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https://doi.org/10.1038/s42005-020-00435-w OPEN Magnon-driven dynamics of a hybrid system excited with ultrafast optical pulses ✉ N. Crescini 1,2 , C. Braggio 2,3, G. Carugno2,3, R. Di Vora3,4, A. Ortolan 1 & G. Ruoso 1 1234567890():,; The potential of -magnon hybrid systems as building blocks for quantum information science has been widely demonstrated, and it is still the focus of much research. We leverage the strengths of this unique heterogeneous physical system in the field of precision physics beyond the , where the sensitivity to the so-called “invisibles” is currently being boosted by quantum technologies. Here, we demonstrate that quanta of , induced by effective magnetic fields, can be detected in a large frequency band using a hybrid system as transducer. This result can be applied to the search of cosmological signals related, for example, to cold Dark , which may directly interact with magnons. Our model of the transducer is based on a second-quantisation two-oscillators hybrid system, it matches the observations, and can be easily extended to thoroughly describe future large-scale fer- romagnetic haloscopes.

1 INFN-Laboratori Nazionali di Legnaro, Viale dell’Università 2, Legnaro, PD 35020, Italy. 2 DFA-Universitá degli Studi di Padova, Via Marzolo 8, Padova 35131, Italy. 3 INFN-Sezione di Padova, Via Marzolo 8, Padova 35131, Italy. 4 DSFTA-Universitá di Siena, Via Roma 56, Siena 53100, Italy. ✉ email: [email protected]

COMMUNICATIONS PHYSICS | (2020) 3:164 | https://doi.org/10.1038/s42005-020-00435-w | www.nature.com/commsphys 1 ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-020-00435-w

n the last decades, precision magnetometry emerged as a devised to probe weak and persistent effective rf fields, which is Ipromising probe of physics beyond the standard model1,2. the physics case of dark matter (DM) axions24. Amongst all, GHz-frequency magnetometers can probe spin- The is a hypothetical introduced as a con- related effects through the use of the spin resonance sequence of the strong CP problem solution found by Peccei and techniques. Recent advances in the field are due to quantum Quinn25–27. It is a light pseudo-Goldstone arising from the computing research3, which allowed to reach the sensitivity to breaking of the Peccei–Quinn symmetry at extremely high ener- 12 detect single quanta of magnetisation in macroscopic samples gies fa ≃ 10 GeV. Since its mass and couplings are proportional 4,5 28–31 using photon-magnon hybrid systems (HSs) . In these devices, to 1/fa the axion is extremely light and weakly interacting . the coherent interaction between and magnons is Relevant quantities of them may have been produced in the early increased to such an extent that they can no longer be considered universe, qualifying the axion as a viable candidate of cold DM 3 as separate entities, and the HS is described within the cavity that would account for the whole density ρDM = 0.45 GeV/cm of quantum electrodynamics framework. Photons are confined the Milky Way’s halo32–36. A suitable mass range for the axion is −(3/5) 37–41 within a microwave cavity hosting a magnetised sample, whose ma = 10 eV , hence its de Broglie wavelength is of the ω Larmor frequency m is adjusted close to the one of a suitable order of metres and allows a coherent interaction with macro- cavity mode ωc. In the strong coupling regime, in which the scopic systems. Together with the high occupation number ρDM/ fi fi coupling strength is much larger than the related linewidths, ma, this allows to treat the axion DM eld as a classical rf eld. photons and magnons are in a hybrid magnon- state The presence of can be tested with earth-based precision induced by magnon Rabi oscillation. Such a cooperative spin measurements42–44 probing observables sensitive to the presence dynamics is governed by the physics of coupled harmonic oscil- of DM axions, i.e., the rf-power in a microwave cavity under a lators with beating periods much shorter than dissipation times6. static magnetic field45–51 or the magnetisation of a sample24,52–55; As under these conditions the two-oscillators exchange , we focus on this last case. The described HSs emerge as a natural the dispersion plot of the system displays the quantum phe- choice to detect the axion effective field as the variation of a ω ≃ ω nomenon of avoided crossing: when c m, instead of two sample magnetisation, and an apparatus measuring the power intersecting lines, the system dispersion relation exhibits an deposited in an HS by DM axions, and hereafter called Pac,is anticrossing curve7,8 as shown in Fig. 1a. A scheme of the HS therefore called ferromagnetic haloscope. As the underlying used in this work is reported in Fig. 1b, and its conceptual interaction is tiniest, the ferromagnetic haloscope relies on max- representation is in Fig. 1c. imising signal power Pac, which translates to hosting in the limited These HSs have been realised and extensively studied in the last volume of GHz-frequency cavities a sample with the highest decade9–14, and applied to the development of quantum mem- possible electron spin density. In addition, the hybrid modes Q- ories15,16, microwave to optical photon conversion17–21, detection factor should not be much lower than the axion figure of merit of single magnons5, or in spintronics22. Furthermore, the inves- (2 × 106). By using yttrium iron garnet (YIG) as magnetic sample, tigation of non-Hermitian quantum mechanics23 is currently these two requirements are well satisfied, owing to its high spin pursued with photon-magnon HSs. Here we focus on an HS density (2 × 1028 spins/m3) and small damping constant (~10−5).

Fig. 1 Schematic representation of a hybrid system and of the setup used in this work. a Avoided level crossing of a photon-magnon hybrid system. The curve is typically obtained by recording the cavity transmission spectrum at several values of the static magnetic field, which determines the ferrimagnetic resonance ωm according to ωm = γB0 ≃ (2π × 28 GHz/T) × B0, with γ of the electron. b Scheme of the apparatus used in this work. The magnetic material (black) is hosted inside a microwave cavity (orange), and polarised with the field generated by an electromagnet (grey, with N and S labels standing for north and south poles). A laser pulse (red) excites the magnetostatic modes of a yttrium iron garnet sphere, including the Kittel mode, allowing for investigating the hybrid system dynamic response to a direct excitation of its magnetic component. c Coupled oscillators representation of the ferromagnetic haloscope, in which the strong coupling is realised between an rf cavity mode c (frequency ωc) with the Kittel mode m (ωm) to improve the setup sensitivity. The axion–Dark Matter field acts as a tiniest classical rf field on the hybrid system, and it is thus represented by a weakly coupled oscillator a.

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In this work we theoretically analyse the response of a photon- where M is the two-component vector M = (〈m〉, 〈c〉). If we ¼ ðÀ ω Þ magnon HS to a direct excitation of the magnetic component rewrite M as a plane M A exp i at , Eq. 4 can be recast using a second-quantisation model, and experimentally test the as "# ! model results with an apparatus involving an optical excitation of  γ  ω 0 ω À i m g pffiffiffiffiffiffi 1 the magnetic material. We obtain the first experimental demon- a À m 2 cm ¼ ¼ ; γ KA gam Na 0 ω ω À i c 0 stration of the broad frequency tunability of the ferromagnetic a gcm c 2 haloscope, a crucial property that previously had only been the- ð5Þ oretically analysed54. This precision detector at the low energy frontier of can probe the axion coupling with yielding the amplitude of M electron spins in mass ranges of few μeV, as large as that probed  pffiffiffiffiffiffi 1 by Primakoff haloscopes45,50,51,56, with the advantage of a much ¼ À1 ð Þ A gam NaK 6 simpler frequency scan system. The latter is, in this case, 0 accomplished by varying the amplitude of the external magnetic fi from which it is possible to extract the (2, 1) component eld, instead of moving tuning rods in ultra-cryogenic environ- describing the coupling between the axion field and the cavity ments. While Primakoff haloscopes are based o the coupling of pffiffiffiffiffi pffiffiffiffiffiffi g g N axions to photons, ferromagnetic haloscopes search for axions ¼ ð À1Þ ¼ cm am a : ð7Þ A21 gam Na K 21 ðω Àω þ γ = Þðω Àω þ γ = ÞÀ 2 through their electron interaction. Since testing both these cou- a m i m 2 a c i c 2 gcm plings is a way to distinguish between different axions, the two The observable of our apparatus is the power deposited in the detectors provide a complementary insight into the different resonant cavity by the axion field, which canpffiffiffiffiffiffiffi be calculated models. through the quadrature operator z ðcþ þ cÞ= 2ω , and results In addition, the knowledge we gain on the dynamic response of c γ γ ω2 γ ω2 2 2 ¼ c h_2i¼ c a j j2 ¼ c a gcmgamNa : this photon-magnon HS, allows for the optimising the spin P z ω A ω iγ iγ g 2 2 ac 2 4 c 21 8 c jðω Àω þ mÞðω Àω þ cÞÀð cmÞ j magnetometer, as we can maximise the collected axion power a m 2 a c 2 2 with different magnetising fields. ð8Þ The obtained expression of P fully describes the dynamics of Results ac Theoretical model the system and can be used to maximise the ferromagnetic . To understand the dynamics of a ferromag- haloscope sensitivity. It is interesting to consider the case of the netic haloscope and optimise its operation, it is useful to model it. power deposited by an axion field on resonance with one of the The system parameters are the resonant frequencies of the two hybrid modes of the system at frequencies cavity mode ω , the one of the Kittel mode ω , and their line- c m rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ γ ω þ ω ω À ω 2 2 widths c and m, respectively. We use a second-quantisation c m c m gcm ð Þ 57–61 ω ¼ ± þ : 9 formalism in natural units to write the Hamiltonian of this ± 2 2 2 system as the one of two coupled oscillators, photon and magnon, and an external axion field a that interacts only with the magnon. Since the system collects power at the hybrid modes frequencies, ω In the rotating-wave approximation it results to infer the bandwidth of the apparatus one needs to recast m in þ þ þ þ terms of ω by inverting Eq. 9. Substituting the expression into H ¼ ω m m þ ω c c þ g ðmc þ m cÞ ± m pffiffiffiffiffiffi c cm ð Þ Eq. 8, Pac results Àiω t þ iω t 1 þ g N ðme a þ m e a Þ; γ ω2 2 2 = ω am a ðω Þj ¼ c ± gcmgamNa 8 c : Pac ± ω ¼ω 2 + + a ± γ γ ω2 Àω ω Àðg =2Þ2 where m (m ), c (c ) are the destruction (creation) operators of ðÞω þi cÀω ω þi mÀ ± c ± cm ÀðÞgcm 2 ± 2 c ± 2 ω ± Àωc 2 magnons and photons respectively, and gcm and gam are the magnon-photon and axion-magnon couplings. A scheme of the ð10Þ considered system is shown in Fig. 1c. The last term in Eq. 1 The deposited power is plotted in Fig. 2 for the parameters of represent the interaction of the axion with the magnon, Na is the = 〈 + 〉 fi ω our experimental apparatus, and is in agreement with previously axion number Na a a and its mass xes the frequency a. reported results54,62. The haloscope collects power on two We are assuming Na > > 1 and treating axions as a classical separated axion-mass intervals, each being one order of external field perturbing the photon-magnon system. The effect fi fl magnitude broader than the resonance linewidth, demonstrating of the axion eld is to deposit energy in the form of spin- ips (i.e. a wideband tunability of the haloscope. Hereafter, we call magnons), which, with the rate gcm, are converted into photons dynamical bandwidth the frequency interval that can be scanned that can be collected as rf power. by changing the Larmor frequency, highlighted in red in Fig. 2 for Starting from the Hamiltonian in Eq. 1, the evolution of the – the lower frequency hybrid mode. system can be derived with the Heisenberg Langevin equa- The parameters used for this calculation are arbitrarily chosen tions57,58 for the mean values of magnon 〈m〉 pffiffiffiffiffiffi as they do not influence the experimental results, but we mention À ω h _ i¼ðω À γ = Þþ þ i at ð Þ i m m i m 2 gcmc gam Nae 2 that they are related to the axion model and to some cosmological 〈 〉 parameters. In particular, the axion number Na depends on the and cavity photon c state axion number density and on the volume of magnetic material; h_i¼ðω À γ = Þþ ; ð Þ i c c i 2 g m 3 under the assumption that DM is entirely composed of axions, at c cm ρ ≃ 13 3 γ the considered frequency we have DM/ma 2×10 axions/cm . where c,m accounts for dissipations to the thermal reservoir. The axions figure of merit depends on their thermal distribution Given the high occupation number of the magnon and photon and, as they are cold DM, the energy dispersion is smaller than its states, we solve the equations in a semiclassical approach, and 6 mean by a factor 2 × 10 . The coupling gam is proportional to the neglect quantum correlations. Eqs. 2 and 3 can be recast as a axion–electron coupling constant g ≃ 3×10−11(m /1 eV), matrix differential equation aee a "# ! which slightly depends on the considered axion model, and is a iγ  ω À m pffiffiffiffiffiffi number inversely proportional to the axion mass. For m gcm À ω 1 _ ¼ 2 þ i at ; ð Þ iM γ gam Nae M 4 more details on the axion-to-magnon conversion scheme see ω À i c 0 gcm c 2 refs. 24,53,63.

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representative signal r(t) shown in Fig. 3b is detected with the heterodyne microwave receiver detailed in the Methods section. The amplitude of the down-converted r(t) signal is comparable with the receiver noise when B0 is such that the system is far from the anticrossing point, ruling out a direct cavity excitation. To obtain the transduction coefficient from magnons to photons realised by the strong coupling, we record r(t) for several values of the hybrid frequency ω−, which is varied through the B0 field. The experimental results are compared to the theoretical prediction in Fig. 4. The transduction curve ðω Þ¼ ðω Þ= ð ðω ÞÞ q À Pac À max Pac À has been derived from Eq. 10, and is the normalised top-left projection of the dispersion plot reported in Fig. 2. Each measured value is instead obtained with the integral: Z þ τ tlaser 4 h ðω Þ¼ j ð Þj2 ; ð Þ qr À r t dt 11 t0

where t0 is given by the laser pulse (orange curve in Fig. 3b). As the laser system exhibits shot-to-shot intensity fluctuation of the Fig. 2 Deposited axion power as a function of the HS normal modes’ output pulses, for each ω− the mean and standard deviation of the ω frequencies (ω±). a The main figure is the anticrossing plot Pac(ω±, ωm) data is obtained by averaging hundreds of qr( −). In Fig. 4 data given by Eq. 8, while the upper and right plots are the projection of the points are plotted normalised with respect to the one with larger deposited power as function of ω± and ωm, respectively. The upper plot, amplitude at 4.67 GHz. It is important to note that no additional fi representing Pac(ω±), shows the apparatus bandwidth in blue for ω− and in elaboration has been carried out on the data, nor tting procedure orange for ω+. The bandwidth is calculated as the full width half maximum on the curve reported in Fig. 4.Theyindeeddisplayagood ω of the ω− curve of Pac(ω±), and its experimentally measured value is agreement for − > 4.65 GHz, with a slight discrepancy at lower reported in red. b In the grey box, the experimental parameters used for the frequencies that can be explained in terms of mismatched calculation are reported in black, gam and Na (values in blue) have been transmission line. In fact, in the model we implicitly assume a arbitrarily chosen, and the dynamical bandwidth is shown in red. condition of critical coupling (i.e. the extracted power is maximum), whereas the coupling between the receiver antenna Experimental validation. To study the system frequency and the cavity mode changes for different values of ω−.When response to a direct excitation of the material like that related to the coupling is optimised (red data points in Fig. 4), the agreement the searched particle in ferromagnetic haloscopes, we photoexcite between experiment and theory significantly improves. The system, the material with 1064-nm wavelength, 11-ps duration laser see Fig. 2, should behave symmetrically for the resonance ω+. pulses and measure the power stored in the HS modes for several values of applied magnetic field. At 1064 nm, the absorption Discussion coefficient of YIG is ~10 cm−1 64–66, corresponding to the Devising a simple and yet predictive model of an HS-based 6 6 → 4 4 transition A1g( S) T1g( G) between electronic levels in the transducer is a key ingredient to understand the behaviour of a octaedral crystal field configuration67. In addition, the laser beam ferromagnetic haloscope. We used a simple second-quantisation is focused within the sphere and we can thus reasonably assume model based on a system of two strongly coupled oscillators, a that the 0.1 mJ energy of the pulse is almost entirely absorbed in microwave cavity mode and a magnetostatic mode, to describe the material. A fraction of this energy is transferred from the the transduction of pure magnetic excitations to microwave excited to magnetic oscillations of the material, photons. The modelled physical system includes an external including the uniform precession mode used in the ferromagnetic power injection through the magnetostatic mode. This magneti- haloscope. The optical pulse corresponds to a broadband exci- sation oscillation can be given e.g. by DM axions, which, inter- tation on the HS, allowing to study the system dynamics by acting coherently with magnons, would resonantly deposit power considering the stimulus as frequency-independent. Therefore, in the HS. The transducer converts magnons into photons which this optical excitation is well represented by the last term in Eq. 1, can be collected by an antenna. From this model we derive the HS mimicking the axion interaction for the present purpose of transduction coefficient as a function of the external magnetic demonstrating the spin magnetometer dynamical bandwidth. field, an easily tunable parameter which changes the resonant We accomplish the strong cavity regime by coupling the Kittel frequencies of the HS, and thus the transduction frequency. mode, i.e., the uniform precession ferromagnetic resonance8,68,to To validate the model and to measure the dynamical band- the TE102 mode of a rectangular cavity, whose resonance width of a photon-magnon HS transducer, we introduce a new ω ≃ π γ ≃ π frequency is c (2 ) 4.7 GHz and the linewidth is c (2 ) 1.1 optical method to realise a selective excitation of the HS magnetic MHz. This magnetic dipole coupling is strengthened when the component, as is the case for axion–DM interactions in the magnetic field amplitude of the chosen cavity mode is maximum haloscope. This is fundamentally different from the calibration fi 50,51 at the location of the spin ensemble. By setting a static eld B0 procedure used in Primakoff haloscopes , wherein rf power ω = γ = ω fi such that m B0 c, we measure the coupling coef cient injection is accomplished through an antenna coupled to the ≃ π γ fi 2gcm (2 ) 57 MHz, and m through the linewidth of the hybrid cavity mode. The axion- eld sensitivity of this setup is not dis- γ γ = γ − γ ≃ π mode h as m 2 h c (2 ) 3.5 MHz (see Fig. 3a). cussed as the apparatus was not optimised to this aim. The The laser pulses deposit energy in the YIG sphere, which is sensitivity of a similar apparatus, and the one of an optimised electromagnetically transduced to cavity excitations. The power haloscope, are reported in refs. 63,69, respectively. We note that γ = τ fi deposited in the hybrid mode is dissipated in a time 1/ h h, the magnetic eld tuning of the apparatus can be combined with and is measured through an antenna coupled to the cavity. the variation of the cavity resonant frequency50,51,70 to further Hence, when the infrared laser pulse excites the YIG sphere, the increase the total bandwidth of the apparatus.

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Fig. 3 Characterisation of the experimental setup and of the optical excitation and microwave response pulses. a Transmission spectrum of the cavity

(B0 = 0) and of the hybrid system (B0 = 168 mT). The frequency band where the signal is down-converted is evidenced in green. The frequency separation between the hybrid modes is related to the coupling strength between cavity photons and magnons. b Oscilloscope traces of the relevant signals. The laser pulse acts as acquisition trigger (orange) for the down-converted microwave pulse r(t) (blue).

Fig. 5 Detailed scheme of the experimental setup. The 2-mm diameter yttrium iron garnet (YIG) sphere is hosted inside a 98 × 12.6 × 42.5 mm rectangular microwave cavity resonator, at the position of maximum magnetic field intensity for the TE102 mode (as indicated by the colour map filling the Fig. 4 Theoretical curve and experimental data. Thebluecurvehasbeen cavity) to allow for maximising the coupling between cavity photons and calculated with the parameters reported in Fig. 2. The maximum measured magnons. An electromagnet, represented by its north (N, blue) ad south (S, signal was taken to be unitary and the other points were scaled consequently. red) poles, provides the static magnetic field. Picosecond-duration, 1064-nm The grey points were collected with a fixed antenna optimised for measure- wavelength laser pulses are used to excite the magnetostatic modes in the ments around 4.67 GHz, while for the red ones its coupling was, to some extent, YIG sphere, and are monitored thanks to a beam splitter (BS) and a adapted for the respective frequency. The data points and relative errorbars are photodiode. The hybrid system dynamics is investigated through a obtained as mean and standard deviation of repeated measurements. heterodyne receiver coupled to the cavity mode through the antenna θ1, where A1, A2, and A3 are its amplifiers. The signal is amplified before mixing In a ferromagnetic haloscope, a spectroscopic characterisation with the output of a local oscillator (LO), further amplified and recorded at the provides the HS parameters, namely its resonant frequencies, oscilloscope for several values of the external magnetic field. A signal linewidths and couplings, that we insert in our analytical model. generator (SG), which is weakly coupled to the cavity mode through an fi By varying the static magnetic eld, we change the transduction inductive loop (θ2), combined with the power splitter (PS) and the spectrum frequency and experimentally reconstruct the dynamical band- analyser are used to obtain cavity and hybrid system transmission spectra. width, that is found to be in agreement with the function obtained fi fi fi by the model. Our ndings represent the rst con rmation of the These optical tests will be fundamental to get end-to-end cali- wide frequency tunability of ferromagnetic haloscopes. Axion bration in precision measurement apparatuses as those previously masses can in principle be probed by an optimised haloscope in a mentioned, provided the conversion factor of the underlying range up to a few GHz, provided that large bandwidth quantum- process is precisely known. For instance, a cw laser whose intensity limited amplifiers, like travelling-wave Josephson parametric fi 71,72 fi fi is modulated at the relevant frequency might be used to mimic the ampli ers , are employed as a rst stage of ampli cation. effective AC magnetic field via the inverse Faraday effect18. The present model can thus be extended to describe more Moreover, similar phenomena are studied for optical manipulation sophisticated ferromagnetic haloscope configurations, namely 17,18 69 of magnons , hence setups like the one presented in this work including ten YIG spheres, as reported in ref. , where the trans- could be useful to understand the conversion capabilities of HSs. duction efficiency has been used to calculate the upper limit on the – The study of HSs is continuously growing and cross-fertilise axion electron coupling constant. Whilst we envision the upgrade multiple physical topics. In the last years non-Hermitian physics of the ferromagnetic haloscope to even larger scales to boost the phenomena are tackled by coupling photons and magnons, and axion signal, we are aware of limitations that might arise in the HS exceptional points and surfaces were experimentally demon- due to the consequent large hybridisation and the corresponding strated73–77. In general, our scheme allows for a selective power broader dynamical bandwidth. The use of larger spheres54,mul- 53 69 injection in HSs, feature which can be used to study them on a tiple samples or both , leads to higher order modes, size-effects fundamental level78. or multi-spheres disturbances that are not taken into account in the modelled two oscillator HS. A consistent description of such Methods complex apparatus can be obtained by adding additional modes in Ultrafast laser. The laser system in Fig. 5 consists of a passively mode-locked fi our model, that can be easily extended to an arbitrary number of oscillator, followed by a quasi-CW, two-stage Nd:YVO4 slabs ampli er. From a coupled oscillators. stable train of 8-ps duration, 1064-nm wavelength pulses at 60 MHz repetition rate,

COMMUNICATIONS PHYSICS | (2020) 3:164 | https://doi.org/10.1038/s42005-020-00435-w | www.nature.com/commsphys 5 ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-020-00435-w an extra-cavity acousto-optic (AO) pulse-picker selects, with adjustable repetition 14. Zhang, D. et al. Cavity quantum electrodynamics with ferromagnetic magnons rate, nJ energy pulses that are amplified up to an energy of 100 μJ. The amplifier in a small yttrium-iron-garnet sphere. npj Quantum Inf. 1, 15014 (2015). slabs are pumped by a 150 W peak power quasi-CW laser diode array, synchro- 15. Heshami, K. et al. Quantum memories: emerging applications and recent nised with the oscillator pulses sampled by the AO pulse picker. This has been advances. J. Mod. Opt. 63, 2005–2028 (2016). accomplished by clocking the laser electronics with a secondary output beam of the 16. Kurizki, G. et al. Quantum technologies with hybrid systems. Proc. Natl Acad. oscillator by means of a photodiode. The amplified pulses duration is 11 ps, and Sci. 112, 3866–3873 (2015). their repetition rate is 30 Hz. 17. Braggio, C., Carugno, G., Guarise, M., Ortolan, A. & Ruoso, G. Optical manipulation of a magnon-photon hybrid system. Phys. Rev. Lett. 118, 107205 Hybrid system. The magnetic material is a 2-mm diameter spherical YIG single (2017). crystal. The sphere is glued to a ceramic rod to suspend it in its position, with the 18. Hisatomi, R. et al. Bidirectional conversion between microwave and light via fi fi 93 easy axis aligned to the static magnetic eld. The B0 eld is supplied by a small ferromagnetic magnons. Phys. Rev. B , 174427 (2016). electromagnet. The cavity is a rectangular copper cavity, whose dimensions are 19. Fernandez-Gonzalvo, X., Chen, Y.-H., Yin, C., Rogge, S. & Longdell, J. J. 98 × 12.6 × 42.5 mm. The TE102 mode resonates at 4.7 GHz and the quality factor Coherent frequency up-conversion of microwaves to the optical 92 is 4300, estimated with an S21 measurement. The sphere is located at the centre of telecommunications band in an er:yso crystal. Phys. Rev. A , 062313 (2015). the cavity, where the rf magnetic field is maximum, and perpendicular to the 20. Williamson, L. A., Chen, Y.-H. & Longdell, J. J. Magneto-optic modulator with fi 113 direction of B0. unit quantum ef ciency. Phys. Rev. Lett. , 203601 (2014). 21. Kimble, H. J. The quantum internet. Nature 453, 1023–1030 (2008). 22. Chumak, A. V., Vasyuchka, V. I., Serga, A. A. & Hillebrands, B. Magnon Readout and heterodyne. Two antennas, θ1 and θ2, are coupled to the rectangular θ spintronics. Nat. Phys. 11, 453–461 (2015). cavity as sketched in Fig. 5. The coupling of 2, connected to the microwave fi θ 23. Moiseyev, N. Non-Hermitian Quantum Mechanics (Cambridge University oscillator SG, is xed, while the coupling of 1 can be varied by changing its θ Press, 2011). position. The weakly coupled 2 is used to inject microwave power, and thus θ 24. Barbieri, R. et al. 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To view a copy of this license, visit http://creativecommons.org/ 124 Phys. Rev. Lett. , 053602 (2020). licenses/by/4.0/. 74. Zhang, D., Luo, X.-Q., Wang, Y.-P., Li, T.-F. & You, J. Q. Observation of the exceptional point in cavity magnon-polaritons. Nat. Commun. 8, 1368–1368 (2017). © The Author(s) 2020

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