Supplementary Materials for

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1 avances.sciencemag.org/cgi/content/full//3/e150186/dc1 The PDF file inclues: Supplementary Materials for Cavity magnomechanics Xufeng Zhang, Chang-Ling Zou, Liang Jiang, Hong X. Tang Publishe 18 March 016, Sci. Av., e (016) DOI: /sciav Magnomechanical interaction through magnetostrictive forces Magnon moes of the YIG sphere Coherent cavity magnomechanical coupling Thermal instability Table S1. Definitions an meanings of symbols. Fig. S1. Magnetically inuce transparency/absorption at various rive frequencies. Fig. S. Magnomechanical resonance line shape an parameter space iagram. Fig. S3. Cascae transparency/absorption. References (43 5)

2 Quantity Symbol Expression an Explanation Planck s constant J s Gyromagnetic ratio γ π.8 MHz/Oe Bias magnetic fiel H - Operator for microwave photon â - Operator for magnon ˆm - Operator for phonon ˆb - Hybriize magnon-photon  ± Eq.(S57) Frequency of microwave cavity photon ω a - Intrinsic issipation of microwave cavity photon κ a,i - External issipation of microwave cavity photon κ a,e - Frequency/issipation of magnon ω m/κ m - Frequency/issipation of phonon ω b /κ b - Frequency of hybri moe ω ± - Microwave rive frequency ω - Signal (probe) microwave frequency ω s - Magnon-photon etuning ma ma = ω m ω a Drive-resonance etuning ± ± = ω ω ± Two-photon (probe-rive) etuning s s = ω s ω Mean magnon number ˆn m ˆn m = ˆm ˆm Magnon-photon coupling strength g ma - Magnon-phonon coupling strength g mb Eq. (S9) Effective magnon-hybri moe coupling strength G ± Eqs.(S65)&(S66) Steay state population A ±,ss Eq.(S67) Cooperativity C C = G ±/κ ±κ b TABLE S1: Definitions an meaning of symbols. I. MAGNOMECHANICAL INTERACTION VIA MAGNETOSTRICTIVE FORCES A. The Hamiltonian About two centuries ago, J. P. Joule [43] iscovere the magnetostrictive effect, which escribes the eformation of a magnetic material in response to an external magnetic fiel. On the other han, eformation of the magnetic material can also change the magnetization. Such magnetostrictive effect can be attribute to three types of interactions epening on the istance between ions: exchange interaction, ipole-ipole interaction an spin-orbital interaction [44]. In general, the magnetoelastic energy reas [45] E me = b 1 [ M MS x ε xx + My ε yy + Mz ε zz] + b MS [M x M y ε xy + M x M z ε xz + M y M z ε yz ] (S1) where b erg/cm 3 an b erg/cm 3 are the magnetoelastic coupling coefficients, M x,y,z are magnetization components, an M S is the saturation magnetization. The energy of the system epens on the strain tensor of the crystal ε ij = 1 ( ui + u ) j (S) x j x i where u is the isplacement. The magnon excitation can be quantize as V m ˆm = (M x im y ) γm S (S3)

3 where V m = 4π 3 ( D ) 3 is the YIG sphere volume an D is the sphere iameter. Therefore, we have an M z M S γms M x = ( ˆm + ˆm ) V m γms M y = i ( ˆm ˆm ) V m γ V mm S ˆm ˆm consiering M z = M S M x M y. Substituting the quantize operator into the Hamiltonian, the first term becomes H 1 = b ˆ 1 MS x 3 [ Mx ε xx + My ε yy (Mx + M y )ε ] zz = b 1 γm S MS [ ˆm ˆm 1 ˆ V m ] x 3 (ε xx + ε yy ε zz ) While the secon term is H = b ˆ MS x 3 [M x M y ε xy + M x M z ε xz + M y M z ε yz ] =i b ˆ γ ( ˆm ˆm ) x 3 ε xy M S V m + b M S γ (MS V m M γ ˆm ˆm) S V m [ ˆ ˆm ] x 3 (ε xy + iε yz ) + h.c. (S4) (S5) (S6) (S7) It can be foun that H 1 correspons to the ispersive interaction between magnon an phonon, where the magnon frequency is shifte by phonon. In contrast, H can lea to either parametric magnon generation if the phonon frequency is twice of the magnon frequency, or linear magnon-phonon coupling when they are on-resonance. In our experiments, the frequency of acoustic phonon is much lower than that of the magnon, so we only concern about the ispersive magnon-phonon interaction. The resulting interaction Hamiltonian can be written as where H I / = g mb ˆm ˆm(ˆb + ˆb ) g mb = 1 b 1 ζ ρs (S8) (S9) is the single magnon coupling strength, ζ = 1 ˆ V m x 3 (ε xx + ε yy ε zz ) (S10) is the single phonon strain fiel overlapping factor, ˆb (ˆb ) is the annihilation (creation) operator of phonon, ρ = m 3 is the spin ensity in YIG an S = 5 is the spin number. B. Phonon Moes of YIG Sphere As YIG is an elastic meium with cubic symmetry, we can analytically solve the phonon moes in a spherical structure. The elastic equation can be written as [46] ρ m t u = (λlamé + µ Lamé ) ( u ) µ Lamé u (S11) where ρ m is the mass ensity, λ Lamé an µ Lamé are Lamé constants. In the spherical coorinator (r, θ, φ), we can represent the isplacement by [47] u = ψ0 + Ψ 1 + Ψ (S1)

4 where Ψ 1, = (rψ 1,, 0, 0) are vector potentials an ψ 0,1, are scalar functions. Substituting the isplacement into Eq. (S11), we obtain the scalar equation where δ i,0 is Kronecker elta function. [ + ρω (λ Lamé + µ Lamé )δ i,0 + µ Lamé ]ψ i = 0 (S13) In the spherical coorinator, this Helmholtz equation has general solutions ψ i = c i j l ( ωr v i )P m l (cosθ)e imaφ e iωt (S14) (λlamé +µ Lamé )δ i,0+µ Lamé where j l (z) is spherical Bessel function, P ma l is the spherical harmonic function, an v i = ρ (v 0 = v l an v 1, = v t ). Here l is the angular moe inex an m a is the azimuthal moe inex. The free surface stress bounary conition gives σ rr = σ rθ = σ rφ = 0 at the surface where r = D/. Substituting the general solutions with coefficients into the bounary conitions, we have the equation ( ( ) λ Lamé + µ Lamé r )ψ 0 0 µ Lamé Λ ψ ( ) ( ) r r ψ0 r ψ1 [ r θ r sinθ r φ r 1 θ r r r (rψ ] c 0 ) ψ c ( ) ( ) ψ0 sinθ r φ r r ψ1 [ 1 r θ r sinθ φ r r r (rψ ] 1 = 0 (S15) ) ψ c ( ) From this matrix, one solution is ψ1 r r = 0 which correspons to the torsional vibration of the sphere. Another solution is c 1 = 0 with the characteristic equation as ( ( )) η4 ξj l+1 (ξ) η + (l 1)(l + ) ηjl+1 (η) + η j l (η) (l + 1) (l 1)(l + 1) + j l (ξ) + ηj l+1(η) ( η l(l 1)(l + ) ) = 0 (S16) j l (η) where ξ = ωd v l an η = ωd v t. This solution correspons to spheroial motion, therefore we label the moes as S q,l,ma, with q is the orer of solution to Eq.(S16). For each angular moe inex l, there are l+1 moes egenerate at the same frequency. In practice, the egeneracy is lifte ue to geometry imperfection or efects, therefore counter-propagating moe pairs, such as S q,l,ma an S q,l, ma for non-zero m a, form staning waves as F(r, θ, t)sin(m a φ + φ 0 ) an F(r, θ, t)cos(m a φ + φ 0 ), where F(r, θ, t) is the fiel istribution function an φ 0 is a constant. The frequency of these two staning wave moes are still very close to each other, but with ifferent symmetry. Since the solution to the characteristic equation η = ωd v is imensionless, the eigenfrequency ω q,l,m t a is inversely proportional to the sphere iameter D. As the YIG sphere is glue on a silica fiber for mechanical support, there is clamping loss ue to the elastic energy coupling to the traveling waves in the supporting fiber. In the main text, Fig. D plots the phonon linewith as a function of the fiber iameter, from which we can see that the S 1,, moe has the narrowest linewith. This can be interprete from the moe fiel istribution: the isplacement of S 1,l,l moe concentrates aroun the equator an the isplacement at the poles where the supporting fiber is attache to is almost zero, therefore the clamping loss is very low. The calculate clamping-loss-limite linewith of the S 1,, moe is only about 0 Hz when the YIG sphere is glue to a 15-µm-iameter fiber. In our experiment, the measure phonon linewith is above 100 Hz, which therefore is not limite by the supporting fiber but instea by other factors such as air amping or geometry imperfection. C. Magnon-Phonon Coupling Strength The well-accepte Young s moulus of the YIG is about Y = Pa [48], an the Poisson ratio is about vy v = 0.9. With these parameters, we can calculate the Lamé constants as λ Lamé = (1+v)(1 v) an µ Lamé = Y (1+v), an the longituinal velocity as v t = µlamé ρ = m/s an transverse velocity as v l = λlamé +µ Lamé ρ = m/s, where mass ensity ρ m = 517 kg/m 3. These results agree well with previously reporte values [49, 50].

5 Particularly, in our experiment we focus on the l = moe, which has an eigenfrequency ω 1,, = 3.6 D π MHz with D being the sphere iameter in unit of mm. The calculate frequencies agree well with the measurement results. For the sphere size D = 50 µm use in our experiment, we get ζ = for a single phonon of S 1 moe. Therefore, the maximum coupling strength is g mb = 9.88 π mhz. In the sphere, the strain for a single phonon is proportional to ω b V, where V b D 3. Therefore, the coupling strength g mb 1/D, inicating that smaller spheres are favorable for stronger magnon-phonon coupling. FigureB in the main text shows the epenence of the coupling strength on the sphere iameter, showing a clear 1/D epenence. The cooperativity is the Figure of Merit to evaluate the coherent magnon-photon-phonon coupling, which is b C = n m g mb κ ± κ b (S17) where n m is the mean magnon excitation number. In experiments, n m is mainly limite by the thermal instability. Assuming the net power absorbe by YIG sphere is constant, 0 Bm for instance, an the hybri moe linewith is.4 MHz, we can estimate n m = π (S18) for 7.8 GHz magnons. In aition, the linewiths of magnon an microwave photon are mainly limite by the material loss, an o not change with the iameter of the YIG sphere. In this case, the upper boun of the coupling strength is given by corresponing to a cooperativity g mb /π 5 8D mhz C D 4 (S19) (S0) for κ b /π = 300 Hz, where D is in unity of mm. Therefore, the cooperativity critically epens on the sphere size, an smaller YIG spheres always give larger cooperativities. For example, if D = 0.1 mm, C can be as large as 80. On the other han, smaller sphere size also have some rawbacks. The increasing phonon frequency with shrinking sphere size causes the rive to be further etune, which means lower riving efficiency an therefore reuces the magnomechanical coupling strength. Moreover, the magnon-microwave photon coupling strength also reuces for small YIG spheres, which may be too small for obtaining the triple-resonance conition. II. MAGNON MODES OF YIG SPHERE When a YIG sphere is place in a uniform bias magnetic fiel, it support various magnon moes. When placing the sphere in a uniform electromagnetic fiel with magnetic fiel unparallel to the bias fiel, a uniform magnon moe (Kittel moe) with all the spins precessing in phase will be excite. The frequency of such uniform magnon moe is simply etermine by the external bias magnetic fiel: ω m = πγh, where γ = π.8 MHz/Oe is the gyromagnetic ratio an H is the bias magnetic fiel. For a saturate magnetization (H > 1750 Oe), the magnon frequency lies in the gigahertz range. Thanks to the extremely large spin ensity in single crystal YIG, strong microwave photon-magnon coupling can be achieve [31 33]. In our experiments, the YIG sphere is excite by the uniform microwave magnetic fiel of the cavity TE 011 moe. The photon-magnon coupling strength can be tune by ajusting the YIG sphere position insie cavity or the angle between the bias magnetic fiel an the microwave magnetic fiel (For more etails, see Supplementary Material for Ref. [3]). As inicate by the avoie crossing in the reflection spectrum when sweeping the bias magnetic fiel (Fig. 3C), the minimum gap between normal moes g ma is much larger than the linewiths. The two branches are the hybriize magnon an photon moes. These new normal moes contains magnon components which can couple with the phonon through magnetostrictive interaction, as well as some photon component which allows accessibility using microwave methos.

6 III. COHERENT CAVITY MAGNOMECHANICAL COUPLING A. System Hamiltonian The Hamiltonian of the whole system, which consists of the photon, magnon an phonon moes an their interactions, is H = ω a â â + ω bˆb ˆb + ω m ˆm ˆm +g ma (â + â )( ˆm + ˆm ) + g mb ˆm ˆm(ˆb + ˆb ) +ε κa,e (âe iω t + â e iω t ) + ε s κa,e (âe iωst + â e iωst ) (S1) Here, the cavity is riven by a microwave source at frequency ω an probe by a weak microwave signal at frequency P ω s. The rive fiels ε = P ω an probe fiels ε s = s ω s, where P an P s are the rive an probe power, an κ a,e is the external coupling rate of microwave cavity. First, we linearize the nonlinear ispersive magnon-phonon coupling term to stuy the coherent magnon-phonon coupling. The intensity of the signal is much smaller than that of the microwave rive, therefore we omitte the signal first to solve the mean fiels of photon an magnon. In the rotating frame of the rive an applying the rotating-wave approximation g ma (â + â )( ˆm + ˆm ) g ma (â ˆm + â ˆm), we have H = (ω a ω )â â + ω bˆb ˆb + (ω m ω ) ˆm ˆm +g ma (â ˆm + â ˆm) + g mb ˆm ˆm(ˆb + ˆb ) +ε κa,e (â + â ) (S) The ynamics of the system satisfy (in the rest of the Supplementary Information, the expect value of operator Ô is represente by the symbol O for simplicity) t a = [ i(ω a ω ) κ a ]a ig ma m i κ a,e ε t b = ( iω b κ b )b ig mb m m t m = [ i(ω m ω ) κ m ]m ig ma a ig mb (b + b )m Here, κ a,b,m is the total issipation rate of microwave cavity photon, the phonon an the magnon. At steay state t a = t b = tm = 0, we have (S3) (S4) (S5) b ss = m ss = ig mb m ss iω b κ m ss b ig ma a ss i(ω m ω ) κ m ig mb (b ss + b ss) (S6) (S7) a ss = i(ω a ω ) κ a + Therefore, we can solve the mean fiel of magnon as i κ a,e ε g ma i(ω m ω ) κ m ig mb (b ss+b ss ) m ss ( κ a,e ε g ma ) = [ i(ω a ω ) κ a ][ i(ω m ω ) + i g mb ω b m ss κ m ] + gma We notice that the strong rive ε may inuce instability, which we will analyze together with the thermal instability in Section V. Substituting the steay state values to the Hamiltonian an omitting the high orer terms of fluctuations, we get the linear coherent interacting Hamiltonian for the system ω b +κ b H = (ω a ω )â â + ω bˆb ˆb + (ωm ω ) ˆm ˆm +g ma (â ˆm + â ˆm) + G mb ( ˆm + ˆm)(ˆb + ˆb ) (S8) (S9) + κ a,e ε s (âe i(ωs ω )t + â e i(ωs ω )t ) (S30)

7 where ω m = ω m ω bgmb m ωb ss an G mb = g mb m ss. By Fourier transform of operators O(ω) = 1 +κ π to(t)e iωt an b O ( ω) = 1 π to (t)e iωt, the ynamics of the system can be written as The full solution to these equations is 0 = [ i(ω a ω ω) κ a ]a(ω) ig ma m(ω) i κ a,e ε s δ(ω ω s + ω ) (S31) 0 = [ i(ω b ω) κ b ]b(ω) ig mb [m ( ω) + m(ω)] (S3) 0 = [ i(ω m ω ω) κ m ]m(ω) ig ma a(ω) ig mb [b(ω) + b ( ω)] (S33) a(ω) = i κ a,e E in X 1 X X 3 + X 4 + χ b χ bp g 4 ma (S34) where E in = ε s δ(ω + ω s ω ) is input power spectrum, X 1 = gma ( ) G mb (χ b χ bp ) χ b χ bp χ m ( ) X = χ ap G mb (χ b χ bp ) (χ m χ mp ) + χ b χ bp χ m χ mp (S35) (S36) X 3 = g ma ( χb χ bp (χ a χ m + χ ap χ mp ) G mb (χ a χ ap ) (χ b χ bp ) ) (S37) X 4 = χ a χ ap ( G mb (χ b χ bp )(χ m χ mp ) + χ b χ bp χ m χ mp ) (S38) an the susceptibilities χ a = i(ω a ω ω) κ a, χ ap = i(ω a ω +ω) κ a, χ b = i(ω b ω) κ b, χ bp = i(ω b +ω) κ b, χ m = i(ω m ω ω) κ m an χ mp = i(ω m ω + ω) κ m. By applying the approximation that G mb χ a, χ m, g ma, we obtain a(ω) The reflection spectrum can be calculate by i κ a,e χ a g ma χ m g ma G mb χ bp χ + g ma G mb m χ b χ m a out = ε s + i κ a,e a ε s (S39) (S40) The full an approximate solutions are teste for various experiment parameters, an they perfectly agree with each other. Therefore, we just simply neglect the counter-rotating terms of magnon-phonon interaction in the following analysis. B. Classical Derivations For the uniform magnon moe (or FMR, the ferromagnetic resonance) in a uniform magnetic fiel H, the Lanau- Lifshitz-Gilbert equation is t M = γµ 0H M α M M t M (S41) The magnetic fiel inclues the contribution of static bias magnetic fiel H 0 ez, the microwave cavity fiel H c ey an the effective magnetic fiel prouce by the magnetoelastic effect h me = Mx,y,z E me = b 1 MS [M x ε xxex + M y ε yyey + M z ε zzez ] (S4) Note here the shear strains are neglecte since they only contribute to the linear magnon-phonon coupling instea of the parametric process, as inicate by Eq.(S1). Treat the microwave an magnetoelastic fiel as perturbations, we have the magnetization as M = M z ez + (M x + S x + A x ) e x + (M y + S y + A y ) e y (S43) where M z M S, M x,y come from the microwave excitation, an A x,y (S x,y ) originate from the magnetoelastic interaction. The M x,y is oscillating at microwave riving frequency ω, while A x,y an S x,y are oscillating at frequencies ω + ω b an ω ω b.

8 Similarly, we have the cavity fiel as H c = H + H S + H A corresponing to the riving fiel at frequency ω, an the two cavity fiels ue to the coupling with magnetoelastically inuce magnetic excitation at frequencies ω + ω b an ω ω b, respectively. Since the parametric magnetoelastic coupling is much weaker than the microwave excitation, we shoul also treat the magnetoelastic effect as a perturbation to the microwave excitation. Then, we obtain the ynamics for the microwave excite magnetization as t M x = γh 0 M y γh M S αγh 0 M x t M y = γh 0 M x + γh M S αγh 0 M y an for the cavity fiel which is riven by an external microwave source as t H = κ a H ω a E + γh 0 M x + κ a,e ε t E = κ a E + ω a H Here, E is the electric fiel which is a conjugate variable to the magnetic fiel. For the ω ω b sieban, we have t S x = γh 0 S y γh S M S γb 1 [(ε yy ε zz )M y ] ω ω M b αγh 0 S x S t S y = γh 0 S x + γh S M S γ γb 1 [(ε zz ε xx )M x ] ω ω M b αγh 0 S y S where [ ] ω ω b represents the ω ω b component of the expression, an For the ω + ω b sieban, we have an t H S = κ a H S ω a E S + γh 0 S x t E S = κ a E S + ω a H S t A x = γh 0 A y γh A M S γb 1 [(ε yy ε zz )M y ] ω +ω M b αγh 0 A x S t S y = γh 0 A x + γh A M S γ γb 1 [(ε zz ε xx )M x ] ω +ω M b αγh 0 A y S t H A = κ a H A ω a E A + γh 0 A x t E A = κ a E A + ω a H A Since there are three frequency components for the magnetization an microwave cavity, an there is a mechanical vibrational moe, the classical erivation becomes cumbersome. However, by introucing the quantize operation for magnon m = 1 M S (M x + im y ), phonon an photon a = 1 (E + ih), the problem can be simplifie as shown in previous sections. (S44) (S45) (S46) (S47) (S48) (S49) (S50) (S51) (S5) (S53) (S54) (S55) (S56) C. Hybriize Magnon-Photon Moe For the strong magnon-photon coupling conition, we have g ma κ a, κ m, an then magnon an photon moes are hybriize. We can solve the normal moes as ( ) ( ) ( ) Â+ cosθ sin θ â = (S57) sinθ cosθ ˆm Â

9 where an θ [0, π ], with frequencies Therefore, Conversely, we have ω ± iκ ± = ω a + ω m ω a + ω m ( g ma tan θ = ω m ω a i κ a + κ m ± ± g ma sin θ i(κ a + κ m κ + = cos θκ a + sin θκ m κ = sin θκ a + cos θκ m ) ( â cosθ sinθ = ˆm sinθ cosθ g ma + (ω a ω m i κ a κ m ) (S58) ± cosθ κ a κ m ) (S59) ) ( Â+  ) (S60) (S61) (S6) When θ = π/4,  ± = 1 ( ˆm ± â) are the maximum hybriize moes. In general, the hybri moe-phonon interaction can be rewritten as H mb = g mb (ˆb + ˆb )(sin θâ +Â+ + cos θâ  + sin θ cosθâ + + sin θ cosθâ Â+). (S63) The first two terms  +Â+ +   enote that the two iniviual hybri moes ispersively couple to the phonon moes, while the last two terms  + +  Â+ inicate a triply resonant coupling. Now we can rewrite the linearize Hamiltonian using the hybri moes as where H/ = (ω + ω ) +Â+ + (ω ω )  + ω bˆb ˆb +(G +  + + G +Â+)(ˆb + ˆb ) +(G  + G  )(ˆb + ˆb ) + κ +,e ε s (Â+e i(ωs ω )t +  + e i(ωs ω )t ) κ,e ε s ( e i(ωs ω )t +  e i(ωs ω )t ) G + = g mb  +,ss sin θ + g mb Â,ss sin θ cosθ G = g mb Â,ss cos θ + g mb  +,ss sinθ cosθ In our system, the hybri moes strongly epen on the etuning between magnon an photon, which can be tune easily in the experiments. In the simple single moe regime, the steay state excitation i κ ±,e ε A ±,ss = i(ω ± ω ) κ ± Here, the hybri moe frequency shift cause by the steay state phonon excitation is negligible for the excitation power we are intereste. The external coupling rate is κ ±,e = 1 ± cosθ The corresponing input-output formalism becomes κ a,e a out = ε s + i κ +,e A + i κ,e A From the Hamiltonian, the system can be treate as two inepenent hybri moes which are riven by an external microwave signal an also couple with the phonon moes. The two hybri moes are well resolve in the spectrum ue to the strong coupling conition ω + ω g ma κ ±, an also the phonon frequency satisfies ω b κ ± which leas to the resolve sieban regime. We can stuy the system with rotating-wave approximation, which means (A + + A + )(b + b ) reuces to (A ±b + A ± b ) or (A ±b + A ±b ) for re or blue etune rive whose frequency satisfies ω ω ± ω b. (S64) (S65) (S66) (S67) (S68) (S69)

10 D. Reflection Spectrum 1. Re Detuning For re etune rive to the lower hybri moe at ω, we have ω ω ω b κ an ω + ω ω b g ma / sinθ κ +, an therefore we ignore the A + moe an get a simplifie Hamiltonian At steay state, H = (ω ω s )Â Â + (ω b + ω ω s )ˆb ˆb +(G Â ˆb + G Â ˆb ) κ,e ε s (Â + Â ) t A = 0 = ( iδ κ )A ig b + i κ,e ε s (S71) t b = 0 = ( iδ b κ b )b ig A (S7) with δ = ω ω s anδ b = ω b + ω ω s. Due to the coupling with magnon, the effective intrinsic loss of the phonon moe becomes G κ b,eff = κ b + δ /κ (S73) + κ which is increase an inicates the cooling of the thermal mechanical vibration. The phonon frequency shift cause by the magnomechanical coupling is Then, the reflectivity of the cavity becomes G ω b = κ /δ + δ (S70) (S74) r = 1 κ,e iδ + κ + G iδ b +κ b (S75). Blue Detuning For blue etune rive to the upper hybri moe at ω +, we have ω ω + ω b κ + an ω ω ω b g ma / sinθ κ +, an therefore we ignore the A moe an get a simplifie Hamiltonian At steay state H = (ω ω s )Â Â + (ω b + ω ω s )ˆb ˆb +(G + Â +ˆb + G +Â +ˆb ) + κ +,e ε s (Â+ + Â + ) t A + = 0 = ( iδ + κ + )A + ig + b i κ +,e ε s (S77) t b = 0 = (iδ b κ b )b + ig +A + (S78) with δ + = ω + ω s an δ b = ω b + ω ω s. (S76) Similar to the re etune case, the intrinsic loss of the phonon moe becomes G + κ b,eff = κ b δ+ /κ + + κ + which amplifies the thermal mechanical vibration. In aition, the phonon frequency shifts by Then, the reflectivity of the cavity becomes G + ω b = κ + /δ + + δ + κ +,e r = 1 iδ + + κ + G+ iδ b +κ b (S79) (S80) (S81)

11 A ω ω s B ω s ω Reflection (B) A - A + Zoom-in s (MHz) s (MHz) Reflection (B) A - A s (MHz) Zoom-in s (MHz) FIG. S1: Magnetically inuce transparency/absorption at various rive frequencies. (A) Measure reflection spectra as a function of the two-photon etuning s for various riving-resonance etuning. Zoom-in shows the etaile spectra of the shae region. (B) Measure reflection spectra as a function of the two-photon etuning s for various rivingresonance etuning +. Zoom-in shows the etaile spectra of the shae region. All black curves are calculate results using the same parameter as in Fig. 3 in the main text, with only a single fitting parameter g mb. 3. Magnomechanically Inuce Transparency, Absorption an Parametric Amplification From the above analysis, we know that the cavity reflection spectrum will be moifie by the phonon resonance. Figures 3A an B in the main text an Fig. S1 show the typical spectra, incluing magnomechanically inuce transparency/absorption (MMIT/MMIA) an magnomechanical parametric amplification (MMPA). These results show great agreement with our theory above. Here, we provie a unifie picture for better analyzing these phenomena. When the rive is off etune from the hybri moe by the phonon frequency, we have the reflectivity as r = κ ±,i G± κ b κ ±,e κ ±,i G± κ b + κ ±,e = 1 κ±,e κ ± C. (S8) 1 C Where the intrinsic loss of hybri moes κ ±,i = κ ± κ ±,e. From this equation, we can see that the parametric rive inuces an effective intrinsic loss as κ ±,i G± κ b. By varying the external coupling an the cooperativity, we can control the lineshape of the magnomechanically inuce resonance to be either absorption, transparency, or amplification, as shown by Fig. SA. The absorption/transparency is efine as the reuction/enhancement of the reflectivity by magnomechanical interaction. The amplification is efine as when the reflectivity is larger than unity. The parameter space iagrams of these interesting phenomena are plotte in Figs. S B an C. For a blue-etune rive, when κ ±,i G± κ b < 0, the transmission r will be larger than 1 an therefore we are in the amplification regime. Since the cooperativity is efine as C = G ± /κ ± κ b, the threshol for amplification is κ ±,i < C(κ ±,i +κ ±,e ), i.e., κ ±,e /κ ± > 1 C. Further increasing ε above the threshol G ± = κ ± κ b (i.e., C = 1) will lea to the singular infinite amplification. There exists a bounary between MMIT an MMIA as κ ±,e /κ ± = 1 C C. For the re etune rive, such bounary conition becomes κ ±,e /κ ± = 1+C +C. Therefore, the spectra can cross ifferent regimes when we increase C by increasing the riving power. Such phenomena also epen on the external coupling rate κ ±,e /κ ±. In our experiments, both the external coupling rate an the cooperativity can be tune, yieling the rich ynamics of our system.

12 A Normalize reflection Absorp on Cri cal coupling 1.0 Transparency B C κ,e /κ + + κ,e /κ Transparency Absorption Blue etuning Crtical coupling Amplification Cooperativity Re etuning Crtical coupling Transparency Absorption Amplifica on Frequency Cooperativity FIG. S: Magnomechanical resonance line shape an parameter space iagram. (A) Representative lineshapes of the magnomechanical resonance in ifferent regimes. The broa resonance ip correspons to the hybri moe resonance, while the narrow ip/peak insie the hybri moe is the magnomechanical resonance. (B an C) Parameter space iagrams for the blue- an re-etune rive, respectively, as a function of external coupling an magnomechanical cooperativity. 4. Magnetic Fiel Depenence of the Cooperativity In our system, the magnon-photon hybriization strongly epens on their etuning, which can be tune easily in the experiment. On the one han, the steay state excitation Eq.(S67) epens on θ. On the other han, the 1±cos θ effective coupling strength g mb varies with θ as the portion of magnon component in the hybri moe changes. Making the approximation that A,ss 0 for the blue-etune rive an A +,ss 0 for the re-etune rive, an combining with the formalism of effective coupling strength, we have an similarly G + κ b κ + = g mb P ω κ b κ + G κ b κ g mb P ω κ b ω b P κ a,e g mb ω ω b κ b P κ a,e g mb ω ω b κ b κ +,e ωb + sin 4 θ κ + κ +,e κ + sin 4 θ κ a,e sin 4 θ cos θ κ a cos θ + κ m sin θ κ a,e cos 4 θ sin θ κ a sin θ + κ m cos θ Therefore, maximum cooperativity can be achieve by tuning the magnetic fiel. The soli lines in Fig. 3D of the main text is calculate using these expressions, which agrees well with experiment results. (S83) (S84) 5. Cascae Transparency/Absorption Our previous research has shown that the great flexibility of our system enables us to work in ifferent magnonphoton coupling regimes. By choosing proper issipation an coupling parameters that satisfy κ m < g ma < κ a an

13 A -8 D -8 Reflection (B) Reflection (B) B Frequency (GHz) E Frequency (GHz) Reflection (B) Reflection (B) C Frequency (GHz) F Frequency (GHz) Reflection (B) Reflection (B) Frequency (GHz) Frequency (GHz) FIG. S3: Cascae transparency/absorption. (A an D), Transparency winow inuce by the magnon-photon coupling. Uner the conition κ m < g ma < κ a, the low loss magnon moe inuces a transparency winow in the broa cavity photon resonance as a result of the magnon-photon coupling. (B an E) Zoom-in of the magnon transparency winow, which shows a secon absorption ip (transparency winow) for a re (blue) etune rive as a result of the magnomechanical coupling. (C an F) Zoom-in of the magnomechanically inuce absorption ip (transparency winow). Visible in the spectrum is a secon (small) resonance that correspons to a non-egenerate phonon moe. Dots are measurement results, while soli lines are numerical fittings. tune the magnon an cavity photon moes on resonance, we obtaine the magnetically inuce transparency (MIT) regime [3], where the magnon resonance shows up in the cavity resonance as a transparency peak (Fig. S3A). Now with a strong rive that is re etune from the resonance frequency ω a by the phonon frequency ω b, we can obtain a magnomechanical absorption ip insie the (magnetically inuce) transparency winow, as shown in Figs. S3B an C. Similar phenomenon also exists when the rive is blue etune, where we obtain a MMIT winow instea of a MMIA ip insie the MIT winow (Figs. S3, D-F). The so-calle cascae transparency phenomena was first reporte in the optomechanical system [18]. Such a cascae system combines two ifferent types of transparency, allowing us to achieve an extremely long elay (0.6 ms, as compare with a 67-ns elay obtaine solely with a MIT winow) in a very lossy microwave cavity with a photon lifetime of only 0. ns. 6. Triple Resonance For the triple resonance conition ω + ω ω b, the single moe approximation is still vali, while the effective coupling strength is rastically enhance ue to the on-resonance rive. The parametric rive at A ± moe leas to an effective phonon-magnon interaction at A moes. Therefore, we make the approximation that A ±,ss 0 for the

14 re(blue) etune rive. In this case, the effective coupling strength κ,e ε g mb G ± = sin θ cosθ (ω ω ) + κ (S85) The cooperativities become G + = g mb P κ a,e sin 3 θ cosθ κ b κ + ω κ b κ + (ω ω ) + κ G κ b κ = g mb P ω κ b κ κ a,e cos 3 θ sinθ (ω + ω ) + κ + (S86) (S87) For triply resonant ω a = ω m an ω b = g ma, we have θ = π/4 an Compare with the previous off-resonance riving situation which has a C = enhance by C = g mb P 4κ a,e ω κ b (κ a + κ m ) 3 (S88) F tr = ( ωb κ a+κ m P g mb ω ω b κ b κ a,e κ a+κ m, the cooperativity is ) (S89) 7. Magnomechanical Parametric Amplification Figure 3 in the main text shows the magnomechanical parametric amplification (MMPA) obtaine in our triply resonant systems for a blue etune microwave rive. In the measurement, the hybri moe is uner-couple, which leas to a very sharp increase of the parametric gain as we increase the rive power. This is also evient from Fig. SB, where the parametric gain only exists in a small C range before iverging for small external coupling rates. In practice, the measurement of such sharp change is quite challenging ue to issues such as thermal instability. As a result the maximum parametric gain obtaine is only 3 B. However, if we change the coupling conition to over-coupling for the hybri moe, the epenence of the parametric gain on the rive power becomes more graual an therefore we can ajust the riving power to obtain a much higher gain. Figure4E plots the obtaine parametric gain as a function of the riving power, shows a gain as high as 5 B with a moerate riving power of 0.3 Bm. The measure cavity reflectivity spectrum for this configuration is plotte in the inset of Fig. 4E. We have two hybri moes that are over-couple (juging from the π phase change which is not shown here) with a 10 B extinction. The strong riving is applie to the upper hybri moe, an then we obtain a rastically amplifie signal at the lower hybri moe, with an extinction of 35 B. Subtracting the 10 B insertion loss cause by the hybri moe itself, we obtain a net gain of 5 B. IV. THERMAL INSTABILITY Due to the nonlinear effects involve in our experiments with high riving power, the system becomes multistable, preventing us from injecting more power into the system. Taking the thermal an magnon-phonon nonlinear effects into account, we obtain m ss ( κ a,e ε g ma ) = [ i(ω a ω ) κ a ][ i(ω m ω ) ikt + i g mb ω b m ss κ m ] + gma ω b +κ b (S90) Here, the coefficient K enotes the thermal magnon frequency shift proportional to the sample temperature T. The thermal effect is mainly attribute to the anisotropic fiel of the YIG sphere, an therefore K may vary by a factor of 10 epening on the sphere orientation insie the bias fiel [51]. In quasi-static processes, the sample temperature is proportional to the absorbe microwave power, which leas to T = q m ss with q is a constant etermine by the

15 thermal issipation of the sample. In aition, the phonon frequency may also change with input power as a result of the sample heating ue to thermal effect [5], but it is negligible in our experiments. We can combine the coefficients of nonlinear frequency shifts of the steay state solution as Kq g mb ω b. In ωb +κ b experiments, the cooperativity which can be approximate by C g mb κ aκ b m ss is limite to the orer of Therefore, with the maximum power we can inject to the system, the shift of the magnon frequency g mb ω b m ωb ss +κ b C κa ω b /κ b 10 4 κ a is negligible. Fitting the parameter Kq to show bifurcation by assuming P is 0 Bm, we obtain Kq π Hz, which is about 3 orers of magnitue larger compare with the coefficient originating from the magnon-phonon coupling ( g mb ω b = Hz). Therefore, in our experiment we only nee to consier the ωb +κ b instability ue to the thermal effect.