Observation of strong tripartite coupling in a cavity-quantum circuit-antiferromagnet platform

MainAt the crossroads of quantum information science and condensed matter physics, the field of quantum magnonics has attracted increasing attention in recent years due to its promising potential for various quantum technologies1. Although numerous hybrid platforms have been proposed to realize quantum magnonic systems, several challenges remain, notably the search for new materials and coupling mechanisms5,6,7. Applications of such systems include achieving ultrastrong coupling8, implementing quantum-sensing protocols9,10,11,12 and engineering non-classical states, such as squeezed states13, entangled states14 and cat states15,16.Several platforms using yttrium iron garnet (YIG) have been extensively explored5,7. In particular, coupling between a YIG sphere and a superconducting qubit has been demonstrated17,18. Previous studies have also reported ultrastrong coupling between superconducting microwave resonators and antiferromagnetic resonances of rare-earth ion spins in GdVO4 crystals19, which underscores the potential of antiferromagnetic materials as platforms for quantum hybrid systems, in particular for pushing up the operating frequencies of quantum devices. Moreover, antiferromagnetic resonances and the associated magnon modes under static magnetic fields have been extensively studied both theoretically20,21,22,23,24 and experimentally25.Bipartite coherent coupling between different quantum systems has become a cornerstone for many architectures. Being able to couple coherently, and in a controlled fashion, more than two elementary quantum systems is one of the general class of tasks needed for scaling up quantum processors. This naturally motivates an interest in tripartite coherent quantum systems in general. Interestingly, tripartite coherent coupling including magnetic modes has been proposed very recently as a resource for quantum simulation, frequency conversion or quantum sensing2,4,12. Therefore, the implementation of tripartite coherent coupling between magnons, cavities and any other type of quantum system is an outstanding challenge, besides the step-by-step construction of elementary quantum processors.Here we demonstrate coherent coupling between a microwave cavity, a magnetic-field-resilient superconducting circuit and a rare-earth antiferromagnet (a GdVO4 crystal). The antiferromagnetic modes are coupled to the cavity mode close to 6 GHz in the ultrastrong coupling regime, thus forming two branches of magnetic-photon modes. By inducing anharmonicity in the system thanks to the superconducting circuit, we are able to demonstrate efficient nonlinear interactions between all three modes over about a 20-GHz frequency span. By bringing into resonance the transition frequency of the superconducting circuit and the lower branch of the magnetic-photon mode at 932.5 mT, we demonstrate strong tripartite coupling as a result of the coherent superposition of quantum states in our platform.We first present the magnetospectroscopic measurements shown in Fig. 1a. The experimental set-up consists of a GdVO4 antiferromagnetic crystal and a granular aluminium (grAl) superconducting circuit in a three-dimensional two-loop-gap microwave cavity (see Extended Data Fig. 1 for an individual spectrum). A static magnetic field is applied along the easy axis of the crystal. By construction, the minimal Hamiltonian describing such a system, omitting for simplicity the baths and the drives, reads:$$\begin{array}{rcl}\hat{H} & = & \hslash {\omega }_{{\rm{a}}}{\hat{a}}^{\dagger }\hat{a}+\hslash {\omega }_{{\rm{m}}}{\hat{m}}^{\dagger }\hat{m}+\hslash {\omega }_{{\rm{q}}}{\widehat{q}}^{\dagger }\widehat{q}+\hslash {K}_{{\rm{q}}}{\widehat{q}}^{\dagger 2}{\hat{q}}^{2}+\\ & & +\hslash {g}_{\mathrm{aq}}({\hat{a}}^{\dagger }\hat{q}+{\hat{q}}^{\dagger }\hat{a})+\hslash {g}_{\mathrm{am}}({\hat{a}}^{\dagger }\hat{m}+{\hat{m}}^{\dagger }\hat{a})+\hslash {g}_{\mathrm{qm}}({\hat{q}}^{\dagger }\hat{m}+{\hat{m}}^{\dagger }\hat{q}),\end{array}$$ (1) where ℏ is the Planck constant, and the operators \(\hat{a}\), \(\hat{m}\) and \(\widehat{q}\) are for one of the cavity modes, one of the magnetic modes and the mode of the superconducting circuit, respectively. The mutual coupling constants between modes â, \(\hat{m}\) and \(\hat{q}\) read gaq, gam and gqm. The mode of the superconducting circuit is anharmonic and characterized by the Kerr constant Kq. One can access frequencies ωa,m,q/2π in the Hamiltonian of equation (1) using conventional microwave spectroscopy, which consists of measuring the transmission of the microwave signal through the cavity as a function of both the frequency (fd) and the external magnetic field B0. Figure 1a is a colour scale plot of the transmission contrast for this measurement (see Supplementary Information Section I.C for the raw transmission map).We observe several characteristic features. At zero field, two cavity modes are observed at 5.065 GHz and 6.193 GHz, corresponding to the bright and dark modes of the loop-gap resonator. We focus on the latter mode, which is set, for example, with ωa ≈ 2π × 6.193 GHz in the Hamiltonian of equation (1) at zero magnetic field. Its linewidth is about 5 MHz. As shown from the theoretical calculation in Fig. 1d, the antiferromagnetic resonance should occur in the range 30–40 GHz (34 GHz is often quoted in the literature19). When a magnetic field is applied along the easy axis of the crystal, this resonance is expected to split into two magnon modes. A spin-flop phase transition is also expected to occur around 1.1 T. Besides the cavity modes, there is a strongly dispersing mode with a large transmission contrast and a linewidth of about 1 MHz. As shown in Fig. 1a, the lower magnon branch exhibits a clear anticrossing with the cavity mode at 6.193 GHz, indicating coherent coupling between the two subsystems. The coupling constant gaq between the cavity and the magnon modes is directly extracted from the anticrossing gap. We estimate gam ≈ 2π × 1.62 GHz, which places our antiferromagnetic cavity system in the ultrastrong coupling regime (Supplementary Information Section II.C). The ultrastrong coupling regime is crucial for efficient frequency conversion, as it enables us to keep strong the hybridization between all the relevant modes on a large frequency span (of several gam), unlike the more conventional strong coupling regime.Fig. 1: Microwave spectroscopy of the hybrid system.Full size imagea, Microwave spectroscopy from transmission contrast \(\bar{S}_{21}\) versus magnetic field showing the symmetric and antisymmetric modes of the two-loop-gap resonator (5.065 GHz and 6.193 GHz at zero field) together with the two gyromagnetic modes. A clear anticrossing with the lower magnon mode is observed. b, Theoretical model of the cavity and gyromagnetic modes. Dispersion branches for the first three couples with mode indices (n, l) = {(1, 0), (1, 1), (2, 0)} are shown in red (modes m(2,0), m(1,1) and m(1,0)). The hybridization of the two empty cavity modes (5.065 GHz and 6.193 GHz) with the two branches of the (n, l) = (0, 0) mode (modes cav1 and cav2) and the grAl circuit are shown in blue. The grAl circuit is represented as a linear dispersion with magnetic field. c, Schematic of the cavity design containing the grAl circuit and the GdVO4 crystal including the orientation of the magnetic field and the three different microwave drives. d, Full theoretical model illustrating the two counterrotating magnon modes of the antiferromagnet (n, l) = (0, 0) and the spin-flop transition at B0 ≈ 1.1 T along with the first three gyromagnetic and the two lowest cavity modes. AF, antiferromagnet; SF, spin-flop.A third mode at 5.86 GHz corresponds to the grAl superconducting circuit, which has a linewidth of about 1 MHz and can be observed up to 2.2 T. The corresponding cut in the transmission where we see both the upper cavity and the grAl circuit modes is shown in Fig. 1c. grAl has been shown recently to be a new material for superconducting circuits due to interesting features such as magnetic field resilience and large anharmonicity26,27,28,29,30,31,32,33. Here we demonstrate a new functionality by including such a circuit in a magnonic platform. The full characterization of this mode, which is like that in ref. 33, allows us to determine gaq ≈ 2π × 30 MHz and Kq ≈ 2π × 200 kHz. Importantly, the frequency ωq decreases linearly with increasing magnetic field, consistent with the expected field dependence of the Josephson energy from the small threaded in-plane magnetic flux.As shown in Fig. 1a, we also see other modes with strong dispersion as a function of the applied field. As we will show below, we attribute these to gyromagnetic modes of the GdVO4 crystal, which can be modelled by combining Maxwell’s equations with the Landau–Lifshitz–Gilbert equation for magnetization dynamics (Supplementary Information Section II and Methods). Although such modes have been extensively studied in small YIG spheres, they have never been addressed for antiferromagnetic materials nor for large magnetic samples placed inside a cavity. In our case, the sample size is 5 × 5 × 5 mm3, such that at zero applied magnetic field, the gyromagnetic modes are within the frequency range for the cavity mode. Interestingly, we can model all the observed modes in the antiferromagnetic phase, as shown in Fig. 1b. The main input parameters of our model are mainly the exchange interaction J = 32 GHz, the anisotropy K = 11.2 GHz, the angle of the external magnetic field, and the coupling gam ≈ 2π × 1.62 GHz between the magnon mode and the upper cavity mode. The full spectrum corresponding to these parameters is shown in Fig. 1d. This allows us to gain further insights into the electrodynamics of the GdVO4 antiferromagnet. Our method could be used to study the electrodynamics of other antiferromagnets or generally magnetic materials.Antiferromagnets display exchange interactions and, therefore, have a typical frequency ωm/2π in the high gigahertz to low terahertz range. How can one access such very high frequency modes if conventional microwave spectroscopy measurements are limited to the bandwidth of the microwave amplifying set-up, which spans 4 GHz to 9 GHz? The anharmonicity of the superconducting circuit in our set-up is a key resource for addressing this problem. Indeed, because we use a tripartite system, the anharmonicity in one of the subsystems can be transferred to all the others, even though they are primarily linear. The superconducting circuit naturally generates nonlinear interactions between all the modes, even though the modes have very different frequencies. This is a priori true also for gyromagnetic modes above the important technological threshold of 20 GHz for quantum circuits34. To summarize this first part, we have built a system consisting of three elements with distinct roles. The cavity enables the read-out of all the modes of the system, whether magnetic or for the superconducting circuit. The magnetic modes open a window in the high-frequency range, above 20 GHz. The superconducting circuit, thanks to its anharmonicity, enables nonlinear interactions between the high-frequency magnetic modes and the low-frequency modes of the cavity.To be more specific on how these interactions work, it is useful to rewrite the Hamiltonian of equation (1) in the dispersive regime where all the frequencies ωa,m,q/2π are non-resonant:$$\begin{array}{rcl}\hat{H} & = & \hslash {\omega }_{{\rm{a}}}{\hat{a}}^{\dagger }\hat{a}+\hslash {\omega }_{{\rm{m}}}{\hat{m}}^{\dagger }\hat{m}+\hslash {\omega }_{{\rm{q}}}{\hat{q}}^{\dagger }\hat{q}+\hslash {K}_{{\rm{q}}}{\hat{q}}^{\dagger 2}{\hat{q}}^{2}+\\ & & +\hslash {K}_{\mathrm{am}}{\hat{a}}^{\dagger } \hat{a}{\hat{m}}^{\dagger }\hat{m}+\hslash {K}_{\mathrm{aq}}{\hat{a}}^{\dagger }\hat{a}{\hat{q}}^{\dagger }\hat{q}+\hslash {K}_{\mathrm{qm}}{\hat{m}}^{\dagger }\hat{m}{\hat{q}}^{\dagger }\hat{q.}\end{array}$$ (2) The ‘cross-Kerr’ nonlinear interaction constants Kam, Kaq and Kqm in the above equation couple the number of quanta in the three different subsystems, as shown in Fig. 2a. A change in the number of quanta in one of the modes changes the resonance frequency of the two other modes. As in cavity quantum electrodynamics, by measuring the phase contrast Δϕ of the microwave signal close to the \(\hat{a}\) mode, we can map the other modes, especially the \(\hat{m}\) mode. A priori, all the magnon modes and their coupling to the cavity and the superconducting circuit can be modelled using equation (2). This is the strength of the dispersive magnetospectroscopy used here. We do not need to have a read-out set-up resonant with the magnon modes, which substantially enlarges the scope of our ‘spectrometer’. The corresponding measurements are shown for two modes in Fig. 2b–d.We first observed the magnetic field dependence of high-frequency gyromagnetic modes m(n,l) between 22 GHz and 23.6 GHz up to 300 mT. The second gyromagnetic resonance m(2,1) was observed at frequencies starting close to 12.1 GHz at zero field, with a clear signal in two-tone spectroscopy up to 0.8 T (Fig. 2c) and again from 1.4 T to 2.5 T (Fig. 2d), beyond the spin-flop transition. Notably, the resilience of the grAl circuit to high magnetic fields enables continuous spectroscopy across the spin-flop phase of the antiferromagnet. In the region between 0.8 T and 1.4 T in the GdVO4 phase diagram, the magnetic modes and all the other modes of the system become strongly damped. This explains why there is a ‘gap’ in the microwave spectroscopy in this field range. Above the gap, the GdVO4 enters the ferromagnetic region. This full phase sequence is not included in our theory, which focuses on the antiferromagnetic phase. This is why the lines corresponding to the coupled magnon–cav2 mode seem to be ‘off’ at high field. Our platform, thus, enables direct downconversion between the magnetic modes well above 10 GHz and the upper cavity mode close to 6 GHz. Our use of grAl circuits, which are magnetic field resilient33, overcomes the limitations of a traditional superconducting circuit, as they allow magnetospectroscopy at high field and high frequency. Nonlinear tripartite coupling between three systems with very different frequencies, as demonstrated here, is an important prerequisite for frequency conversion, as shown in ref. 4. Our findings thus open the path to frequency conversion between high- and low-frequency modes.Fig. 2: Nonlinear interaction of gyromagnetic modes and cavity modes.Full size imagea, Diagram of the system and the different couplings between the cavity, the circuit and the gyromagnetic modes. b, Probing high-frequency gyromagnetic modes. Spectroscopy of the high-frequency gyromagnetic modes between 22 GHz and 23 GHz as a function of excitation frequency fq and at zero field revealed their dispersion with magnetic field from 0 T to 0.3 T. The phase signals in b–d were unwrapped along the frequency axis and subsequently filtered using a median kernel (width 1,001) to enhance the large-scale phase features and reduce noise. c, Evolution of the second gyromagnetic mode m(2, 0) at 12.3 GHz at zero field and its dispersion with the B field between 0 T and 0.8 T. d, Measurement of mode m(2, 0) at high field, showing its dispersion with the B field between 1.4 T and 2.5 T. Insets of panels c and d: individual phase curves at the specified magnetic field corresponding to the maps in the two panels.We next probed the effect of gyromagnetic excitation on the superconducting circuit. To map accurately the frequency of the grAl circuit, we performed Ramsey interferometry using two \({\rm{\pi }}/2\) pulses of duration τ = 25 ns separated by a time τR = 150 ns, which results in a fringe pattern33. As shown in ref. 33 by some of us, Ramsey fringes can survive in the extreme weak anharmonic regime where the Rabi and Ramsey oscillations correspond to about 60 quanta of excitations in the grAl circuit. These quantum oscillations, which are related to squeezing dynamics, correspond to the critical regime of a quantum weak anharmonic oscillator with an anharmonicity (200 kHz) close to the dissipation rate (~1 MHz). The immediate consequence is that we can operate our weak anharmonic circuit like a qubit and use it to detect small changes in the number of photons, as in a conventional quantum-sensing set-up. Hence, we mention ‘π/2’ pulses in our pulse sequence as a reminder of qubit dynamics, although our superconducting circuit is not a qubit.Figure 3a shows the phase contrast of the Ramsey fringes obtained as a function of the superconducting drive frequency (fRamsey) and the gyromagnetic drive frequency (fm) for different numbers of photons in the gyromagnetic mode nm at 600 mT. The resonance frequency of the m(2,1) mode at 600 mT is 11.702 GHz, as seen in Fig. 2d, corresponds to the largest detuning observed in the Ramsey fringes. The Ramsey fringes were fitted using a semiclassical model and an input–output formalism35. The result is plotted in Fig. 3b. The detuning of the Ramsey fringes Δf is plotted in Fig. 3c as the number of photons in the gyromagnetic mode is increased. This results in a linear trend \(\Delta f={\widetilde{K}}_{\mathrm{qm}}{n}_{{\rm{m}}}\), with \({\widetilde{K}}_{\mathrm{qm}}\approx -0.3\,\mathrm{Hz}\), which is confirmed by Fig. 3d. This highlights the direct effect of the gyromagnetic mode on the circuit, which provides a pathway for tunable quantum-sensing schemes12,36.Fig. 3: Nonlinear interaction of antiferromagnetic modes and the superconducting circuit mapped from Ramsey interferometry.Full size imagea, Evolution of the Ramsey fringes for different numbers of photons in the gyromagnetic mode m(2, 0) at 600 mT (fm = 11.702 GHz at 600 mT). The Ramsey fringes are strongly renormalized in frequency as the gyromagnetic mode m(2, 0) is excited. b, Ramsey fringes with a fit using semiclassical theory (red line). c, Evolution of the frequency shift Δf as a function of the frequency fm excited, for different nm (B0 = 600 mT). d, Evolution of the frequency shift Δf as a function of nm, the number of photons in the gyromagnetic mode (B0 = 600 mT). The plot has a semi-logarithmic x axis.The ultrastrong coupling observed between modes \(\hat{a}\) and \(\hat{m}\) in Fig. 1a implies that there is a large B0 field range and frequency range where there is strong hybridization between these two modes. The large avoided crossing visible between the upper cavity mode and the magnon mode in Fig. 1a reveals that the modes at this magnetic field are coherent superpositions of cavity-like and magnetic-like modes with similar weights. The ultrastrong coupling regime implies that coherent superposition survives over a large magnetic field range along the large avoided crossing. This opens an interesting possibility for tripartite strong coupling, as it relaxes the constraint of needing to have the three systems exactly in resonance.To investigate this phenomenon, we performed high-resolution single-tone spectroscopy as a function of the magnetic field. As the lower magnon–photon polaritonic branch disperses faster than the grAl circuit with the magnetic field, the three subsystems of our platform can be brought into resonance at a given magnetic field. The corresponding plot is shown in Fig. 4a. This plot reveals a clear anticrossing between the two modes, for which we extract the coupling g = 25 MHz. This coupling is consistent with the estimates gaq ≈ 30 MHz and gqm ≈ 15 MHz, as found by our modelling (Supplementary Information Section II.C.).Figure 4b–d represents individual cuts at specific magnetic fields. At B = 0.917 T, the transmission spectrum reveals two distinct resonances corresponding to the circuit and the magnon–cavity polariton. As the magnetic field is increased to B = 0.9325 T, a clear anticrossing between the two modes is observed, with two polaritonic branches at f = 5.763 GHz and f = 5.815 GHz (Fig. 4c). Interestingly, the asymmetric line shapes of the tripartite polaritonic branches are inherited from the anharmonicity of the grAl circuit. At this particular field, the coherent superposition of all the three modes is a maximum.A detailed analysis, presented in Supplementary Information Section II.C., reveals the composition of the polaritonic eigenvectors at the anticrossing:$$\begin{array}{rcl}| {\psi }_{-}{\rangle }_{B=0.9325{\rm{T}}} & = & 0.68\,| \mathrm{grAl}\rangle +0.18\,| \mathrm{AF}\rangle -0.7\,| {\mathrm{cav}}_{{\rm{2}}}\rangle +0.08\,| {\mathrm{cav}}_{{\rm{1}}}\rangle ,\\ {| {\psi }_{+}\rangle }_{B=0.9325{\rm{T}}} & = & -0.68\,| \mathrm{grAl}\rangle +0.18\,| \mathrm{AF}\rangle -0.7\,| {\mathrm{cav}}_{{\rm{2}}}\rangle +0.08\,| {\mathrm{cav}}_{{\rm{1}}}\rangle .\end{array}$$ (3) This state structure shows that substantial hybridization occurs between the antiferromagnetic mode, the grAl circuit and the cavity 2 mode. There is also a residual contribution from cavity mode 1. The structure of these modes is clearly reminiscent of the ‘anticrossing within an anticrossing structure’ of our tripartite coupling. Interestingly, our observation of tripartite strong coupling is further corroborated by the power dependence of the spectrum shown in Supplementary Information Section III.B. When the field was increased to B = 0.946 T, the circuit shifted to f = 5.797 GHz, while the hybridized mode continued to disperse down to f = 5.718 GHz (Fig. 4d). These observations confirm the formation of coherent tripartite hybridization between the magnon, the cavity and the superconducting circuit.Fig. 4: Tripartite strong coupling.Full size imagea, High-resolution microwave spectroscopy as a function of magnetic field, zoomed around the anticrossing between the cavity and the superconducting circuit. The horizontal bright lines correspond to the resonant modes of the cavity, which are already hybridized with a magnon mode. The dashed line is the four-mode theory presented in Supplementary Information Section II.C, which allows us to extract the mode structure at the anticrossing in particular, as labelled here. The α, β, γ and δ mode weights are given in the text. The symbols and vertical dashed lines indicate the resonances and the cut position respectively for the corresponding panels b, c and d. b, Frequency cut at B = 0.917 T. The circuit appears at f = 5.766 GHz, and the (magnon-hybridized) cavity mode at f = 5.48 GHz. c, Frequency cut at B = 0.929 T, showing clear polaritonic splitting with two peaks at f = 5.763 GHz and f = 5.815 GHz. d, Frequency cut at B = 0.946 T. After the anticrossing, the circuit frequency shifts to f = 5.797 GHz. The cavity (still hybridized with the magnon) continues to disperse and reaches f = 5.718 GHz at B = 0.946 T.In conclusion, we have realized a hybrid quantum platform that combines a grAl superconducting circuit with an antiferromagnet. Our device enables nonlinear interactions of modes over a large frequency range as well as strong coupling in a tripartite quantum system comprising a magnetic material, a superconducting circuit and a microwave cavity. Our work opens the way towards quantum devices exploiting antiferromagnetic spin dynamics, including memory devices37, quantum gate operations38 and precision measurements technologies, such as magnetometers39 or axion dark matter detectors36,40. Finally, our use of antiferromagnets enables the potential for frequency conversion between gigahertz and terahertz fields in the quantum regime4.MethodsExperimentsAll measurements were made in a dilution refrigerator at 18 mK. The magnetic field B was applied following the easy axis of the antiferromagnetic crystal. The one-tone spectroscopy sequence presented in Fig. 1 has a total duration of tsequence = 1 μs. The cavity was populated with photons during tsequence and then measured during tmeas = 600 ns.The two-tone spectroscopy sequence presented in Fig. 2 has a total duration of tsequence = 1 μs. Both the cavity and the superconducting circuit were driven with continuous tones during the sequence time, and the cavity was measured during tmeas = 600 ns. The preprocessing methods used for Figs. 1 and 2 are described in Supplementary Information. Theoretical calculations of the gyromagnetic modes were performed using the programming language Julia.The Ramsey sequence presented in Fig. 3 consists of a Ramsey pulse sequence with a total duration of tsequence = 4 μs. In the initial stage, the cavity was populated with photons at its resonance frequency during a pulse of duration tpulse = 2 μs. At \(t={t}_{\mathrm{pulse}}-{t}_{Ramsey}-2{t}_{{\rm{\pi }}/2}\), a first \(\pi /2\) pulse was applied to the superconducting circuit for a duration \({t}_{{\rm{\pi }}/2}\) to place the circuit into a superposition of two distinct states. The first π/2 pulse was followed by free evolution during tRamsey and by another π/2 pulse for measurement. The system was then measured via the cavity during t = tmeas = 0.3 μs. A further postprocessing delay (1 μs) ensured that the circuit returned to its initial state, thus completing the sequence at tsequence. Both timing and amplitude parameters were optimized for this measurement, and the number of circuit photons remained fixed at nq = 3 × 104. Typically, tRamsey = 150 ns and tpulse = 25 ns. The pulse drives for the cavity and superconducting circuit were controlled via an arbitrary waveform generator and single-sideband radio-frequency components. The cavity drive was modulated at 20 MHz and the circuit drive at 50 MHz. The output signal was then demodulated at 20 MHz. The fit presented in Fig. 3d was made using Python library SciPy.Modelling the gyromagnetic modesOne important aspect of our work is the modelling of the gyromagnetic modes that arise from the off-diagonal structure of the magnetic permeability tensor in magnetic insulators. Those are well known in ferro- or ferrimagnetic materials such as YIG. The direct observation and characterization of gyromagnetic modes arising from the magnetic permeability tensor in antiferromagnets have, until now, remained elusive. Although magnon modes are often described using a quantum formalism, we show that a proper understanding of these gyromagnetic modes can also be obtained from a semiclassical treatment that explicitly accounts for the permeability tensor structure36. In the main text, we present single-tone and two-tone spectroscopy, which provided the direct observation of the lowest gyromagnetic modes in GdVO4. The precession of the two sublattices in an external magnetic field yields the off-diagonal structure of the permeability tensor, whose eigenfrequencies correspond to the gyromagnetic modes, which are depicted in Fig. 1d. This model is valid in the antiferromagnetic regime. To capture the interaction with the bright and dark cavity modes, we focused on the m(0, 0) mode, which corresponds to the two fundamental magnon branches, and we analysed the eigenvalues of the coupled system. This approximation provides a good description, especially in the antiferromagnetic regime. The remaining discrepancies between theory and experimental data in the spin-flop phase are discussed in Supplementary Information. Theoretical modelling based on a semiclassical approach successfully reproduces the gyromagnetic modes observed in the antiferromagnetic phase. The hybridization with cavity modes is well reproduced using a minimal coupling model involving the (n = 0, l = 0) magnon branch. In the spin-flop regime, discrepancies arise due to the breakdown of the simplified permeability tensor. As discussed in Supplementary Information, a full tensorial treatment would be required to accurately describe this phase.