Abstract
The potential of photonmagnon 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 standard model, where the sensitivity to the socalled “invisibles” is currently being boosted by quantum technologies. Here, we demonstrate that quanta of spin waves, 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 Matter, which may directly interact with magnons. Our model of the transducer is based on a secondquantisation twooscillators hybrid system, it matches the observations, and can be easily extended to thoroughly describe future largescale ferromagnetic haloscopes.
Introduction
In the last decades, precision magnetometry emerged as a promising probe of physics beyond the standard model^{1,2}. Amongst all, GHzfrequency magnetometers can probe spinrelated effects through the use of the electron spin resonance techniques. Recent advances in the field are due to quantum computing research^{3}, which allowed to reach the sensitivity to detect single quanta of magnetisation in macroscopic samples using photonmagnon hybrid systems (HSs)^{4,5}. In these devices, the coherent interaction between photons and magnons is increased to such an extent that they can no longer be considered as separate entities, and the HS is described within the cavity quantum electrodynamics framework. Photons are confined within a microwave cavity hosting a magnetised sample, whose Larmor frequency ω_{m} is adjusted close to the one of a suitable cavity mode ω_{c}. In the strong coupling regime, in which the coupling strength is much larger than the related linewidths, photons and magnons are in a hybrid magnonpolariton state induced by magnon Rabi oscillation. Such a cooperative spin dynamics is governed by the physics of coupled harmonic oscillators with beating periods much shorter than dissipation times^{6}. As under these conditions the twooscillators exchange energy, the dispersion plot of the system displays the quantum phenomenon of avoided crossing: when ω_{c} ≃ ω_{m}, instead of two intersecting lines, the system dispersion relation exhibits an anticrossing curve^{7,8} as shown in Fig. 1a. A scheme of the HS used in this work is reported in Fig. 1b, and its conceptual representation is in Fig. 1c.
These HSs have been realised and extensively studied in the last decade^{9,10,11,12,13,14}, and applied to the development of quantum memories^{15,16}, microwave to optical photon conversion^{17,18,19,20,21}, detection of single magnons^{5}, or in spintronics^{22}. Furthermore, the investigation of nonHermitian quantum mechanics^{23} is currently pursued with photonmagnon HSs. Here we focus on an HS devised to probe weak and persistent effective rf fields, which is the physics case of dark matter (DM) axions^{24}.
The axion is a hypothetical particle introduced as a consequence of the strong CP problem solution found by Peccei and Quinn^{25,26,27}. It is a light pseudoGoldstone boson arising from the breaking of the Peccei–Quinn symmetry at extremely high energies f_{a} ≃ 10^{12} GeV. Since its mass and couplings are proportional to 1/f_{a} the axion is extremely light and weakly interacting^{28,29,30,31}. Relevant quantities of them may have been produced in the early universe, qualifying the axion as a viable candidate of cold DM that would account for the whole density ρ_{DM} = 0.45 GeV/cm^{3} of the Milky Way’s halo^{32,33,34,35,36}. A suitable mass range for the axion is m_{a} = 10^{−(3/5)} eV^{37,38,39,40,41}, hence its de Broglie wavelength is of the order of metres and allows a coherent interaction with macroscopic systems. Together with the high occupation number ρ_{DM}/m_{a}, this allows to treat the axion DM field as a classical rf field. The presence of axions can be tested with earthbased precision measurements^{42,43,44} probing observables sensitive to the presence of DM axions, i.e., the rfpower in a microwave cavity under a static magnetic field^{45,46,47,48,49,50,51} or the magnetisation of a sample^{24,52,53,54,55}; we focus on this last case. The described HSs emerge as a natural choice to detect the axion effective field as the variation of a sample magnetisation, and an apparatus measuring the power deposited in an HS by DM axions, and hereafter called P_{ac}, is therefore called ferromagnetic haloscope. As the underlying interaction is tiniest, the ferromagnetic haloscope relies on maximising signal power P_{ac}, which translates to hosting in the limited volume of GHzfrequency cavities a sample with the highest possible electron spin density. In addition, the hybrid modes Qfactor should not be much lower than the axion figure of merit (2 × 10^{6}). By using yttrium iron garnet (YIG) as magnetic sample, these two requirements are well satisfied, owing to its high spin density (2 × 10^{28} spins/m^{3}) and small damping constant (~10^{−5}).
In this work we theoretically analyse the response of a photonmagnon HS to a direct excitation of the magnetic component using a secondquantisation model, and experimentally test the model results with an apparatus involving an optical excitation of the magnetic material. We obtain the first experimental demonstration of the broad frequency tunability of the ferromagnetic haloscope, a crucial property that previously had only been theoretically analysed^{54}. This precision detector at the low energy frontier of particle physics can probe the axion coupling with electron spins in mass ranges of few μeV, as large as that probed by Primakoff haloscopes^{45,50,51,56}, with the advantage of a much simpler frequency scan system. The latter is, in this case, accomplished by varying the amplitude of the external magnetic field, instead of moving tuning rods in ultracryogenic environments. While Primakoff haloscopes are based o the coupling of axions to photons, ferromagnetic haloscopes search for axions through their electron interaction. Since testing both these couplings is a way to distinguish between different axions, the two detectors provide a complementary insight into the different models.
In addition, the knowledge we gain on the dynamic response of this photonmagnon HS, allows for the optimising the spin magnetometer, as we can maximise the collected axion power with different magnetising fields.
Results
Theoretical model
To understand the dynamics of a ferromagnetic haloscope and optimise its operation, it is useful to model it. The system parameters are the resonant frequencies of the cavity mode ω_{c}, the one of the Kittel mode ω_{m}, and their linewidths γ_{c} and γ_{m}, respectively. We use a secondquantisation formalism^{57,58,59,60,61} in natural units to write the Hamiltonian of this system as the one of two coupled oscillators, photon and magnon, and an external axion field a that interacts only with the magnon. In the rotatingwave approximation it results
where m (m^{+}), c (c^{+}) are the destruction (creation) operators of magnons and photons respectively, and g_{cm} and g_{am} are the magnonphoton and axionmagnon couplings. A scheme of the considered system is shown in Fig. 1c. The last term in Eq. 1 represent the interaction of the axion with the magnon, N_{a} is the axion number N_{a} = 〈a^{+}a〉 and its mass fixes the frequency ω_{a}. We are assuming N_{a} > > 1 and treating axions as a classical external field perturbing the photonmagnon system. The effect of the axion field is to deposit energy in the form of spinflips (i.e. magnons), which, with the rate g_{cm}, are converted into photons that can be collected as rf power.
Starting from the Hamiltonian in Eq. 1, the evolution of the system can be derived with the Heisenberg–Langevin equations^{57,58} for the mean values of magnon 〈m〉
and cavity photon 〈c〉 state
where γ_{c,m} accounts for dissipations to the thermal reservoir. Given the high occupation number of the magnon and photon states, we solve the equations in a semiclassical approach, and neglect quantum correlations. Eqs. 2 and 3 can be recast as a matrix differential equation
where M is the twocomponent vector M = (〈m〉, 〈c〉). If we rewrite M as a plane wave \(M=A\exp (i{\omega }_{a}t)\), Eq. 4 can be recast as
yielding the amplitude of M
from which it is possible to extract the (2, 1) component describing the coupling between the axion field and the cavity
The observable of our apparatus is the power deposited in the resonant cavity by the axion field, which can be calculated through the quadrature operator \(z\equiv ({c}^{+}+c)/\sqrt{2{\omega }_{c}}\), and results
The obtained expression of P_{ac} fully describes the dynamics of the system and can be used to maximise the ferromagnetic haloscope sensitivity. It is interesting to consider the case of the power deposited by an axion field on resonance with one of the two hybrid modes of the system at frequencies
Since the system collects power at the hybrid modes frequencies, to infer the bandwidth of the apparatus one needs to recast ω_{m} in terms of ω_{±} by inverting Eq. 9. Substituting the expression into Eq. 8, P_{ac} results
The deposited power is plotted in Fig. 2 for the parameters of our experimental apparatus, and is in agreement with previously reported results^{54,62}. The haloscope collects power on two separated axionmass intervals, each being one order of magnitude broader than the resonance linewidth, demonstrating a wideband tunability of the haloscope. Hereafter, we call dynamical bandwidth the frequency interval that can be scanned by changing the Larmor frequency, highlighted in red in Fig. 2 for the lower frequency hybrid mode.
The parameters used for this calculation are arbitrarily chosen as they do not influence the experimental results, but we mention that they are related to the axion model and to some cosmological parameters. In particular, the axion number N_{a} depends on the axion number density and on the volume of magnetic material; under the assumption that DM is entirely composed of axions, at the considered frequency we have ρ_{DM}/m_{a} ≃ 2 × 10^{13} axions/cm^{3}. The axions figure of merit depends on their thermal distribution and, as they are cold DM, the energy dispersion is smaller than its mean by a factor 2 × 10^{6}. The coupling g_{am} is proportional to the axion–electron coupling constant g_{aee} ≃ 3 × 10^{−11}(m_{a}/1 eV), which slightly depends on the considered axion model, and is a number inversely proportional to the axion mass. For more details on the axiontomagnon conversion scheme see refs. ^{24,53,63}.
Experimental validation
To study the system frequency response to a direct excitation of the material like that related to the searched particle in ferromagnetic haloscopes, we photoexcite the material with 1064nm wavelength, 11ps duration laser pulses and measure the power stored in the HS modes for several values of applied magnetic field. At 1064 nm, the absorption coefficient of YIG is ~10 cm^{−1} ^{64,65,66}, corresponding to the transition ^{6}A_{1g}(^{6}S) → ^{4}T_{1g}(^{4}G) between electronic levels in the octaedral crystal field configuration^{67}. In addition, the laser beam is focused within the sphere and we can thus reasonably assume that the 0.1 mJ energy of the pulse is almost entirely absorbed in the material. A fraction of this energy is transferred from the excited electrons to magnetic oscillations of the material, including the uniform precession mode used in the ferromagnetic haloscope. The optical pulse corresponds to a broadband excitation on the HS, allowing to study the system dynamics by considering the stimulus as frequencyindependent. Therefore, this optical excitation is well represented by the last term in Eq. 1, mimicking the axion interaction for the present purpose of demonstrating the spin magnetometer dynamical bandwidth.
We accomplish the strong cavity regime by coupling the Kittel mode, i.e., the uniform precession ferromagnetic resonance^{8,68}, to the TE102 mode of a rectangular cavity, whose resonance frequency is ω_{c} ≃ (2π) 4.7 GHz and the linewidth is γ_{c} ≃ (2π) 1.1 MHz. This magnetic dipole coupling is strengthened when the magnetic field amplitude of the chosen cavity mode is maximum at the location of the spin ensemble. By setting a static field B_{0} such that ω_{m} = γB_{0} = ω_{c}, we measure the coupling coefficient 2g_{cm} ≃ (2π) 57 MHz, and γ_{m} through the linewidth of the hybrid mode γ_{h} as γ_{m} = 2γ_{h} − γ_{c} ≃ (2π) 3.5 MHz (see Fig. 3a).
The laser pulses deposit energy in the YIG sphere, which is electromagnetically transduced to cavity excitations. The power deposited in the hybrid mode is dissipated in a time 1/γ_{h} = τ_{h}, and is measured through an antenna coupled to the cavity. Hence, when the infrared laser pulse excites the YIG sphere, the representative signal r(t) shown in Fig. 3b is detected with the heterodyne microwave receiver detailed in the Methods section. The amplitude of the downconverted r(t) signal is comparable with the receiver noise when B_{0} 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 B_{0} field. The experimental results are compared to the theoretical prediction in Fig. 4. The transduction curve \(q({\omega }_{})={P}_{ac}({\omega }_{})/\max ({P}_{ac}({\omega }_{}))\) has been derived from Eq. 10, and is the normalised topleft projection of the dispersion plot reported in Fig. 2.
Each measured value is instead obtained with the integral:
where t_{0} is given by the laser pulse (orange curve in Fig. 3b). As the laser system exhibits shottoshot intensity fluctuation of the output pulses, for each ω_{−} the mean and standard deviation of the data is obtained by averaging hundreds of q_{r}(ω_{−}). In Fig. 4 data points are plotted normalised with respect to the one with larger amplitude at 4.67 GHz. It is important to note that no additional elaboration has been carried out on the data, nor fitting procedure on the curve reported in Fig. 4. They indeed display a good agreement for ω_{−} > 4.65 GHz, with a slight discrepancy at lower frequencies that can be explained in terms of mismatched transmission line. In fact, in the model we implicitly assume a condition of critical coupling (i.e. the extracted power is maximum), whereas the coupling between the receiver antenna and the cavity mode changes for different values of ω_{−}. When the coupling is optimised (red data points in Fig. 4), the agreement between experiment and theory significantly improves. The system, see Fig. 2, should behave symmetrically for the resonance ω_{+}.
Discussion
Devising a simple and yet predictive model of an HSbased transducer is a key ingredient to understand the behaviour of a ferromagnetic haloscope. We used a simple secondquantisation model based on a system of two strongly coupled oscillators, a microwave cavity mode and a magnetostatic mode, to describe the transduction of pure magnetic excitations to microwave photons. The modelled physical system includes an external power injection through the magnetostatic mode. This magnetisation oscillation can be given e.g. by DM axions, which, interacting coherently with magnons, would resonantly deposit power in the HS. The transducer converts magnons into photons which can be collected by an antenna. From this model we derive the HS transduction coefficient as a function of the external magnetic field, an easily tunable parameter which changes the resonant frequencies of the HS, and thus the transduction frequency.
To validate the model and to measure the dynamical bandwidth of a photonmagnon HS transducer, we introduce a new optical method to realise a selective excitation of the HS magnetic component, as is the case for axion–DM interactions in the haloscope. This is fundamentally different from the calibration procedure used in Primakoff haloscopes^{50,51}, wherein rf power injection is accomplished through an antenna coupled to the cavity mode. The axionfield sensitivity of this setup is not discussed as the apparatus was not optimised to this aim. The sensitivity of a similar apparatus, and the one of an optimised haloscope, are reported in refs. ^{63,69}, respectively. We note that the magnetic field tuning of the apparatus can be combined with the variation of the cavity resonant frequency^{50,51,70} to further increase the total bandwidth of the apparatus.
In a ferromagnetic haloscope, a spectroscopic characterisation provides the HS parameters, namely its resonant frequencies, linewidths and couplings, that we insert in our analytical model. By varying the static magnetic field, we change the transduction frequency and experimentally reconstruct the dynamical bandwidth, that is found to be in agreement with the function obtained by the model. Our findings represent the first confirmation of the wide frequency tunability of ferromagnetic haloscopes. Axion masses can in principle be probed by an optimised haloscope in a range up to a few GHz, provided that large bandwidth quantumlimited amplifiers, like travellingwave Josephson parametric amplifiers^{71,72}, are employed as a first stage of amplification.
The present model can thus be extended to describe more sophisticated ferromagnetic haloscope configurations, namely including ten YIG spheres, as reported in ref. ^{69}, where the transduction efficiency has been used to calculate the upper limit on the axion–electron coupling constant. Whilst we envision the upgrade of the ferromagnetic haloscope to even larger scales to boost the axion signal, we are aware of limitations that might arise in the HS due to the consequent large hybridisation and the corresponding broader dynamical bandwidth. The use of larger spheres^{54}, multiple samples^{53} or both^{69}, leads to higher order modes, sizeeffects or multispheres disturbances that are not taken into account in the modelled two oscillator HS. A consistent description of such complex apparatus can be obtained by adding additional modes in our model, that can be easily extended to an arbitrary number of coupled oscillators.
These optical tests will be fundamental to get endtoend calibration in precision measurement apparatuses as those previously mentioned, provided the conversion factor of the underlying process is precisely known. For instance, a cw laser whose intensity is modulated at the relevant frequency might be used to mimic the effective AC magnetic field via the inverse Faraday effect^{18}. Moreover, similar phenomena are studied for optical manipulation of magnons^{17,18}, hence setups like the one presented in this work could be useful to understand the conversion capabilities of HSs.
The study of HSs is continuously growing and crossfertilise multiple physical topics. In the last years nonHermitian physics phenomena are tackled by coupling photons and magnons, and exceptional points and surfaces were experimentally demonstrated^{73,74,75,76,77}. In general, our scheme allows for a selective power injection in HSs, feature which can be used to study them on a fundamental level^{78}.
Methods
Ultrafast laser
The laser system in Fig. 5 consists of a passively modelocked oscillator, followed by a quasiCW, twostage Nd:YVO_{4} slabs amplifier. From a stable train of 8ps duration, 1064nm wavelength pulses at 60 MHz repetition rate, an extracavity acoustooptic (AO) pulsepicker selects, with adjustable repetition rate, nJ energy pulses that are amplified up to an energy of 100 μJ. The amplifier slabs are pumped by a 150 W peak power quasiCW laser diode array, synchronised with the oscillator pulses sampled by the AO pulse picker. This has been accomplished by clocking the laser electronics with a secondary output beam of the oscillator by means of a photodiode. The amplified pulses duration is 11 ps, and their repetition rate is 30 Hz.
Hybrid system
The magnetic material is a 2mm diameter spherical YIG single crystal. The sphere is glued to a ceramic rod to suspend it in its position, with the easy axis aligned to the static magnetic field. The B_{0} field is supplied by a small electromagnet. The cavity is a rectangular copper cavity, whose dimensions are 98 × 12.6 × 42.5 mm. The TE102 mode resonates at 4.7 GHz and the quality factor is 4300, estimated with an S_{21} measurement. The sphere is located at the centre of the cavity, where the rf magnetic field is maximum, and perpendicular to the direction of B_{0}.
Readout and heterodyne
Two antennas, θ_{1} and θ_{2}, are coupled to the rectangular cavity as sketched in Fig. 5. The coupling of θ_{2}, connected to the microwave oscillator SG, is fixed, while the coupling of θ_{1} can be varied by changing its position. The weakly coupled θ_{2} is used to inject microwave power, and thus perform spectroscopic measurements of the HS. The position of θ_{1} is chosen by doubling the hybrid uncoupled linewidth when ω_{c} = ω_{m}, so that the coupling is close to critical, and remains the same within the measurement. The signal collected by θ_{1} is amplified by A1 and A2, two high electron mobility field effect transistors lownoise amplifiers. It is then split between a spectrum analyser, useful to acquire the transmission measurements of the HS, and a heterodyne that downconverts the signal to lower frequencies using a mixer and a local oscillator (LO). The downconverted signal, whose band is shown in green in Fig. 3a, is further amplified by A3 and acquired by an oscilloscope triggered by the signal of a photodiode, as shown in Fig. 3b. The frequency of the LO is adjusted whenever the B_{0} field is changed, in order to keep the hybrid mode frequency ω_{−} within in the downconverted band. The frequency of r(t) in Fig. 3b is the difference between the ones of the LO and of the hybrid mode. The total gain of the amplification chain is of order 71 dB, and the noise temperature of A1 is about 40 K, thus the noise of the apparatus is essentially due to room temperature thermodynamical fluctuations.
Data availability
The data collected during this study are available from the corresponding author upon reasonable request.
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Acknowledgements
The authors would like to thank Riccardo Barbieri for the help in the development of the theoretical model and Federico Pirzio for the advice on the laser system. We acknowledge the work of Enrico Berto, Mario Tessaro, and Fulvio Calaon, which contributed in the realisation of the mechanics and electronics of the experiment, and INFNLaboratori Nazionali di Legnaro for hosting the experiment.
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N.C. conceived the experiment and performed the measurements with the help of R.D.V., C.B. G.C. G.R. contributed to the construction of the setup, and N.C., R.D.V., A.O. analysed the data. N.C., C.B. wrote the manuscript. All the author discussed the results and the manuscript.
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Crescini, N., Braggio, C., Carugno, G. et al. Magnondriven dynamics of a hybrid system excited with ultrafast optical pulses. Commun Phys 3, 164 (2020). https://doi.org/10.1038/s4200502000435w
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Further reading

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