Antiferromagnetic cavity optomagnonics

Currently, there is a growing interest in studying the coherent interaction between magnetic systems and electromagnetic radiation in a cavity, prompted partly by possible applications in hybrid quantum systems. We propose a multimode cavity optomagnonic system based on antiferromagnetic insulators, where optical photons couple coherently to the two homogeneous magnon modes of the antiferromagnet. These have frequencies typically in the THz range, a regime so far mostly unexplored in the realm of coherent interactions, and which makes antiferromagnets attractive for quantum transduction from THz to optical frequencies. We derive the theoretical model for the coupled system, and show that it presents unique characteristics. In particular, if the antiferromagnet presents hard-axis magnetic anisotropy, the optomagnonic coupling can be tuned by a magnetic field applied along the easy axis. This allows to bring a selected magnon mode into and out of a dark mode, providing an alternative for a quantum memory protocol. The dynamical features of the driven system present unusual behavior due to optically induced magnon-magnon interactions, including regions of magnon heating for a red detuned driving laser. The multimode character of the system is evident in a substructure of the optomagnonically induced transparency window.

Exciting developments have also emerged very recently in the THz regime, a frequency range which has been historically challenging due to the absence of efficient sources and detectors (the "THz gap") [21].Strong lightmatter coupling in the THz regime has been achieved employing cavities [22][23][24], including strong coupling to magnons in antiferromagnets (AFM) [25,26], opening the door for quantum applications in the THz domain.Antiferromagnetic materials support magnons that can be described as excitations of a spin anti-aligned ground state (the Nèel state) [27].Their frequencies are typically in the THz range, making AFMs ideal candidates to incorporate in THz platforms [28][29][30][31].The physics of AFMs in combination with electromagnetic cavities is just starting to be explored.Besides the mentioned ex- periments in the THz regime [25,26], strong coupling between MW photons and AFM magnons has been reported [32], while magnon dark modes [33] and coupling to ferromagnets [34] via a MW cavity have been proposed theoretically.In turn, methods involving light to probe and control AFMs are being developed [35,36].These developments are a great incentive to study the coupling of AFM magnons to optical cavities, which could lead to quantum transducers from the THz to the optical regime.
We show that for an AFM new phenomenology emerges.We derive the Hamiltonian governing the system and show that, in the presence of hard-axis anisotropy, the optomagnonic coupling to both supported homogeneous magnon modes can be tuned by an external magnetic field.In particular, we show that the magnon modes can be selectively decoupled from the cavity, rendering them dark.This is unique to AFM cavity optomagnonics.
Based on this tunability, we sketch a quantum memory protocol.We further characterize the dynamical response of the system and show that the cavity-mediated interaction between the AFM magnon modes leads to unusual optically induced magnon cooling and heating.
Model.-We consider an AFM insulator with two magnetic sublattices A and B of opposite spin.The AFM hosts two homogeneous magnon modes α and β, see Fig. 1, and we assume it acts also as an optical cavity by total internal reflection, analogous to dielectric optomechanical [48] or optomagnonic [1][2][3] cavities.AFMs with a high index of refraction and low absorption in the optical range, such as NiO (n ≈ 2.4) [49] would serve the purpose, or heterostructures containing MnF 2 (n ≈ 1.4) [50] or FeF 2 (n ≈ 1.5) [51].The Hamiltonian of the coupled system is with Ĥph and ĤAFM the free photonic and AFM Hamiltonians, respectively.ĤOM contains the coupling between the AFM magnons and the cavity photons, and is our main result in this section.
The quantized optical field in the cavity is ĤAFM consists of (i) the exchange interaction between nearest-neighbor spins J i =j Ŝi • Ŝj (J > 0), (ii) the Zeeman interaction between spins and an external DC magnetic field B 0 along e z , |γ| B 0 i Ŝz i (γ gyromagnetic ratio), and (iii) easy − axis anisotropy in the e z and e x directions respectively.For small magnetization fluctuations around the Nèel ordered state, the Holstein-Primakoff (HP) transformations [52,53] can be used to express ĤAFM in terms of bosonic operators âk and bk associated with the sublattices A and B. To first order, the HP transformations are given in terms of spin ladder operators as , where N is the total number of sites per sublattice and S the spin on each site.ĤAFM is diagonalized via a 4D Bo-goliubov transformation to the bosonic operators αk , with ω α,βk the respective eigenfrequencies.We restrict our analysis to the two homogeneous (k = 0) AFM magnon modes, hence from now onwards we drop the index k.The corresponding magnon frequencies ω α,β are functions of the characteristic frequencies ω E = JSN , ω ,⊥ = SN K ,⊥ , and 55,56].Note that ω α ≥ ω β and hence α (β) labels the upper (lower) mode.Whereas ω α increases with the magnetic field, ω β decreases and goes to zero at the onset of the spin-flop phase at ω H = ω SF ≈ 2ω E ω [57].
The interaction between light and magnetization is described by the magneto-optical coupling , where ε µν (µ, ν = x, y, z) is the spin-dependent part of the permittivity tensor.In this letter we consider simple cubic and rutilestructure AFMs, other more complex structures will be considered elsewhere.For these materials, within linear response in the deviations from the magnetic equilibrium, H OM reduces to [54,58,59] where we have discretized the interaction and , with E ± = E x ± iE y and S ± = S x ± iS y .The linear magneto-optic coefficients K ± correspond to processes in which the two sublattices scatter the light in-phase (+) or out-of-phase (−).For our purposes, one-magnon processes coming from quadratic terms in the spin (e.g.∝ Ŝz Ŝ± ) can be absorbed in the definition of K ± .This model applies e.g. to the uniaxial AFMs MnF 2 or FeF 2 [60,61], and for the simple cubic AFM NiO for which K − = 0.
We obtain the optomagnonic coupling Hamiltonian ĤOM by quantizing Eq. ( 2) assuming that the electric field varies smoothly, such that P ± i ≈ P ± j for nearest neighbors.We focus on the interaction between the homogeneous AFM magnon modes α and β with a single optical mode ĉ with frequency ω c .For an optical mode with circular polarization in the yz plane, from Eq. (2) we obtain [54] with (ε is the AFM dielectric constant) The AFM optomagnonic coupling depends on the Bogoliubov coefficients through where K = K − /K + quantifies the intrinsic magnetooptical asymmetry between the sublattices, and we have defined u ± α(β) = u a,α(β) ± u b,α(β) and v ± α(β) = v a,α(β) ± v b,α(β) .Hence, the two AFM magnon modes α and β couple, in general, with different strength to the cavity mode.
Optomagnonic Coupling.-Theconstant G describes the coupling to the magnetization's fluctuations sector and is consistent with the one derived in Ref. [44] for the optomagnonic coupling in a ferromagnetically ordered system.Assuming equivalent sublattices with Faraday rotation per unit length θ F , then K + = c √ εθ F /(ω c S) (with c the speed of light) and thus G = 1/ √ 2N S (cθ F /4 √ ε).The 1/ √ N dependence indicates that the density of excitations is relevant for the coupling, favoring small magnetic volumes.Due to the lack of data on absolute values for K + (or θ F ) for simple AFMs, we take as an estimate for G the value for (1µm) 3 YIG (diffraction limit volume), G YIG = 0.1MHz [44].Some measurements indicate that the Faraday rotation coefficient in AFMs can be quite large, e.g.similar values as for YIG have been reported for BiFeO 3 [62].Note that G given in Eq. ( 21) assumes perfect mode matching.Imperfect mode overlap can be accounted for by a mode-volume ratio factor [44] and it is responsible for a suppression of the coupling in current experiments with YIG [19,63].The second term in Eq. ( 5) gives a contribution proportional to KG and describes the coupling to fluctuations of the Nèel vector.Typical values of K are K ≈ 0.01 (e.g. for MnF 2 or FeF 2 [60]).
The reduced couplings g α,β can be found analytically, but the general solution is lengthy.Simple expressions can be given in certain cases.Since the exchange energy is usually the largest energy scale in the AFM, the condition ω ⊥, ω E holds.For an easy-axis AFM (ω ⊥ = 0), we obtain [54] Eq. ( 6) is independent of the magnetic field B 0 , a consequence of the axial symmetry of the system in this case [54].In the absence of magneto-optical asymmetry (K = 0) both modes couple equally to the light field, while for finite K, g β = 0 and β is a dark mode, completely decoupled from the cavity.Whereas this requires fine tuning, it could be achievable in cold atoms realizations where the relevant parameters can be tuned [64][65][66][67][68].The situation nevertheless changes in the presence of hardaxis anisotropy, where the coupling to the modes can be       Easy-axis dominated regime !?< !k < l a t e x i t s h a 1 _ b a s e 6 4 = " t U L a p Z q D w U a   tuned externally by the magnetic field as we show below.From Eq. ( 6) we obtain g MnF2 α,β ≈ 0.5, 0.4 (ω E = 9.3 THz, ω = 0.15 THz, K = 0.007 [60,69]) and g FeF2 α,β ≈ 0.6, 0.7 (ω E = 9.5 THz, ω = 3.5 THz, K = 0.01 [59,70]).
In the absence of a magnetic field, ĤAFM is invariant under âk ←→ b−k .For finite hard-axis anisotropy (ω ⊥ = 0), imposing this symmetry we obtain [54] and hence for K = 0 α is a dark mode (g α = 0) while β is independent of K.The case ω ⊥ = B 0 = 0 is however pathological, since α and β are degenerate.Then Eq. ( 6) holds, with g α = g β = 0 (the Bogoliubov coefficients present a discontinuity at ω ⊥ = 0).Fig. 2 shows |g α | and |g β | as a function of B 0 and K for representative finite anisotropy values ω ⊥ and ω .In Fig. 2 (a) we took these as for NiO [71], and in Fig. 2 (b) we exchanged them such that ω > ω ⊥ .In both cases the coupling strengths g α,β can be tuned by B 0 , although with some qualitative differences.For K = 0 the α-mode can be tuned from dark to bright by increasing B 0 , with a slow linear increase for ω < ω ⊥ and rapidly but saturating for ω > ω ⊥ .For both cases there is a threshold K th such that for K > K th , there exists a finite B 0 for which the β-mode is rendered dark (g β = 0).In the regime considered for Fig. 2, g α,β < 1 for all fields, since the maximum B 0 is limited by the spin-flop transition.This suppresses the corresponding optomagnonic coupling (Gg α,β ).g α increases nevertheless rapidly with K, so materials with a larger magneto-optical asymmetry would be favorable for larger coupling values.Our calculations indicate that K 0.1 would be sufficient for g α > 1 [54].
-We now consider a cavity driven by a strong control laser with amplitude s d and frequency ω d , and a weak probe laser with amplitude s p and frequency ω p , see Fig. 1.Correspondingly, we add a driving term ĤD = i √ ηκ(ĉ † ξ s in + H.c.) to the Hamiltonian in Eq. (1), where s in = s d e −iω l t + s p e −iωpt .The total loss rate of the optical cavity is κ = κ ex + κ 0 , where κ ex and κ 0 correspond to the loss rates due to external coupling and intrinsic dissipation, respectively.The coupling efficiency η = κ ex /κ 0 is adjustable in experiments [74,75].
The cavity leads to the modification of both the magnon resonance frequency and the magnon damping.Both effects are quantified through the magnon self energy, which also includes a cavity-mediated coupling between the two magnon modes.This term becomes relevant in the strong coupling regime (g j √ n c > Γ j , κ, with j = α, β and Γ j the magnon linewidth of mode j) for near degenerate magnon modes |ω α − ω β | < Γ j , see Sup.
Mat. [54].Together with hybridization effects [76][77][78] and counter-rotating terms that cannot be neglected in this regime, the optically induced magnon-magnon interaction is responsible for unusual behavior, for example amplification in the red detuned regime, see Fig. 3.The AFM cavity provides a unique platform to probe such regimes for materials that exhibit degenerate modes at zero magnetic field (such as MnF 2 ), since |ω α − ω β | can be tuned via an external magnetic field.
We turn now our attention to the transmission and reflection properties of the AFM optomagnonic cavity.Following the standard procedure (see Sup. Mat.[54]) we obtain the cavity mode spectra δc[ω] in the frame rotating at the control light frequency where ω = ω p −ω d is the pump-probe detuning, ∆ = ∆+ 2G(g α Re[ α ] + g β Re[ β ]) is the renormalized detuning due to the magnon induced cavity frequency shift with ĵ = iGg j n c / (iω j + Γ j /2) (for j = α, β), and The transmission and reflection spectra are obtained from Eq. ( 7) by using the input-output boundary conditions δc out (ω) = δc in (ω) + (κ ex /2) 1/2 δc(ω).In Fig. 4 we plot the reflection spectra for the fast-cavity regime (Γ < g α G √ n c < κ and for simplicity we assume Γ α = Γ β ≡ Γ).Due to destructive interference between the up-converted control field and the probe field, an optomagnonically induced transparency (OMIT) window opens in the transmission spectrum around the corresponding magnon resonance.In the near degenerate regime, depicted Fig. 4 (a) for representative parameters of MnF 2 (easy-axis AFM), the OMIT window has an additional structure due to the closeness of the α and β modes' sidebands.Increasing B 0 increases |ω β − ω α | (but has no effect on the optomagnonic coupling, see Eq. 6) and the usual OMIT behavior is recovered.In Fig. 4 (b) we show results for representative parameters of NiO (finite hard-axis anisotropy), for which the magnon frequencies are well separated even at zero magnetic field.In this case the OMIT behavior can be tuned by B 0 through the optomagnonic coupling, see Fig. 2. Finally, the dark-to-bright tunability of the magnon modes can be used for a quantum memory protocol.Driving the system with a strong control red detuned laser, the cavity-magnon coupling can be controlled by B 0 such that the (linearized) Hamiltonian is ∼ g(t)(δĉ † α + δĉα † ) [48].An arbitrary initial cavity state can then be stored in the magnon mode by bringing g(t) from its initial value g 0 to g(T ) = 0 such that T 0 dtg(t) = πg 0 (analogous to a π-pulse protocol [79]).This swaps the state of the cavity with the magnon mode, which is then rendered dark for t > T .The state can be transferred with high fidelity for strong coupling and T 1/κ, and stored up to the magnon lifetime.Alternatively, the OMIT could be employed, in a similar fashion to memories implemented in cold atoms [80][81][82][83][84].The AFM permits to tune the OMIT window via B 0 , allowing a broad bandwidth storage.
Conclusions.-We proposed a solid state optomagnonic cavity system in which optical photons are coupled to long wavelength AFM magnons and derived its governing Hamiltonian and dynamical features.We showed that the AFM system presents unique characteristics, such as tunability of the coupling with a magnetic field, and unusual dynamical effects due to cavity-induced interactions between the two homogeneous magnon modes.We estimated the values for the coupling and showed that, although challenging, the strong coupling regime could be reached in micron sized single-domain AFMs cavities [85,86].AFMs optical cavities could therefore provide a new platform to study light-matter interaction, and possibly a new tool to probe AFMs due to the enhanced light-magnon coupling.The tunability with a magnetic field, in particular for tuning a magnon mode from dark to bright, shows promise for quantum protocols for quantum information storage and retrieval.The coherent coupling of THz magnons to optical photons could allow the implementation of a quantum transducer [87].In this first work we focused on one-magnon processes, two-magnon processes will be treated elsewhere.
For completeness we present here how to diagonalize the antiferromagnetic Hamiltonian following Ref.[34].Considering small fluctuations around equilibrium, which we set to be in the e z direction, the spin ladder operators Ŝi ± = Ŝi x ± i Ŝi y , and Ŝi z are given by where âk and bk are the collective bosonic operators associated to the sublattices A and B respectively, satisfying the commutation relations âk , â † S is the total spin per lattice site and N is the number of sites in each sublattice.Using Eq. 10 and keeping only terms up to two bosonic operators, ĤAFM is written in k-space as The coefficients A, B, C k and D are given in terms of the characteristic angular frequencies ω E = SJZ (with Z the number of nearest neighbors), ω = SK , ω ⊥ = SK ⊥ , and ω H = |γ| B 0 as where f (k, δ) = j e −ik•δj , with the sum carried over all nearest-neighbor vectors δ j .For long wavelength magnons C k ∼ ω E .In order to diagonalize the Hamiltonian, we use the four-dimensional Bogoliubov transformation [34,55] where αk and βk are the destruction operators of the bosonic magnon modes which satisfy [α k , ĤAFM ] = ω αk αk and [ βk , ĤAFM ] = ω βk βk .These commutation relations lead to an eigenvalue problem that, together with the bosonic commutation rules for αk and βk , determine the coefficients of ( 13) and the eigenfrequencies ω α,β [55].The diagonalized form of ( 11) is given by

B. Magneto-optical Hamiltonian for Antiferromagnets
Here we present the details on the derivation of Eq. ( 3) in the main.The starting point is the time-average energy of the electromagnetic field on a magnetized material [88] where the integration is performed over all the cavity volume.We discretize the integral as a sum over all lattices as To first order in the spins, the permittivity tensor is given by with K µνζ the magneto-optical coefficients which are in general restricted by symmetry conditions.Further terms can be included to describe second order magnetooptical effects [58,59], which we do not consider in this work.Following Ref. [58], we assume system with a rutile crystal structure, where the two magnetic sublattices are arranged in a body-centered cubic geometry.One of the sublattices occupies the central sites, while the other occupies the corner sites.The components of the permittivity tensor are then given in terms of three imaginary constants K 1 , K 2 and K 3 [58], such that for sublattice A Given the considered geometry, K µνζ can be obtained from where R is the π/2 rotation matrix relating the symmetry of the sublattice A to the symmetry of the sublattice B. Therefore, for sublattice B If the sublattices are equivalent, We are interested in the optomagnonic coupling Hamiltonian, which represents the coupling of an optical field to the magnon excitations on top of the static ground state spin configuration.In correspondence with the setup of Section A, we assume that the Nèel equilibrium of the AFM takes place along the e z direction.Therefore, to first order in the magnon excitations, the coupling between light and the deviations from the magnetic equilibrium is encoded in the terms ∝ S x,y , which correspond to scattering processes involving one magnon.We thus do not consider the terms ∝ S z , which would correspond to higher order processes.Substituting ( 16), ( 17) and ( 18) in ( 15), the optomagnonic Hamiltonian can be written as which includes contributions from each sublattice.Defining K ± = i(K 1 ± K 2 )/4, we obtain Eq. ( 2) in the main text.Quantizing it we obtain , with E βs,± = E x ± iE y , and K = K − /K + .For modes polarized in the yz plane (as in Fig. 1 in the main text) P i+ βγ,ss = −P i− βγ,ss = G i βγ,ss , where the last equality defines the coefficients G i βγ,ss ≡ G βγ,ss (r i ).
i∈A,B G i βγ,ss In general, the total coupling coefficients will be determined by taking the continuum limit of Eq. ( 19), such that the sum over the lattice i,j V N is replaced by the corresponding volume integrals.The total coupling will therefore also depend on the degree of overlap of the corresponding optical and magnon modes [45].
We further specify the model by considering that only one relevant cavity mode interacts with AFM magnons with k = 0.Moreover, we assume for simplicity that the cavity mode with polarization χ has a plane wave profile E χ (r) = i ωc 2εV e ikc•r e χ , where ε is the electric permittivity of the material, V is the cavity volume, k c is the wave vector of the mode, and e R(L) = (e y ∓ ie z )/ √ 2 denotes right and left circularly polarized modes.For this case, using the Holstein-Primakoff approximation for the spin operators (see Eq. 10), the Hamiltonian for the right circular polarization component reads (from now on âk=0 ≡ â and bk=0 ≡ b) with We notice that the integration over the plane wave factors gives a volume factor that cancels the volume dependence on the denominator of the electric field modes.For the left circular polarization component ĤL MO = − ĤR MO .Since the system is diagonal in the R/L basis, we further consider only one circular polarization component of the optomagnonic Hamiltonian (R for definiteness) and drop the index R of (20).We also notice that if the two sublattices are equivalent and since we defined K 1,2 via the expansion of the permittivity tensor in terms of the spin in each lattice, see eq. ( 16), the relation of K + to the Faraday rotation angle per length θ F is given by which is equivalent to the coupling in a ferromagnetically ordered system [44].
In order to express the optomagnonic Hamiltonian in terms of the magnon modes α ≡ αk=0 and β ≡ βk=0 , we use the inverse of the transformation (13) β † .Substituting these expressions in Eq. ( 20) we obtain Eqs. ( 4) and ( 5) of the main text.
In the absence of hard axis anisotropy (ω ⊥ = 0) the non-vanishing Bogoliubov coefficients are independent of the external magnetic field and given by [55] In the limit ω ω E the couplings g α,β take the simple form given in Eq. ( 8) of the main text.
In figure 5 we show the exact g α,β as a function of K for ω ⊥ = 0 and representative values of ω (left plot) and as a function of ω H for different values of K (right plot).In the left plot, ω ω E and we observe a linear behavior with K, in agreement with Eq. ( 8) of the main text.For zero external magnetic field, the antiferromagnetic Hamiltonian is invariant under the transformation âk ←→ b−k , see Eqs. (11) and (12).For k = 0, this corresponds simply to swapping the sublattices A and B. Under this transformation Ŝ : â → b , the Bogoliubov modes read that for our case the Bogoli-ubov coefficients are all real) where the prime denotes the transformed modes.Since (11) is invariant under Ŝ, the transformed Bogoliubov modes fulfill Considering Eqs. ( 23) and ( 24), for non-degenerate modes ω α = ω β (this requires ω ⊥ = 0) we obtain the following conditions on the Bogoliubov coefficients: u j,b = ±u j,a and v j,b = ±v j,a (for j = α, β).In our case, α ( β) corresponds to the antisymmetric (symmetric) mode under the transformation: and the inverse transformation reads where we have used that 2 and hence for K = 0 the α-mode is decoupled from the light, while the β-mode coupling is independent of K.
2. Easy axis AFM case (ω ⊥ = 0) In the absence of hard axis anisotropy, the Hamiltonian 9 is invariant under rotations around the e z axis and 11 reads (for k = 0, and = 1) A rotation by θ around the e z axis is given by R : Ŝ+ → e iθ Ŝ+ , thus at the level of the bosonic operators â → e iθ â and b → e −iθb The Bogoliubov modes transform as We conclude that, in order for ĤAFM = ω α α † α + ω β β † β to be invariant we have (for j = α, β) u j,a = v j,b = 0 or u j,b = v j,a = 0.In our case, R: α → e iθ α and β → e −iθ β, We can then write ĤAFM in terms of the Bogoliubov coefficients (besides a constant term) as Comparing the above expression with (26) we have To obtain further information on the form of the Bogoliubov coefficients we use the eigenvalue equations to obtain Using (28) we can rearrange Eq. ( 30) into Comparing with (29) we conclude that of ω H , and hence the Bogoliubov coefficients U and V are independent of ω H . Therefore the couplings are independent of the magnetic field for ω ⊥ = 0 and at K = 0 they have equal strength g α = g β .In this derivation the important fact was u α,b = u β,a = v α,a = v β,b = 0, which is consequence of the invariance under rotations around the e z axis.For ω H = ω ⊥ = 0, under Ŝ : â → b we have Ŝ α Ŝ−1 = β, and the diagonalized Hamiltonian Eq. ( 24) is invariant since ω α = ω β .This falls into the previous case and the couplings g α,β are given by Eqs.(31).
Note that the Bogoliubov coefficients present a discontinuity at ω ⊥ = 0 and therefore also the g α,β .In particular, g α (ω H = 0, K = 0) = 0 for ω ⊥ = 0 as we showed in Subsection A, but it is finite for ω ⊥ = 0, see Subsection B and Eq. ( 8) in the main text.

D. Equations of motion for the control-probe pump scheme
In this section we derive the cavity spectra of the antiferromagnetic optomagnonic system by solving the linearized quantum Langevin equations.The total Hamiltonian of the driven optomagnonic cavity, under the simplifications of the previous sections, reads The intracavity field is driven by input lasers which can be described by the Hamiltonian given in terms of the drive powers P d,p .The total cavity loss rate is denoted by κ = κ 0 + κ ex .where κ 0 denotes the intrinsic loss rate and κ ex the external loss rate.The dimensionless parameter η ≡ κ ex /(κ 0 + κ ex ), can be continuously adjusted in experiments [89,90].In this control-probe scheme, the control laser has a stronger intensity than the probe laser s d s p [48,74,91].In a frame rotating with the control laser frequency, the total Hamiltonian reads where ∆ = ω d − ω c is the detuning between the cavity and the control laser frequency.The Langevin equations of motion for the operators α, β and ĉ are thus with Γ α,β is the intrinsic magnon damping rates which we assume to be equal Γ α = Γ β = Γ.We have disregarded, for now, the effects of the weak probe pump since s d s p .Here and in what follows, the noise operators describe random fluctuations of the system and have vanishing expectation values [92].We consider the equations for the expectations values α , β and ĉ , disregarding quantum fluctuations, such that for example ĉ † ĉ = | ĉ | 2 .The steady state is obtained by setting ċ ≡ ċ = 0, α = 0 and β = 0, such that .
We linearize the dynamics of the system by considering the fluctuations over the steady state values α(t) = α + δ α(t), β(t) = β + δ β(t) and ĉ(t) = c + δĉ(t).Correspondingly for the input field s in (t) = s+δs in (t), where we identify s = s d as the amplitude of the control field due to s d s p .The linearized Langevin equations for the fields' fluctuations read In these equations ∆ = ∆ + 2G(g α Re[ α ] + g β Re[ β ]) takes into account the cavity frequency shift due to the coupling to the magnon modes.Finally, we obtain the mode spectra via the Langevin equations for the average values of the fluctuations in frequency space.Defining the Fourier transform as δX The cavity spectrum is thus , where with Σ(ω) the self-energy term given by Since we are working in a rotating frame with the control laser, the frequency ω is the sideband shift of the probe ω p from the control light frequency, ω = ω p − ω d .In the following and in the main text we restrict our results to the resolved sideband regime case, ω α,β κ).For a red detuned control drive ( ∆ < 0), Stoke's processes are far off-resonance and the relevant resonance is ∆ ≈ −ω ≈ −ω α,β .In this case we can approximate F (ω) ∼ Σ(ω)/2i ∆ and, if ∆ Σ(ω), then the cavity spectrum is given by where Σ AS = − Analogously, by calculating the self energy term of the magnon mode j, Σj [ω] (j = α, β), we can obtain the corresponding linewidth Γj = Γ + 2Re[ Σj ] and frequency shift δω j = Im [ Σj ].The magnon self energy is given explicitly by (for k = j, and omitting the dependences on ω) with

E. Cavity-induced magnon-magnon interactions
The full self-energy expression (33) includes contributions due to bare Stokes and anti-Stokes processes involving the magnon mode and the cavity mode, as one can see directly in the terms ξ j , as well as contributions due to indirect magnon-magnon interactions mediated by the cavity.The effects of these different terms are not immediately clear from (33), thus, we now consider a framework to understand the effects of each contribution of the self-energy by comparing the full expression (33) with the magnon self-energy obtained by considering a fixed detuning and performing the rotating wave approximation.Let's consider that the control laser is red-detuned (the same procedure applies to the blue detuning regime with equivalent conclusions).Employing the rotating wave approximation we obtain the following equations of motion In contrast to the full equations of motion, the above system of equations does not include terms ∝ δα † , δβ † , δc † , since for a red-detuned control laser those are counterrotating terms.From now own we also assume that c * = c and G * = G.By following the same procedure leading to (33), i.e. consider the frequency domain system of equations for the average values of the operators, we obtain the rotating wave approximation self-energy of the magnon mode j = α, β, ΣRW j [ω]: In this scenario (red detuned control light), the counter-rotating contribution to the self energy ΣCR j [ω] can be obtained as the difference between ΣRW j [ω] and the full self-energy term (33): This term includes all counter-rotating contributions due to Stokes process and the induced counter-rotating magnon-magnon interactions generated by them.We can now define the decay rates and frequency shifts associated with each of those terms (A = Bare, MM, CR): We show in Figure 6 ΓBare [ω], ΓMM [ω] and ΓCR [ω] for the α mode and same parameters as in Figure 3 of the main text for (a) the main plot and (b) the inset.We see that in the case (a), in which the magnon frequencies are closer than the effective decay rate (the near degenerate case), both magnon-magnon interactions (green curve) and counter-rotating terms (blue curve) contribute substantially to the decay rate.In particular the contribution due to magnon-magnon indirect interactions is as important as the bare term.The importance of those terms depends on the strength of the coupling, the closeness of the magnon modes and the cavity decay rate, and these are also related to the formation of hybrid modes between the cavity and the magnon modes (see [76][77][78]).The combination of those three terms gives the unusual behavior depicted in the Figure 3 of the main text.In case (b) the frequencies are well separated and both magnon-magnon indirect interactions and counterrotating terms have a small contribution to the overall decay rate.Finally, we notice that under the same assumptions leading to (32), (33) is given by Magnon Linewidth (units of ) which corresponds to the bare self energy term in (34).
In this case the magnon linewidth frequency shift are given by
t e x i t s h a 1 _ b a s e 6 4 = " I I E q l z 8 w s A r s e j q o B 1 0 a o W C + x k E = " > A A A B 7 3 i c d V D J S g N B E K 1 x j X G L e v T S G A R P w 0 w U N b e g F 4 8 R z A L J E G o 6 P U m T n p 6 x u 0 c I IT / h x Y M i X v 0 d b / 6 N n U V w f V D w e K + K q n p h K r g 2 n v f u L C w u L a + s 5 t b y 6 x u b W 9 u F n d 2 6 T j J F W Y 0 m I l H N E D U T X L K a 4 U a w Z q o Y x q F g j X B w O f E b d 0 x p n s g b M 0 x Z E G N P 8 o h T N F Z q 9 j p t F G k f O 4 W i 5 5 Y 9 v 3 z q k 9 / E d 7 0 p i j B H t V N 4 a 3 c T m s V M G i p Q 6 5 b v p S Y Y o T K c C j b O t z P N U q Q D 7 L G W p R J j p o P R 9 N 4 x O b R K l 0 S J s i U N m a p f J 0 Y Y a z 2 M Q 9 s Z o + n r n 9 5 E / M t r Z S Y 6 D 0 Z c p p l h k s 4 W R Z k g J i G T 5 0 m X K 0 a N G F q C V H F 7 K 6 F 9 V E i N j S h v Q / j 8 l P x P 6 i X X P 3 Z L 1 y f F y s U 8 j h z s w w E c g Q 9 n U I E r q E I N K A i 4 h 0 d 4 c m 6 d B + f Z e Z m 1 L j j z m T 3 4 B u f 1 A 3 j m k E Q = < /l a t e x i t > g < l a t e x i t s h a 1 _ b a s e 6 4 = " U G q I H 9 + p q 6 p J4 W v 4 G O f M c n J g j 5 4 = " > A A A B 7 n i c d V D J S g N B E K 2 J W 4 x b 1 K O X x i B 4 G m a i q L k F v X i M Y B Z I h t D T6 U m a 9 P Q M 3 T V C C P k I L x 4 U 8 e r 3 e P N v 7 C y C 6 4 O C x 3 t V V N U L U y k M e t 6 7 k 1 t a X l l d y 6 8 X N j a 3 t n e K u 3 s N k 2 S a 8 T p L Z K J b I T V c C s X r K F D y V q o 5 j U P J m + H w a u o 3 7 7 g 2 I l G 3 O E p 5 E N O + E p F g F K 3 U 7 H c 7 I U f a L Z Y 8 t + L 5 l T O f / C a + 6 8 1 Q g g V q 3 e J b p 5 e w L O Y K m a T G t H 0 v x W B M N Q o m + a T Q y Q x P K R v S P m 9 b q m j M T T C e n T s h R 1 b p k S j R t h S S m f p 1 Y k x j Y 0 Z x a D t j i g P z 0 5 u K f 3 n t D K O L Y C x U m i F X b L 4 o y i T B h E x / J z 2 h O U M 5 s o Q y L e y t h A 2 o p g x t Q g U b w u e n 5 H / S K L v + i V u + O S 1 V L x d x 5 O E A D u E Y f D i H K l x D D e r A Y A j 3 8 A h P T u o 8 O M / O y 7 w 1 5 y x m 9 u E b n N c P s H G P 0 A = = < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " u 1 y u y W M R J A 1 8 h d F 5 q z a w K l U f Q 1 s = " > A A A B 7 n i c b V B N S 8 N A E J 3 4 W e t X 1 a O X Y B E 8 l a Q K e i x 6 8 V j B f k A b y m Y 7 b Z d u N m F 3 I p T Q H + H F g y J e / T 3 e / D d u 2 x y 0 9 c H A 4 7 0 Z Z u a F i R S G P O / b W V v f 2 N z a L u w U d / f 2 D w 5 L R 8 d N E 6 e a Y 4 P H M t b t k B m U Q m G D B E l s J x p Z F E p s h e O 7 m d 9 6 Q m 6 t y 7 T a P o w C n c A Y X 4 M M 1 1 O A e 6 t A A D m N 4 h l d 4 c x L n x X l 3 P h a t a 0 4 + c w J / 4 H z + A D 9 5 j 4 I = < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " O 3 F l / H Y 1 C s W D t l e L e Z B W K k 4 1 t P 0 = " > A A A B 7 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m q o M e i F 4 8 V 7 A e 0 o U y 2 m 3 b p Z h N 3 N 0 I J / R N e P C j i 1 b / j z X / j t s 1 B W x 8 M P N 6 b Y W Z e k A i u j e t + O 4 W 1 9 Y 3 N r e J 2 a W d 3 b / 8 H 7 o / 2 < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " I I E q l z 8 w s A r s e j q o B 1 0 a r A Y A j 3 8 A h P T u o 8 O M / O y 7 w 1 5 y x m 9 u E b n N c P s H G P 0 A = = < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " u 1 y u y 6 t y 7 T a P o w C n c A Y X 4 M M 1 1 O A e 6 t A A D m N 4 h l d 4 c x L n x X l 3 P h a t a 0 4 + c w J / 4 H z + A D 9 5 j 4 I = < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " O 3 F l / H Y 1 C s W D t l e L e Z B W K k 4 1 t P 0 = " > A A A B 7 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m q o M e i F 4 8 V 7 A e 0 o U y 2 m 3 b p Z h N 3 N 0 I J / R N e P C j i 1 b / j z X / j t s 1 B W x 8 M P N 6 b Y W Z e k A i u j e t + O 4 W 1 9 Y 3 N r e J 2 a W d 3 b / x 3 p 2 P R W v B y W e O 4 Q + c z x 8 H 7 o / 2 < / l a t e x i t > dominated regime !?> !k < l a t e x i t s h a 1 _ b a s e 6 4 = " P I M 0 e a A Y I Z 2 W K + 6 9 g L e N H p + 1 1 I r S e r T f r f d p a s G Y z + + g P r I 9 v y V 6 Z U w = = < / l a t e x i t > 9 1 X 8 9 5 Y / M / r Z D q + C H I q 0 k y D I N O P 4 o w 5 O n H G m T h d K o F o N j Q E E 0 n N r g 7 p m w i I N s m V T A j e / M m L p F m t e K e V 6 u 1 Z u X Y 1 i 6 O I D t A R O k E e O k c 1 d I P q q I E I e k B P 6 A W 9 W o / W s / V m v U 9 b C 9 Z s Z h / 9 g f X x D c Y 2 m V E = < / l a t e x i t > Reduced couplings Magneto -optical asymmet ry K < l a t e x i t s h a 1 _ b a s e 6 4 = " H Q f a d D Q r o z r 2 P p v G 9 2 8 H Q m j c O j I = " > A A A B 6 H i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e x G Q Y 9 B L 4 K X B M w D k i X M T n q T M b O z y 8 y s E E K + w I s H R b z 6 S d 7 8 G y f J H j S x o K G o 6 q a 7 K 0 g E 1 8 Z 1 v 5 3 c 2 v r G 5 l Z + u 7 C z u 7 d / U D w 8 a u o 4 V Q w b L B a x a g d U o + A S G 4 Y b g e 1 E I Y 0 C g a 1 g d D v z W 0 + o N I / l g x k n 6 E d 0 I H K b 8 6 j 8 + K 8 O x + L 1 p y T z R z D H z i f P 6 N T j N M = < / l a t e x i t > Magneto -optical asymmet ry K < l a t e x i t s h a 1 _ b a s e 6 4 = " H Q f a d D Q r o z r 2 P p v G 9 2 8 H Q m j c O j I = " > A A A B 6 H i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e x G Q Y 9 B L 4 K X B M w D k i X M T n q T M b O z y 8 y s E E K + w I s H R b z 6 S d 7 8 G y f J H j S x o K G o 6 q a 7 K 0 g E 1 8 Z 1 v 5 3 c 2 v r G 5 l Z + u 7 C z u 7 d / U D w 8 a u o 4 V Q w b L B a x a g d U o + A S G 4 Y b g e 1 E I Y 0 C g a 1 g d D v z W 0 + o N I / l g x k n 6 E d 0 I H

3 .
Degenerate case ωH = ω ⊥ = 0 is the amplitude of the drive normalized to the input photon flux and ω d (ω p ) is the control (probe) laser frequency, with s d,p = 2P d,p ω d,p
with j = k.In the self-energy ΣRW j [ω], the first term ΣBare j[ω] describes the effects of the anti-Stokes processes converting a magnon j into a cavity photon, while the second term ΣMM j[ω] is associated with the magnonmagnon interactions mediated by a cavity photon.

Figure 6 .
Figure 6.Contribution from each term of the magnon self energy (34) and (35) in the red-detuning regime.Parameters as in Figure 3 of the main text for (a) the main figure and (b) the inset.In the inset we show the full magnon decay rate.