CCoherent control of magnon radiative damping with local photon states
Bimu Yao,
1, 2
Tao Yu, ∗ Y. S. Gui, J. W. Rao, Y. T. Zhao, W. Lu, † and C.-M. Hu ‡ State Key Laboratory of Infrared Physics, Shanghai Institute of Technical Physics,Chinese Academy of Sciences, Shanghai 200083, People (cid:48) s Republic of China Department of Physics and Astronomy, University of Manitoba, Winnipeg, Canada R3T 2N2 Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ Delft, The Netherlands (Dated: September 11, 2019)The collective excitation of ordered spins, known as spin waves or magnons, can in principle radi-ate by emitting travelling photons to an open system when decaying to the ground state. However,in contrast to the electric dipoles, magnetic dipoles contributed by magnons are more isolated fromelectromagnetic environment with negligible radiation in the vacuum, limiting their application incoherent communication by photons. Recently, strong interaction between cavity standing-wavephotons and magnons has been reported, indicating the possible manipulation of magnon radiationvia tailoring photon states. Here, with loading an yttrium iron garnet sphere in a one-dimensionalcircular waveguide cavity in the presence of both travelling and standing photon modes, we demon-strate an efficient photon emissions from magnon and a significant magnon radiative damping withradiation rate found to be proportional to the local density of states (LDOS) of photon. By modulat-ing the LDOS including its magnitude and/or polarization, we can flexibly tune the photon emissionand magnon radiative damping on demand. Our findings provide a general way in manipulatingphoton emission from magnon radiation for harnessing energy and angular momentum generation,transfer and storage modulated by magnon in the cavity and waveguide electrodynamics.
I. INTRODUCTION
Magnon is an elementary excitation of magnetic struc-ture that is utilized as information carriers in magnonicsand magnon spintronics [12–15], as it carries polariza-tion or “spins” because the magnetization precesses an-ticlockwise around the equilibrium state [12–15]. Inter-play between magnon and other quasiparticles enrichesthe functionality of information transfer in spintronicdevices. Magnons can excite electron spins by the in-terfacial exchange interaction (spin pumping) [16, 17],phonons by the magnetostriction [18, 19], magnons in aproximity magnet through the dipolar or exchange inter-action [20, 21] and microwave photons by the Zeemaninteraction [22]. The range of spin-information transferby quasiparticles is restricted by their coherence lengththat strongly depends on disorder. Photons are therebyattractive to lift the constraint due to their long coher-ence time or length in high-quality optical device includ-ing the cavity and waveguide. Very recently, pioneeringworks have combined the best feature of cavity photonsand long-lifetime magnon in yttrium iron garnet (YIG)[23, 24], demonstrating the cavity magnon-polariton dy-namics [25–30]. Such high-cooperativity hybrid dynam-ics stimulates the ideas of coherent information process-ing with magnon. So far, these works mainly focused onthe coherent coupling between magnons and standing-wave photons [25–32] in confined boundary. However, theefficient delivery of coherent information needs a waveg-uide [33, 34], in which the magnon radiation at continu-ous wave range [35–37] remains relatively unexplored.Due to the anguler momentum conservation, the emit-ted photon by magnon radiation carries the spin current.The accompanying pumping of energy causes the magnonradiative damping [35–37], which reflects the efficiency of such photon emission process [38–40]. However, weakcoupling between the magnetic dipole and photon leadsthe radiation of magnon to be relatively difficult [38–40].The magnon-photon interaction is hopefully enhanced byconfining photons in a cavity [25–32] or waveguide. Itraises the hope to flexibly control the magnon lifetime onthe basis of intrinsic Gilbert damping, helping to over-come the difficulties in controlling the damping and de-phasing rates of magnon in conventional solid state withunmanageable elements such as disorder. Moreover, thegreat tunability of photonic environment in a microwavewaveguide could tune the efficiency of pumping the pho-ton spin current from magnon radiation in informationprocessing [33, 34]. We envision that in case the mecha-nism of tuning magnon radiation by local photon statescould be demonstrated, various mechanisms that wereused to tune photon emission by, for instance, metamate-rials, antennas and superconducting circuits could be im-plemented with magnon to add functionality in magnonicapplications [38, 41–43].In this work, we address a general way to controlthe photon emission from magnon and magnon radia-tive damping by tuning the local electromagnetic envi-ronment. The radiative damping rate is demonstrated tobe proportional to the local density of states (LDOS) ofphoton in a coupled magnon-photon system. We placean YIG sphere into a circular waveguide cavity that en-sembles to a “clarinet” in shape [see Fig. 5(a)]. Sim-ilar to the sound physics of “clarinet”, standing wavesare constructed with the superposition of continuous-wave background [44, 45], highlighting crucial differencewith confined cavity in normal coupling scheme. Thestanding-wave component causes coherent exchange be-tween magnon and photon and induces a splitting gapin the dispersion, while the superposed travelling-wave a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p component play the key role of transferring radiated spin-information to open system. By simultaneously involv-ing both standing and continuous waves, magnon radi-ation is thereby effectively controlled by photon statesand clearly characterized by magnon linewidth ∆ H fromphoton transmission. A relative suppression of the ra-diative damping at cavity resonance is observed thatseems to be different from the conventional Purcell effect[40, 46, 47], yet is unforeseen in coupled magnon-photondynamics. These measurements are well explained as wetheoretically establish the relation between macroscopicmagnon radiative damping and the microscopic LDOSof microwave photons in a quantitative level. Our resultopens opportunities to tune the LDOS involving the mag-nitude and/or polarization to control the photon emis-sion from magnon and magnon radiative damping. Tothe best of our knowledge, our work is the first convinc-ing observation of the LDOS-tunable magnon radiativedamping in a coupled magnon-photon system, providingthe possibility of photon-mediated spin transport withpreserved coherence. Due to the linearity nature of ourwork, we also anticipate that our method offers a generalapproach to other prototype photonic system or on-chipintegrated devices for advancing the manipulation anddelivery of radiated spin-information. II. RESULTSA. Photon states construction
For clarifying the magnon radiative damping con-trolled by photon states, we first introduce the local elec-tromagnetic environment inside the circular waveguidecavity as shown in Fig. 5(a). This waveguide consists ofa 16 mm-diameter circular waveguide and two transitionsat both ends that are rotated by an angle of θ =45 ◦ . Thetwo transitions can smoothly transform TE mode ofrectangular port to TE mode of circular waveguide, andvice versa. Specifically, the microwaves polarized in theˆ x - and ˆ x (cid:48) -directions are totally reflected at the ends of thecircular waveguide, forming the standing waves aroundspecific microwave frequencies. While the microwavespolarized in ˆ y - and ˆ y (cid:48) -directions can travel across thetransitions and therefore form a continuum of travel-ling waves. Therefore, in our device the standing wavescan form around particular wavevectors or frequenciesthat are superposed on the continuous-wave background[44, 45]. The continuous waves help transfer the informa-tion to open system and the standing waves provide theingredient to form the cavity magnon polariton. Thus,different from discrete modes in the conventional well-confined cavity, our circular waveguide cavity enables toadd the ingredient of continuous modes to modify thephotonic structure [44].The modes in our device can be characterized by mi-crowave transmission using a vector network analyzer(VNA) between port 1 and 2. A standing wave or “cav- ity” resonance mode at ω c / π =12.14 GHz is clearly re-vealed in S with a loaded damping factor of 9 × − , asillustrated by blue circles in Fig. 5(b). This damping ofphoton (109.3 MHz) is larger than the coupling strengthbetween standing-wave photon and magnon, and thusour system lies in the magnetically induced transparency(MIT) rather than the strong regime [29]. Accordingly,even the standing wave allows the delivery of energy outof the waveguide through its damping. It is observed inthe transmission spectrum that standing waves confinedin the waveguide cause the dip in transmission spectrumat cavity resonance [44], while the travelling continuouswaves that deliver photons from port 1 to 2 contribute ahigh transmission close to 1. Since continuous waves arenot negligible in our device, photon modes thereby can-not be described by a single harmonic oscillator as shownin previous works[25, 27–30]. Hence the electromagneticfields in our waveguide cavity are described by a largenumber of harmonic modes [48–50] in a wide frequencyrange and each mode has a certain coupling strength withmagnon mode.The following Fano-Anderson Hamiltonian describesthe interaction between magnon and photon as [22, 48] H / (cid:126) = ω m ˆ m † ˆ m + (cid:88) k z ω k z ˆ a † k z ˆ a k z + (cid:88) k z g k z ( ˆ m † ˆ a k z + ˆ m ˆ a † k z ) , (1)where ˆ m † ( ˆ m ) is the creation (annihilation) operator formagnon in Kittel mode with frequency ω m , ˆ a † k z (ˆ a k z )denotes the photon operator with wavevector k z and fre-quency ω k z , and g k z represents the corresponding cou-pling strength between the magnon and microwave pho-ton mode. We visualize magnon Kittel mode as a sin-gle harmonic oscillator in Eq. (1). Magnon and photonmodes have intrinsic dampings originated from inherentproperty, but our cavity establishes coherent coupling be-tween them [35–37] as schematically shown in Fig. 5(c).Due to the coherent coupling between magnon andphoton, the energy of excited magnon would radiate tothe photons that travel away from the magnetic sphere.This can be pictured as the “auto-ionization” of magnoninto the propagating continuous state that induces thephoton emission from magnon and hence magnon radia-tive damping [51, 52]. Such “additional” magnon dissi-pation induced by photon states can be rigorously cal-culated by the imaginary part of self-energy in magnonGreen’s function that is expressed as ∆ E m = δ m + π (cid:126) | (cid:126) g ( ω ) | D ( ω ). Here, δ m is the intrinsic dissipation rateof magnon mode, D ( ω ) represents the global density ofstates for the whole cavity that is a count of the num-ber of modes per frequency interval. We note that aboveradiative damping is established when the on-shell ap-proximation is valid with energy shift of magnon (tensto hundreds of MHz) is much smaller than its frequency(several GHz) (see Supplementary Note 1) [51, 52]. Byfurther defining the magnon broadening in terms of mag-netic field ∆ E = (cid:126) γµ ∆ H , the magnon linewidth is ex- FIG. 1. (Color online)
Magnon radiative damping controlled by LDOS. (a) Experimental set-up of coupled magnon-photon system in a circular waveguide cavity. (b) Transmission coefficient | S | from measurement (circles) and simulation(solid lines), with insets showing normalized LDOS distribution for standing-wave resonance 12.14 GHz and continuous wave11.64 GHz. (c) By coupling magnons with photons in a waveguide cavity, the radiative damping of magnon can be the dominantenergy dissipation channel compared to its intrinsic damping. (d) Dispersion map for coupled magnon-photon states. | S ( H ) | spectra is measured at fixed frequencies 11.64 GHz (e), 12.14 GHz (f) and 12.64 GHz (g), respectively, with the x-axis offset H m being the biased static magnetic field at magnon resonance. Source data are provided as a Source Data file. pressed as (Supplementary Note 1) µ ∆ H = µ ∆ H + αωγ + 2 πκγ R | ρ l ( d, ω ) | , (2)in which γ is modulus of the gyromagnetic ratio and µ denotes the vacuum permeability. In Eq. (2), the firsttwo terms are the linewidth related to inherent dampingof magnon in which µ ∆ H and αω/γ come from the in-homogeneous broadening at zero frequency [53] and theintrinsic Gilbert damping, respectively. The last term de-scribes the radiative damping induced by photon states inwhich | ρ l ( d, ω ) | represents the LDOS of magnetic fieldswith d and l denoting the position and photon polar-ization direction. Basically, | ρ l ( d, ω ) | counts both thelocal magnetic field strength and the number of electro-magnetic modes per unit frequency and per unit volume. κ = γM s V s (cid:126) c with M s and V s being the saturated magne- tization and volume of the loaded YIG sphere. R rep-resents a fitting parameter that is mainly influenced bycavity design and cable loss in the measurement circuit.Based on above theoretical analysis, we find that theradiative damping is exactly proportional to the LDOS ρ l ( d, ω ). To observe radiation as a dominant channel forthe transfer of magnon angular momentum, it requiresboth low inherent damping of magnon and a large tun-able | ρ l ( d, ω ) | . In the following experiment, both condi-tions are satisfied by introducing a YIG sphere with lowGilbert damping, as well as by modifying photon modedensity through tuning LDOS magnitude (Sec. II B 1),LDOS polarization (Sec. II B 2) and global cavity geom-etry (Sec. II B 3). B. Magnon radiation tunned by photon LDOS
A highly polished YIG sphere with 1 mm diameter isloaded into the middle plane of waveguide cavity. Beforeimmersing into experimental observations, it is instruc-tive to understand the two-dimensional (2D) spatial dis-tribution of LDOS in the middle plane, which is numer-ically simulated by CST (
Computer simulation technol-ogy ) at the center cross section that can well-reproduce | S | as shown in Fig. 5(b). It can be seen that the hotspots for the continuous waves (11.64 GHz) and stand-ing wave (12.14 GHz) are spatially separated, providingthe possibility to control LDOS magnitude by tuning thepositions of magnetic sample inside the cavity.In our first configuration, we focus on the local posi-tion with d =6.5 mm as marked in Fig. 5(b). Such posi-tion enables the magnon mode not only to have overlap-ping [29] with standing waves but also to couple to thecontinuous ones. More interestingly, as indicated by theinsets in Fig. 5(b), LDOS at d =6.5 mm has smaller quan-tity at cavity resonance compared with the ones in thecontinuous-wave range, which is opposite to the LDOSenhancement at resonance in conventional well-confinedcavity [40, 46, 47]. Therefore, according to Eq. (2), incontrast to magnon linewidth enhancement in previousworks, we expect a totally different linewidth evolutionby varying frequency in measurement with linewidth sup-pression at cavity resonance ω c .Concretely, the magnon linewidth can be measuredfrom | S | spectra in ω - H dispersion map. In our mea-surement, a static magnetic field µ H is applied along theˆ x -direction to tune the magnon mode frequency (close toor away from the cavity resonance), which follows a lineardispersion ω m = γµ ( H + H A ) with γ = 2 π ×
28 GHz/Tand µ H A =192 Gauss being the specific anisotropy field.For our YIG sphere, the saturated magnetization is µ M s =0.175 T, the Gilbert damping α is measured tobe 4 . × − by standard waveguide transmission withthe fitted inhomogeneous broadening µ ∆ H being 0.19Gauss. As ω m is tuned to approach the cavity resonance ω c , a hybrid state is generated with the typical anti-crossing dispersion as displayed in Fig. 5(d). A couplingstrength of 16 MHz can be found from rabi splitting atzero detuning condition that indicates the coherent en-ergy conversion between magnon and photon.Magnon linewidth (HWHM) is characterized by a line-shape fitting of | S ( H ) | that is obtained from the mea-sured transmission at fixed frequency and different mag-netic fields. Here, we focus on | S ( H ) | at three differ-ent frequencies with one being at the cavity resonance ω c and the other two chosen at continuous wave frequenciesabove and below ω c (11.64 GHz and 12.64 GHz, respec-tively). As photon frequency is tuned from continuous-wave range to the cavity resonance ω c / π =12.14 GHz,we observe that the lineshape of | S ( H ) | varies fromasymmetry to symmetry, accompanied by an obviouslinewidth suppression from 2.0 Gauss/1.5 Gauss to 1.0Gauss as shown in Fig. 5(e)-(g). It is worth noticing that the magnon linewidth µ ∆ H shows a clear suppression at cavity resonance ratherthan the linewidth enhancement in conventional coupledmagnon-photon system in the cavity [30, 54]. Such sup-pression of magnon linewidth qualitatively follows theLDOS magnitude, which also shows smaller quantity atcavity resonance. This qualitatively agrees with our the-oretical expectation from Eq. (2). In the following sub-sections, it is necessary to study the relation betweenlinewidth and LDOS in a quantitative level by using boththeoretical calculation and experimental verification. FIG. 2. (Color online)
LDOS magnitude dependence. (a)and (b), Simulated ρ x and ρ ⊥ at d =6.5 mm. (c) Measured µ ∆ H - ω relation (squares) with calculated lines from model(the green line) at d =6.5 mm. (d) and (e), Simulated ρ x and ρ ⊥ at d =0 mm. (f) Measured µ ∆ H - ω relation (squares)with calculated lines from model (the green line) at d =0 mm.Black circles and lines indicate the measured and fitted in-trinsic linewidth, respectively. (g) µ ∆ H evolution with tun-ing positions for different frequencies, with circles and solidlines representing the measured magnon linewidth and thelinewidth computed from LDOS, respectively. Source dataare provided as a Source Data file. FIG. 3. (Color online)
LDOS polarization dependence. (a) Schematic of tuning orientation of external magnetic field H relative to the ˆ x -direction in the plane of waveguide cross section (b) Simulated photon LDOS perpendicular to externalmagnetic field H with relative angle ϕ = 0 ◦ , 45 ◦ and 90 ◦ , respectively. (c) Measured magnon linewidth spectra, i.e., µ ∆ H - ω relation (squares) and calculated results (solid lines) for different angles ϕ = 0 ◦ , 45 ◦ and 90 ◦ , respectively. Source data areprovided as a Source Data file.
1. Magnon radiation controlled by LDOS magnitude
In this subsection, we show a quantitative control ofmagnon radiative damping by tuning the LDOS magni-tude over a broadband frequency range. The spacial vari-ation of the magnetic field in our waveguide cavity allowsus to realize different LDOS spectra by simply choosingdifferent positions. Similar to the experimental settingsin the above section with d =6.5 mm, we display a broad-band view of LDOS for per polarization by using simula-tion in Fig. 6. Although ρ x ( ω ) in Fig. 6(a) shows a typicalresonance behaviour, its contribution to the magnon ra-diation is negligible here according to the well-known factthat only photon polarization that perpendicular to theexternal static magnetic field H drives the magnon dy-namics. Following this consideration, we further simulate ρ ⊥ = (cid:113) ρ y + ρ z that plays a dominant important role inthe magnon-photon interaction as displayed in Fig. 6(b). ρ ⊥ ( ω ) shows a dip at the cavity resonance in the fre-quency dependence.It is clearly seen that due to the enhancement ofglobal density of states at the mode cut-off of the waveg-uide, continuous wave LDOS becomes more and moresignificant when frequency goes lower to approach thecut-off frequency (around 9.5 GHz). This phenomenoncan be viewed as a Van Hove singularity effect in thedensity of states for photons (see independent observa-tion via a standard rectangular waveguide in Supplemen-tary Note 2). As such singularity effect is involved inthe coupled magnon-photon dynamics, we obtain largerlinewidth at detuned frequency range that causes a clearlinewidth suppression at cavity resonance. In sharp con-trast to linewidth enhancement from typical Purcell ef-fects, results in Fig. 6(c) provides a new linewidth evo-lution process in a broadband range. Furthermore, tocompare with our theoretical model, we perform calcu-lation by Eq. (2) with κR = 4 . × m / s by usingthe fitting parameter quantity R ∼ .
8. It can be ob-served in Fig. 6(c) that the measured µ ∆ H agrees well with the computed ones from our theoretical model. Thereasonable agreement between experiment and theory ob-tained with R close to unity suggests that the linewidthis coherently controlled by LDOS magnitude, especiallyshowing that radiative power emission induced by con-tinuous waves can unambiguously exceed that inducedby standing waves.To create a different LDOS magnitude to tune themagnon radiation, the magnetic sphere is moved to thecenter of the cross section with d =0 mm. The simulatedLDOS ρ x and ρ ⊥ are illustrated in Fig. 6(d) and (e),respectively. The effective LDOS ρ ⊥ shows an enhance-ment at cavity resonance but suppressions at continuous-wave range. Similar to the frequency dependence ofthe LDOS magnitude, the magnon linewidth is clearlyobserved to be enhanced at cavity resonance but sup-pressed at continuous waves. This relation between themagnon width and LDOS is again quantitatively veri-fied by the good agreement between measurement andcalculated results from Eq. (2) as shown in Fig. 6(f).Particularly, as continuous wave LDOS is suppressed tonearly zero, the radiative damping from LDOS therebybecomes negligibly small. In this case, it can be foundthat the magnon linewdith exactly returns to its intrin-sic damping µ ∆ H + αω/γ mearsured in an independentstandard waveguide.Finally, at a detailed level, to continuously tune theratio of standing/continuous-wave LDOS magnitude, theposition of the YIG sphere is moved with d varied from0 mm to 6.5 mm. Typically for three different frequencydetunings with 0 MHz, -100 MHz and -440 MHz, our re-sults in Fig. 6(g) shows that the magnon linewidth canbe controlled with enhancement, suppression or negligi-ble variation in the position dependence. As shown inFig. 6(g), these results showing good agreement with thetheoretical calculation suggests that magnon linewdithcan be controlled on demand by tuning the LDOS mag-nitude. Moreover, the photon emission efficiency frommagnon radiation can in principle be significantly en-hanced with a larger magnetic sphere and smaller waveg-uide in cross section. For example, a magnetic sphere in2-mm diameter and a waveguide with half radius wouldenhance the radiation rate by 16 times (SupplementaryNote 1).
2. Magnon radiation controlled by LDOS Polarization
Having shown the relation between the magnon radia-tive damping in µ ∆ H and the LDOS magnitude, herewe would like to introduce LDOS polarization as a newdegree of freedom to control the magnon radiation. In ourexperiment, by placing the YIG sphere at d =2.3 mm, thetuning of effective LDOS polarization ρ ⊥ for the magnonto feel can be simply achieved by varying the direction ofexternal static magnetic field H with a relative angle ϕ to the ˆ x -direction as shown in Fig. 3(a). Please note thatcompared with the complicated operation to vary the po-sition of YIG sphere inside a cavity, here the LDOS wascontrolled continuously in a large range simply by rotat-ing the orientation of the static magnetic field. Basedon the orthogonal decomposition of LDOS for photons, ρ ⊥ is simulated for three typical angles ϕ , i.e., 0 ◦ , 45 ◦ ,and 90 ◦ as shown in Fig. 3(b). For ϕ = 0 ◦ with H beingexactly in the ˆ x -direction, the LDOS is dominated bystanding-wave component, which could provide largestcoupling with magnon at cavity resonance. While as ϕ goes close to 90 ◦ , continuous waves become more andmore dominant in the contribution to the LDOS, caus-ing a peak-to-dip flip for LDOS around ω c in Fig. 3(b).Accordingly, in our experiment, we obtain a magnonlinewidth enhancement at ϕ = 0 ◦ as shown in Fig. 3(c)by red squares. As ϕ is tuned towards 90 ◦ , we therebyanticipate and indeed obtain a linewdith suppression atcavity resonance with blue squares, showing good agree-ment with the linewidth scaling of ρ ⊥ in Eq (2). The the-oretically calculated linewidth µ ∆ H is plotted for each ϕ in Fig. 3(c) with κR being consistent with the previ-ous subsection. The good agreement between experimen-tal and theoretical findings suggests a flexible control ofmagnon radiation via LDOS polarization. Moreover, notrestricted to tune relative angle between H and LDOSpolarization in 2D plane, more possibility of magnon ra-diation engineering may be realized by pointing H toarbitrary direction in the whole 3D space.
3. Magnon radiation controlled by cavity geometry
Our device allows us to tune the LDOS magnitudeand polarization together by simply rotating the rela-tive angle θ between the two transitions [44], i.e., theglobal geometry of our circular waveguide cavity. Thiscan again validate and enrich our observations that thesame magnon harmonic mode radiates a different amountof power depending on the surrounding photon environ-ments. In this subsection, we insert a rotating part in FIG. 4. (Color online)
Cavity geometry dependence. (a)Cavity mode transmission profile when rotating θ . (b) Rabisplitting spectra for different angles θ . (c) Simulated ρ ⊥ fordifferent θ . (d) Measured µ ∆ H - ω relation when tuning θ . (e)and (f) Comparison between theoretical results and measure-ment at cavity resonance 11.79 GHz (e) and continuous wavefrequency 11.45 GHz (f). Dashed lines are intrinsic linewidthof YIG sphere. Source data are provided as a Source Datafile. the middle plane of cavity, so that the relative angle θ between two transitions can be smoothly adjusted. Bytuning angle θ from 45 degree to 5 degree, our systemshows a significant change in photon transmission as il-lustrated in Fig. 4(a), accompanied by significant en-hancements in cavity quality factor and global density ofstates [55, 56]. In addition, cavity resonance shows redshift to 11.79 GHz due to the increase of cavity length.The YIG sphere is placed at cavity center cross sectionwith d =6 mm and the external magnetic field is appliedin the ˆ x -direction. Such experimental conditions pro-vide stable magnon-photon coupling strength when θ istuned, as shown by the nearly unchanged mode splittingin Fig. 4(b).Our hybrid system now readily allows us to the investi-gate the magnon radiation controlled by cavity geometry.In particular, tuning θ from 45 degree to 5 degree leadsto a redistribution of photon states in cavity that greatlyenhances the LDOS near ω = ω c and controls the contin-uous wave LDOS in an opposite way, as illustrated bythe simulated ρ ⊥ in Fig. 4(c). Based on the theoreticalmodel, we expect magnon linewidth can quantitativelyfollow the geometry-controlled ρ ⊥ . Results from mea-surements under different θ are shown in Fig. 4(d) andwe indeed obtain linewidth µ ∆ H with similar behav-ior to the simulated ρ ⊥ . As is evident in Fig. 4(e) and(f), we find that the linewidth is well reproduced by ourtheoretical model with κR adjusted to 4 . × m / s .By tuning LDOS via θ , the experimental linewidth is en-hanced by twenty-fold at cavity resonance in comparisonwith the intrinsic damping of magnon as illustrated bythe dashed lines. III. DISCUSSION
Understanding and controlling magnon radiativedamping is essential in tuning the magnon lifetime aswell as transporting spin information by travelling pho-ton in spintronic or magnonic applications [38–40, 57, 58].With revealing the quantitative relation between magnonradiative damping and photon LDOS for the first time,our work brings three perspectives for better exploringand utilizing the magnon radiation in future research.(i)
Flexible control of magnon lifetime.
PhotonLDOS construction can flexibly tune the magnon life-time on the basis of intrinsic Gilbert damping. Althoughlong magnon lifetime is useful for information storage andmemory, suppressed magnon lifetime would bring advan-tageous impact for realizing fast repetition rate in device[40]. Our work explores some techniques including LDOSmagnitude, polarization and global environment to con-trol magnon lifetime, which could open paths for variousLDOS control method to tune magnon radiation in a flex-ible and precise fashion, and provids a new ingredient toadvanced communication processing [59].(ii)
Delivering coherent information of cavity-magnon polariton to open system.
Although theweak interaction between magnetic dipole and photonscan be enhanced by confining the photon mode in a cav-ity, the confinement restricts the magnon radiated infor-mation to be efficiently transferred to the open systemand vice versa. Our constructed magnon-photon systemcan combine both standing and traveling photon modesto couple the magnon. The traveling channel allows todeliver the coherent information out and advances theefficient tuning on the dynamics of the cavity-magnonpolariton. Manipulation of magnon radiation to opensystem, as we demonstrated, is very attractive to ex-plore new physics related to magnon dissipative proces-sion [60], such as dissipative coupling in magnon-basedhybrid system [61].(iii)
Stimulating the advancement of hybridmagnonics.
Controllable magnon radiation could stim-ulate hybrid magnonic systems to access new fron- tiers. Recent study shows coherent information at singlemagnon level can be coherently transferred to photon orsuperconducting qubit through radiation at millikelvintemperatures [25, 62], bringing quantum nature to thehybridized magnonic system. At room temperature, gen-erally the magnon-photon coupling is restricted in lin-ear harmonic dynamics, while our recent research breaksthis harmonic restriction by using feedback mechanism,exhibiting nonlinear triplet spectra similar to quantumdots [32]. The magnon radiation in these new regimescould stimulate the advancement of hybrid magnonics.In conclusion, we observe and show the ability to con-trol photon emission from magnon and magnon radiativedamping in the hybrid magnon-photon system, bridgingtheir relation to the tunable photon LDOS. Comparedwith conventional enhancement of magnon damping atcavity resonance in well-confined magnon-photon system,we report that the magnon linewidth at cavity resonancecan be relatively suppressed by photon LDOS engineer-ing. One quantitative method to design and tune theradiation efficiency of magnon is provided based on tai-loring photon LDOS including LDOS magnitude and/orpolarization, thereby leading to a general technique oftuning magnon relaxation on demand. Our measure-ments are mainly performed in the MIT regime with largephoton damping, which causes the radiative damping byphoton dissipation; while travelling-wave photon can di-rectly transfer the magnon energy to open system. Over-all, our study introduces a novel mechanism to coherentlymanipulate magnon dynamics by local photon states andsuggests a promising potential towards the developmentof magnon-based hybrid devices and related coherent in-formation processing.
IV. METHODSA. Device description
Our waveguide cavity is made up of a cylindricalwaveguide and two circular rectangular transitions coax-ially connected at both ends. Through the transition, asmooth change between TE mode of rectangular waveg-uide port and TE one of cylindrical waveguide can beestablished. Via coaxial cables, the cavity is connectedto the input/output ports of an vector network analyzer(VNA). With an input power of 0 dBm, the transmissionsignals can be precisely picked up by VNA. YIG sphereis fixed firmly inside the cavity with scotch tape, withits location tunable on demand to couple with differentmicrowave magnetic fields. YIG and the scotch tape, asdielectric materials, can slightly influence the microwavefields distribution in our experiment. We neglect thesmall dielectric influence in our theoretical treatment. B. Theoretical description
In Supplementary Note 1, the theory of magnon spon-taneous radiation in the waveguide including the deriva-tion of the magnon linewidth induced by the LDOS isprovided.
V. DATA AVAILABILITY
The source data underlying Figs. 1-4 and Supplemen-tary Figs. 1-2 are provided as a Source Data file. Thedata that support the findings of this study are availablefrom the corresponding authors upon reasonable request. ∗ [email protected] † [email protected] ‡ [email protected] Lenk, B., Ulrichs, H., Garbs, F., & Mnzenberg, M., Thebuilding blocks of magnonics.
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VI. ACKNOWLEDGEMENT
This work was funded by NSERC, the NationalNatural Science Foundation of China under GrantNo.11429401 and No.11804352, the Shanghai PujiangProgram No.18PJ1410600, the Science and Technol-ogy Commission of Shanghai Municipality (STCSM No.16ZR1445400), and SITP Innovation Foundation (CX-245). T.Y. was supported by the Netherland Organi-zation for Scientific Research (NWO). We would like tothank Y. Zhao, J. Sirker, L. H. Bai, P. Hyde, Y. M.Blanter and G. E. W. Bauer for useful discussions.
VII. AUTHOR CONTRIBUTIONS
B.M.Y. and Y.S.G. set up the hybrid system, con-ducted the experiment as well as analyzed the data. T.Y.,in discussions with B.M.Y. and C.-M.H., developed thetheory part. J.W.R. and Y.T.Z. contributed to the de-sign of the cavity parts. B.M.Y. and T.Y. prepared allthe figures as well as the supplementary material part.T.Y., L.W. and C.-M.H. together supervised the work.All authors contribute to the paper writing.
VIII. SUPPLEMENTARY NOTE 1.THEORY OF MAGNON SPONTANEOUS RADIATION
In this part, we describe the model about the magnon spontaneous radiation in a waveguide. We first set up theFano-Anderson Hamiltonian for the magnon-photon coupling in a waveguide [1, 2], and then calculate the energybroadening due to the magnon spontaneous radiation.
A. Hamiltonian
The free energy of the coupled system is written as [3] F = F ( M ) + (cid:15) (cid:90) d rE ( r ) · E ( r ) + µ (cid:90) d rH ( r ) · H ( r ) + µ (cid:90) d rH ( r ) · M ( r ) , (3)where (cid:15) and µ are the vacuum dielectric and permeability constants, respectively. F ( M ) describes the free energyof magnetization M , the second and third terms denote the free energy of electromagnetic waves due to the electricfield E and magnetic field H in a waveguide, and the last term represents the Zeeman interaction between themagnetization and the magnetic field. The Hamiltonian can be obtained by quantizing the free energy.Due to the long-wavelength nature of the electromagnetic waves in the waveguide we are considering, only theuniform precession of the magnetization, i.e., the Kittel mode [4], needs to be considered. We express M = − γ (cid:126) S interms of the spin operators [4–6], where γ is modulus of the gyromagnetic ratio. The spin operators are expanded by0magnon operator ˆ α K in the Kittel mode [4–6],ˆ S Kβ = √ S (cid:2) M Kβ ˆ α K ( t ) + ( M Kβ ) ∗ ˆ α † K ( t ) (cid:3) , (4)in which β = { z, x } assuming that the equilibrium magnetization is along the ˆ y -direction. S = M s / ( γ (cid:126) ) with M s being the saturated magnetization; M Kβ is the amplitude of the Kittel mode that is normalized according to [7, 8] (cid:90) d r (cid:104) M Kz ( r )( M Kx ( r )) ∗ − ( M Kz ( r )) ∗ M Kx ( r ) (cid:105) = − i/ . (5)Because the magnetization is uniform in the magnetic sphere and the Kittel mode is circularly polarized with M x = iM z , we obtain M Kz = 1 / (2 (cid:112) V s ) , (6)in which V s is the volume of the magnetic sphere.In the waveguide, the electromagnetic field operators are expanded by the photon operator a k,λ in which k > z -direction, and λ ( µ below) labels the eigenmodes in thewaveguide [3], E ( r ) = (cid:88) λ,k (cid:2) E λk ( x, y ) e ikz ˆ a k,λ + E λ − k ( x, y ) e − ikz ˆ a − k,λ + h . c . (cid:3) , H ( r ) = (cid:88) λ,k (cid:2) H λk ( x, y ) e ikz ˆ a k,λ + H λ − k ( x, y ) e − ikz ˆ a − k,λ + h . c . (cid:3) , (7)where the eigenmodes of electric and magnetic fields are expressed as [3] E λk ( x, y, z ) = 1 √ L (cid:2) E tλk ( x, y ) + E λkz ( x, y )ˆ z (cid:3) e ikz , H λk ( x, y, z ) = 1 √ L (cid:2) H tλk ( x, y ) + H λkz ( x, y )ˆ z (cid:3) e ikz , (8) E λ − k ( x, y, z ) = 1 √ L (cid:2) E tλk ( x, y ) − E λkz ( x, y )ˆ z (cid:3) e − ikz , H λ − k ( x, y, z ) = 1 √ L (cid:2) − H tλk ( x, y ) + H λkz ( x, y )ˆ z (cid:3) e − ikz . (9)Here, “ t ” means “transverse”, and L is the length of the waveguide. The eigenmodes satisfy the following orthonormalrelations [3], (cid:90) E tλk · E tµk (cid:48) da = δ kk (cid:48) δ λµ A λk , (cid:90) H tλk · H tµk (cid:48) da = 1( Z λk ) δ kk (cid:48) δ λµ A λk , (cid:90) E λk,z E µk (cid:48) ,z da = − γ λ k δ kk (cid:48) δ λµ A λk , (TM) (cid:90) H λk,z H µk (cid:48) ,z da = − γ λ k ( Z λk ) δ kk (cid:48) δ λµ A λk , (TE)12 (cid:90) ( E tλk × H tµk (cid:48) ) · ˆ z da = 12 Z λk δ kk (cid:48) δ λµ A λk , (10)where Z λk = µ ω λk /k and k/ ( (cid:15) ω λk ) are the impedances for the TE and TM modes, A λk = (cid:126) ω λk / (2 (cid:15) ) and (cid:126) / (2 (cid:15) ω λk ) forthe TE and TM modes with ω λk being the eigen-energy, and γ λ = µ (cid:15) ( ω λk ) − k = ( ω λk ) /c − k . (11)Please note that ω λk = c (cid:112) k + γ λ in which γ λ only depends on the band but not the momentum is understood. Withthese orthonormal relations, we demonstrate (cid:15) (cid:90) d rE ( r ) · E ( r ) + µ (cid:90) d rH ( r ) · H ( r ) = (cid:88) k,λ (cid:126) ω λk (ˆ a † k,λ ˆ a k,λ + ˆ a †− k,λ ˆ a − k,λ ) . (12)With the established magnon and photon operators, the coupling Hamiltonian is constructed,ˆ H int = µ (cid:90) d rH ( r ) · M ( r ) = − µ γ (cid:126) (cid:90) d r ( ˆ H x ˆ S x + ˆ H z ˆ S z ) . (13)1By assuming the magnetic sphere is small located at ( x , y , z ), this Hamiltonian is calculated to beˆ H int = (cid:126) (cid:88) k z = ± k,λ (cid:0) g k z ,λ ˆ a k z ,λ ˆ α † K + h k z ,λ ˆ a k z ,λ ˆ α K + h . c . (cid:1) , (14)where the coupling strength reads g k z ,λ = µ (cid:115) γM s V s (cid:126) LA λk (cid:2) i H λk z ,x ( x , y ) − H λk z ,z ( x , y ) (cid:3) e ik z z . (15)as we can see here, as the static field H is assumed at y direction, the coupling effect between magnon and cavityphoton states is determined by a combination of eigenmodes that perpendicular to H . B. Spontaneous radiation
From above calculation, the whole Hamiltonian for a magnetic sphere in the waveguide is written as H / (cid:126) = ω K ˆ α † K ˆ α K + (cid:88) k z ω k z ˆ a † k z ˆ a k z + (cid:88) k z g k z (ˆ α † K ˆ a k z + ˆ α K ˆ a † k z ) , (16)in which k z = ± k and the band index for photon is disregarded when only the lowest band is considered here. ThisHamiltonian is known as the Fano-Anderson Hamiltonian with an exact solution being possible [1, 2]. The lifetimeof magnon can be calculated from the imaginary part of the self-energy [2] that is interpreted to come from thespontaneous radiation of magnon [1]. The Green function of Kittel magnon is exactly calculated to be G m ( ω ) = (cid:110) (cid:126) ω − (cid:126) ω K + iδ K − (cid:88) k z | (cid:126) g k z | (cid:126) ω − (cid:126) ω k z + iδ k z (cid:111) − , (17)where δ K = αω K is the intrinsic Gilbert damping of Kittel magnon with α being the intrinsic Gilbert dampingcoefficient. The imaginary part of the self-energy, i.e., (cid:80) k z | g kz | (cid:126) ω − (cid:126) ω kz + iδ kz , contributes to the broadening of the magnonspectra. Therefore the total broadening of magnon reads∆ E K = δ K + π (cid:88) k z (cid:12)(cid:12) (cid:126) g k z (cid:12)(cid:12) δ ( (cid:126) ω K − (cid:126) ω k z ) = δ K + π (cid:126) (cid:12)(cid:12) (cid:126) g ( ω K ) (cid:12)(cid:12) D ( ω K ) , (18)where D ( ω K ) is the global density of states (DOS) of photon in the waveguide.By further defining the width in terms of the magnetic field ∆ E = (cid:126) γµ ∆ H , we obtain µ ∆ H = αω K γ + πγ | g ( ω K ) | D ( ω K ) = αω K γ + 2 πγ ξ | g CST ( ω K ) | . (19)For TE mode in the experiment, parameter ξ is expressed as ξ = 12 (cid:16) µ (cid:114) γM s V s (cid:126) (cid:114) (cid:126) ω K (cid:15) c √ πµ (cid:17) . (20)In Eq. (19), another coupling strength g CST is defined when the normalization with unit power flow in waveguide isused in the
Computer simulation technology (CST) [3], i.e.,12 (cid:90) (˜ E tλk × ˜ H tµk (cid:48) ) · ˆ z da = 12 δ kk (cid:48) δ λµ . (21)The relation between { ˜ E k , ˜ H k } here and {E k , H k } in Eqs. (10) reads { ˜ E k , ˜ H k } = {E k , H k } × c √ πµ (cid:112) D / ( A k L ) . (22)It is seen that g CST is proportional to the local field strength and √D .2 C. Relation to local density of states
Although the CST does not directly calculate the local density of states (LDOS), we can show | g CST | is indeedproportional to the LDOS. To define the magnetic-field LDOS in the TE mode [9], we define the new eigenmodes withdifferent orthonormal conditions from Eqs. (10), (cid:90) ( H tλk ) ∗ · H tµk (cid:48) da = (cid:90) H tλk · H tµk (cid:48) da = δ kk (cid:48) δ λµ ( B λk ) /µ , (cid:90) ( H λk,z ) ∗ H µk (cid:48) ,z da = − (cid:90) H λk,z H µk (cid:48) ,z da = γ λ k δ kk (cid:48) δ λµ ( B λk ) /µ , (23)where B λk = ck/ω λk . This means H λk = H λk × Z λk B λk √ µ (cid:113) A λk . (24)With these conditions, for the waveguide with uniform section, i.e., with translation symmetry, the magnetic-fieldLDOS per unit length is defined by [9] ρ α ( x, y, ω ) = 1 L µ (cid:88) k,λ δ ( ω − ω λk ) |H λk,α ( x, y ) | , (25)with α = { x, y, z } . From the normalization condition in Eqs. (23), it is checked that (cid:80) α (cid:82) dxdyρ α ( x, y, ω ) = D ( ω ) /L .When λ is the lowest TE mode, we find that ρ α ( x, y, ω ) = 1 L µ D ( ω ) |H k,α ( x, y ) | . (26)This leads to | g ( ω ) | D ( ω ) = ξ | g CST ( ω ) | = (cid:16) µ (cid:17) γM s V s (cid:126) LA λk (cid:2) |H kz ( x , y ) | + |H kx ( x , y ) | (cid:3) D ( ω )= γM s V s (cid:126) c ρ ⊥ ( x , y , ω ) ≡ κρ ⊥ ( x , y , ω ) , (27)where κ = γM s V s (cid:126) c and ρ ⊥ ( x , y , ω ) = ρ z ( x , y , ω ) + ρ x ( x , y , ω ). Thus, for waveguide with uniform section, ρ ⊥ ( x , y , ω ) = ξ | g CST | /κ, (28)that can be determined by CST simulation approach. Finally, we can reach µ ∆ H = µ ∆ H + αω K γ + 2 πκγ R | ρ ⊥ ( x , y , ω ) ||LDOS| , (29)in which |LDOS| ≡ − · s explicitly represents the unit of LDOS for photons, µ ∆ H is the inhomogeneousbroadening of magnon linewidth at zero frequency and R is a fitting parameter mainly influenced by cavity designand cable loss in practical measurement circuit.Above theory is established for the ideal waveguide with the same cross section along the waveguide. When thecross section slowly varies as the case in our experiment, this treatment can be still a good approximation, with thefitting parameter quantity R ∼ . IX. SUPPLEMENTARY NOTE 2.MAGNON LINEWIDTH ENHANCEMENT IN RECTANGULAR WAVEGUIDEA. Enhanced DOS near waveguide cut-off frequency
In this part, the enhancement of DOS of photon near the cut-off frequency of a standard rectangular waveguide ismeasured. A rectangular waveguide that works at Ku band ( a = 15 . b = 7 . FIG. 5. (Color online) (a) Measured S -matrix parameters for Ku-band rectangular waveguide. (b) Caculated microwave photonDOS for the rectangular waveguide (dots) with fit line (solid lines). photon states, as schematically shown in Fig. 5. The transmission | S | as well as the reflection | S | signals aremeasured by an Vector Network analyzer (VNA) with an input power of 0 dbm, as shown in Fig. 5(a). S amplitudeis observed to decrease below the cut-off frequency ω cut / π =9.8 GHz of our device because the microwave becomesevanescent with larger wavelength.Physically, near the cut-off frequency, the magnitude of the momentum for the microwaves tends to be zero alongthe propagation direction, leading to an enhancement of global DOS D ( ω ) near cut-off frequency. This effect can beverified by obtaining the global DOS from S -matrix parameters via the inversion relation [10, 11], D ( ω ) = − i (1 / πV )Tr( S † dS/dω ) , (30)where S ( ω ) = (cid:18) S ( ω ) S ( ω ) S ( ω ) S ( ω ) (cid:19) represents the scattering matrix. From the measured S parameters, we can plotDOS in Fig. 5(b), in which an enhancement of DOS near the cut-off frequency is clearly observed.Analytically, the global DOS for photons in the TE mode in a rectangular waveguide is calculated to be [3] D ( ω ) = (cid:88) k z δ ( ω − ω (1 , k z ) = (cid:88) k z δ ( ω − c (cid:112) ( π/a ) + k z ) = Lπ ωc (cid:112) ω − ( cπ/a ) . (31)For further considering the energy loss caused by the finite conductivity in the waveguide wall, the DOS spectrumcan be well fit by taking consideration of the damping factor β . As shown in Fig. 5(b) , the DOS spectrum is wellreproduced by D ( ω ) = Lπ ωc √ ω − ω + iβωω cut , with the damping β and propagation distance L fitted by 0 . .
03 m, respectively.
B. Magnon linewidth enhancement near mode cut-off
In this part, the magnon linewidth enhancement near waveguide cutoff frequency can be directly tested in ourstandard rectangular waveguide, in which the section is uniform along the waveguide. This allows a direct comparisonbetween the theory [Eq. (29)] and experimental observation without the approximation that the section varies slowlyalong the waveguide.We introduce a 1 mm highly-polished YIG sphere into the rectangular waveguide to display the damping controlvia photon states. We place the YIG sphere in two different positions, i.e., at the cross-section center and close to theconductive wall of the waveguide. By fitting the linewidth of | S ( H ) | spectra at different microwave frequencies ω ,we can plot the magnon linewidth µ ∆ H as a function of ω in Fig. 6 for two different positions, respectively.4 FIG. 6. (Color online) (a) Simulated LDOS ρ ⊥ when placing YIG sphere at waveguide center and near waveguide wall,respectively. (b) Measured magnon linewidth for two different positions with the YIG sphere placed in the center x = a/ x ∼ a (red dots). Red solid curve in (b) is calculated from Eq. (29) in supplementarynote 1 and the green solid curve is fitted for intrinsic linewidth. At the cross-section center with x = a/
2, the microwave LDOS is found to be nearly zero [see the green curvein Fig. 6(a)] and therefore LDOS-induced radiation damping can be approximately neglected here. As a result,intrinsic magnon linewidth can be obtained from our measurement, with a typical linear ∆ H − ω revealed in Fig.6(b) by the green circles. Using linear fitting, the Gilbert damping coefficient can be obtained as 4.3 × − with theinhomogeneous broadening µ ∆ H fitted by 0.19 Gauss.As shown in Fig. 6(a) by the red curve, LDOS near waveguide wall displays larger magnitude with an obviousenhancement near the mode cut-off frequency. Hence by moving the YIG sphere to near the conductive wall, weexpect and indeed obtain a magnon linewidth enhancement according to Eq. (29). Specifically, we plot the measuredmagnon linewidth as a function of frequency in Fig. 6 (b) by the red circles. Compared to ∆ H - ω relation when YIGis placed at center, here we observe significant magnon linewidth enhancement especially at the frequency near themode cut-off. Such linewidth broadening can be well reproduced by our calculation from Eq. (29) with the fittingparameter R close to unity. ∗ [email protected] † [email protected] ‡ [email protected] Fano, U. Effects of Configuration Interaction on Intensities and Phase Shifts.
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