Quantum engineering with hybrid magnonics systems and materials
D. D. Awschalom, C. H. R. Du, R. He, F. J. Heremans, A. Hoffmann, J. T. Hou, H. Kurebayashi, Y. Li, L. Liu, V. Novosad, J. Sklenar, S. E. Sullivan, D. Sun, H. Tang, V. Tiberkevich, C. Trevillian, A. W. Tsen, L. R. Weiss, W. Zhang, X. Zhang, L. Zhao, C. W. Zollitsch
QQUANTUM ENGINEERING WITH HYBRID MAGNONICS SYSTEMSAND MATERIALS (Alphabetical order)
D. D. AWSCHALOM,
Member, IEEE ,
1, 2
C. H. R. DU, R. HE, F. J. HEREMANS,
Member, IEEE ,
1, 2
A. HOFFMANN,
Fellow, IEEE , J. T. HOU,
Member, IEEE , H. KUREBAYASHI, Y. LI,
Member, IEEE , L. LIU,
Member, IEEE , V. NOVOSAD,
Member, IEEE , J. SKLENAR,
Member, IEEE , S. E.SULLIVAN, D. SUN, H. TANG,
Member, IEEE , V. TIBERKEVICH,
Member, IEEE , C. TREVILLIAN,
Member, IEEE , A. W. TSEN, L. R. WEISS, W. ZHANG,
Member, IEEE , X. ZHANG, L. ZHAO, andC. W. ZOLLITSCH Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL,USA Materials Science Division and Center for Molecular Engineering, Argonne National Laboratory, Lemont, IL,USA Department of Physics, University of California, San Diego, La Jolla, CA 92093,USA Department of Electrical and Computer Engineering, Texas Tech University, Lubbock, TX 79409,USA Materials Research Laboratory and Department of Materials Science and Engineering,University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA,USA London Centre for Nanotechnology, University College London, 17-19 Gordon Street, London, WCH1 0AH,UK Materials Science Division, Argonne National Laboratory, Lemont, IL, USA Department of Physics and Astronomy, Wayne State University, Detroit, MI 48201,USA
Department of Physics, North Carolina State University, Raleigh, NC 27695,USA
Department of Electrical Engineering, Yale University, New Haven, CT, USA
Department of Physics, Oakland University, MI 48309, USA
Institute for Quantum Computing and Department of Chemistry, University of Waterloo, Waterloo, ON,Canada
Center for Nanoscale Materials, Argonne National Laboratory, Lemont, IL,USA
Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA (Dated: 8 February 2021)
Quantum technology has made tremendous strides over the past two decades with remarkable advancesin materials engineering, circuit design and dynamic operation. In particular, the integration of differentquantum modules has benefited from hybrid quantum systems, which provide an important pathway forharnessing the different natural advantages of complementary quantum systems and for engineering newfunctionalities. This review focuses on the current frontiers with respect to utilizing magnetic excitatons ormagnons for novel quantum functionality. Magnons are the fundamental excitations of magnetically orderedsolid-state materials and provide great tunability and flexibility for interacting with various quantum modulesfor integration in diverse quantum systems. The concomitant rich variety of physics and material selectionsenable exploration of novel quantum phenomena in materials science and engineering. In addition, the relativeease of generating strong coupling and forming hybrid dynamic systems with other excitations makes hybridmagnonics a unique platform for quantum engineering. We start our discussion with circuit-based hybridmagnonic systems, which are coupled with microwave photons and acoustic phonons. Subsequently, weare focusing on the recent progress of magnon-magnon coupling within confined magnetic systems. Next wehighlight new opportunities for understanding the interactions between magnons and nitrogen-vacancy centersfor quantum sensing and implementing quantum interconnects. Lastly, we focus on the spin excitations andmagnon spectra of novel quantum materials investigated with advanced optical characterization.
I. INTRODUCTION
Quantum technology combines fundamental quantumphysics and information theory with an overarching goalto develop a new generation of computing, sensing, andcommunication architectures based on quantum coher- ent transfer and storage of information. Spurred by thediscovery of quantum properties in novel materials andstructures, a variety of dynamic systems including pho-tons, acoustic excitations, and spins have been cultivatedin diverse platforms such as superconducting circuits,nanomechanical devices, surface acoustic waves, and in- a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b dividual electrons and ions. Many of these systems havebeen employed for implementing artificial two-level sys-tems as qubits, enabling their coherent interaction as wellas manipulating their states. For nascent quantum tech-nologies to reach maturity, a key step is the develop-ment of scalable quantum building blocks: from quantuminterconnects and transducers, to sensors at the singlequasiparticle level. The development of scalable archi-tectures for quantum technologies not only poses chal-lenges in understanding the coupling between disparatequantum systems, but also presents technical and engi-neering challenges associated with developing chip-scalequantum technology.A rapidly growing subfield of quantum engineeringis associated with magnons. Magnons, or the quantaof spin waves, are the collective excitation of exchangecoupled spins hosted in magnetic materials. Similarto electromagnetic and acoustic waves, spin waves canpropagate and interfere, meaning that they can deliverphase information for coherent information processing.Due to the high spin density in magnetic materialscompared with individual spins, large magnetic dipolarcoupling strengths in the sub-gigahertz regime can beeasily achieved between magnons and microwave pho-tons, which means fast operation and transduction be-fore decoherence. In addition, their frequency can bereadily manipulated by magnetic fields, and thus eitherspatially or temporally varying magnetic fields enablestraightforward adiabatic modifications. More impor-tantly, magnons can provide a wide range of interac-tions. For example, magnons have been demonstratedto mediate coupling between microwave and optical pho-tons via magneto-optic Faraday effects. Magnons alsocan interact strongly with phonons due to magnetoelas-tic coupling. The inherent strongly nonlinear dynam-ics of magnons makes them also very susceptible to in-teractions with other magnons. Meanwhile, the inter-action of magnons and spins such as nitrogen-vacancy(NV) centers in diamond brings new ideas for quantumsensing of magnons and coherent manipulation of spinqubits. From a materials perspective, the complex in-teractions between electric charge currents and magne-tization dynamics, which are key to modern spintronicsconcepts, provides additional pathways of magnon gen-eration and evolution. While most current research ef-forts are focused on quasi-classical magnon coupling toother quantum systems, recent studies of non-classicalmagnon states and the discovery of atomically thin mag-netic materials show promise for observing and utiliz-ing new genuinely quantum effects in magnon dynamics.In this review, we explore the evolving boundary be-tween quasi-classical behavior and quantum interactionsenabled by the rich physics of magnonics. We delineatethe ongoing materials and device engineering efforts thatare critical to enabling the next generation of quantumtechnologies based on magnons, which will significantlybroaden the scope of fundamental quantum research andaugment the functionality in quantum information pro- cessing.While many similar interactions may also be possiblewith individual spins, the collective dynamics of the mag-netically ordered systems provides distinct advantageswith easily detectable signals and high coupling efficien-cies while maintaining reasonable quality factors up to ∼ . At the same time, due to the aforementionedbeneficial scaling properties, hybrid magnon systems arevery well suited to on-chip integration, as will be dis-cussed in greater detail in Sec. II. In addition, their dy-namics can provide pronounced non-reciprocities, whichare beneficial for the unidirectional flow of quantum in-formation.In Section III, we discuss how layered antiferromag-nets are exemplary materials for the study of magnon-magnon interactions. Both synthetic antiferromagnetsand van der Waals magnets have an antiferromagneticinterlayer exchange interaction that enables acoustic andoptical magnons. Unique to these materials is that theinteraction between acoustic and optical magnons (oreven amongst optical and acoustic magnons) can be con-tinuously tuned via field orientation, wave number, or magnetic damping. As a consequence, the magnonenergy spectrum can be manipulated via the tunablemagnon-magnon interaction strength. The examples wehighlight have weaker antiferromagnetic interactions thatenable optical magnons to exist at GHz frequencies, com-patible with the characteristic frequency range of existingqubit technologies. The strategies used in these materi-als can be potentially deployed in antiferromagnets withcharacteristic THz magnons.In Section IV, we explore optically addressable solid-state qubits derived from atomic defects in semiconduc-tors and their coupling to magnons in magnetic mate-rials. In particular, color centers like the NV center indiamond combine the ability to optically detect and ma-nipulate the qubit spin state with high sensitivity to prox-imal magnetic fields, making them excellent sensors forprobing magnons and potential quantum transducers forupconverting GHz qubit excitations to optical photons. In addition to sensing with solid-state defects, we discussthe important materials engineering considerations foron-chip quantum hybrid systems based on magnons andsolid-state defects, from theoretical to materials consider-ations for both the defect material itself and the magneticmaterials that host spin wave excitations.In Section V, we discuss novel magnetic excitationsand effects in quantum materials and their detection viaoptical spectroscopy. In particular, we highlight threefamilies of magnetic quantum materials that have drawnextensive research interest, including two-dimensional(2D) magnetic atomic crystals, strong spin-orbit cou-pled (SOC) J eff = 1 / d transitionmetal oxides, and assembled molecular chiral spin andmagnon systems. We show the recent development ofhigh-sensitivity, symmetry-resolved magneto-Ramanspectroscopy in detecting magnetic excitations in 2Dmagnets and SOC magnets and the progress of ultra-sensitive magneto-optical Kerr effect (MOKE) of Sagnacinterferometers in probing the chirality induced spinselectivity (CISS) effect in molecular spin and magnonsystems. For each highlighted development, we furthercomment on the prospects of controlling the magneticproperties and integrating them with spintronic devices. II. DIRECTIONS FOR “ON-CHIP” QUANTUMPLATFORMS WITH HYBRID MAGNONICSA. Magnon-Photon Coupling and Superconductingresonators
Magnon-photon hybrid dynamic systems have re-cently attracted great attentions due to their richphysics, strong coupling and convenience for microwaveengineering.
Specifically, the emerging field of quan-tum magnonics aims to realize the potential offeredby magnons for quantum information, with the re-cent demonstration of coherent coupling between a sin-gle magnon and a superconducting qubit and single-shotreadout of magnon numbers. As collective excita-tions of exchange coupled spins, magnons enable largecoupling strengths to microwave photons due to thehigh spin density in magnetic materials. Furthermore,magnons exhibit excellent frequency tunability and caninteract with different excitations such as optical pho-tons, phonons and spins, making them suitable for ex-ploring fundamental physics in coherent dynamics as wellas constructing highly tunable quantum transducers.On-chip implementation and integration of quantummagnonic systems are highly desirable for circuit-basedapplications and building up complex networks of cou-pled magnonic systems.
One direct approach is tointegrate magnetic devices with coplanar superconduct-ing resonators, which can carry long-coherence mi-crowave photons and couple with superconducting qubitsfor building up circuit quantum electrodynamics (cQED).The coupling strength between magnons and mi-crowave photons can be expressed as: g = γ (cid:114) N µ ¯ hω p V c . (1)where ω p denotes the photon frequency of the microwavecavity, N denotes the total number of spins and V c de-notes the effective volume of the microwave cavity. Notethat N can be converted to the effect magnetic volume V M by N µ B = M s V M where M s is the magnetizationand µ B = γ ¯ h is the single Bohr magneton momentum.The factor of 1 / g with a limited magnetic device volume V M (or N ), it is impor-tant to have a small V c , which leads to a large couplingstrength per Bohr magneton defined as g = g/ √ N .Recently, Li et al. and Hou et al. have demon-strated strong magnon-photon coupling between permal-loy (Ni Fe , Py) thin-film devices and coplanar super-conducting resonators. As shown in Figs. 1(a) and (b),in both works a Py device was fabricated in the mid-dle of a superconducting resonator, which is a copla-nar waveguide with two capacitive couplers on the twoside defining the half wavelength and the resonant fre-quency. With the microwave transmission data shownin Figs. 1(c) and (d), an in-plane magnetic field was ap-plied to modify the magnon frequency of the Py device.Clear avoided crossings are observed when the magnonmode intersects with the resonator photon mode. Li et al. have achieved a coupling strength of 152 MHzwith a 900 − µ m × − µ m × − nm Py stripe and Hou etal. have reported a coupling strength of 171 MHz for a2000 − µ m × − µ m × − nm stripe. In addition, both re-ports have shown that the coupling strength scales with √ V M by changing the thickness or the length of the Pydevice, agreeing with the prediction of Eq. (1).The key for reaching sub-gigahertz coupling strengthwith small Py devices, as compared with the macro-scopic yttrium iron garnet (YIG) crystals is thatthe effective volume of the coplanar microwave resonatoris much smaller than 3D cavities. Li et al. have esti-mated V c = 0 . , leading to efficient coupling of g / π = 26 . et al. have obtained a similarsensitivity of g / π = 18 Hz. In addition, a high qualityfactor of the superconducting resonator also ensures largecooperativity which is important for coherent operationand transduction. At a temperature of 1.5 K, the twoworks show quality factors of Q = 7800 from Ref. 25 and1520 from Ref. 26 for their superconducting resonator.Although the quality factor will be reduced when the Pydevice is fabricated, a large cooperativity of C = 68 and 160 can still be achieved and the strong couplingregime is obtained.The beauty of coplanar superconducting resonators isthat they can be designed with great flexibility whilestill maintaining a high quality factor. To further re-duce the effective volume, a lumped element resonator(LER) design has been used by McKenzie-Sell et al. and Hou et al. . As illustrated in Fig. 2(a), the LERconsists of a small wire inductor and a large interdig-itated capacitor, which minimize the inductive volumefor large magnetic coupling efficiency while maintaininga balanced resonant frequency with a large capacitance.The inverse design with minimized capacitor and max-imized inductor has been also used for increasing theelectrical coupling efficiency. In Fig. 2(b), by cover-ing the middle wire inductor with a small piece of 2- µ m-thick YIG film, a magnon-photon coupling strengthof 300 MHz has been achieved. In Fig. 2(c), a couplingstrength of 74.5 MHz has been achieved with a 40 − µ m × − µ m × − nm Py device. The latter corresponds to a M a g n o n P h o t o n H yb ri d m ode S (dB) (a) (c)(b) (d) FIG. 1. Strong magnon-photon coupling between Ni Fe (permalloy, Py) devices and superconducting resonators. (a-b)Optical microscope images of half-wavelength superconducting resonators with Ni Fe devices fabricated at the center, with(a) fabricated from NbN thin film and (b) from Nb film. (c-d) Mode anti-crossing spectra between Ni Fe magnon modes andthe photon modes of superconducting resonators. (a) and (c) are adapted from Ref. 25; (b) and (d) are adapted from Ref. 26. sensitivity of g / π = 263 Hz. This value agrees with aprior report of LER, with g / π = 150 Hz. In Ref. 29the quality factor of the loaded LER can reach Q = 1400,which is sufficient for a clear observation of avoided cross-ing between the magnon and photon modes. (a)(b) (c) FIG. 2. Lumped element resonator (LER) design. (a-b) LERresonator coupled to a flip-chip YIG film grown on a GGGsubstrate, (a) without and (b) with a YIG film. (c) LERcoupled to a Py device fabricated at the center inductive wire.(a) and (b) are adapted from Ref. 24; (c) is adapted fromRef. 26.
For all-on-chip integration of magnon-photon hybrid systems with superconducting resonator, in addition tothe resonator design, the search for low-damping andfabrication-friendly magnetic materials is also crucial.While Py is a classical ferromagnet with convenient depo-sition and fabrication, its relatively large Gilbert damp-ing is not optimal for long-coherence magnon-photon in-teraction. For low-damping YIG thin films, one issue isthat they are typically grown on gadolinium gallium gar-net (GGG) substrates, which possess a complex magneticbehavior at cryogenic temperature and will increasethe loss of excitations if the superconducting circuits arefabricated on GGG substrate. To address this issue, free-standing YIG thin films or YIG films grown on Sisubstrate may provide a solution. Conversely, sinceflip-chip arrangements of YIG/GGG sample have beenused to achieve strong coupling with superconducting res-onators, it is also possible to separate the fabrication ofsuperconducting resonators and the YIG magnonic struc-tures and then mount the two systems together by aflip-chip technique, which has been applied in quantumacoustics. B. Magnetomechanics and Magnon-Phonon CoupledDevices
The study of magnon-phonon interaction dates back tothe 1950s and 1960s, when the strong coupling betweenspin waves (magnons) and acoustic waves (phonons),as well as the resulting magnetoelastic waves (hybridmagnon-phonon states) have been investigated in boththeory and experiments.
In recent years, with theincreasing interests in utilizing phonons as an infor-mation carrier for coherent and quantum informationprocessing, magnon-phonon coupled systems havere-emerged as a promising platform for hybrid magnon-ics. Phonons exhibit very long lifetimes in solid stateplatform such as silicon, quartz, diamond, aluminum ni-tride, lithium niobate, etc., which satisfy the require-ments of a large variety of coherent applications. How-ever, the frequency of phonon modes, although they canbe conveniently tailored by proper geometry engineer-ing, are usually fixed. On the other hand, as a coherentinformation carrier, magnons possess superior tunabil-ity but suffer from their finite lifetimes. Therefore, thereare emerging needs for hybridizing magnons and phononsto combine their respective advantages for coherent andquantum information processing.The most prominent underlying mechanism formagnon-phonon coupling in magnetic media such as YIGis the magneostrictive effect, which links magnetizationwith strain in magnetic materials. Although magnonsrepresent small, typically weak perturbations of the mag-netization, the associated dynamical spin precession canstill efficiently couple with the long-lived phonon modesthanks to the excellent mechanical properties of the ma-terial. In particular, as in magnon-photon coupling, themagnon-phonon coupling is greatly enhanced by the largespin density. In general, the interaction between magnons andphonons, which is referred to as magnetomechanical cou-pling, can take two different forms, considering that themagnon modes usually fall into the GHz frequency rangewhile the phonon modes can be tailored in a broad fre-quency range from kHz or even lower to GHz. In thefirst scheme, the interacting magnon mode and phononmode have identical frequencies and the interaction isdescribed by a beam-splitter type Hamiltonian H int =¯ hG ( mb † + m † b ), where m and m † ( b and b † ) are the cre-ation and annihilation operator of the magnon (phonon)mode, respectively, and G is the coupling strength that isenhanced by the large spin number. In the other scheme,the phonon mode is at a much lower frequency comparedwith the magnon mode, and they interact with eachother with the assistance of a parametric drive, whichcan be described by a radiation-pressure type Hamilto-nian H int = ¯ hGm † m ( b + b † ). In this case, the parametricmagnomechanical interaction is not only enhanced by thelarge spin density but can also be boosted by the strongdrive power.The magnetomechanical devices can take variousforms. In the most commonly used YIG sphere res-onators, radiation-pressure type magnetomechanical in-teraction is experimentally demonstrated by Zhangand co-workers in 2016. Benchmark coherent phe-nomena, including magnetomechanically induced trans-parency/absorption (MMIT/MMIA), parametric ampli-fication, and phonon lasing has been observed, reveal-ing the great applied potential of cavity magnetome-chanics. Compared with optomechanical or electrome-chanical systems, magnomechanical systems benefit fromthe intrinsic magnon properties and exhibit unprece-dented tunability. Moreover, such magnetomechani-cal systems support mode hybridization not only be-tween magnons and phonons but also between magnonsand microwave photons, which further enriches thesystem dynamics. Since then, various novel physi- cal phenomena or applications based on magnetome-chanical interactions have been proposed or experimen-tally studied, including magnon blockade, phonon-mediated magnon entanglement, magnetomechanicalsqueezing, magnon-assisted ground state cooling ofphonon state, etc.Beam-splitter type magnetomechanical interactionsare usually observed in planar structures with YIG thinfilms grown on GGG substrates. Compared with the me-chanical modes in spherical geometries which are typi-cally in the MHz range, acoustic modes in planar struc-tures can be excited at GHz frequencies and thereforecan directly coupled with magnons. Among many acous-tic excitation forms, bulk acoustic waves (BAWs) are of-ten used for hybridizing with magnons. BAW is a type ofvolume excitation with the mechanical displacements dis-tributed in the whole substrate including both the mag-netic YIG layer and the non-magnetic GGG layer. Con-sidering that magnons only reside in the top YIG layer,their interaction is far from optimal. However, because ofthe excellent material property of single crystalline YIGand GGG, the dissipation of BAWs is extremely low. Asa result, the magnetomechanical coupling can still ex-ceed the system dissipations and reach the strong cou-pling regime. Thanks to the non-localized nature ofBAWs, the magnon-phonon strong coupling can be uti-lized to mediate long-range interactions between magnonmodes. In addition to BAWs, surface acoustic waves (SAWs)have also been adopted for coupling with magnons.
Unlike BAW-based magnetomechanical interactions,SAW-based magnon-phonon interactions have been car-ried out on more diverse material platforms. AlthoughSAWs normally suffer from lower frequencies and higherlosses, and usually requires heterogeneous integrationwith piezoelectric materials such as lithium niobate orgallium nitride, they exhibit better mode matching andconsequently improved coupling with magnons becausetheir mechanical displacements are mainly localized onthe surface of the device, which is distinctively differentfrom BAWs. In addition, the unique behaviors of SAWalso enabled the experimental observation of a novel cou-pling mechanism for magnon-phonon interactions thathas been predicted decades ago: the magneto-rotationcoupling, which may lead to novel physical phenomenaand applications.The interaction between magnons and phonons has ledto many novel functionalities. On the one hand, magnonlifetimes are usually on the order of 100 nanosecond while phonon lifetimes can be several orders of magni-tude longer, thus magnon systems benefit from theirhybridization with phonons resulting in significantly im-proved coherence. On the other hand, phonons also gainnew properties that are intrinsically missing in the me-chanical degree of freedom. For instance, through hy-bridization with magnon modes, phonons can inherittheir magnetic characteristics and carry spins or sup-port unidirectional propagation. (a) (b)(c)(d) (e) (f) FIG. 3. (a) Magnetomechanical device consisting of a YIG sphere inside a 3D microwave cavity. The Kittel magnon modein YIG is modulated by the mechanical deformation of the sphere. Adopted from Ref. 48. (b) Planar YIG device supportingmagnon-BAW phonon interaction. BAW phonons in the GGG substrate couple remote magnons modes in the YIG filmson the opposite sides of the substrate. Adopted from Ref. 53. (c) Planar device with a magnetic thin film deposited ontop of a piezoelectric substrate (lithium niobate), supporting magnon-SAW phonon interaction. Adopted from Ref. 59. (d)Scanning electron microscope (SEM) image of a suspended monocrystalline freestanding YIG bridge fabricated using pulsedlaser deposition and lift-off. Adopted from Ref. 63. (e) SEM image of a YIG microbridge structure fabricated using focused ionbeam etching. Adopted from Ref. 64. (f) SEM image of 330 × ×
30 nm Ni nanomagnet which supports strong magnon-phononcoupling. Adopted from Ref. 65.
As of today, most magnetomechanical interactions arereported in macroscopic devices. Recently there havebeen increasing efforts in pushing magnetomechanicaldevices towards smaller scales for better device perfor-mances or integration. In spite of the challenges in YIGfabrication, single-crystal free-standing YIG microbeamshave been fabricated based on unconventional fabrica-tion approaches such as patterned growth and an-gled etching using focused ion beam. As the massand footprint of the resulting magnetomechanical de-vices are drastically reduced, one may potentially furtherenhance the magnon-phonon interaction. By adoptingother easy-to-fabricate magnetic materials, magnetome-chanical strong coupling has been reported in nanoscaledevices.
Moreover, it is proposed that in magneticnanoparticles, the coupling of magnons with the rota-tional degree of freedom of the particle can significantlyaffect the magnon properties, leading to novel opportu-nities for coherent signal processing. Developing novel fabrication techniques will dramat-ically facilitate novel device designs and break many existing experimental restrictions. Therefore, it rep-resents one important future directions for the studyon magnon-phonon interactions. Another promising di-rection involves exploring different materials, such asmultiferroics and antiferromagnets, as new mag-nomechanical platforms. In the context of coherent infor-mation processing, it is critical that these materials pos-sess low losses for both magnons and phonons to ensurecoherent operations can be performed. Currently mostdemonstrations of magnetomechanical interactions arestill in the classical regime at room temperatures. Onestraightforward future direction is to bring the magnon-phonon interactions into the quantum regime at cryo-genic temperatures, to observe quantum magnetome-chanical interactions and perform quantum operationssuch as entanglement, ground state cooling, squeezing,etc.Another distinct advantage of microwave magnonmodes is their short wavelength and low propagationspeed of the dynamic excitations at microwave frequen-cies compared to electromagnetic waves. This makesmagnons highly beneficial for an efficient miniaturizationof microwave components. Phonon based microwave de-vices that incorporate piezoelectric materials have similaradvantages for miniaturization. Traditionally, quartz( α -SiO ) has been the material of choice for piezoelectrictransducers, but more recently materials like LiNbO orLiTaO are often used for high quality peizoelectric trans-ducres, since they can reach mechanical quality factorsapproaching 10 . In addition, SiC and AlN have gar-nered high interest for piezoelectric devices, since thesematerials can be readily integrated with thin film growthprocesses. Using any of these materials, phonons, such assurface acoustic waves, can be coupled to electromagneticradiation via interdigitated transducers, such as the onesshown in Figs. 3(c). Such surface acoustic wave deviceshave formed the basis for many contemporary investiga-tions of magnon-phonon coupling.In fact, more than a century ago the deep connectionbetween spin and mechanical angular momentum has al-ready been established by a pair of pioneering measure-ments. Namely, Barnett showed that the mechanical ro-tation can result in a net magnetization, while concomi-tantly Einstein and de Haas demonstrated the inversephenomenon of mechanical rotation due to magnetizationreversal. This provides in principle a pathway for cou-pling phonons to magnetization and its dynamics. Re-cently, experiments with elastically driven ferromagneticresonance in Ni Fe on LiNbO suggest that such directangular momentum coupling is indeed possible. How-ever, a more efficient pathway for magnon-phonon inter-actions is due to magneoelastic coupling and it was al-ready pointed out early on that the classical understand-ing of this coupling provides the same dynamic equationsas a quantum mechanical description. This magnetoe-lastic coupling gives rise to hybrid magnetoelastic modesor fully mixed magnon-polarons, when the energy of thephonons coincides with the corresponding magnon en-mergies. Since both phonons and magnons also coupledirectly to optical photons, details of their interactionscan therefore be directly investigated via spatially re-solved inelastic light scattering. Furthermore, magnon-polarons may also be important for understanding spincurrents interacting with temperature gradients, e.g. ,for spin Seebeck effects. More importantly, magnetoelastic coupling providesnew opportunities for coherent interactions betweenmagnons and phonons. An important starting pointwas the demonstration that ferromagnetic resonance inNi can be directly excited via surface acoustic waves inLiNbO .. Theoretically, it was shown that the coherentelastic interfacial excitation of the magnetization dynam-ics may be associated with the emergence of evanescentinterface states, which depend on the relative orienta-tion of the magnetization with respect to the phononpropagation. In addition it was shown that the phononpropagation can be modulated due to the magnetoelas-tic interactions in complex ways. Namely, the couplingto the magnons may give rise to a mixing between differ- ent phonon modes, but the precise interactions dependvery sensitively on processing conditions that change theinterfacial microstructure. As the complex modulation of phonon propagation inthe presence of magnetoelastic interaction shows, it isalso possible to generate phonons from magnons. Thiswas directly demonstrated by combining excitations offerromagnetic resonance in Ni wire through rf magneticfields with detection through surface acoustic waves inadjacent LiNbO via interdigitated transducers. Inter-estingly, using inealstic light scattering it was possibleto directly measure the angular momentum associatedwith phonons generated from magnons. Using an adi-abatic conversion of magnons into phonons via spatiallyvarying magnetic fields enabled to separate the magnonand phonon components of the hybridized modes. Fur-thermore, the higher group velocity of phonons can bedetected in pulsed time-dependent measurements andprovides thereby direct verification of the magnon tophonon conversion. Similarly, hybrid magnon-phononmodes may therefore provide significantly faster magnonpropagation. Thus phonons provide new pathwaysfor fast angular momentum transport. Similarly, due totheir long coherence length, phonons may also mediatemagnon coupling over long distances. This was demon-strated by coupled magnon modes for two 200-nm thickyttrium iron garnet films deposited on opposite sides ofa 0.5-mm thick gadolinium gallium garnet substrate. Gadolinium gallium garnet has a very long phonon meanfree path, such that the phonon decay length is about2 mm and therefore exceeds the thickness of the sub-strate. This results in standing phonon modes that giverise to constructive or destructive interference betweenthe ferromagnetic resonance of the two yttrium iron gar-net layers depending on whether the phonon mode num-ber is odd or even. This shows that phonons can providea coherent long distance pathway for angular momen-tum communication, which may have direct relevance forquantum transduction.Another important aspect for quantum devices is non-reciprocity, which allows phase coherent information pro-cessing with a well defined directionality. In magneticsystems, non-recirpocal phenomena are a direct conse-quence of the inherently broken time-reversal symme-try, and has been harnessed for many microwave andoptical devices, such as circulators and isolators. How-ever, if these systems are based on coupling electromag-netic waves to the magnetic system they become invari-ably bulky. Here the coupling of magnons to phononsprovides a distinct opportunity for combining a com-pact device structure with high efficiency performance.Already in the 1970’s surface acoustic wave insulatorswere demonstrated by using yttrium iron garnet filmson gadolinium gallium garnet. More recently, simi-lar non-reciprocal propagation of surface acoustic waveshas also been observed for LiNbO combined with Ni. It turns out that interfacial chiral exchange coupling,the Dzyaloshinskii-Moriya interaction, can shift the hy-bridization gaps relative to each other for the magnon-phonon modes with opposite momenta and thus allowsto optimize the non-reciprocity. Further improvementsare possible by increasing the magnetoelastic coupling,and towards this end an isolation of close to 50 dBnear 1.5 GHz was recently demonstrated by combiningLiNbO with FeGaB. FIG. 4. Hybridization of magnon and phonon modes in thestrong coupling regime. The color map shows the spectraldensity of transient Kerr rotation signals as a function of theexternal magnetic field. The dispersing mode is the ferromag-netic resonance of the Fe Ga crossing the quasi transverse(QTA) and quasi longitudinal acoustic (QLA) modes of thenanograting. The inset shows the clear hybridization gap ofthe ferromagnetic resonance with the QTA mode. Adaptedfrom Ref. 67. The fact that strongly magnetostrictive materials pro-vide superior performance for non-reciprocal devices, ex-emplifies the opportunities that still remain in optimizingmagnon-phonon coupled devices. The majority of inves-tigations have focused on Ni based devices. Ni has mod-erately high magneto-elastic coupling, but also very highmagnetic damping. This means that it is very difficult tofabricate Ni-based systems in the strong magnon-phononcoupling regime. Nevertheless, even with Ni it is possi-ble to have magnon-phonon coupled systems where thecooperativity approaches and start to exceed 1. How-ever, it turns out that Fe Ga can outperform Ni signif-icantly, since it combines strong magnetoelastic couplingwith reasonably low magnetic damping. Using periodicthickness modulation for increasing the magnetoelasticcoupling to specific phonon modes, Godejohann et al. showed that clear magnon polarons in the strong couplingregime can be observed, see Fig. 4. These measure-ments showed relatively long coherence times with deco-herence rates of 30 MHz for the phonons and 170 MHzfor the magnons. Together with the experimentally mea-sured coupling of 200 MHz this results in a cooperativ- ity exceeding 7.8. This shows that a judicious choiceof magnetic materials offers still plenty of opportunitiesto improve device performance. In that respect, towardsthe goal of getting stronger magnetoelastic coupling witha simultaneously reduced magnetic damping, amorphousB- and C-doping of FeGa and FeCo alloys appears to bea promising direction. C. A Brief Theoretical Consideration: Magnon-MediatedOperations via Dynamic Tuning
Hybrid magnonic systems benefit from the wide-ranged tunability of the resonant magnon frequency us-ing changes in bias magnetic field strength. Though ex-isting systems achieving high coupling rates encompass awide variety of geometries ranging from millimeter-scalemicrowave cavities to micrometer-scale supercon-ducting coplanar resonators, all of these systemsfunction via resonant coupling between the electromag-netic and magnonic resonators. This resonant couplingoccurs when the resonant magnon frequency is near theresonant frequency of the photonic resonator. Chang-ing the strength of the bias magnetic field and, thus, themagnon resonant frequency allows one to continuouslytune the hybrid system on and off the resonance and ob-serve the famous anti-crossing behavior of the coupledoscillator’s frequencies, shown schematically in Fig. 5(a).The behavior of a hybrid magnon-photon system canbe mathematically described by a simple system of equa-tions for complex amplitudes of the photonic a ( t ) andmagnonic a m ( t ) resonator modes: da dt + iω a = − iκ a m ,da m dt + iω m a m − Γ m a m = − iκ ∗ a . (2)Here ω and ω m are the resonant frequencies of the pho-tonic and magnonic modes, respectively; the magnonicfrequency ω m depends on the applied bias magnetic fieldand can be easily tuned in a wide range. Γ m is thedamping rate of the magnonic resonator; the correspond-ing term was, for simplicity, neglected in the equationfor photonic mode a since the photon damping rate Γ p is usually much lower than Γ m . The parameter κ inEqs. (2) is the coupling rate between the magnon andphoton modes and can exceed 100 MHz. Model Eqs. (2) describe linear coupling of two resonantmodes. One can easily find eigen-frequencies of two cou-pled oscillations formed as a result of mode interaction: ω ± = ω + ω m ± (cid:115)(cid:18) ω − ω m (cid:19) + κ . (3)The solutions for Eq. (3) for varying ω m (which, physi-cally, corresponds to varying the bias magnetic field) areshown in Fig. 5(a) and demonstrates the formation ofa mode anti-crossing at the point of the strongest res-onant coupling ( ω = ω m ), where the modes becomestrongly hybridized to prevent degeneracy of their fre-quencies. The gap between the coupled mode frequenciesat the anti-crossing point is proportional to the couplingstrength, ω + − ω − = 2 | κ | .All the existing studies of hybrid magnon-photonic sys-tems focus on quasi-static tuning of the magnon reso-nance frequency ω m , in which the characteristic time τ ofmagnetic field variation is much longer than the magnonlifetime, τ (cid:29) / Γ m . In this case, the hybrid system be-haves as an usual resonance system with statically tun-able resonance parameters described by Eq. (3).The strongly coupled ( | κ | (cid:29) Γ m ) magnon-photon sys-tems, however, may exhibit much richer and much moreinteresting behavior in case of dynamic tuning of the res-onance frequency ω m . If the characteristic time of themagnetic field variation τ satisfies 1 / Γ m (cid:29) τ > ∼ / | κ | ,the parameters of the magnon mode change during thelifetime of a single magnon; at the same time, magnonand photon modes interact over a sufficiently long timeinterval ( τ | κ | > ∼
1) to ensure efficient energy and infor-mation exchange between the modes.In the simplest case of a linearly ramped bias mag-netic field B ( t ) = B + ( dB/dt ) t the behavior of thedynamically-tuned hybrid system can be qualitativelyunderstood from Fig. 5(a), in which the horizontal axisnow plays the role of time. As the bias field B ( t ) comescloser to the resonance value, the two modes hybridize,but the excitation initially localized within the higher(lower) mode, will continue to stay in that mode. Theoverall result of the “passage” shown in Fig. 5(a) isthe transfer of energy and information from the pho-tonic mode to the magnonic one [along the upper branch ω + ( t )], and vice versa [along the lower branch ω − ( t )].Note, however, that this simple picture of an adiabaticpassage is, strictly speaking, quantitatively correct forrelatively slow ramps τ (cid:29) | κ | ; in a more realistic non-adiabatic setting τ ∼ / | κ | a more elaborate model ofhybrid system dynamics, which is described below, mustbe utilized.Mathematically, dynamically-tuned hybrid systemscan be described by the system of equations similar toEqs. (2), but with time-dependent magnon frequency ω m ( t ), determined by the time profile of the bias mag-netic field. By shaping the profile of the pulsed magneticfield one can control how strongly, when, and for howlong the resonant coupling occurs, which, in turn, willdetermine the overall outcome of the “passage”.Below, we illustrate some of the capabilities ofdynamically-tuned hybrid magnon-photon systems us-ing, as an example, a system that consists of two pho-tonic resonators (amplitudes a and a , resonant frequen-cies ω and ω ) and one magnonic resonator (amplitude a m , frequency ω m , damping rate Γ m ). The photonic res-onators are not coupled directly, but both interact withthe magnonic resonator with coupling rates κ and κ , respectively. The equations describing this system is adirect generalization of Eqs. (2): da dt + iω a = − iκ a m ,da m dt + iω m ( t ) a m − Γ m a m = − iκ ∗ a − iκ ∗ a ,da dt + iω a = − iκ a m . (4)In the following simulations we used conservative es-timate for the coupling rates, κ , = κ c = 2 π ·
20 MHz,which is much lower than the experimentally achievablevalues. We assumed the damping value Γ m = 6 . µ s − typical for magnonic resonators made of yttrium iron gar-net.The dynamics of the hybrid magnon-photon system isdetermined by the profile of the pulsed bias magneticfield, i.e., by the time dependence of the magnonic reso-nance frequency ω m ( t ). Here we consider a parabolicallyshaped profile, ω m ( t ) − ω = − λt + ∆ , (5)where the parameters ∆ and λ determine the maxi-mum value and the curvature λ of the pulse as shownin Fig. 5(b). The characteristic passage time in this caseis τ ∼ (cid:112) κ c /λ .In the adiabatic limit τ (cid:29) | κ c | the parabolic “passage”has no effect on the final populations of the interactingmodes, as the magnonic frequency ω m ( t ) crosses each ofthe photonic modes twice. The situation, however, is dif-ferent in the non-adiabatic regime τ ∼ /κ c , in which thefinal populations of each mode depend on the pulse pa-rameters λ and ∆. This means that by simply changingthe shape of the pulsed magnetic field profile, differentoperations can be realized in the same physical system.One such operation useful in quantum computing is thecoherent exchange of information between the two pho-tonic modes (SWAP operation). Fig. 5(d) shows a simu-lation of this SWAP operation realized using λ = 2 . κ =2 π · .
63 MHz/ns , ∆ = − . κ c = − π · . ω , = ω − ω = 2 π · . ω , betweenthe photonic modes. This asymmetry in the SWAP op-eration can clearly be seen in the content of the magnonicmode during passage, shown in the bottom panels ofFig. 5(d). It is important to note, that this SWAP oper-ation is also possible in the case of degenerate photonicmodes.It is also important to note, that the population of themagnon mode is zero after the passage, i.e., the magnonicserves only as a mediator of energy transfer between thephotonic subsystems. Since the passage occurs during0a relatively short time τ (cid:28) / Γ m , the magnon dissipa-tion does not play a crucial role in this process, and theaccuracy of the SWAP operation is close to 100%.Another useful operation in quantum computing is theSPLIT operation, where one coherent information stateis split into a coherent superposition of multiple infor-mation states. Such SPLIT operation can be performedusing the same parabolic-shaped pulsed magnetic fieldprofile Eq. (3) with different values of the curvature λ and maximum frequency ∆. Simulation of this opera-tion in Fig. 5(e) used λ = 1 . κ = 2 π · .
513 MHz/ns ,∆ = − . κ c = − π · . ω , = ω − ω =2 π · . ω m ( t ) fromEq. (5). The dependence of the instantaneous eigen-frequencies of the hybrid modes for this case is shown inFig. 5(c). If the magnetic field profile is non-adiabatic, τ ∼ | κ c | , then only a partial (say, 50%) exchange of quan-tum state will occur at each of the two points of resonanthybridization. Then, in the time interval between the hy-bridization points the system will exist in an entangledmagnon/photon state. The two components of this statewill interfere at the second hybridization point, and thefinal population of each mode will depend on details ofsingle-magnon dynamics. Experiments of this type willbe useful to investigate single-magnon non-dissipative de-coherence processes, which are difficult to study usingother means.Finally, we note, that similar magnon-mediated co-herent operations, described here for the case of pho-tonic modes, are also possible with phonons and othersolid-state excitations that efficiently couple to magnons.Thus, magnons represent a promising candidate for a uni-versal mediator of coherent quantum information trans-duction in heterogeneous quantum systems. III. DIRECTIONS IN USING LAYERED “SYNTHETICMAGNETS” FOR QUANTUM MAGNONICS
Magnon-photon and magnon-phonon devices inher-ently feature a two-dimensional circuit layout due tothe physical nature of their hybridizing mechanisms.Another emerging direction in the context of quantummagnonics takes advantage of the magnon-magnon cou-pling mechanism induced by exchange-coupled, synthetic magnetic layers.Recently, strong light-matter interaction became akey basis for many experiments in areas of quantuminformation technologies, including processing, storageor sensing.
Magnetically-ordered materials arepromising candidates for reaching the strong couplingregime of coherent excitation exchange, due to theirhigh spin densities. Antiferromagnetic systems intrin-sically possess two magnon modes, typically referred toas acoustic and optical modes due to their sub-lattice natures. Their dynamics is characterised by thestrength of the exchange coupling and antiferromagneticcrystals tend to show a large coupling strength, bring-ing the frequency of their dynamics into THz regions. Inthis regard, synthetic antiferromagnets and layeredantiferromagnets are especially appealing in terms ofaccessibility since their weaker interlayer exchange cou-pling nature allows GHz resonances which can alsobe easily tuned by growth/material parameters.
Asillustrated in Fig. 6, this Section will broadly be con-cerned with providing an overview of how the magnon-magnon coupling mechanism can be controlled layeredantiferromagnetic materials.
A. Magnon-Magnon Coupling in SyntheticAntiferromagnets
We here review recent progress of magnon-magnoncoupling of uniform spin-wave modes in syntheticantiferromagnets. To hybridize the acoustic and opti-cal modes, the twofold rotational symmetry of the twomagnetisation vectors about the dc magnetic field needsto be broken. This is , e.g., by applying a dc magneticfield ( B ) at an angle θ B with respect to the film plane;see the schematic coordinate system in Fig. 7(a). Thedemagnetization field ( B s ) arising from the thin-film na-ture prevents a full alignment of the magnetisation tothe applied dc magnetic field. This breaks the rotationalsymmetry and enables an interaction between the twomagnon modes. Furthermore, the strength of the inter-action increases as the out-of-plane angle increases. Thisintroduces an in-situ control over the coupling strengthbetween the two antiferromagnetic modes.
In the case of a small interlayer exchange field B ex orfor the in-plane condition, the acoustic and optic modes[see Fig. 7(a)] cannot interact strongly at their point de-generate a energy, showing a mode crossing. This degen-eracy can be lifted by having large B ex and a small θ B ,where an avoided crossing starts to emerge as a signa-ture of mode hybridisation, as shown in Fig. 7(b). Inthis regime, the energy transfer between the two modestakes place at the rate of g /2 π within the magnon sys-tem. Here, the engineering of B ex and in-situ θ B tuningacts as a valve of the mode coupling strength as depictedin Fig. 7(a).Sud et al. experimentally show strong magnon-magnon coupling in a CoFeB/Ru/CoFeB synthetic1 Bias magnetic field strength Time Time C oup l e d fr e qu e n c i e s λ ∆ Time (ns)(a) (b) (c)(d) (e) (f)
FIG. 5. (a) Dependence of the coupled mode frequencies ω ± Eqs. (3) on the bias magnetic field strength (i.e., magnonic modefrequency ω m ). Dashed lines show the frequencies of uncoupled photonic ω and magnonic ω m modes. The color of the branchesdenotes the distribution of the coupled mode energy between the magnonic (green) and photonic (blue) resonators. (b) Timedependence of the instantaneous eigen-frequencies of Eqs. (4) for a parabolic profile Eq. (5) of the bias magnetic field. Theparameter ∆ determines the maximum value of the magnon frequency ω m , while λ is proportional to the curvature of theparabola. The color of the branches denotes the distribution of the coupled mode energy between the interacting resonators.(c) Time dependence of the instantaneous eigen-frequencies of Eqs. (2) for a parabolic profile Eq. (5) of the bias magnetic field.The color of the branch denotes the content of the mode (green – magnonic mode a m , blue – photonic a mode). (d) Temporalprofiles of the asymmetric SWAP operation using a parabolic pulsed magnetic field profile with λ = 2 . κ , ∆ = − . κ c , and∆ ω , = ω − ω = 2 π · . a mode (blue) is coherently transduced to a mode (red). Right panels: information originally contained in a mode (red) is coherently transduced to a mode (blue).Bottom panels show the population of the magnonic mode a m during the process. (e) Temporal profiles of the populations ofthe interacting modes during the SPLIT operation realized with a parabolic field profile using λ = 1 . κ , ∆ = − . κ c , and∆ ω , = 2 π ·
12 MHz. Left panels: information originally contained in a (blue) mode is split into a and a (red) modes. Rightpanels: information originally contained in a mode is split between a and a modes. Bottom panels show the population ofthe magnonic mode a m . (f) Temporal profiles showing the reversibility of the SPLIT operation while using the same parametersof the field pulse as in Fig. 5(e). Initial states in the simulations correspond to the final states of the simulations shown inFig. 5(e). In both cases (left and right panels) the initial state is an entangled state with different phase relations betweenpartial a and a photonic modes for two panels. Repeating the SPLIT operation successfully disentangles the states into apure a (left panel) or a (right panel) state. antiferromagnets. Figure 7(b) represents spin-wavespectra measured by using a broad-band microwavetransmission line with the sample placed on top of it,as a function of microwave frequency and dc magneticfield. For smaller θ B a clear gap is showing at the de-generacy point of the acoustic and optic modes. The ex-tracted coupling strength g as a function of the magneticfield angle is shown in Fig. 7(c) for two different thick-ness of the non-magnetic Ru layer, which shows excellentagreement with the following equation derived from thethe LLG equation for coupled magnetic moments at themacrospin limit: g = γB ex B B s + 4 B ex cos θ B . (6)As shown in this equation, g is maximised for small θ B and large B ex , which are well-demonstrated by their ex-periments.As the resonance frequencies of the modes experi- ence a level splitting during coupling, the magnetic fieldlinewidth ∆ B shows an attraction. Figure 7(d) plots ∆ B as a function of the dc magnetic field for an angle of 40 ◦ .At the point of interaction ( B ≈
275 mT) the linewidthare moving together. Similarly, the frequency linewidthor loss rates (1 /τ ac ) and (1 /τ op ) are attracted and eventu-ally average at the point of degeneracy, as the couplingbecomes stronger [see Fig. 7(e)]. This reflects that inthe coupled state the acoustic and optic mode no longerare individual and form a single hybrid system with acombined loss rate. Similar results were also found inmagnetic hybrid structures.
The magnon-magnon coupling can be modelled by con-sidering two coupled Landau-Lifshitz-Gilbert equations,where specifically a mutual spin pumping is assumed.
The mutual spin pumping describes a spin current ex-change between the two ferromagnetic layers and addsan additional damping to the dynamics, apart from theGilbert damping. This term is crucial to model the ex-2
Out-of-plane In-plane H ext H ext H ext OPTICALACOUSTIC OPTICAL λ λ
Increase surface damping
INTERIOR-OPTICALSURFACE-OPTICAL
Increasing:
Out-of-plane field -or- λ Decreasing:
Surface damping F r e q u e n c y Field H ext ACOUSTIC (a)(b)
FIG. 6. (a) Illustration of three pairs of magnonic excitations that can be coupled together within a layered antiferromagnet.In the context of synthetic antiferromagnets, blue arrows represent magnetization directions two magnetic layers. Alternatively,in a bulk sample, each arrow represents the magnetization of two magnetic sublattices. The red arrows are used in the caseof a layered magnetic material with four magnetic layers, and represent the interior layers while the blue arrows represent thesurface layers. From left-to-right the following pairs of magnons can be coupled together: 1) A spatially uniform optical andacoustic magnon, 2) A finite wavenumber optical and acoustic magnon, 3) An optical magnon residing on the interior andsurfaces of a layered system comprised of four magnetic layers. (b) In these three instances, the magnon-magnon interactioncan be adjusted by rotating an external field out-of-plane, changing the wavenumber, or changing the magnetic damping onthe surface layers relative to the interior layers. perimental findings accurately, especially the loss ratesof the hybrid system. Notably, extended simulation worksuggests that the magnon-magnon coupling is, partially,mediated by spin currents, as shown in Fig. 7(f). Thesimulated plots depict resonance frequencies and lossrates as a function of dc magnetic field at an angle of27 ◦ for different values of the damping contribution fromspin pumping α sp . The results suggest that an increaseddamping due to spin pumping can lead to a resonancefrequency crossing and a loss rate splitting, similar asin cavity magnonics. However, further experimentalstudies are required to fully confirm this.In conclusion, synthetic antiferromagnets are highlypromising for novel spin-wave states in spintronicdevices, high speed information processing andstorage, due to their precisely engineerableproperties and in-situ tuning capabilities. Together, these material systems are adaptive for future applica-tions.
B. Magnon-Magnon Coupling in 2D van der WaalsMaterials
Strong magnon-magnon coupling has been achievedat the interface of two adjacent magnetic layers.
To realize magnon-magnon coupling within a singlematerial, antiferromagnetic or ferrimagnetic materialswith magnetic sublattice structures are required. How-ever, conventional antiferromagnetic resonances lie inTHz frequencies which require specialized techniques toprobe.
Here, we review the recent observation ofmagnon-magnon coupling in the layered van der Waalsantiferromagnetic (AFM) insulator CrCl . Fig. 8(a) il-3 θ B (°)90 70 50 30 101.41.00.60.2 Lo ss R a t e / π ( G H z ) τ op τ ac (e) ω op ω ac g ≥ / τ ac , / τ op g ≈ small B ex or θ B = π /2 large B ex and θ B ≪ π /2 B x y θ B z B B B (T)8121620 f ( G H z ) f ( G H z ) B (T)0.5 0.7 0.160.120.080.040.02 P ( a . u . ) P ( a . u . ) (b)(a) (f) B (T)0.2 0.3 0.4 0.5 0.60.50.40.31.00.52.51.50.5 Lo ss R a t e / π ( G H z ) R e s onan c e F r equen cy ( G H z ) α sp α sp α sp α sp α sp α sp B (T)0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 B (T) 0.40204060 Δ B ( m T ) TheoryTheoryAcoustic DataOptic Data θ B = 40° (d) Theory0.5 nm Ru0.6 nm Ru g / π ( G H z ) θ B (°)90 70 50 30 10 (c) FIG. 7. (a) Schematic of the coupling process between acoustic and optic magnon modes. The coupling strength is controlledby engineering the exchange coupling between the magnetic layers and by dc magnetic field orientation. The coordinate systemsshows the definition of the magnetic field angle θ B . (b) Measured and simulated microwave transmission through a CPW with aSAF on top as a function of microwave frequency and dc magnetic field. The two columns are presenting the results for differentmagnetic field angles. (c) Extracted coupling strength g as a function of the dc magnetic field angle θ B for two different spacerlayer thickness between the ferromagnetic layers of the SAF. (d) Magnetic field linewidth ∆ B of the acoustic and optic magnonmodes as a function of dc magnetic field, applied at an angle of 4 ◦ . At the point of degeneracy at B ≈
275 mT the linewidthare merging towards a single value. (e) Calculated loss rates of the acoustic and optic magnon modes as a function of the dcmagnetic field angle θ B . The solid symbols are results from calculations assuming a coupling, while the empty symbols showresults without a coupling. (f) Simulated resonance frequencies and loss rates of the acoustic and optic modes as a function ofthe dc magnetic field for θ B = 27 ◦ . Parameters are the same as presented in Ref. 5, where only the damping contribution frommutual spin pumping α sp is changed. The simulation results suggest that mutual spin pumping is partially contributing to thecoupling process. lustrates the magnetic structure of CrCl below the Ne´eltemperature T N ≈
14 K, which shows parallel intralayeralignment and antiparallel interlayer alignment of mag-netic moments. The alternating magnetization across thelayers can be modeled as two sublattice magnetizationunit vectors ˆ m A and ˆ m B in the macrospin approxima- tion. When the external field is applied in the crystalplane, the system is symmetric under twofold rotationaround the applied field direction combined with sublat-tice exchange. This symmetry results in two decoupledmodes, the optical mode and the acoustic mode, witheven and odd parity under the symmetry, respectively.4 FIG. 8. (a) Magnetic structure of bulk CrCl below the Ne´el temperature, and without an applied magnetic field. The bluespheres represent the Cr atoms. The red arrows represent the magnetic moment of each Cr atom with parallel intralayeralignment and antiparallel interlayer alignment. The net magnetization direction alternates between layers, having directionˆ m A ( ˆ m B ) on layers in the A (B) magnetic sublattice. (b) Schematic illustrations of the precession orbits for the two sublatticemagnetizations in the optical mode and the acoustic mode. (c) Experimental schematic featuring a coplanar waveguide (CPW)with a CrCl crystal placed over the signal line. The DC magnetic field is applied along the H || direction. (d)(e)(f) Microwavetransmission as a function of frequency and applied field at 1.56 K; the field is applied at an angle of (d) ψ = 0 ◦ , (e) 30 ◦ , and(f) 55 ◦ from the sample plane. (g) Microwave transmission versus applied field at ψ = 55 ◦ for various frequencies, showing thecoupling gap. (h) The coupling strength g increases with ψ , and can be tuned from 0–1.37 GHz. Adapted from Ref. 4. Fig. 8(b) illustrates the oscillation orbits of the two sub-lattices magnetization in these two modes. To detectthe magnetic resonance, a CrCl crystal is placed on acoplanar waveguide (CPW), with crystal c axis normalto the CPW plane and field applied in H || direction, asshown in Fig. 8(c). The microwave transmission signalas a function of frequency and in-plane applied field at T = 1 .
56 K is plotted in Fig. 8(d), which shows an opticalmode with finite frequency at zero field, and an acous-tic mode with frequency linear in field. The dispersionof the two modes can be understood using the two sub-lattice Landau-Lifshitz-Gilbert (LLG) equations, whosesolutions result in blue and red dashed lines for the op-tical and the acoustic modes, respectively. By fitting thefrequency dispersion, the interlayer exchange couplingfield µ H E = 101 mT and the saturation magnetization µ M s = 409 mT are obtained, which are consistent withmagnetometry measurement results. A slope changeof the acoustic mode dispersion occurs at field H = 2 H E because the two sublattices are aligned with the appliedfield when H > H E . In this field range, the acousticmode transforms into uniform ferromagnetic resonancemode, and the Kittel formula is utilized to fit the res- onance dispersion. Note that the acoustic and opticalmodes cross without interaction, and this mode crossingis protected by the two-fold rotation symmetry when fieldis applied in the crystal plane. In principle, breaking thissymmetry can hybridize the two modes and induce ananti-crossing gap, which can be realized by tilting the ap-plied field direction at an angle ψ with respect to the crys-tal plane. Fig. 8(e) shows the microwave transmissionsignal with field applied at angle ψ = 30 ◦ , which demon-strates a coupling gap generated by the out-of-plane field.The size of the gap increases at angle ψ = 55 ◦ , as shownin Fig. 8(f). The modes evolution can be understood asan eigenvalue problem of a two by two matrix derivedfrom the coupled LLG equations, whose solutions resultin black dashed lines. The coupling strength g/ π is de-termined as half of the minimal frequency spacing of themodes dispersion, with g/ π ≈ . ψ = 55 ◦ . This is larger than the dissipation rates of theupper and the lower branches, with values κ U / π ≈ . κ L / π ≈ . ψ = 55 ◦ . The angular dependence of g is shown inFig. 8(h). By rotating the crystal alignment in an ex-5ternal field, the system can be tuned from a symmetry-protected mode crossing to the strong coupling regime.In summary, AFM resonances in CrCl have beenobserved with frequencies within the range of typicalmicrowave electronics ( <
20 GHz) because of the weakanisotropy and interlayer exchange coupling. This es-tablishes CrCl as a convenient platform for studyingAFM dynamics. Moreover, strong magnon-magnon cou-pling within a single material is realized by symmetrybreaking induced by a finite out-of-plane field. BecauseCrCl is a van der Waals material which can be cleavedto produce air-stable monolayer thin films, these re-sults open up the possibility to realize magnon-magnoncoupling in magnetic van der Waals heterostructures bysymmetry engineering. While we discuss spin dynamicsand mode hybridization of antiferromagnetically-coupledmoments in CrCl , where the exchange interaction is rel-atively weak due to the interlayer exchange coupling, itis also possible to study magnon modes arising from in-tralayer exchange coupling within a CrCl layer. Due tothe much stronger exchange strength, these modes residein high energy states, out of reach of microwave tech-niques, yet accessible by optical techniques. This areaof development is summarized in Sec. V in this reviewpaper. C. Magnon-Magnon Coupling Mechanisms in LayeredSystems
As we have now discussed, the magnon-magnon in-teraction between acoustic and optical magnons in bothSAFs, and layered van der Waals antiferromagnets, can be controlled by application of a symmetry break-ing external field. There are other avenues to controlthe coupling between magnons in these types of lay-ered magnets that are being both experimentally andcomputationally explored. To begin this discussion,we first note that the earlier summarized works rely onthe generation of spatially uniform optical and acousticmagnons. In other words, these modes are antiferromag-netic resonances corresponding to a magnon wavenumbernear k = 0. To excite finite wavenumber magnons, onestrategy is to use micro/nanofabrication tools to litho-graphically pattern “meandering” antennae directly ontop of the magnets. In this way, finite wavenum-ber magnons can be excited with an in-plane wavevectorconsistent with the period of the meandering antenna.For k (cid:54) = 0 magnons, the dynamic dipolar magnetic fieldassociated with an optical(acoustic) mode can couple themagnon to the other acoustic(optical) magnon. Shiota etal. realized this dynamic-dipolar coupling of optical andacoustic magnons in SAFs recently in 2020. For a finitevalue of k , the magnon-magnon coupling was observed tobe dependent upon the relative orientation between thein-plane external field and the magnon wavevector. Ad-ditionally, the coupling strength between the optical andacoustic magnons was found to increase monotonically with the wavenumber.It is most often the case that when SAFs are fabri-cated there are only two ferromagnetic layers separatedby a nonmagnetic layer. We now discuss layered magnetswith additional layers, so we will differentiate by countingthe number of magnetic layers. With this definition, themajority of measurements in the literature involve bilay-ers. Thus,the optical and acoustic magnon modes thatare usually reported in SAFs are from bilayers. The uniform optical and acoustic antiferromagnetic res-onances frequencies of bilayers can modeled and obtainedby linearizing a coupled pair of equations that are basedon the Landau-Lifshitz-Gilbert (LLG) equation: d ˆm A dt = − µ γ ˆm A × [ H ext − H E ˆm B − M s ( ˆm A · ˆ z )ˆ z ] , (7) d ˆm B dt = − µ γ ˆm B × [ H ext − H E ˆm A − M s ( ˆm B · ˆ z )ˆ z ] . (8)Models based on these coupled equations are called“macrospin” models, since each layer is treated as a singlelarge magnetic moment. Here, ˆm A and ˆm B refer to thedirection the moment of each layer points. H ext and H E are the external and interlayer exchange fields, and M s is the magnetization of each layer. The above macrospinmodel of a bilayer can be expanded upon. For example,a tetralayer can be modeled as: d ˆm A dt = − µ γ ˆm A × [ H ext − H E ˆm B − M s ( ˆm A · ˆ z )ˆ z ] , (9) d ˆm B dt = − µ γ ˆm B × [ H ext − H E ˆm A − H E ˆm C − M s ( ˆm B · ˆ z )ˆ z ] , (10) d ˆm C dt = − µ γ ˆm C × [ H ext − H E ˆm B − H E ˆm D − M s ( ˆm C · ˆ z )ˆ z ] , (11) d ˆm D dt = − µ γ ˆm D × [ H ext − H E ˆm C − M s ( ˆm D · ˆ z )ˆ z ] , (12)By solving for the eigenvalues of these four coupled equa-tions, it can be shown that two additional optical andacoustic magnon modes each exist in layered magnets. We now discuss how this enables ways to control magnon-magnon interactions in SAFs and van der Waals magnets.It was shown using both a macrospin model and mi-cromagnetic simulations that two optical and two acous-tic magnons can be excited in a tetralayer. The simula-tions performed used the material parameters for CrCl ,but the results can be generalized to include SAFs.Each pair of modes resides in a different spatial regionof the tetralayer. For example, a low frequency opti-cal mode can be excited on the surface layers, while ahigh frequency mode resides on the interior layers. Amagnon-magnon interaction exists between the pair ofoptical modes or the pair of acoustic modes, even with-out a symmetry-breaking external field. Using the optical6 FIG. 9. The optical mode spectra of a SAF-tetralayer is cal-culated via micromagnetic simulations in the top panel. Anavoided energy level crossing, indicative of a magnon-magnoninteraction between two optical branches, is observed. In themiddle and bottom panels it is shown that by increasing thedamping on the surface layers relative to the interior layers, bya factor of 10 and 1000 respectively, the avoided energy levelcrossing closes. In all plots, dark red color indicates a re-gion in field-frequency space where the magnons are stronglyexcited. Dark blue indicates regions where no magnons areexcited. modes as an example, the optical magnon branch on thesurface layers is found to interact with the optical branchof the interior layers due to the exchange field that isgenerated when magnetization dynamics are present oneither surface or interior layers. This magnon-magnon in-teraction is evidenced by the avoided energy level cross-ing present in the optical magnon spectrum shown inFig. 9. Here, we have performed calculations for a SAF-tetralayer assuming that the magnetic layers are permal-loy, and that there is an interlayer exchange coupling of4 . × − A/m. Because the magnon-magnon inter-action is mediated through the dynamic exchange field,there are appealing strategies to control the interaction.In a SAF, it is simple to deposit additional “capping”layers made of a spin Hall metal like Pt, Ta, or W.
In fact, current-induced spin-torque excitations of bothacoustic and optical modes in SAFs have been demon-strated by Sud et al. very recently.
By using a DCcurrent bias, a damping like torque can then be appliedto the surfaces layers.
This damping like torque effec-tively can control the strength of the magnon-magnoninteraction, by suppressing the dynamic exchange field.Thus, a strategy to electrically tune the magnon-magnoninteraction in layered magnets was proposed. To summarize, beyond the use of a symmetry break- ing external field, there are other approaches to tune themagnon-magnon interaction in layered magnets. By ad-justing the magnon wavenumber, and the in-plane ori-entation of an external field relative to the wavevector,the dipolar interaction can couple acoustic and opticalmagnons together. Alternatively, one may consider lay-ered systems with four or more layers. In this case, byadjusting the damping on the surface layers, relative tothe interior layers, magnon-magnon interactions can becontrolled. D. Exceptional points in Synthetic magnets
A more comprehensive overview of exceptional pointsin magnonic systems takes place in the next section.Here, we restrict our discussion to SAFs as an outlookfor future efforts with these materials.In quantum mechanics, non-Hermitian Hamiltonians,invariant under the combination of Parity and TimeReversal operations (PT-symmetric), can harbor excep-tional points (EP) in the eigenvalue structure of theHamiltonian.
To achieve a PT-symmetric Hamilto-nian, a system described by the given Hamiltonian shouldhave an equal balance of both gain and loss. In 2015, itwas theoretically proposed that synthetic magnets wereideal macroscopic systems to look for EPs.
Becausethe LLG equation of motion incorporate damping (loss)through the Gilbert damping parameter, these equationsof motion can be thought of as a classical analog tonon-Hermitian Hamiltonians, having a complex eigen-value spectrum. If two LLG equations are coupled to-gether through an exchange field, it was proposed thatspin-torque effects could be used to modify the damp-ing on one magnetic layer relative to another.
Inother words (anti)damping-like torques can be used toadd (gain)loss to one magnetic layer relative to an adja-cent layer.Theoretical works calculate the EPs by fixing the val-ues of the damping ratio between layers, and varyingthe interlayer exchange field.
In this way, it canbe shown that for a critical exchange field, two com-plex eigenvalues will bifurcate into two real eigenval-ues. These two real eigenvalues are unique from oneanother, and correspond to the optical and acousticmagnon modes that have been discussed throughoutthis section. In 2019, an experimental breakthroughattempted to mimic this theoretical approach.
InPermalloy/Pt/Cobalt magnetic bilayers, the damping ra-tio was fixed and set by the mismatch between thedamping of Permalloy and Cobalt. By changing thePt thickness between samples, both the magnitude andsign of the interlayer exchange field, set via the Rud-erman–Kittel–Kasuya–Yosida (RKKY) interaction, wascontrolled. By experimentally measuring the optical andacoustic spectra across a series of devices, there wasenough sampling of interlayer exchange fields to indicatean exceptional point existed in the Permalloy/Pt/Cobalt7structure.The theoretical and experimental works discussed inthe preceding paragraphs clearly demonstrate how thefrequencies of the optical and acoustic magnons can betuned in the vicinity of an EP. More recent theoreticalwork suggests that other attributes of magnons, such asnon-reciprocal spin wave propagation, can be manipu-lated in the vicinity of an EP.
An unexplored direc-tion to consider involves the magnon-magnon interac-tion between modes near an EP. Once an EP is reached,the acoustic and optical branches coalesce, and only onemagnon mode is present. This would appear to precludea magnon-magnon interaction from existing, as only onemode is present. The evolution of the magnon-magnoninteraction, towards this limit, may offer a new way tocontrol the magnon-magnon interaction through the us-age of EPs. It is also important to note that althoughtheory and experiment tend to focus on bilayers, higherorder EPs have been calculated in trilayers.
Consid-ering that additional magnon-magnon interactions existin layered structures beyond bilayers, the potential inter-play between higher order EPs and these magnon cou-plings is vastly unexplored.
E. Non-Hermitian Physics and Exceptional Points
Non-Hermitian physics, with a focus on open systemswhere energy conservation does not apply, has beenattracting intensive attentions in recent years.
Although non-Hermiticity usually leads to complexeigenvalues and therefore is not favorable when char-acterizing quantum systems, it has been recognizedas the origin of many novel physics that extendbeyond quantum mechanics. As a ubiquitous phe-nomenon, non-Hermiticity has been studied in manydifferent physical realizations based on platforms inelectronics, microwave, acoustics, optics/photonics, optomechanics andmagnetics.
Exceptional point (EP) is one of the most in-tensively studied novel phenomena in non-Hermitiansystems.
An EP, sometimes also referred to as abranch point, is a singularity point on an Riemann sur-face for a system with two or more coupled modes. Itis a special type of degeneracy point, where not onlythe eigenfrequencies but also the eigenmodes are degen-erate. The mathematical singularity at EP is accom-panied by a long list of anomalous physical phenomenaand applications. At the EP, novel behaviors such as chi-ral states, unidirectional lasing and high-sensitivitysensing can be achieved. The peculiar behaviorscan be observed beyond the EP itself. By encircling theEP in the parameter space, asymmetric mode conversioncan be achieved.
Most importantly, such modeconversions are topologically protected and therefore isinsensitive to the specific looping paths as long as theEP is encircled. In order to observe the EP, a non-Hermitian systemhas to satisfy the following requirements. First, the sys-tem should contain two or more coupled modes. Thesemodes can have the same physical origin ( e.g. , two opti-cal resonances), or they can be different types of modes(one microwave mode and one magnon mode, e.g. , as willbe shown below). Second, the two coupled modes shouldhave identical frequencies. Alternatively, one mode mayhave a tunable frequency which can vary across the fre-quency of the other mode. Third, strong coupling shouldbe achievable. An EP can be observed at the phase tran-sition from weak coupling to strong coupling, and there-fore strong coupling is a prerequisite for EP.Hybrid magnonic systems consist ofinteracting magnons and at least one other type of in-formation carriers (microwave photons, optical photons,mechanical phonons), and therefore it is natually a non-Hermitian platform, as indicated by the Hamiltonian ofthe system H = (cid:18) ω c − iκ c / gg ω m − iκ m / (cid:19) , (13)where ω c and κ c ( ω m and κ m ) represent the resonancefrequency and dissipation rate of the photon (magnon)mode, respectively, and g is the coupling strength of themagnetic dipole-dipole interaction between magnons andmicrowave photons. Since the magnon frequency can beeasily tuned by an external bias magnetic field, the on-resonance condition can be conveniently achieved. More-over, when magnons are hybridized with microwave pho-tons, the coupling strength is significantly enhanced bythe large spin density in the commonly used magneticmedia (YIG, permalloy, etc.) and exceed the dissipa-tion of both the magnon and microwave photon mode,enabling the strong coupling condition. With all the es-sential requirements satisfied, it is straightforward to ob-serve EPs in a hybrid magnon-microwave photon system.Exceptional point in hybrid magnonics was first exper-imentally observed in 2017. In this work by Zhang etal. the coupling strength between the magnon mode ina YIG sphere and the microwave resonance in a three-dimensional (3D) copper cavity is adjusted by changingthe position of the YIG sphere. This changes the fieldoverlap between the two modes and accordingly affectsthe coupling strength. Scanning the sphere position in-duces a phase transition from weak coupling to strongcoupling and the unambiguously reveals an EP. In addi-tion, it is also proposed that multiple EPs or high-orderEPs can be supported on this type of hybrid magnonicsystems, which are highly desired for topological modeconversion or EP-based sensing.There is increasing interests in recent years in pur-suing high-dimensional EPs. The concept of syntheticspace has been utilized to convert the requirementsfor higher dimensions to a larger parameter space. How-ever, increased dimensions are accompanied by increasedtunability, which is not readily achievable in many phys-ical platforms. This severely limits the experimental8demonstration of high dimensional EPs, restricting pre-vious reports to just a handful of investigations on ex-ceptional rings and practically preventing the di-rect observation of exceptional surfaces (ES) – a surfaceformed by a collection of EPs.
Taking advantageof the excellent flexibility in hybrid magnon-microwavephoton systems, a four-dimensional synthetic space isconstructed in the work by Zhang et al. , which leads tothe first experimental observation of an ES in a non-Hermitian system.
Unique anisotropic behaviors areobserved on a special exceptional saddle point on the ES,which can enable multiplexed EP sensing as previouslyshown in lower dimension systems.The aforementioned EP observations treat the hy-brid magnonic systems as effective parity-time (PT)symmetric systems, with EPs representing the onsetof the PT symmetry breaking. Interestingly, anti-PT-symmetry –the counterpart of PT symmetry–hasalso been demonstrated in hybrid magnonics.
Inthese demonstrations, purely imaginary coupling be-tween two interacting modes can be obtained throughdissipative coupling, resulting in properties that are con-jugate to PT-symmetric systems. In anti-PT-symmetrichybrid magnonic systems, EPs are also observed at thetransition from the anti-PT-symmetric phase to the anti-PT-symmetry-broken phase. In addition, such systemsallow the observation of other intriguing phenomena suchas bound-state-in-continuum (BIC), where the hybridmodes exhibits maximal coherence as well as slow lightcapability.Time reversal symmetry breaking is another featureof interest in non-Hermitian systems.
In a hybridmagnonic system, this can be conveniently achieved dueto the magnetic nature of magnons. The precessional mo-tion of magnons indicates that they can only coupled withone polarization but not the orthogonal one if microwavephotons are circularly polarized. As a result, time rever-sal symmetry can be broken on a cavity supporting circu-larly polarized microwave resonances, which can furtherresult in nonreciprocal transmission if the photon polar-ization is port-dependent.
In an alternative approach,time reversal symmetry can be broken using a linearlypolarized λ/ Reversing the signal propaga-tion direction changes the phase of the cavity field atthe position of the YIG sphere, which in turn inducesnonreciprocity in the cavity transmission. Through cou-pling strength engineering (in the first case) of dissipationengineering (in the second case), tunable nonreciprocityand unidirectional invisibility can be achieved in hybridmagnonic systems.The study of non-Hermitian physics in hybridmagnonic systems is still in its very early stage. Withits remarkable diversity and flexibility, there are enor-mous opportunities for further exploring novel non-Hermitian physics and applications in hybrid magnon-ics. For instance, the combination of the large tunabilityof magnons and their excellent compatibility with otherinformation carriers, together with their rich nonlinear- ity and capability of breaking time reversal symmetry,may lead to extraordinarily novel behaviors. But alongthe way there are a number of technical obstacles, suchas the limited lifetime and semi-static coupling strength,need to be addressed. The expected outcomes of non-Hermitian hybrid magnonics will not only be primarilyin the classical regime, but may also be readily appliedto quantum information science.
IV. DIRECTIONS FOR QUANTUM SENSING ANDHYBRID SOLID-STATE QUANTUM SYSTEMSA. Introduction to optically addressable solid-state defects
Atomic defects in semiconductors, such as the neg-atively charged nitrogen-vacancy (NV − ) center in di-amond, are a promising platform for quantum tech-nologies due to their long coherence times, atom-ically defined localization, and optical interface forspin initialization and readout. Over the pastdecade, extensive experimental and theoretical effortshave been devoted to realizing quantum networks, registers, and memories in NV-based hybrid quan-tum systems.
The coupling between solid-statequbits and magnetic materials, in particular, un-derlies varied applications in quantum informationscience.
In the weak coupling regime, the sen-sitivity of single spin qubits to local magnetic fields pro-vides a non-invasive, nanoscale probe of magnon physicsover a wide range of temperatures.
As in-teractions between spin qubits and magnons increase, ahybrid system is formed that enables possible coherenttransfer of information between the two systems. In thisstrong coupling regime, magnon-mediated spin interac-tions and driving become accessible, providing a futureroute toward on-chip coherent quantum control andentanglement of atomically localized spin qubits.
Here we address the properties of solid-state qubits andtheir interactions with magnetic fields that lend them-selves to integration and coupling with magnonic systemsfor both quantum sensing and coherent control of solid-state spin qubits.To demonstrate the utility of these defect systems assensors and in magnonic hybrid quantum systems, wewill briefly introduce the origin of the spin-photon in-terface and coupling with nearby magnetic excitations,focusing on the diamond NV − center as a canonical ex-ample. An NV center is formed by a nitrogen atom ad-jacent to a carbon atom vacancy in one of the nearestneighboring sites of a diamond crystal lattice. The nega-tively charged NV − state has a spin triplet ground statewith robust quantum coherence and remarkable versa-tility over a broad temperature range. The energy-level structure of the NV − center and its associated op-tical transitions are shown in Fig. 10(a). Photons withenergy hν ≥ .
945 eV excite an electron from tripletground-state (GS) to triplet excited state (ES) in a spin-9
GSES ±1m S = ±1m S = 00radiativespin-conserving nonradiativespin-altering1.945 eV (a) S = 1 S = 02.87 GHz (b) (e)
Frequency (GHz) P h o t o l u m i n e s c e n c e ( a r b . u n i t s ) I n c r ea s i ng B e x t (c)(d) NV SpinB ac m s = 0m s = −1 FIG. 10. The diamond NV center spin-photon interface and coupling with magnetic fields. (a) The energy level structureshowing the spin-dependent transitions of negatively charged NV center, including the spin-conserving radiative transitionsand spin-altering nonradiative transitions. Optical excitation preferentially drives the defect into the bright m s = 0 state,unless an ac magnetic field on resonance with D is applied to mix the m s = ± m s = 0 sublevels, resulting in reducedphotoluminescence. (b) An energy diagram of the NV ESR transitions as a function of external field B ext applied along theNV-axis. In the presence of a magnetic field, the Zeeman effect splits the ground state m s = ± m s = 0 ↔ ± conserving manner (total-spin S = 1 and spin-projectionquantum numbers m s = ± m s = ± m s = 0 sub-level. Because the nonradia-tive transition rates depend on the spin state of the GS,repeating this optical excitation cycle will preferentiallydrive the system into an m s = 0 polarization. This mech-anism provides both 90% fidelity optical spin initializa-tion and spin-dependent optical readout. The GS is split by the diamond crystal field, such thatthe m s = ± D = 2 .
87 GHz fromthe m s = 0 state. Applying a GHz-range ac magneticfield on resonance with D mixes the m s = ± m s = ± has enabled both non-invasive initialization and read-out of qubits for sensing and long-range entanglementexperiments between distant NV centers, a promisingstep toward the development of quantum communicationnetworks. Coherent control of the defect spin state can beachieved with ac-magnetic fields (as described above),but also by strain, and ac-electric fields.
Stronginteractions with magnetic excitations and strain0waves in both the host lattice and nearby materialsmake solid-state quantum defects additionally attractivefor applications in hybrid quantum systems where trans-duction through magnons and phonons occurs at the on-chip level. The addition of their robust spin-photon in-terface positions quantum defects as prime candidates forbridging the gap between GHz-level qubit excitations andoptical photons at the scale of hundreds of THz, whichcould enable long-distance optical quantum communica-tion.In the presence of a local magnetic field applied alongthe NV axis, the Zeeman effect splits the m s = ± γB where the gyromagnetic ratio γ is28 MHz/mT, as shown in Fig. 10(b). The correspond-ing (ODMR) spectrum is thereby split into two field-dependent peaks, as demonstrated in Fig. 10(c). Thisfrequency splitting depends on the orientation of the NVcenter relative to an external magnetic field, whichalso sets the coupling with magnonic systems for bothhybrid systems and NV-based magnetometers.For sensing applications the ultimate dc magnetic fieldsensitivity of NV centers is determined by their electronspin resonance (ESR) linewidths. In addition to dc fields,NV centers also serve as a sensitive probe of ac mag-netic fields. When a microwave field is applied with afrequency matching the spin energy-level-splitting of theNV center, an NV spin will periodically oscillate betweentwo different spin-states in the rotating frame, which isusually referred to as Rabi oscillation as illustratedin Fig. 10(d). The amplitude B ac ⊥ of the microwavefield perpendicular to the NV-axis can be measured bythe Rabi oscillation frequency. Any fluctuating magneticfields at the NV resonance frequency will induce these NVESR transitions. To date, NV centers have been used todetect weak magnetic fluctuations, e.g. Johnson noisegenerated by fluctuating electric charges and magneticnoise associated with spin excitations. Figure 10(e) shows the calculated field dependenceof magnetostatic surface spin wave modes of a yttriumiron garnet film (contained within the shaded blue region,black and red circles indicate experimentally observedfrequencies of surface spin waves and ferromagnetic res-onance modes respectively).
The green lines indicatethe field dependence of the NV − m s = 0 ↔ ± When the external field istuned such that the two transitions overlap, the interac-tion between the NV center and spin waves in the YIGfilm is maximized. This coupling with magnons can bedetected in so-called relaxometry experiments measuringlongitudinal spin relaxation, parametrized by Γ = 1 /T ,the rate at which the nonequilibrium m s = ± T is shortened in the presence ofmagnetic field noise from nearby spin waves at the ESRfrequency, thereby affecting the emitted photolumines-cence. NV centers exhibit millisecond-long spin relax-ation times, enabling field sensitivity down to 10 − Tesla to local static and oscillating magnetic fields.
B. Quantum sensing of magnons in spintronic systemsusing NV centers
In this section, we review the recent progress of NVcenter based quantum sensing platform and its appli-cation to detect magnons in functional spintronic sys-tems. Due to their single-spin sensitivity, NV centershave been demonstrated to be a powerful sensing toolto detect magnetic domains, spin transport, anddynamic behaviors in a range of emergent mag-netic materials. A unique advantage of NV centers re-sults from a combination of the high field sensitivity andnanoscale spatial resolution. The spatial resolution ofan NV center is mainly determined by NV-to-sampledistance.
There are a couple of methods to ensurenanoscale proximity of NV centers to studied materials.The most straightforward way is to deposit materials ona single-crystalline diamond substrate with NV centersimplanted a few nanometers below the surface.
Forcertain materials requiring epitaxial growth on specificsubstrates, patterned diamond nanostructures containingindividual NV centers will be used and transferred ontothe surface of samples.
Thirdly, by employ-ing scanning NV microscopy, where a micrometer-sized diamond cantilever containing individual NV cen-ters is attached to an atomic force microscope, the NV-to-sample distance can be systematically controlled withnanoscale resolution.Next, we briefly review a recent work using NV cen-ters to detect the spin chemical potential in a magneticinsulator.
Figure 11(a) illustrates the schematicof the device structure for the measurements, where a10-nm-thick Pt and a 600-nm-thick Au strip are fabri-cated on a 20-nm-thick ferrimagnetic insulator Y Fe O film. A diamond nanobeam containing individually ad-dressable NV centers is transferred on the surface of theYIG film. The Au stripline provides microwave con-trol of the NV spin states and the local spin excitationsof YIG. The Pt strip provides electrical spin injectionthrough spin Hall effect.
The NV center is posi-tioned ∼ µ m away from the Au strip and the NV-to-YIG distance is estimated to be ∼
100 nm. With amoderate external magnetic field applied in the sampleplane, the minimum magnon energy band of the YIGfilm is below the NV ESR frequencies as illustrated inFig. 11(b). YIG thermal magnons at the NV ESR fre-quencies will induce NV relaxation according to the for-mula: Γ ± = k B Tf ± h (cid:82) D ( f ± , k ) f ( k , d ) d k . Here, f ± and Γ ± are the NV ESR frequencies and relaxation rates of the m s = 0 ↔ ± T is temperature, D ( f ± , k ) is magnon spectral density, k is the magnonwave vector and f ( k , d ) is a transfer function describingthe magnon-generated fields at the NV site. In absence of external spin excitation, NV centers canbe used to probe dispersion relationship of the YIG ther-1
FIG. 11. (a) A scanning electron microcopy image showing diamond nanobeams containing individually addressable NV centerspositioned on top of a YIG film. A 600-nm-thick Au stripline (false-colored yellow) provides microwave control of the magnonchemical potential in the YIG film and the NV spin states. A 10-nm-thick Pt stripline (false-colored gray) provides spin injectionthrough the spin Hall effect. (b) Sketch of the magnon dispersion and the magnon density with zero chemical potential. NVspin probes the magnon density at the NV ESR frequencies. (c) Field dependence of the measured NV relaxation rate withoutexternal spin excitation. (d) Spin excitation effectively increases the magnon chemical potential in YIG. (e) The measured spinchemical potential µ as a function of microwave power and external field. µ saturates at the minimum magnon band set bythe FMR frequency. (f) The variation of NV spin relaxation rate Γ and spin chemical potential as a function of the electriccurrent density flowing through the Pt layer. All the figures are taken from Ref. 9. mal magnons. The top panel of Fig. 11(c) shows the mea-surement sequence of the NV relaxometry measurement.A microsecond scale green laser pulse is first applied toinitialize the NV spin to the m s = 0 state. During thetime delay, YIG magnons at the NV ESR frequencieswill couple to the NV spin and accelerate its relaxation.After a delay time t , a microwave π pulse on the cor-responding ESR frequencies is applied to measure theoccupation probabilities of the NV spin at the m s = 0,and m s = ± − shown in the bottompanel of Fig. 11(c) exhibits a maximum in the region thatthe corresponding ESR frequency crosses the minimummagnon band of the YIG film, demonstrating the sensi-tivity of NV centers to noncoherent thermal magnons.When driving the magnetic system to ferromagneticresonance (FMR), spin chemical potential will be estab-lished in the YIG film [Fig. 11(d)], leading to an increasedmagnon density at the NV ESR frequencies and enhanced NV relaxation rates. By measuring the variation of theNV relaxation rates as a function of the driving power,spin wave density and associated spin chemical potentialcan be directly measured, which is independent of manydetails of both the quantum sensor and the magneticmaterial: µ = hf ± (1 − Γ ± (0)Γ ± ( µ ) ). The measured magnonchemical potential under resonant condition saturates tothe minimum magnon energy band set by the FMR fre-quency in the large microwave power regime as shownin Fig. 11(e), demonstrating Bose-Einstein statistics of thermal magnons in a magnetic insulator. In the lowmicrowave power regime, the local magnon chemical po-tential can be systematically controlled by magnetic res-onance, in agreement with the theoretical analysis of theunderlying multi-magnon processes. In addition to magnetic resonance, spin chemical po-tential could also be electrically established by spin Halleffect.
When a charge current J c is applied inthe Pt strip, spin currents induced by the spin Hall ef-fect are injected across the YIG/Pt interface, leadingto variation of the number of the magnons at the NVESR frequencies. When the polarization of the spin Hallcurrent is antiparallel to the YIG magnetization, the in-2jected spin current effectively increases the spin chemicalpotential and the corresponding magnon density at theESR frequencies. When the polarity of J c reverses, spinpolarization is parallel to the YIG magnetization, lead-ing to a reduced magnon density. This electrically tun-able spin chemical potential gives rise to a linear varia-tion of the NV relaxation rate Γ (∆Γ − ) on the appliedelectric current density J c as shown in Fig. 11(f). For J c = 1 . × A/m , Γ (∆Γ − ) is ∼ . µ s − , whichchanges sign when the current polarity reverses, in con-sistent with the spin Hall current injection model. C. Magnonic systems for quantum interconnects
While quenching of T by the broad spectrum of mag-netic noise from spin waves is a useful tool for spin wavesensing via noise spectroscopy, it limits the number of co-herent operations that can be performed on the coupledqubits. Placing the lower branch ( m s = −
1) of the NVtransition spectrum in resonance with a magnetostaticsurface spin wave mode of a YIG film [Fig. 10(e)], itwas demonstrated that directly driving these spin wavescould coherently drive NV centers over a distance of sev-eral hundred microns up to several millimeters,.
In particular, it was found that reducing the microwavepower reduced noise from off-resonant spin wave excita-tions and revealed a considerable enhancement of the lo-cal microwave driving field mediated by spin waves com-pared to the driving field directly coupled to the antenna.As observed in Fig. 12(a), this field enhancement can beas high as a factor of several hundred, coherently driv-ing Rabi oscillations in an NV center at a distance ofover 200 microns away from the antenna strip line.
These strong interactions point to magnons in magneticfilms as a potential on-chip quantum interconnect be-tween qubits.In principle, the magnon-defect interaction is bidirec-tional. A | (cid:105) ↔ | − (cid:105) transition in the NV center cancreate or annihilate a magnon in the proximal magneticfilm. When two qubits coupled to a magnon waveg-uide are tuned to be nearly in resonance with long wave-length magnons in the magnet, the two qubits can theo-retically be coherently coupled over long distances withthe interaction mediated by virtual magnons. Thisscheme was outlined theoretically considering dipolar in-teractions with magnons in a one dimensional ferromag-netic waveguide, as well as along an antiferromag-netic domain wall, shown in Fig. 12(b) and (c). Itis anticipated that two-qubit gate operations should beachievable over distances on the order of 1 µ m at low( ≤ yzxx M (b)(c) Microwave pulse length ( µ s) R e l a t i v e pho t o l u m i ne sc en c e ( a . u . ) µ W, Spin-Wave Driving µ W, Antenna Driving (a)
FIG. 12. Coupling schemes for distant quantum defect spinsmediated by magnons. (a) Microwave driving of surface spinwaves in a YIG film coherently drive Rabi oscillations in NVcenters over 200 µ m away from the antenna, requiring consid-erably less power compared to microwave driving of the NVcenters coupled through the vacuum. Figure taken from. (b) A schematic of the geometry outlined in where two NVspins are entangled by virtual magnons in a one-dimensionalferromagnet. (c) Quantum defect spins mediated by magnonsconfined to a domain wall of width λ in an antiferromagnet.Reprinted figure with permission from Ref. 252. Copyright(2019) by the American Physical Society. the quantum defect and the magnonic material is im-portant for both sensing applications and coherentlycoupled interactions. Early studies typically relied onthe relatively random placement of defects such asthose in drop-cast diamond nanoparticles. Ad-vances in deterministic placement of defects, includingtemplated nanoparticle transfer, nitrogen deltadoping of diamond, local laser annealing, nano-implantation, and placement with scanning probe3microscopy techniques now allow for improved con-trol of qubit positioning.
D. Materials outlook for magnonic hybrid quantumsystems
The materials science challenges of developing opti-mized hybrid systems present an opportunity to exploreemerging spin qubit candidates and magnonic materials.Prior studies of NV center-magnon coupling have focusedon YIG owing to its low Gilbert damping coefficient onthe order of 10 − for thin films, resulting in a longmagnon spin diffusion length, as well as a convenientoverlap of its magnon spectrum with the NV transitionsat modest magnetic fields. YIG films are often grownepitaxially on lattice-matched gadolinium gallium garnet(GGG), which is a paramagnet at room tempera-ture, and can present integration challenges in fabricationof diamond/YIG hybrid systems. Further, the magneticproperties of GGG are complex in the millikelvin tem-perature range relevant to single quasiparticle coherentinteractions, resulting in considerable damping. These materials challenges motivate the search for ad-ditional low-damping magnetic thin films that can oper-ate as quantum interconnects. Some potential candidatesincluding Co-Fe alloys, (Ni, Zn, Al) ferrite, and or-ganic vanadium tetracyanoethylene (V[TCNE] x ). Beyond diamond NV centers, developing quantum de-fects in other host materials, opens up opportu-nities for designing defect-host systems with propertiesspecifically tailored for magnonic hybrid systems. Par-ticularly appealing are defects in semiconductors withmature wafer-scale processing infrastructure, such as sil-icon carbide (SiC). For instance, excellent optical andspin coherence properties have been demonstrated in di-vacancy complexes integrated into SiC classical electronicdevices.
Divacancies in SiC have demonstrated smallerzero-field splitting values than diamond NV centers, which could allow their ESR transitions to more easilycouple to magnetic materials with small magnon gaps atsmall magnetic fields. Further afield, qubits consisting oforganic molecules containing transition metal ions haverecently shown very promising spin and optical proper-ties that can be tuned with ligand chemistry.
Cou-pling such qubits with low-damping organic magnets likeV[TCNE] x could enable all-organic quantum spintron-ics with qubits separated from magnon waveguides byatomic-scale distances with facile deposition and chemi-cal tunability. Overcoming material and fabrication chal-lenges both in more traditional systems (e.g., YIG, dia-mond) and emerging materials is a key priority in the de-velopment of integrated magnon/spin-qubit hybrid tech-nologies. V. DIRECTIONS FOR NOVEL MAGNONICEXCITATIONS IN QUANTUM MATERIALSA. A Brief Background of Magneto-Raman Spectroscopy
Magneto-Raman spectroscopy has been shown to bea powerful tool for characterizing collective excitationsin magnetic materials.
Although the excitationsprobed are restricted to those with zero total momentum,the optical magneto-Raman scattering process has a de-cently large scattering cross-section ( ∼ − cm sr − ),well-defined symmetry selection rules (dominated by theleading-order electric-dipole approximation), ultra-fineenergy resolution ( ∼ . − ), and optical diffraction-limited spatial resolution ( ∼ sub- µ m). As a result, ithas made major contributions to the recent bloomingfields of two-dimensional (2D) van der Waals (vdW)magnetism and magnetism in systems with spin-orbit-coupling (SOC). In this section of this review article, wewill both summarize key recent progress and provide anoutlook for optical magneto-Raman spectroscopy appliedto these two fields.The scattering Hamiltonian by a magnetic system is expressed as H (cid:48) = (cid:88) α,β (cid:88) R E α E β χ αβ ( R ) , (14)where E and E are the electric fields of the incident andscattered light, respectively, at the magnetic site R inthe crystal, and χ αβ ( R ) is the magnetization-dependentpolarizability tensor element at site R , which can be ex-panded in terms of the magnetization operators M R , (cid:18) NV (cid:19) χ αβ ( R ) = (cid:88) µ K αβµ ( R ) M µ R + (cid:88) µ,ν G αβµν ( R ) M µ R M ν R + (cid:88) µ,ν, r L αβµν ( R , r ) M µ R M ν R + r + · · · (15)There are N magnetic sites within the volume of V . Thefirst and second terms include operations on the samemagnetic site R corresponding to single-magnon excita-tions as well as the elastic scattering off the static mag-netic order. The third term contains pairs of operationsat adjacent sites R and R + r representing two-magnonexcitations. The tensor forms of coefficients K αβµ ( R ), G αβµν ( R ), and L αβµν ( R , r ) are subject to the crystallo-graphic symmetry of the magnetic system. Below, we will focus on three types of magnetic ex-citations in 2D vdW and SOC magnetic systems: (a)phonons coupled to the static magnetic order that cor-respond to time-reversal-symmetry (TRS) broken anti-symmetric Raman tensors, (b) single magnon ex-citations that contain contributions from both the firstand second terms in Eq. (15), and (c) two-magnonexcitations involving pairs of magnons from the Brillouinzone boundaries and center.
2D vdW magnetism isan emergent field that started in 2017 with the success-ful isolation of atomically thin magnetic crystals and the4definitive confirmation of ferromagnetism in the mono-layer limit.
While early studies have focused onmagnetic ordering in single- and few-layer samples usingstatic magneto-optics (i.e., magneto-opticalKerr effect (MOKE) and magnetic circular dichroism(MCD)) as well as tunneling magnetoresistance, optical magneto-Raman spectroscopy has contributedunique insights into the dynamic magnetic excitations in2D vdW magnets.
SOC magnetism refers to mag-netic systems where SOC is at an energy scale compa-rable to that of electronic correlations, and it isprimarily realized in 5 d transition metal oxides (TMOs),such as iridium oxides. While the spin dynamicsof 5 d TMOs have been primarily probed by hard x-ray scattering and neutron scattering (if thesample volume is large enough to overcome the challengesposed by the strong neutron absorption by Ir), opticalmagneto-Raman spectroscopy has provided complemen-tary information thanks to its superior energy resolutionand clean selection rules.
B. Example in Phonons Coupled to Static MagneticOrder in a 2D Layered Antiferromagnet
While ordinary phonons and phonons that are zone-folded by magnetic orders are commonly seen in Ra-man spectroscopy, they typically correspond to symmet-ric Raman tensors with TRS.
The observation of TRS-broken phonon modes with antisymmetric Raman ten-sors in bulk and few-layer CrI adds a new vari-ation of phononic excitations, which is now understood asfinite-momentum phonons coupled to the static magneticorder.
Few-layer CrI breaks the out-of-plane translationalsymmetry, and therefore a singly degenerate phononmode in monolayer [e.g., A g (D d ) modes] gets splitinto N modes in a N -layer CrI ( N >
1) as a resultof Davydov splitting [Fig. 13(a) and (b)]. In the ab-sence of magnetic order, the crystal structure of N -layerCrI is always centrosymmetric, such that the N -splitphonon modes have alternating parities under spatialinversion. The highest frequency mode with in-phaseatomic displacement between layers is parity-even, andso can be detected in the parallel linear polarizationchannel. Within the layered antiferromagnetic (AFM)state, the magnetic order of N -layer CrI is centrosym-metric (parity-even) for odd N and noncentrosymmet-ric (parity-odd) for even N , and so selects parity-evenphonons for odd N and parity-odd phonons for even N to restore the centrosymmetry for the product of layeredAFM and selected phonons. Such layered-AFM-coupledphonon modes are selected in the crossed polarizationchannel with a spectral weight proportional to (cid:126)U i · (cid:126)M ,where (cid:126)U i is the eigenvector of the i th phonon mode and (cid:126)M = (1 , − , · · · , ( − N − ) is the axial vector for the lay-ered AFM order of N -layer CrI .Due to the involvement of layered AFM order in the scattering process for the crossed polarization channelunder zero magnetic field, it can be used to track the evo-lution of the static magnetic order across magnetic phasestransitions where (cid:126)M ( B ) changes from layered AFM toferromagnetic (FM) at B c = 0 . N = 2 and 3, B c = 0 . B c = 1 . N ≥ and B c = 2 . . In-terestingly, because of the differences in the eigenvector (cid:126)U i among the N phonon modes, individual modes cou-ple differently to the magnetic order and show distinctmagnetic field dependencies [Fig. 13(c) and(d)]. Sincethe layered magnetism in 2D magnets can also be tunedby external pressure, carrier doping, and elec-tric field, the observed magneto-Raman responsecan be further controlled through device engineering andintegrated with spintronic applications.The rich magneto-Raman behavior of the layered-magnetism-coupled phonons and their utility in distin-guishing different magnetic ordering across layers providea new avenue to probe magnetic structures in 2D mag-nets. They also hold promise for 2D magnetism-basedspintronic applications. C. Example in Single-Magnon Excitations in 2D van derWaals Magnets
Magnons or spin waves are fundamental excitationsfor magnetically ordered systems. Single-magnon Ra-man spectroscopy detects the presence of zone-center(zero-momentum) magnons, whereby the Stoke’s or anti-Stoke’s energy shift is matched with the magnon en-ergy. The selection rule is determined by the first (lin-ear) and/or the second (quadratic) expansions of onsitemagnetization operators in Eq. (15) above. Inelasticx-ray and neutron scattering spectroscopies are widelyused in studying magnon excitations in three-dimensional(3D) bulk magnetic materials and can access magnonswith momenta across the entire Brillouin zone. Ramanspectroscopy, although being restricted to zone-centermagnons, is uniquely suitable for probing small-sizedsamples, such as 2D vdW magnets, and further has afinite penetration depth for detecting surface magnetismin 3D magnets.
3D CrI was thought to be a ferromagnet as bulkmagnetization measurements show a classic FM hys-teresis loop and inelastic neutron diffraction re-solves single acoustic and optical magnon branches ex-pected for the honeycomb lattice [Fig. 14(a)]. Incontrast to bulk crystals, few-layer CrI is revealedto host a layered AFM state through magnetic-field-dependent MOKE and MCD, as well as tunnel-ing magnetoresistance. The discrepancy betweenthe bulk and few-layer magnetic phases has been rec-onciled by the observation that the surface layers ofbulk CrI has layered AFM order, as with few-layerCrI , while the interior layers show FM order that isconsistent with bulk probes. This finding was made5 FIG. 13. (a) Layer-number-dependent polarized Raman spectra taken on 1-4L CrI in both linear parallel (red) and crossed(blue) channels at 10 K. (b) Calculated eigenvectors for the phonon modes in 1-4L CrI . (c) Magnetic-field-dependent Ramanspectra taken on 4L CrI in co-circular polarized channel at 10 K. (d) Calculated magnetic field dependence of the four phononmodes of 4L CrI in co-circular polarized channel. Figures a, c, and d are adapted from Ref. 279.FIG. 14. (a) Calculated spin wave dispersion for a honeycomb ferromagnet (i.e., 3D bulk or 1L CrI ), and acoustic andoptical spin wave modes. (b) Magnetic-field-dependent acoustic single-magnon Raman spectra taken on 3D bulk CrI with theproposed coexisting surface AFM (SAFM) and deep bulk FM (BFM) modes to interpret the experimental data. (c) Magneticfield dependent acoustic single-magnon energy in 1L and 2L CrI . (d) Magnetic-field-dependent optical single-magnon excitationin 2L CrI . Figures a, b, and c and d are adapted from Refs. 278, 283, and 284, respectively. by magnetic-field-dependent single-magnon Raman spec-troscopy measurements [Fig. 14(b)]. Both Stoke’s andanti-Stoke’s Raman spectra successfully capture the sin-gle magnon excitations from the acoustic branch in 3DCrI and reveal a sudden change at B c = 2 T that corre-sponds to the layered AFM to FM transition in thick (i.e., >
10 nm) CrI flakes. Below B c , three magnon branchesare detected in total, a pair with opposite Zeeman shiftsfor the two spin wave branches with opposite angular mo-menta in the surface layered AFM and the third branchwith a positive Zeeman shift for the spin wave in theinterior bulk FM. Above B c , all three branches collapseonto a single mode with a positive Zeeman shift. Boththe critical field value and the opposite Zeeman splitting are indicative of the layered AFM state being present inthe surface layers of bulk CrI , which is consistent withmagnetic force microscopy measurements on thick CrI flakes. Raman spectroscopy is successful in detecting singlemagnon excitations from the acoustic branch down to thebilayer and monolayer limit, although the signal level ismuch reduced [Fig. 14(c)]. It is of particular inter-est that the layered AFM order in bilayer CrI breaksthe spatial inversion symmetry, making the optical spinwave branch Raman-active [Fig. 14(d)]. Compared withthe acoustic branch whose frequency at the zone centercorresponds to the magnetic anisotropy (i.e., ∝ J z − J x ),the optical branch is at a much higher energy in the tera-6hertz (THz) regime that scales with the average magneticexchange coupling (i.e., ∝ ( J z + J x ) / D. Example in Zone-Center and Zone-BoundaryTwo-Magnon Excitations in SOC Iridates
Two-magnon excitations are commonly observed inAFMs where the virtual spin-flip on two adjacent mag-netic sites with opposite spins is both permissible andefficient. This process generally happens for the casethat the pair of magnons involved are from the Brillouinzone boundaries (i.e., zone-boundary magnons), as illus-trated in strongly correlated electron systems like cupratesuperconductors, vdW AFMs like NiPS bulk andflakes, and traditional bulk magnets like MnF . This common scenario of two-magnon excitations re-quires a revision when there is more than one pair of op-positely aligned spins per magnetic unit cell, for examplein chiral AFMs like Mn X, AFM double-layer perovskiteiridates like Sr Ir O , or pyrochlore iridates like R Ir O .Taking double layer perovskite iridate Sr Ir O as an ex-ample, the AFM order happens both within each layerand between the two layers within a bilayer unit, andmoreover, the intralayer and interlayer AFM exchangecoupling between J eff = 1 / Ir O , the virtual spin flip on the two adjacent sitescould be either within the layer or between the two layerswithin the bilayer. Indeed, this has been shown to be thecase in the two-magnon Raman spectra of Sr Ir O . Two broad Raman features were detected in the Ra-man spectra of Sr Ir O , with the one at lower energy( ∼
800 cm − ) being fully symmetric with respect to theunderlying crystal lattice [i.e., A g (D h )] and the otherat higher energy ( ∼ − ) possessing lower sym-metry than that of the lattice [Fig. 15(b)]. The higherenergy feature is also observed at a similar energy andshows the same reduced symmetries in the Raman spec-tra of the single layer counterpart Sr IrO , and thereforeis attributed to the zone-boundary two-magnon scatter-ing same as that in Sr IrO . The lower energy broadfeature, which appears only below the magnetic onsettemperature T N = 250 K, is proven of two-magnon na-ture, but is a new addition in the two-magnon Ramanspectra of Sr Ir O that is not present in Sr IrO . More-over, its full symmetry is unexpected as no magnetic ex-citations reported so far have shown such high symme-tries. From the symmetry perspective, the fully sym- metric two-magnon scattering Hamiltonian of Sr IrO shares the same form as (and commutes with) the spinHamiltonian, and thus no A g type two-magnon exci-tations are allowed in Sr IrO . In contrast, the A g two-magnon scattering Hamiltonian of Sr Ir O differsfrom its spin Hamiltonian, allowing for the presence ofa fully symmetric two-magnon scattering process. Fromthe magnon dispersion perspective, the doubling of theunit cell in Sr Ir O from Sr IrO doubles the numberof spin wave branches, while the interlayer and intralayerexchange coupling being comparable, creates an appre-ciable energy difference between the acoustic and the op-tical branch. This fully symmetric two-magnon scatter-ing process therefore involves pairs of magnons from thezone-center optical magnon branch and has an energynearly twice the optical magnon gap in Sr Ir O (corre-sponding to ∼
400 cm − or ∼
12 THz) [Fig. 15(c)].The symmetry-resolved two-magnon excitations in thebilayer perovskite iridate Sr Ir O addresses the debateon the nature of its magnetism and sheds light on themysterious spin wave excitation spectra seen in resonantinelastic x-ray scattering (RIXS). The ultimate potentialof well-defined symmetry selection rules in two-magnonscattering is yet to be fully exploited in exploring mag-nets with strong SOC.Overall, magneto-Raman spectroscopy has been apowerful experimental technique that is complementaryto inelastic x-ray and neutron scattering spectroscopy.It has led to discoveries of new types of magnetic exci-tations and revealed the nature of novel magnetism inmodern magnetic systems. We discuss a few additionalpossibilities for future applications of magneto-Ramanspectroscopy. First, the current lowest detectable Ramanshift energy is ∼ − , which is still higher than thespin wave gap energy for many soft magnets (i.e., mag-nets with small exchange anisotropy), such as XY mag-nets, magnetic skyrmions, etc. Combining Raman spec-troscopy with Brillouin scattering spectroscopy would beone promising avenue to access these low energy magneticexcitations. Second, Raman selection rules allow only forthe detection of parity-even modes in centrosymmetricsystems, making the high-energy, parity-odd optical spinwave modes often inaccessible by Raman spectroscopy.It may be possible to break the centrosymmetry by ap-plying external parity-odd electrtic fields or by providingcontrollable non-centrosymmetric defect scattering cen-ters at low symmetry sites in order to access the other-wise forbidden optical magnons. Third, while diffraction-limited magneto-Raman spectroscopy using free-spaceoptics has been primarily used to detect magnetic ex-citations in low area/volume systems (i.e., 2D film orsurface of 3D bulk), tip-enhanced magneto-Raman spec-troscopy could be a powerful technique to detect topolog-ical edge spin waves locally. Finally, while it is impossibleto achieve both high-energy and high-time resolutions,there may be a window to obtain reasonable resolutionfor both. We propose to access this window by perform-ing time-resolved magneto-Raman spectroscopy for these7 FIG. 15. (a) Illustration of intra and interlayer exchange coupling between J eff = 1 / Ir O . (b) Symmetry-resolved two-magnon excitation Raman spectra taken on Sr Ir O at 80 K, showing two two-magnon scattering features. (c) Calculated magnon dispersion, magnon density of state (DOS), and two-magnon scatteringcross section for a bilayer square-lattice AFM. Figures a–c are adapted from Ref. 285. broad, high-energy two-magnon excitations in Sr Ir O and to further control the magnetic exchange couplingusing light. On the other hand, from a materials scienceperspective, we also note that the study and understand-ing of new layered magnets and exotic spin configura-tions could provide more suitable and interesting can-didates to explore the magnon-magnon coupling physicsthat are addressed in Sec. III. For example, it couldbe possible to identify other layered magnetic systemsat higher temperatures than that of CrCl . For anotherexample, it could also be possible to identify layered mag-netic systems with unconventional magnetic excitationssuch as Dirac magnons in honeycomb ferromagnets orflat magnon bands in kagome ferromagnets, and explorethe magnon-magnon coupling in these systems. E. Example in Quantum Molecular Chiral Spin andMagnon Systems
Designing novel magnetically ordered system at a low-dimensional limit beyond the current 2D layered magnetswould offer an alternative route for understanding the 2Dmagnonic excitation and their rich interactions betweenmicrowave photons, optical photons, and phonons.
The Chirality-Induced Spin Selectivity (CISS) effect, aunique ‘spin filtering’ effect arising from the chirality oflow-cost non-magnetic organic materials and their assem-blies that lack inversion symmetry, suggests a promisingpathway to study the low-dimensional magnetism andtheir spin wave excitation in a hybrid organic-inorganicquantum system.
Chirality is a geometrically distinguishable property ofa system that does not possess inversion symmetry, i.e., amirror plane or glide plane symmetry [Fig. 16(a)]. Twotypes of chirality (right- and left-handed, labeled as Rand S) can be produced in the molecular structure, ascalled chiral enantiomers. The CISS effect utilizes chi-rality to generate a spin-polarized current from chiral(left- or right-handed) enantiomers without the need formagnetic elements.
The CISS effect has been currently understood as a helicity-dependent spin polarization pro-cess via the quantum transport theory:
The spinpolarization is produced by the interplay between thespin moment of an electron ( (cid:126)S
CISS ) and an effectivemagnetic field ( (cid:126)B
CISS ) induced by electron propagationthrough the molecular helix, thereby aligning the electronspin parallel (right-handed helicity) or antiparallel (left-handed helicity) to the direction of the electron’s linearmomentum ( (cid:126)S
CISS (cid:107) ± (cid:126)k ) [Fig. 16(a)]. This process canbe phenomenologically described as: (cid:126)S
CISS ∝ (cid:126)B CISS = (cid:126)v/c × (cid:126)E helix ( r, ∆ d, L ) , (16)where c is the speed of light, v is an electron’s veloc-ity moving through a helical electrostatic field ( (cid:126)E helix ),and r , ∆ d , and L are the radius, the pitch of the helix,and the number of helical turns, respectively, that de-termine the magnitude of (cid:126)E helix . Remarkably, whileacting as a magnet with large perpendicular magneticanisotropy (PMA) to produce an out-of-plane spin cur-rent, protected by reduced chiral symmetry, the CISSeffect seems not possess an ordering temperature since itdoes not rely on the itinerant ferromagnetism present intypical magnets, thereby bypassing the challenges associ-ated with thermal fluctuations.
The robust spin (mag-netic) signal it generates can be equivalent to an effectivemagnetic field of up to tens of Tesla.
Successful exam-ples of materials exhibiting the CISS effect include quan-tum dots, small molecules,
DNAs, andalso in 2D chiral hybrid metal halide perovskites (chiral-HMHs).
The 2D chiral-HMHs consist of alternatinglayers of an inorganic framework of corner-shared leadhalide octahedra and chiral organic compounds.
The 2D chiral-HMH inherently lacks bulk inversion sym-metry and, in principle, is capable of producing an out-of-plane spin current at elevated temperatures via theCISS effect.Here we show the experimental observation of theCISS-induced spin (magnetic) signals in the 2D-chiral-HMH/ferromagnet interface under photoexcitation using8an ultrasensitive magneto-optical Kerr effect (MOKE)detection scheme called a Sagnac interferometer.
MOKE has been systematically used for sensing sub-tle magnetic signals in the semiconductors andmetals, and is particularly suitable for layered 2Dmagnetism.
Compared to the conventional steady-state MOKE tool that has a Kerr rotation angle reso-lution of microRadians, a modified fiber Sagnac inter-ferometer is employed based on the design conceivedby J. Xia et al. with dramatically-enhanced sensi-tivity. The built Sagnac interferometer has a Kerr an-gle resolution of 50 nanoRadians.
It is thereforereasonable that such a non-invasive spatially resolvedSagnac MOKE approach can locally probe the CISS-induced magnetic signals at the molecular level, provid-ing a direct readout to monitor and to convert light sig-nals into a magnetic response. Figure 16(b-c) shows aschematic illustration of photoinduced magnetism in theITO/2D-chiral-HMH/ferromagnet trilayer structure de-tected by the Sagnac MOKE approach. Under light il-lumination, photo-excited charge carriers become spin-polarized by the CISS effect as they propagate throughthe chiral cations in the structure.
The direction ofspin polarization is determined by the chirality, result-ing in a change of local magnetization at the 2D-chiral-HMH/ferromagnet interface.
Figure 16(d) shows the successful observation oflight-driven CISS-induced magnetism in the 2D-chiral-HMH/NiFe bilayer detected by Sagnac MOKE. The sam-ple was illuminated at a low laser intensity ( ≈ . mW )to suppress the laser-induced heating. By applying apositive out-of-plane magnetic field ( B z : ± mT ),the Kerr signal in the ITO/(S-Phenylethylamine,PEA) PbI /NiFe sample shows a decrease on the or-der of microRadians (∆ θ Kerr ≈ − . µrad ), support-ing our statement that an ultrasensitive detection toolis required to probe this Kerr signal. Whereas there isno similar change found in the A-chiral-HMH sample,the R-chiral-HMH sample exhibits a surprising increaseof the Kerr signal under the same illumination. By re-versing the magnetic field, the sign of ∆ θ Kerr is inverteddepending on the chirality, mimicking the nature ofmagnetization in typical ferromagnets.The photoinduced magnetism is further corroboratedby the magnetic field dependence as shown in Fig. 16(e)–(g). The change in the Kerr angle ∆ θ Kerr (B z ) exhibitsa linear response with the external magnetic field, show-ing no saturation up to 0.3 T. The sign of the slopedepends on the chirality, while there is no field depen-dence of Kerr signal found in the A-chiral HMH sam-ple. Compared to the Kerr signal obtained in the sameNiFe only sample ( ≈ µrad ), the light-driven ∆ θ Kerr ( ≈ . µrad ) suggests that the generated interfacial mag-netization, ∆ M , is around 0 .
7% of the total magnetiza-tion of the NiFe layer, corresponding to an effective fieldof ≈ ± which may be attributed to limited photogenerated carriers under the low light in-tensity ( < . mW ). Our work demonstrates the rich characteristics ofchiral-HMHs for bridging opto-spintronic applicationswith the CISS effect.
Magneto-optic Kerr ro-tation measurements prove that linearly polarized exci-tation of chiral-HMHs can change the magnetization ofan adjacent ferromagnetic substrate. Owing to syntheti-cally tunable optoelectronic properties, the implementa-tion of chiral molecules also offers a novel mutual inter-conversion between photons, charges, and spins for future‘opto-magnetism’ applications. The convergence of theCISS-induced large spin polarization, room temperaturemagnetism, and fascinating optoelectronic/photovoltaicproperties of these hybrid layered semiconductors wouldoffer a paradigm shift to transform the field of spintronicsusing solution-processed hybrid 2D materials, to enablefuture quantum information technologies.
VI. CONCLUSION
In summary, quantum technologies are promising forthe next-generation of computing, sensing, and communi-cation architectures, based on quantum coherent transferand storage of information. It is an emerging field com-bining fundamental quantum physics, information the-ory, materials science, and a variety of engineering efforts.For nascent quantum technologies to reach maturity, akey step is the development of scalable quantum build-ing blocks. Currently, the development of scalable archi-tectures for quantum technologies not only poses chal-lenges in understanding the coupling between disparatequantum systems, but also presents technical and engi-neering challenges associated with developing chip-scaletechnologies.As a rapid-growing subfield of quantum engineering,the developments of hybrid quantum systems have re-ceived great attention in recent years. Indeed, the inte-gration of different quantum modules has benefited fromhybrid quantum systems, which provide an importantpathway for harnessing the different inherent advantagesof complementary quantum systems, and for engineeringnew functionalities. This review article has attemptedto summarize and focus on the current frontiers with re-spect to utilizing magnetic excitations for novel quantumfunctionality. We have briefly reviewed recent achieve-ments and discoveries in the subject of circuit-based hy-brid magnonics systems, layered magnon-magnon sys-tems, quantum-defect sensing of magnons, and novel spinexcitations in quantum materials. From each Section, wehave attempted to discuss topics spanning the physicsfundamentals, technical aspects, and examples of engi-neered devices and/or material systems. We have alsoprovided a brief outlook on the future directions of eachindividual area discussed.Overall, magnonics based hybrid systems provide greattunability and flexibility for interacting with various9 K e rr s i gn a l ( m r a d ) R-chiral-HMH
Light off Light on Light off
S-chiral-HMH
Time (seconds) r∆d k(z)x yS
CISS (a)
SagnacbeamIllumination (b) (d) 𝑩 𝑪𝑰𝑺𝑺 = 𝑽𝒄 × 𝑬 𝒉𝒆𝒍𝒊𝒙 (𝒓, ∆𝒅, 𝑳) -0.3 -0.2 -0.1 0 0.1 0.2 0.3-0.9-0.6-0.300.30.60.9 K e rr A ng l e ( μ r a d ) B z (T) S-chiral-HMH/NiFe -0.3 -0.2 -0.1 0 0.1 0.2 0.3-0.9-0.6-0.300.30.60.9 K e rr A ng l e ( μ r a d ) B z (T) R-chiral-HMH/NiFe (e) (f) (c) -0.3 -0.2 -0.1 0 0.1 0.2 0.3-0.9-0.6-0.300.30.60.9 K e rr A ng l e ( μ r a d ) B z (T) A-chiral-HMH/NiFe (g)
FIG. 16. (a) Schematic illustration of the electron (green dots) propagation through a chiral molecular helix and the CISSeffect. The red arrows represent the evolution of spin polarization during the transmission. The helical field ( (cid:126)E helix ) inducesa magnetic field (cid:126)B
CISS and thus alters their spin states. (b) A sketch of light-driven CISS-induced interfacial magnetizationin a trilayer structure, illuminated by a 405-nm diode laser from the substrate. The changed magnetic response (∆ M ) at the2D-chiral-HMH/NiFe interface is detected by Sagnac MOKE. (c) Schematic illustration of the photoinduced magnetism at theNiFe/chiral-HMH interface. A net magnetization mediated by spin-dependent photocarriers via the CISS effect is formed inthe chiral-HMH structure next to the NiFe layer under the laser illumination. (d) Time trace of measured Kerr (t) signal uponthe illumination in the S- and R-chiral-HMH/NiFe heterostructure, respectively. The red line is an adjacent average smoothingof the data. (e)-(g) show the change in light-induced Kerr angle (black, squares) as a function of external magnetic field.Panels (e), (f), and (g) correspond to S-HMH, A-chiral-HMH and R-HMH respectively. The red line is a linear fit to the data.The Kerr signals have been averaged by four cycles of the illumination to improve the signal-to-noise (Adapted with permissionfrom (358). Copyright (2020) American Chemical Society.) quantum modules for integration in diverse quantum sys-tems. The concomitant rich variety of physics and mate-rial selections enable exploration of novel quantum phe-nomena in materials science and engineering. The rela-tive ease of generating strong coupling and forming hy-brid dynamic system with other excitations also makeshybrid magnonics a unique platform for quantum engi-neering. ACKNOWLEDGMENT
W.Z. is grateful for the encouragements and supportreceived from his colleagues on the organization andcoordination of this perspective article. Work on themanuscript preparation by W.Z. was partially supportedby U.S. National Science Foundation under award No.ECCS-1941426. Work on manuscript preparation byD.D.A., F.J.H. and S.E.S. acknowledge support from theU.S. Department of Energy, Office of Science, Basic En-ergy Sciences, Materials Sciences and Engineering Divi-sion. L.R.W. acknowledges support from the Univer-sity of Chicago/Tohoku University Advanced Institutefor Materials Research (AIMR) Joint Research Center. Work on the manuscript preparation by A.H. was sup-ported as part of Quantum Materials for Energy Effi-cient Neuromorphic Computing, an Energy Frontier Re-search Center funded by the U.S. DOE, Office of Science,under Award
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