that excitation at frequency f produces a detectable signal at 2 f under high drive power,\nrf rf\nconsistent with nonlinear up-conversion of acoustic magnons into optic magnons.\nThese reports indicate that, contrary to the reviewer’s concern, nonlinear interactions between magnons of\ndifferent frequencies can indeed appear in single-frequency measurements through three-magnon processes.\nOur results should therefore be viewed in this broader context: while we excite at a single frequency, the\nmeasured spectra can nevertheless reflect interactions involving magnons at multiple frequencies.\nTaken together, these recent advances highlight that the understanding of nonlinear magnon interactions is\nstill developing, and we believe that our work contributes valuable experimental evidence toward building a\ncomprehensive picture of such processes in SAFs. In particular, our results extend beyond the conventional\nfocus on coherent k = 0 Kittel magnons by also capturing the crucial role of propagating magnons, thereby\nproviding a more comprehensive view of nonlinear magnon dynamics.\nb) Let us now consider the hysteresis (e.g., Fig. 3f). It appears that the state of the system found at 20 mT\ndepends on whether previous measurements were made at larger or smaller field. However, all magnons\nexcited in these previous measurements relax completely before the measurement at 20 mT is started.\nTherefore, the observed hysteretic behavior cannot be due to the magnon accumulation. Obviously, it is\ncaused by a process characterized by a much longer relaxation time. For example, it can be caused by heating\nthe sample with intense microwaves.\nResponse: We thank the reviewer for this critical comment. As the reviewer points out, the observation of\nhysteresis in our SAF system implies a memory effect, which is a hallmark of nonlinear dynamics. In such\nsystems, multiple stable states can coexist, and the selected state depends on the sweep direction of the control\nparameter.\nIn our case, strong nonlinear magnon–magnon coupling can give rise to multistable states associated with\neither the optic or the acoustic mode. The presence of hysteresis therefore indicates that some form of memory\neffect is at play. In our numerical calculations, different initial conditions were imposed depending on the\nsweep direction, and hysteresis naturally emerged from this procedure. However, in our experiment, the origin\nof the memory effect is not clear, because, as the reviewer mentioned, the magnons are completely relaxed\nbefore the next measurement. Although the exact microscopic mechanism is not yet fully clear, one plausible\nscenario is that the magnetization configuration at a given field differs slightly depending on the sweep\ndirection, and this difference biases the system toward a particular magnon mode.\nTo examine this possibility, we performed additional experiments by varying the in-plane field angle, since\ndifferent angles correspond to different magnetization configurations. The results (Fig. R4.1) show that the\nthreshold conditions for mode hopping are strongly angle dependent: at 𝜃 = 30o, no clear threshold was\nk\nobserved; at 𝜃 = 45o, the threshold power is about 2.5 dBm; and at 𝜃 = 60o, the threshold is reduced to\nk k\nnearly 0 dBm. Likewise, at fixed power (P = 10 dBm), the threshold field shifts from ~20 mT at 𝜃 = 45o,to\nk\n~15 mT at 𝜃 = 60o.\nk\nThis angle dependence suggests that the dominant magnon mode depends sensitively on the magnetization\nstate: at smaller 𝜃 angles the optic mode dominates (see Fig. 1c of the main text), whereas at larger angles\nk\napproaching 90° the acoustic mode is favored (Fig. 1d of the main text). This implies that even subtle\ndifferences in the magnetization configuration, depending on the sweep direction, can determine which of the\ntwo modes becomes stabilized and thus give rise to hysteresis in mode hopping.\nWe note that similar hysteretic memory effects have been reported in other nonlinear and hybrid magnonic\nsystems (e.g., Yao et al., PRL 130, 146702 (2023); Wang et al., PRL 120, 057202 (2018)), highlighting this\nas an actively debated topic in the field. Our experimental evidence adds a valuable data point to this ongoing\ndiscussion and may help elucidate the physics of nonlinear magnon–magnon coupling. Thus, while the exact\nmechanism behind the memory effect in our SAF remains an open and potentially controversial question, our\nresults provide meaningful experimental insight into nonlinear magnon dynamics.\nFinally, we carefully evaluated the possibility of microwave-induced heating. If heating were significant,\none would expect observable changes in magnetization or anisotropy, detectable via MOKE. However, our\nMOKE measurements of MH loops taken directly next to the CPW electrode under RF drive (Fig. R4.2) show\nnegligible modification. Moreover, heating would be expected to produce a gradual redshift of resonance\nfrequencies, whereas our data exhibit abrupt redshifts at well-defined thresholds (Fig. 2g of the main text).\nThese observations indicate that Joule heating does not play a significant role in our experimental conditions.\nFigure. R4.1 | Angle dependence of threshold power and field for nonlinear mode hopping. (a–c)\nMeasured magnon spectra as functions of in-plane magnetic field μ 𝐻 and frequency at different in-\n0 in\nplane field angles: (a) 𝜃 = 30o, (b) 𝜃 = 45o, and (c) 𝜃 = 60o. Data are shown for representative\nk k k\nRF drive powers of –10, 0, 2.5, and 10 dBm.\nFigure R4.2 | Longitudinal Magneto-optical Kerr Effect (L-MOKE) hysteresis loop measurement\nunder RF excitation. (a) Schematic illustration of the MOKE set-up for measuring magnetization\nhysteresis loops. (b)-(d) MOKE hysteresis loops measured under various RF excitation powers applied\nvia the signal generator. The RF frequency is set as (b) 5GHz, (c) 7GHz, (d) 9GHz."}, {"round": 2, "reviewer_comments": {"reviewer_1": "After revision, this manuscript has been improved by addressing the reviewers' comment. Regarding to my comments, the authors have addressed properly. In summary, I would like to recommend its publication in NC.", "reviewer_2": "We have carefully reviewed the authors’ rebuttal and thoroughly examined the revised manuscript. The authors have done an excellent job addressing all reviewers’ concerns in a clear and comprehensive manner. In particular, their detailed response to the points raised regarding four-magnon processes is compelling. It satisfactorily resolves our original question. The newly incorporated material also improves the accuracy and overall strength of the work. Lastly, the manuscript is now written in a balanced and polished tone, with language suitable for publication in Nature Communications. We therefore support acceptance of this revised submission.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "Unfortunately, I cannot conclude that the authors responded satisfactorily to my criticism. The main points in my initial report remain unaddressed:\na) I have no doubt that magnons at different frequencies can interact via three- and four-magnon processes. My point is that the description of the mode hoping and hysteresis in terms of the interaction of coherent magnons is misleading. One gets the impression that two magnons are simultaneously excited by a coherent source and interact with each other, which is not the case. The authors write (lines 93-94): “both modes are simultaneously excited, resulting in their coupling”. This is not correct. These two modes have different frequencies and are never excited simultaneously by a swept microwave source. It takes at least several tens of milliseconds for the source to change its frequency from 6 to 9 GHz. Therefore, the first mode and all its scattering products completely relax before the second mode is excited. How then one can speak about an interaction of two coherent modes?\nThe authors use many times the expression such as “coherent magnon mode”, “coherent magnon coupling”, “interaction between coherent magnons”, etc. Given the above, it is not clear what “coherent” means in this context.\nb) The authors agree with me that the hysteretic effects observed in their experiment cannot be caused by the magnon accumulation. However, they base their theoretical interpretation on magnon accumulation effects. In other words, the used model, even if it shows behavior resembling that observed in the experiment, does not reflect real physics. This makes the paper misleading."}, "author_response": "We sincerely thank all the reviewers for their thoughtful feedback and for their positive evaluations\nfrom the majority. Below we provide our detailed responses to all remaining questions and comments.\nResponse to Reviewer #1\nAfter revision, this manuscript has been improved by addressing the reviewers' comment. Regarding to\nmy comments, the authors have addressed properly. In summary, I would like to recommend its\npublication in NC.\nResponse: We sincerly appreciate the reviewer’s positive evaluation and encouraging recommendation for\npublication. We are grateful that the reviewer recognized the improvements made during the revision and the\nadequacy of our responses.\n----------------------------------------------------------------------------------------\nReviewer #2\nWe have carefully reviewed the authors’ rebuttal and thoroughly examined the revised manuscript.\nThe authors have done an excellent job addressing all reviewers’ concerns in a clear and comprehensive\nmanner. In particular, their detailed response to the points raised regarding four-magnon processes is\ncompelling. It satisfactorily resolves our original question. The newly incorporated material also\nimproves the accuracy and overall strength of the work. Lastly, the manuscript is now written in a\nbalanced and polished tone, with language suitable for publication in Nature Communications. We\ntherefore support acceptance of this revised submission.\nResponse: We appreciate the reviewer’s careful evaluation of our rebuttal and revised manuscript. We are\ngrateful for the recognition that our responses and revisions—including the clarification of the four-magnon\nprocesses—adequately addressed the concerns. We also appreciate the reviewer’s supportive\nrecommendation for acceptance.\n----------------------------------------------------------------------------------------\nReviewer #3\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the\nNature Communications initiative to facilitate training in peer review and to provide appropriate recognition\nfor Early Career Researchers who co-review manuscripts.\nResponse:\nWe are grateful to Reviewer #3 for co-reviewing our manuscript and for supporting its publication.\n----------------------------------------------------------------------------------------\nReviewer #4\nUnfortunately, I cannot conclude that the authors responded satisfactorily to my criticism. The main\npoints in my initial report remain unaddressed:.\nComment (a). I have no doubt that magnons at different frequencies can interact via three- and four-magnon\nprocesses. My point is that the description of the mode hoping and hysteresis in terms of the interaction of\ncoherent magnons is misleading. One gets the impression that two magnons are simultaneously excited by a\ncoherent source and interact with each other, which is not the case. The authors write (lines 93-94): “both\nmodes are simultaneously excited, resulting in their coupling”. This is not correct. These two modes have\ndifferent frequencies and are never excited simultaneously by a swept microwave source. It takes at least\nseveral tens of milliseconds for the source to change its frequency from 6 to 9 GHz. Therefore, the first mode\nand all its scattering products completely relax before the second mode is excited. How then one can speak\nabout an interaction of two coherent modes?\nThe authors use many times the expression such as “coherent magnon mode”, “coherent magnon coupling”,\n“interaction between coherent magnons”, etc. Given the above, it is not clear what “coherent” means in this\ncontext.\nResponse: We appreciate the reviewer’s thoughtful clarification regarding the use of the term “coherent”. We\nagree with the reviewer’s comment that our manuscript did not sufficiently define what we meant by\n“coherent”, which may have led to misunderstanding.\nIn our original usage, “coherent magnons” referred to magnons excited by the microwave antenna, whose\nphases are well defined, in contrast to thermally generated magnons that have random phases and are generally\nregarded as incoherent—a behavior commonly observed in nonlinear magnon interactions probed by\nBrillouin light scattering. From this perspective, the acoustic and optic modes observed in our experiment\nwere described as “coherent” because they were antenna-driven and phase-defined.\nHowever, we fully appreciate the reviewer’s point that this definition differs from the conventional meaning\nused when discussing “coherent coupling”—namely, the simultaneous excitation of two modes by a common\ncoherent source, with well-defined relative phase, allowing direct phase-sensitive interaction. As the reviewer\ncorrectly notes, in our swept-frequency experiment the two modes are not simultaneously driven, and the\nrelaxation time between frequency sweeps prevents any phase-matched excitation of the two modes. Thus,\nour data do not directly demonstrate “coherent coupling” in this stricter sense.\nWe agree that our earlier terminology could be misleading, and we have revised the manuscript accordingly.\nSpecifically: 1) we now clearly define our usage of the term “coherent”; 2) we removed instances where\n“coherent” might imply phase-preserving magnon–magnon coupling; and 3) we deleted the phrase\n“simultaneously excited,” which could give an incorrect impression about the excitation process. The detailed\nrevisions are listed below.\nWe appreciate the reviewer for raising this important conceptual point, which helped us significantly improve\nthe clarity and accuracy of the manuscript.\n----------------------------------------------------------\n- List of changes\np.3, L10: “…, such as magnon Bose-Einstein condensation (BEC)15–17. More recently, magnon nonlinearity\nhas been explored in hybrid systems18,19, where coherent magnons couple with other quasiparticles,…”\n “…, such as magnon Bose-Einstein condensation (BEC)15–17, which is typically observed in\nincoherent magnon interactions where the magnons have random phases. More recently, magnon\nnonlinearity has been explored in hybrid systems18,19, where phase-defined coherent magnons couple\nwith other quasiparticles,…”\np.3, L15: “Direct nonlinear interactions between two coherent magnons,…”\n “Direct nonlinear magnon-magnon interactions,…”\np.3, L16: “…recent reports have demonstrated nonlinear coupling phenomena between coherent magnon\nmodes12,29–32,…”\n “…recent reports have demonstrated nonlinear magnon-magnon coupling phenomena12,29–32,…”\np.3, L19: “—an abrupt transition between multistable states with coherent magnon modes…”\n “—an abrupt transition between two magnon modes…”\np.3, L21: “The SAF supports two distinct coherent magnon modes:…”\n “The SAF supports two distinct magnon modes:…”\np.4, L5: “…stem from interactions between four coherent magnon modes:…”\n”… stem from interactions between four different magnon modes:…”\np.4, L10: “These findings advance the understanding of nonlinear coherent magnon dynamics…”\n “These findings advance the understanding of nonlinear magnon dynamics”\np.5, L9: “…, where both modes are simultaneously excited,…”\n “…, where both modes are excited,…”\np.5, L20: “We next investigate the nonlinear behavior of coherent magnon coupling.”\n” We next investigate the nonlinear behavior of the magnon-magnon coupling.”\np.12, L25: “…nonlinear mode hopping between two coherent magnon modes …”\n “…nonlinear mode hopping between optic and acoustic magnon modes …”\np.13, L6: “…nonlinear interactions between coherent magnons …”\n “…nonlinear magnon-magnon interactions …”\n----------------------------------------------------------\nComment (b). The authors agree with me that the hysteretic effects observed in their experiment cannot be\ncaused by the magnon accumulation. However, they base their theoretical interpretation on magnon\naccumulation effects. In other words, the used model, even if it shows behavior resembling that observed in\nthe experiment, does not reflect real physics. This makes the paper misleading.\nResponse: We believe that part of the reviewer’s concern may stem from an insufficient explanation in the\nprevious version. Both our experimental observations and theoretical calculations indicate that the observed\nmode hopping and hysteresis are indeed connected to magnon accumulation, and our model reflects the\nrelevant physics of the system. Below, we elaborate on each point more clearly.\n1. Mechanism of Mode Hopping\nThe mode hopping originates from the well-known three-magnon scattering process, in which one Kittel\nmagnon at frequency f splits into two magnons at f/2. This nonlinear process requires a sufficiently large\npopulation of Kittel magnons. Once this condition is met, mode hopping occurs when the relation 𝜔 =\nop\n2𝜔 is satisfied.\nac\nIn our experiment, however, the magnons initially excited by the antenna are propagating magnons, not Kittel\nmagnons. Therefore, for mode hopping to occur, propagating magnons must first be converted into Kittel\nmagnons—another nonlinear process whose efficiency increases with RF power. In our theoretical model,\nthis conversion is parameterized by G . Increasing RF power in experiment corresponds to increasing G in\n1 1\nthe model, and both lead to a larger Kittel-magnon population, making mode hopping more likely.\nA key question is whether RF power indeed drives the transition from propagating to Kittel magnons. In the\noriginal manuscript, we presented frequency redshift and gap closing as indirect evidence. In the revised\nversion, we provided more direct support: (1) Micromagnetic simulations show that higher RF power\nexplicitly drives the transition toward Kittel-dominated states (Fig. R4.1, Supplementary Note 6). (2) k-\nresolved MOKE measurements, performed during revision and published recently (Fig. R4.2), directly\ndemonstrate that RF power increases the population near k = 0, consistent with enhanced Kittel magnons.\nThus, the observed mode hopping arises from a combination of two nonlinear processes: (a) propagating-to-\nKittel conversion, and (b) three-magnon splitting of Kittel magnons.\nFigure R4.1 | Mode hopping via micromagnetic simulation. (a) The f-H relation at the low CPW\npower. Conventional acoustic and optic magnon signals are detected. The red (blue) circular point\nindicates the analysis point for the (c)-(f). (b) The f-H relation at the high CPW power. The clear\ndiscontinuous mode hopping at the 30 mT is observed. (c) and (d) describe the magnon population with\nrespect to wavevector in the 𝜇 𝐻 = 40 mT, 𝑓 ~ 4 GHz acoustic mode condition for the low and\n0 in\nhigh CPW powers. (e) and (f) describe the magnon population with respect to wavevector in the\n𝜇 𝐻 = 20 mT, 𝑓 ~ 6 GHz optic mode condition for the low and high CPW powers.\n0 in\nFigure. R4.2 | k-resolved spectra measured by the heterodyne MOKE, showing the evolution from\npropagating magnons to Kittel magnons with increasing RF power. As the RF power increases, spectral\nweight progressively shifts toward k = 0, indicating a growing population of Kittel magnons. [D. Hayashi et\nal., Appl. Phys. Lett. 126, 082401 (2025)].\n2. Origin of the Hysteresis\nAs described above, mode hopping in our system requires sufficient Kittel magnons to accumulate. Even if\nthe frequency condition of 𝜔 = 2𝜔 is met during a field sweep, insufficient Kittel-magnon population\nop ac\ndelays the onset of mode hopping until a higher density is reached. This is the first source of hysteresis.\nTo verify this, we varied G in our theoretical model. As shown in Fig. R4.3 (also Fig. 4 in the main text),\n1\nincreasing G (i.e., increasing the efficiency of propagating-to-Kittel conversion) narrows the hysteresis gap\n1\nand sharpens the transition—mirroring the effect of higher RF power in the experiment (Figs. 3f-3h in the\nmain text).\nFigure. R4.3 | Acoustic magnonic gain G dependence on the mode hopping hysteresis in the numerical\n1\nsimulation. The hysteresis gap becomes narrower and saturated at some point as increasing G.\nThe magnonic\n1\ngain G is set as (a) 0.68, (b) 0.70, (c) 1.00.\nthe field-sweep directions. In our calculations, for positive field sweeps we set the initial amplitude of the\noptic mode much larger than that of the acoustic mode, whereas for negative sweeps the opposite was assigned.\nThis choice reflects the field-dependent magnon population: at low fields the optic mode is more efficiently\nexcited, while at high fields the acoustic mode becomes dominant. Consequently, in positive sweeps, optic\nmagnons dominate at low fields, and mode hopping occurs when the acoustic amplitude grows large enough\nto overtake it. In negative sweeps, acoustic magnons dominate at high fields, and the transition occurs when\nthe optic amplitude becomes larger. If this initial condition asymmetry is removed, hysteresis disappears\nentirely (Fig. R4.4), demonstrating that initial conditions play a crucial role for the observed hysteresis.\nExperimentally, we believe that the initial condition is influenced by the underlying magnetization\nconfiguration. As demonstrated in Supplementary Fig. S5, even a slight change in magnetization\nconfiguration leads to a noticeable shift in the mode-hopping threshold, providing an evidence for this\ndependence. Based on this observation, our experimental results can be interpreted as follows: magnon\naccumulation—encompassing both propagation-to-Kittel and optic-to-acoustic mode conversions—first\ndevelops, and this accumulation modifies the magnetization configuration. The modified configuration then\nchanges the initial condition for the subsequent state. In other words, when the field is held fixed and a\nfrequency sweep induces mode hopping, as in our experimental protocol, the associated redistribution of\nmagnons slightly alters the magnetization configuration. As a result, the final state of that sweep no longer\nmatches the original initial state. The next field-sweep cycle therefore begins with a modified initial condition,\nnaturally giving rise to the observed hysteresis.\nWhile this process—where magnon accumulation leads to a mode hopping and slight change in the\nmagnetization configuration that causes the hysteresis—provides a plausible and physically consistent\nexplanation, we note that it is one of several possible scenarios and has not been rigorously proven at the\ncurrent stage. Nevertheless, this framework clarifies our earlier point that such magnon accumulation-induced\nmodifications of the magnetization configuration can give rise to the observed hysteresis.\nIn summary, we believe our theoretical model appropriately captures the physics of our experimental system.\nG corresponds to the RF-driving power and governs the conversion from propagating to Kittel magnons.\n1\nmeasurements. The initial-condition asymmetry used in our model can be interpreted as being consistent with\na sweep-direction-dependent magnetization configuration in the experiment.\nFigure. R4.4 | Initial condition dependence of the mode hopping hysteresis in the numerical simulation.\nThe hysteresis gap is strongly affected by initial condition setting which stems from the initial magnetization\nconfiguration. The experimental condition of magnetization would be optic dominant for the forward field-\nsweep and acoustic dominant for the negative field-sweep (a). If the initial conditions are adjusted,\nthe\nhysteresis gap becomes zero as acoustic and optic amplitudes comparable (b)-(c)."}, {"round": 3, "reviewer_comments": {"reviewer_4": "Although I find it difficult to imagine that the accumulation of magnons could lead to a persistent change in the static magnetic configuration, I agree with the authors that such an explanation is not completely excluded. Therefore, I recommend publication of the revised manuscript."}, "author_response": ""}]

Decision letter