Interaction between magnon and skyrmion: Toward quantum magnonics

Since the discovery of giant magnetoresistance in 1988,¹ spintronic devices have achieved considerable progress. Over the last decade, as the information technology utilized by static properties of magnets has been adequately investigated, the dynamic properties have gradually attracted increasing attention. Collective excitations in magnetically ordered systems are called spin waves, whose quanta are magnons. During the exploration of the physical properties and potential application of magnon, a new branch of spintronics, magnonics, has been formed.^2–5 Magnon is expected to be an appropriate candidate for a novel information carrier due to the advantages of high frequency, short wavelength, convenient modulation, and low energy consumption.^3,6–12 Despite the above advantages, there are a few problems that still need to be solved for applications based on magnon propagation, for instance, the main obstacle is the large damping of skyrmion hosting magnetic metals.^13–18 Recently, inspired by quantum optics, the quantum phenomena of magnon have been fierily studied, for example, Bose-Einstein condensation (BEC)^19–24 and superfluidity.^25–29 In addition, the coupling between the magnon and other quantum systems has been realized, such as cavity photons,^30–32 phonons,^33–39 and superconducting qubits.^40–43 Furthermore, the coupling between different magnon modes in a single magnet and multilayer has been predicted and realized, such as the coupling between the ferromagnetic resonance mode and the standing spin waves mode^44–46 and the coupling between the optical mode and the acoustic mode of antiferromagnets.^47–51 These achievements of quantum phenomena and hybrid quantum systems open up the usage of magnons in the field of quantum technology and information.^52–54 

Skyrmion, the particle-like solution, was first proposed in nuclear physics by Skyrme in 1962.⁵⁵ This concept has been extended into condensed matter physics by predicting the existence of skyrmions in magnetic materials by Bogdanov et al.^56–59 Magnetic skyrmion was first observed in chiral magnet MnSi at about 29 K by neutron scattering in 2009.⁶⁰ Since then, magnetic skyrmion attracted widespread academic attention and led to the renaissance of topological magnetism. Since then, beyond skyrmion, increasing numbers of topological nontrivial spin textures have been discovered, including biskyrmions,^61–63 skyrmionium,^64–66 meron,^67–70 bimeron,^71–73 chiral bobber,^74–76 and hopfion.^77–80 Due to the properties of nanoscale, topological protection, robustness, and dynamic properties,^81–86 skyrmion and other nontrivial spin textures have a huge potential for information storage, transmission, and processing.^87–89 

As the coexisting fundamental excitations in chiral magnets, magnon and skyrmion could be connected by direct exchange interactions, and the collision between them gave birth to many interesting physical phenomena.⁹⁰ In this review, we discuss the interactions between magnon and topologically nontrivial skyrmion. First, the characteristics of magnon and skyrmion are briefly introduced. Second, three kinds of interactions between magnon and skyrmion are discussed: the scattering of magnon, the motion of skyrmion, and the coupling between them. Finally, the potential magnonic devices utilizing these magnon–skyrmion interactions are discussed as well as the perspective for the development of quantum magnonics.

In the 1930s, the concept of spin waves was first proposed by Bloch.⁹¹ Spin waves are the collective excitation in the spin lattice; and magnons are quanta of spin waves, a kind of boson. To understand the characteristic of magnon, it is vital to introduce the dynamics of magnetic moments. The static and dynamic properties of magnetizations can be derived from the Landau–Lifshitz–Gilbert equation,⁹² 

where m is the normalized magnetic moment, γ is the gyromagnetic ratio, and α is the Gilbert damping. H_eff is the effective magnetic field, where H_ext, H_A, H_demag, H_k, and H_DM represent the external magnetic field, the exchange interaction field, the demagnetization field, the anisotropy field, and the Dzyaloshinskii–Moriya (DM) interaction field, respectively. The DM interaction is a chiral interaction originating from the inversion symmetry breaking in the lattice or at the interface of magnetic films, which is corresponding to the bulk and the interfacial DM interaction, respectively.^93–95 Its energy is expressed as $EDM=Dij⋅(Si×Sj)$⁠,¹⁶ where $Dij$ is the DM interaction vector and $Si$ and $Sj$ denote adjacent spins. Based on interactions, each of the individual spins is coupled together and forms a wave-like pattern. Spin waves can be classified into short wavelength exchange spin waves mainly dominated by exchange interaction and long wavelength magnetostatic waves mainly dominated by dipole interaction. Magnetostatic waves have been proposed and demonstrated as delay lines,^96,97 filters,^98,99 and resonators.^100,101 Benefiting from its short wavelength, exchange spin wave has been applied as a logic operation for the miniaturization of magnonic devices.^102–113 

In recent years, many breakthroughs in excitation,^114–117 detection,^3,118,119 and modulation^120–123 of magnon promote its wide applications. Figure 1 shows conceptual magnonic devices for processing information and emerging phenomena in quantum magnonics. For information processing, magnon exhibits numerous merits, including short wavelength, high frequency, flexible modulation, and no carrier movement involved.^4,5 As the magnetic analog of photonic crystals, magnonic crystals with periodic magnetic properties has been studied to manipulate magnons. The periodicity of magnonic crystals could be introduced in the manner of spatial geometry or spin textures.^124–128 The flexibility of manipulating band structures of magnon in these artificial crystals enriches the information processing functions.^89,129,130 So far, various magnonic devices on the scale of nanometer have been demonstrated, for instance, multiplexers,¹⁰⁵ transistors,^102,110,111 majority gates,^{103,106,109,113} logic gates,^{104,107,108,112,131} diodes.¹³² Like photons, magnons exhibit most of the properties inherent in waves or quasiparticles and have the potential to replicate the success of the mature field of quantum optics.⁵⁴ Several quantum phenomena achieved by magnon instead photon are explored, i.e., BEC,^{19–22,24,133} spin superfluidity,^25–29 squeezed states,^134–140 and quantum entanglement.^141–143 In addition, the connection of magnon to other quantum platforms is also important. For instance, the hybridization between magnon and cavity microwave photon, optical photon, and phonon have been realized by magnetic dipole, magneto-optical, and magnetostrictive interactions, respectively.^52,144 Furthermore, the magnonic system has experimentally demonstrated its coherent coupling with two-level superconductor qubit, which is mediated by cavity photons.^41–43 Several applications based on the hybrid quantum platform based on magnonic system are proposed, such as quantum memories,¹⁴⁵ single magnon detection,¹⁴⁶ and dark matter searches.^147,148 Such achievements gained in both quantum phenomena and hybridization based on magnon could promote the development of quantum computing and the exploration of quantum magnon states, which leads to the rapid development of an emerging field: quantum magnonics. As the interdisciplinary subject of spintronics and quantum information, quantum magnonics provides a prospective platform for the study of quantum phenomena and a new direction for the development of spintronics, i.e., precise measurements.⁵⁴ Although it is still challenging experimentally to realize quantum magnon states, their conceptual usages have been theoretically proposed, i.e., single magnon states¹⁴⁹ and Schrödinger cat states.¹⁵⁰ Very recently, the exploration of quantum magnonics based on spin textures is emerging, and the hybridization between confined magnon states in spin textures and photonic mode has been studied both theoretically and experimentally.^17,151–157

Magnetic skyrmion is a kind of nano-sized spin texture with quasiparticle properties.^{82,84,85,158–165} Two typical skyrmions are schematically shown in Figs. 2(a) and 2(b): the Bloch type and the Néel type. Bloch skyrmion has been experimentally observed in bulk magnets.^{60,75,93,166–169} While Néel skyrmion has been experimentally observed in heavy metal/ferromagnetic metal multilayer thin-films.^82,170–175 The distinct configurations of skyrmion are due to different kinds of DM interaction: the bulk DM interaction prompts the emergence of Bloch type, and the interfacial DM interaction prompts the emergence of Néel type, as shown in Figs. 2(c) and 2(d). The nucleation of skyrmions is resulting from the competition between various magnetic interactions, i.e., DM interactions, exchange interaction, anisotropy, and dipole interaction. In addition, the size of the skyrmion is proportional to the strength of DM interaction.^176–179 The nontrivial topological properties of skyrmion in magnetic films are determined by the topological charge Q,¹⁸⁰ 

The Q of one single skyrmion is ±1, which means that when all magnetic moments are translated to a point, their pointing can cover the entire sphere. The value of Q is also used to present the topological characters of other spin textures, like Q = 2 for biskyrmion^61,62 and Q = 0 for skyrmionium.^64–66 The robustness against defects and quasiparticle properties of skyrmion are indeed derived from the nontrivial topology.^16,180

The striking points about skyrmions consist not only the topological protection but also the collective dynamics.^181–185 Mochizuki theoretically determined the existence of three eigen-modes of skyrmion crystals:¹⁸³ two in-plane gyroscopic modes and one out-of-plane breathing mode. The counterclockwise (CCW) and clockwise (CW) modes manifest the counterclockwise and clockwise rotation of the skyrmion core. The breathing mode shows the periodical expansion and contraction of the skyrmion. The profile of the above three fundamental modes is shown in Fig. 3(a). Experimentally, the existence of three modes was confirmed in chiral magnet Cu₂OSeO₃.^186,187 In addition to the three eigen-modes, Lin et al. discovered the higher-order gyroscopic modes by numerically diagonalizing dynamical matrices for a single skyrmion,¹⁸⁴ which are marked by the weak deformations for the center of skyrmion, as shown in Fig. 3(b). In fact, the complete energy spectrum of the skyrmion-based magnonic crystals has a complex structure, which consists of energy bands formed by the scattering of magnons by periodically arranged skyrmions and multiple eigen-modes of skyrmions,^85,188 as shown in Fig. 3(c). Compared with conventional magnonic crystals, skyrmion-based magnonic crystals have excellent controllability and application. Furthermore, it is shown that topological magnons and chiral boundary states with low dissipation can exist in skyrmion-based magnonic crystals,^189–193 which implies a great potential application of there in information transfer.

In the last decade, remarkable achievements of magnetic skyrmion have been gained with their observation in ferromagnetic,^{60,166,167,170,171,173} ferrimagnetic,^194–197 and antiferromagnetic materials.^198–203 Other spin textures with nontrivial topological properties have been discovered in rapid sequence, which constitutes a large family of skyrmions.⁸⁶ Studies on the nontrivial topological properties and dynamics of skyrmion and skyrmion-like spin textures are forming a new branch—skyrmionics, which aims to promote the devices and applications based on topological spin textures.

As two promising fields, magnonics and skyrmionics are of great significance to the innovation and development for information storage and processing. The combination of two fields allows us to design pioneering magnonic devices and to explore novel phenomena arising from the collision of nontrivial topology with magnon quantum states.

The influence of spin textures acting with propagating magnons could pave a way for modulation of magnon, such as using magnetic domain walls to control magnon amplitude and phase,²⁰⁴ and designing magnonic crystals to modulate magnonic band structures.^205,206 Typically, the method to investigate the scattering of magnons by spin texture by replacing the effects of nonuniform spin textures with equivalent magnetic fields^207–210 and by transforming the scattering to a wave-particle interaction model,^211,212 and the above method is also suitable for dealing with magnon–skyrmion scattering.

In 2014, Iwasaki et al. systematically studied the scattering mechanism of magnons and the skyrmion through micromagnetic simulations.²¹³ As shown in Fig. 4(a), there are two distinct characteristics of the scatting process: (i) skyrmion could remain a stable shape for large amplitude magnon due to its topological protection; (ii) the trajectory of magnon is deflected with skew angle Φ, which exhibits wavenumber dependence. With the wavenumber of the magnon equivalent to the reciprocal of skyrmion diameter, the angle reaches maximum; while the wavenumber tends to be far away from the reciprocal of skyrmion diameter, the skewed angle tends to zero. It originates from the topological nontrivial properties of skyrmion. Under the effect of Berry curvature of the skyrmion, an emergent magnetic field will be generated. It is the emerging magnetic field located on the core of skyrmion that causes skewed scattering. In other words, the emerging magnetic field leads to the formation of the fictional Lorentz force acting on propagating magnon. In the same year, Schütte and Garst treat the skyrmion as local exponentially decayed scattering potential, resulting in the effective flux density with a singularity at the skyrmion center.¹⁸⁵ Note that they not only observe skewed scattering but also calculated asymmetrical scattering cross section with multiple peaks, the latter is known as rainbow scattering, see Fig. 4(b). The origin of rainbow scattering was also revealed by their early theory of high-energy magnon scattering.²¹⁴ According to their theoretical results, the direction of deflection is determined by the sign of the emerging magnetic flux. Since the magnetic flux density is rotationally symmetric, all outgoing trajectories have the same deflection angle and interfere to form an oscillating differential cross section. Additionally, the skew scattering can be investigated by multifarious approaches, such as Holstein–Primakoff Hamiltonian²¹⁵ and collective coordinate model.²¹⁶ 

The magnon–skyrmion scattering can be utilized for information transfer and processing functions. Hu et al. designed a voltage-controlled magnon–skyrmion switch by exploiting the dependence of the outgoing magnon deflection angle on the skyrmion size.²¹⁷ Wang et al. theoretically studied the magnonic analog of an optical frequency comb by using amplitudes above a threshold to drive a single skyrmion in a two-dimensional plane.²¹⁸ A discrete set of equally spaced frequencies has been formed, and the frequency interval is equal to the breathing mode frequency of the skyrmion. Rodrigues et al. numerically gained the frequency multiplier by applying fractions of the eigenfrequency to excite eigen-modes of a single skyrmion.²¹⁹ At present, these designs of devices based on skyrmion are still in the theoretical stage, but it is believed that experimental breakthroughs will soon be achieved based on these theoretical works.

Due to the conservation of momentum and energy during the scattering, there will be a momentum-transfer force acting on the skyrmion, causing a finite velocity toward the exciton source and large topological Hall effect.^185,213,214 It means that magnons could act as a way to drive the motion of skyrmion in magnetic insulator and might broaden the horizon of magnonic devices.

The skyrmion motion in nanowires driven by magnon has been numerically studied in recent years.^220–222 Instead of moving to the excitation source with a large skew angle, skyrmion shifts along the direction of magnon propagation under the combination of force from magnon and edge repulsion, and the skyrmion Hall effect has been suppressed by the constricted geometries. The driving efficiency of skyrmion motion is mainly affected by three factors: size of the skyrmion, damping of the material, and manner of magnons excitation. For the size of skyrmion, the larger size leads to the larger surface interaction with magnon and, hence, the larger driving force.^220,221 This means that the driving efficiency could be controlled by means such as external magnetic field and voltage. The displacements of a skyrmion in nanowires under longitudinal and transverse driving with different damping α, respectively, are shown in Figs. 5(a) and 5(b). It is obvious that the moving distance decreases with an increase in damping. The skyrmion velocity as a function of time under two different driving manners in nanowires is shown in Figs. 5(c) and 5(d). For longitudinal driving, it goes through the processes of acceleration and deceleration successively, which conforms to the functional form $v(t)=ate−bt$⁠. The deceleration process occurs due to the skyrmion-edge repulsion. For transverse driving, the skyrmion moves faster than longitudinal driving and without the process of deceleration after reaching the maximum velocity. However, for excessively large amplitude for transverse driving, the skyrmion is pushed toward the drive region and eventually annihilates. Additionally, the motion of skyrmions in shape-confined geometries is investigated, including L-corners and T- and Y-junctions, as shown in Figs. 5(e) and 5(f).²²⁰ The skyrmion can smoothly turn left or right to pass through the L-corner. For T- and Y-junction, the trajectory of Q = 1 skyrmion always turns left due to the skyrmion Hall effect;^180,223 while for skyrmion with Q = −1 will turn right.

The strong coupling phenomenon between two exciting modes implies entanglement and fast energy exchange, which could be applied to the field of quantum information. The strong coupling could be divided into coherent coupling and dissipative coupling. Both kinds of coupling could be understood by two coupled harmonic oscillators with different connections, as shown in Figs. 6(a) and 6(b). The coherent coupling can be described by the spring-connected harmonic oscillator model, which manifests as an energy-level repulsion phenomenon at the intersection in the dispersion relation, also known as anti-crossing, as shown in Fig. 6(c), where Δk is the detuning between two harmonic oscillators. The dissipative coupling is depicted by a damper-connected harmonic oscillator model, which appears as an energy-level attraction phenomenon at the intersection of the dispersion relation, as shown in Fig. 6(d). The coupling could be applied to quantum information,^224,225 quantum sensing,²²⁶ and nonreciprocal photon transmission.^227,228

In 2010, Soykal and Flatté proposed and theoretically demonstrated the coupling between the ferromagnet resonance mode of a single magnetic sphere and the photonic mode of a cavity.^229,230 This work not only provides a strong theoretical basis for the experiment but also stimulates the upsurge of experimental studies on photon–magnon coupling. In 2013, Huebl et al. first observed the magnon–photon coupling by measuring the microwave transmission of a superconducting coplanar microwave resonator by placing a YIG film on top.²³¹ Since then, the magnon–cavity photon coherent coupling has been realized in different experimental setups: YIG sphere in three-dimensional microwave cavity^232–234 and nanometer thick permalloy film on superconducting coplanar waveguides.^30,31 Furthermore, the transformation of the coherent coupling to the dissipative coupling has been realized by varying the relative phase.^32,235 Beside the above studies in uniformly magnetized materials, significant progress has been made in the theoretical and experimental studies of the coupling between cavity photons and spin textures, e.g., chiral domain wall,^153,156 vortex,^151,154 and skyrmion.^17,152,155 These studies of the interaction of both magnon and spin textures with cavity photons are leading to the field of cavity magnonics.⁵³ 

However, the size of the microwave cavity for cavity magnonics is usually in the order of millimeters, which hinders the application of magnon–photon coupling in integrated devices. The shortcomings could be avoided by replacing the microwave cavity by studying the coupling between magnetic excitations in a single magnet, i.e., magnon–magnon coupling. The coupling between magnon modes in various kinds of structures has been reported, including the monolayer magnetic structures,^47,236–238 multilayer magnetic structures,^{44–46,239,240} cavity photon-mediated separated magnets,^241–243 and synthetic antiferromagnetic structures.^48,244,245

Inspired by the admirable studies of the coupling between magnon and cavity photon as well as magnon and magnon, the coupling of skyrmion to magnon is gradually extracting interesting. As two fundamental excitations that coexist in magnetic systems, magnon and skyrmion are promising to achieve strong coupling through direct exchange interactions. Liu et al. discovered that propagating magnon mode can couple with the skyrmion higher-order mode in the one-dimensional skyrmion crystal by numerical simulation.²⁴⁶ As shown in Fig. 7(a), not only a magnonic analogy of polariton gap is observed in the f-k_x dispersion, but also Rabi splitting is observed in the f-H_z dispersion. Six different mode profiles are listed in Fig. 7(b) corresponding to ①–⑥ as in Fig. 7(a). The integer l is introduced to describe the different orders of skyrmion modes, the value represents the confined magnon modes with different node numbers along the perimeter of skyrmion and the plus or minus sign corresponds to the clockwise or counterclockwise rotation. The mode $|l=−1⟩$ represents propagating magnon mode, as shown in Fig. 7(b) ① and ②. $|l=2⟩$ corresponds to the higher-order gyrotropic mode of skyrmion, as shown in Fig. 7(b) ③ and ④. The mode $|ψ−1,2⟩$ is the hybridization between $|l=−1⟩$ and $|l=2⟩$⁠, as shown in Fig. 7(b) ⑤ and ⑥, and the contributions of the two modes are comparable. Note that $|l=2⟩$ higher-order gyrotropic mode is difficult to be stimulated alone; therefore, the profile of the $|l=2⟩$ mode contains a small part of the $|l=−1⟩$ component. Figure 7(c) shows the tunability of this coupling. The width of the Rabi splitting is determined by the wavenumber of the spin-wave traveling wave, while the width of the polariton bandgap depends on the external magnetic field. For the elected wavenumber of propagating magnons, the coupling could access the regime of strong coupling, which is the coupling strength greater than the dissipation rate of the two modes.

Higher-order modes localized in the skyrmion core can couple not only with propagating magnons but also with skyrmion eigen-modes. Recently, hybridization modes CCW-Sextupole, Breathing-Octupole, and CW-Dectupole are experimentally observed in the chiral magnetic material Cu₂OSeO₃ at low temperatures.^157,247,248 By field cycling or rapidly cooling the high-temperature skyrmion phase to gain the low-temperature skyrmion crystal state, the hybridization of three fundamental modes and higher-order gyrotropic modes in f-H dispersion has been observed, as shown in Fig. 8(a). Intuitively, the profiles of the above hybrid modes are shown in Fig. 8(b). Further, the observed hybridization modes originate from cubic anisotropy and disappear with increasing temperature, and the gap width of the anti-crossing could be modulated by varying the cubic anisotropy.

In recent years, the study on magnon–skyrmion interaction has theoretically yielded favorable results. On the classical level, several information devices based on magnon–skyrmion scattering have been proposed.^217–219 Propagating magnon could act as an efficient way to drive skyrmion motion.^220–222 On the quantum level, the coupling between propagating magnon and skyrmion results in the hybrid magnon states.²⁴⁶ These studies demonstrate the possibility of realizing hybrid quantum systems based on magnon, which bridges the skyrmionics and quantum magnonics. However, there are still challenges to be solved. From the theory point of view, the dissipative coupling between magnon and skyrmion is a meaningful topic. The reported coupling or hybridization between magnons and skyrmion lattice all exhibits level repulsion. However, the level of attraction has not been realized between magnon and skyrmion. This kind of coupling could be gained by introducing a third-party highly dissipative mode as what for magnon–magnon coupling.²⁴⁹ Considering the important potential applications of dissipative coupling in quantum entanglement states and quantum sensing, it is necessary to carry out investigation on this kind of coupling in the near future. Additionally, most of quantum phenomena of magnon, like BEC and entanglement, have been studied in collinear magnetic configuration, i.e., ferromagnet, while it is unclear in noncollinear counterparts, especially for topologically nontrivial magnetic textures. The combination of topological nontriviality and quantum magnon states remains mysterious. From the experimental point of view, skyrmion hosted materials with perpendicular magnetic anisotropy are characterized by high damping, which makes it difficult for magnon propagation and hinders the applicability for magnon–skyrmion coupling. Therefore, the realization of the coupling needs a new type of material with low damping and suitable for skyrmion stabilization. From the application point of view, the utilization of the magnon–skyrmion coupling could enrich the magnonic devices for quantum information, such as information storage¹⁴⁵ and coherent gate operations.^250,251 Moreover, skyrmions have been theoretically proposed as qubits for the realization of quantum logic elements, with information preserved in the quantum helicity of the skyrmion and logical states modified by electric and magnetic fields.²⁵² The readout and processing of quantum information from skyrmion qubits via magnon–skyrmion coupling is also worth exploring.

In this review, we provide a brief review on the concept of magnon and skyrmion and discuss three kinds of interaction between them. As a quasiparticle, skyrmion exhibits two simultaneous interactions with propagating magnons: the skew scattering of magnon under the emergent magnetic field generated by skyrmion and the motion of skyrmion driven by magnons due to the conservation of energy and momentum. As a resonator, skyrmion could provide gyrotropic modes to couple with propagating magnons or even with eigen-mode of themselves. The first two interactions contribute to the design of magnonic devices for information transmission and processing, while the last one broadens the platform for quantum information. Finally, we give a brief perspective on the potential for magnon–skyrmion interaction in the field of quantum information.

This work was supported by the National Natural Science Foundation of China (NNSFC) (Grant No. 12074189).

The authors have no conflicts to disclose.

Zhengyi Li: Writing – original draft (lead). Mangyuan Ma: Writing – original draft (equal). Zhendong Chen: Writing – review & editing (equal). Kaile Xie: Writing – original draft (equal). Fusheng Ma: Conceptualization (lead); Writing – original draft (lead); Writing – review & editing (lead).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.