Magnons at low excitations: Observation of incoherent coupling to a bath of two-level-systems
Marco Pfirrmann, Isabella Boventer, Andre Schneider, Tim Wolz, Mathias Kläui, Alexey V. Ustinov, Martin Weides
MMagnons at low excitations: Observation of incoherent coupling to a bath of two-levelsystems
Marco Pfirrmann, ∗ Isabella Boventer,
1, 2
Andre Schneider, TimWolz, Mathias Kl¨aui, Alexey V. Ustinov,
1, 3 and Martin Weides
1, 4, † Institute of Physics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany Institute of Physics, Johannes Gutenberg-University Mainz, 55099 Mainz, Germany Russian Quantum Center, National University of Science and Technology MISIS, 119049 Moscow, Russia James Watt School of Engineering, University of Glasgow, Glasgow G12 8LT, United Kingdom (Dated: November 26, 2019)Collective magnetic excitation modes, magnons, can be coherently coupled to microwave photonsin the single excitation limit. This allows for access to quantum properties of magnons and opens upa range of applications in quantum information processing, with the intrinsic magnon linewidth rep-resenting the coherence time of a quantum resonator. Our measurement system consists of a yttriumiron garnet sphere and a three-dimensional microwave cavity at temperatures and excitation powerstypical for superconducting quantum circuit experiments. We perform spectroscopic measurementsto determine the limiting factor of magnon coherence at these experimental conditions. Using theinput-output formalism, we extract the magnon linewidth κ m . We attribute the limitations of thecoherence time at lowest temperatures and excitation powers to incoherent losses into a bath ofnear-resonance two-level systems (TLSs), a generic loss mechanism known from superconductingcircuits under these experimental conditions. We find that the TLSs saturate when increasing theexcitation power from quantum excitation to multiphoton excitation and their contribution to thelinewidth vanishes. At higher temperatures, the TLSs saturate thermally and the magnon linewidthdecreases as well. Strongly coupled light-spin hybrid systems allow forcoherent exchange of quantum information. Such sys-tems are usually studied either classically at room tem-perature [1] or at millikelvin temperatures approachingthe quantum limit of excitation [2–4]. The field of cav-ity magnonics [5–9] harnesses the coherent exchange ofexcitation due to the strong coupling within the systemand is used to access a new range of applications suchas quantum transducers and memories [10]. Nonlinear-ity in the system is needed to gain access to the controland detection of single magnons. Because of experimen-tal constraints regarding required light intensities in apurely optomagnonic system [11], hybridized systems ofmagnon excitations and non-linear macroscopic quantumsystems such as superconducting qubits [12, 13] are usedinstead, which opens up new possibilities in the emergingfield of quantum magnonics [14, 15]. An efficient interac-tion of magnonic systems and qubits requires their life-times to exceed the exchange time. Magnon excitationlosses, expressed by the magnon linewidth κ m , translateinto a lifetime of the spin excitation. Identifying its lim-iting factors is an important step toward more sophisti-cated implementations of hybrid quantum systems usingmagnons. Studies in literature show the losses in magnonexcitations from room temperature down to about liquidhelium temperatures [9]. The main contribution changeswith temperature from scattering at rare-earth impu-rities [16, 17] to multi-magnon scattering at imperfectsample surfaces [18, 19]. For a typical environment ofsuperconducting quantum circuit experiments, temper-atures below 100 mK and microwave probe powers com-parable to single-photon excitations, temperature sweeps show losses into TLSs [3]. In this paper, we presentboth temperature- and power-dependent measurementsof the magnon linewidth in a spherical yttrium iron gar-net (YIG) sample in the quantum limit of magnon ex-citations. We extract the critical saturation power andpresent on- and off-resonant linewidth that is mappedto the ratio of magnon excitation in the hybrid system.For large detuning, the fundamental linewidth can beextracted, thereby avoiding unwanted saturation effectsfrom the residual cavity photon population. This rendersthe off-resonant linewidth a valuable information on thelimiting factors of spin lifetimes.The magnetization dynamics inside a magnetic crys-tal is described by bosonic quasiparticles of collectivespin excitation, called magnons. These magnons man-ifest as the collective precessional motion of the partic-ipating spins out of their equilibrium positions. Theirenergies and spatial distribution can be calculated ana-lytically using the Walker modes for spherical samples[20, 21]. We focus on the uniform in-phase precessionmode corresponding to the wave vector k = 0, calledthe Kittel mode [22], treating it equivalently to one sin-gle large macro spin. The precession frequency (magnonfrequency) of the Kittel mode in a sphere changes lin-early with a uniform external magnetic bias field. Theprecessional motion is excited by a magnetic field os-cillating at the magnon frequency perpendicular to thebias field. We use the confined magnetic field of a cav-ity photon resonance to create magnetic excitations in amacroscopic sample, biased by a static external magneticfield. Tuning them into resonance, the magnon and pho-ton degree of freedom mix due to their strong interaction. a r X i v : . [ qu a n t - ph ] N ov Ext. magnetic field (mT) P r o b e f r e q u e n c y / ( G H z ) Probe frequency / (GHz) R e f l e c t i o n a m p li t u d e || ( d B ) /= . Coil current (A) a b
Refl. amplitude | | (arb. units)
FIG. 1. (a) Color coded absolute value of the reflection spectrum plotted against probe frequency and applied current at T = 55 mK and P = −
140 dBm. The resonance dips show the dressed photon-magnon states forming an avoided level crossingwith the degeneracy point at I = 2 .
09 A, corresponding to an applied field of B = 186 .
98 mT (dashed vertical line). The insetdisplays the squared gradient of the zoomed-in amplitude data. The kink in the data represents a weakly coupled magnetostaticmode. This was also seen in Ref. [9]. (b) Raw data of the cross section at the center of the avoided level crossing and fit tothe input-output formalism. The fit gives a magnon linewidth of κ m / π = 1 . ± .
18 MHz. The data are normalized by thefield independent background before fitting and is multiplied to the fit to display it over the raw data.
This creates hybridized states described as repulsive cav-ity magnon polaritons, which are visible as an avoidedlevel crossing in the spectroscopic data with two reso-nance dips at frequencies ω ± (see Supplemental Material[23]) appearing in the data cross section. The interactionis described by the macroscopic magnon-photon couplingstrength g . The system is probed in reflection with mi-crowave frequencies using standard ferromagnetic reso-nance techniques [24]. We use the input-output formal-ism [25] to describe the reflection spectrum. The complexreflection parameter S , the ratio of reflected to inputenergy with respect to the probe frequency ω p , reads as S ( ω p ) = − κ c i ( ω r − ω p ) + κ l + g i( ω m − ω p )+ κ m , (1)with the cavity’s coupling and loaded linewidths κ c and κ l , and the internal magnon linewidth κ m (HWHM).For our hybrid system we mount a commercially avail-able YIG (Y Fe O ) sphere with a diameter d = 0 . − to 10 − [27–29] and a high netspin density of 2 . × µ B / cm [30]. The single crys-tal sphere comes pre-mounted to a beryllium oxide rodalong the [110] crystal direction. The 3D cavity has aTE mode resonance frequency of ω barer / π = 5 .
24 GHz and is equipped with one SMA connector for reflectionspectroscopy measurements. For low temperatures andexcitation powers, we find the internal and couplingquality factors to be Q i = ω r / κ i = 7125 ±
97 and Q c = ω r / κ c = 5439 ±
29, combining to a loaded qualityfactor Q l = (1 /Q i + 1 /Q c ) − = 3084 ±
24 (see Supple-mental Material [23]). We mount the YIG at a magneticanti-node of the cavity resonance and apply a static mag-netic field of about 187 mT perpendicular to the cavityfield to tune the magnetic excitation into resonance withthe cavity photon. The magnetic field is created by aniron yoke holding a superconducting niobium-titaniumcoil. Additional permanent samarium-cobalt magnetsare used to create a zero-current offset magnetic field ofabout 178 mT. The probing microwave signal is providedby a vector network analyzer (VNA). Microwave attenu-ators and cable losses account for −
75 dB of cable atten-uation to the sample. We apply probe powers between −
140 dBm and −
65 dBm at the sample’s SMA port. To-gether with the cavity parameters, this corresponds to anaverage magnon population number (cid:104) m (cid:105) from 0 . [23] in the hybridized case. The probesignal is coupled capacitively to the cavity photon usingthe bare inner conductor of a coaxial cable positioned inparallel to the electric field component. The temperatureof the sample is swept between 55 mK and 1 . Temperature (K) M a g n o n li n e w i d t h / ( M H z ) ==
140 120 100 80 60
Probe power (dBm) ba == Temperature / (GHz) Average magnon number
FIG. 2. (a) Temperature dependence of the magnon linewidth κ m at the degeneracy point. For low probe powers, κ m follows a tanh (1 /T ) behavior (crosses), while for high probe powers (circles) the linewidth does not show any temperaturedependence. (b) Power dependence of the magnon linewidth κ m for T = 55 mK and 200 mK at the degeneracy point. Bothtemperature curves show a similar behavior. At probe powers of about −
90 dB m κ m drops for both temperatures, followingthe (1 + P/P c ) − / trend of the TLS model. All linewidth data shown here are extracted from the fit at matching frequencies. A typical measurement is shown in Fig. 1(a), mea-sured at T = 55 mK with an input power level of P = −
140 dBm. Figure 1(b) shows the raw data and the fit ofthe cavity-magnon polariton at matching resonance fre-quencies for an applied external field of B = 186 .
98 mT.We correct the raw data from background resonances andextract the parameters of the hybridized system by fit-ting to Eq. (1). The coupling strength g/ π = 10 . κ l / π = ω r Q l / π = 0 .
85 MHz and the internal magnonlinewidth κ m / π = 1 .
82 MHz, thus being well in thestrong coupling regime ( g (cid:29) κ l , κ m ) for all temperaturesand probe powers. The measured coupling strength is ingood agreement with the expected value g th = γ e η (cid:114) µ (cid:126) ω r V a (cid:112) N s s, (2)with the gyromagnetic ratio of the electron γ e , the modevolume V a = 5 . × − m , the Fe spin number s = 5 /
2, the spatial overlap between microwave field andmagnon field η , and the total number of spins N s [9]. Theoverlap factor is given by the ratio of mode volumes inthe cavity volume and the sample volume [1]. We findfor our setup the overlap factor to be η = 0 . d = 0 . N s = 1 . × spins. We find the expected couplingstrength g th / π = 12 .
48 MHz to be in good agreementwith our measured value. Even for measurements at highpowers, the number of participating spins of the order of10 is much larger than the estimated number of magnonexcitations ( ∼ ). We therefore do not expect to see the intrinsic magnon nonlinearity as observed at excita-tion powers comparable to the number of participatingspins [32].The internal magnon linewidth decreases at highertemperature and powers (Fig. 2) while the couplingstrength remains geometrically determined and does notchange with either temperature or power. This behav-ior can be explained by an incoherent coupling to a bathof two-level systems (TLSs) as the main source of loss inour measurements. In the TLS model [33–36], a quantumstate is confined in a double-well potential with differentground-state energies and a barrier in-between. TLSs be-come thermally saturated at temperatures higher thantheir frequency ( T (cid:38) (cid:126) ω TLS /k B ). Dynamics at low tem-peratures are dominated by quantum tunneling throughthe barrier that can be stimulated by excitations at sim-ilar energies. This resonant energy absorption shifts theequilibrium between the excitation rate and lifetime ofthe TLSs and their influence to the overall excitation lossvanishes. Loss into an ensemble of near-resonant TLSsis a widely known generic model for excitation losses insolids, glasses, and superconducting circuits at these ex-perimental conditions [37]. We fit the magnon linewidthto the generic TLS model loss tangent κ m ( T, P ) = κ tanh ( (cid:126) ω r / k B T ) (cid:112) P/P c + κ off . (3)Directly in the avoided level crossing we find κ / π =1 . ± .
15 MHz as the low temperature limit of thelinewidth describing the TLS spectrum within the sam-ple and κ off / π = 0 . ± .
11 MHz as an offset linewidth
Ext. magnetic field (mT) M a g n o n li n e w i d t h ( M H z ) Ext. magnetic field (mT)01 R a t i o photonmagnon R a t i o Magnon frequency / (GHz) Magnon frequency / (GHz) a b = = FIG. 3. Magnetic field (magnon frequency) dependence of the the magnon linewidth κ m for different probe powers at T = 55 mK (a) and T = 200 mK (b). The shown probe powers correspond to the ones at the transition in Fig. 2 (b). Thenumber of excited magnons depends on the detuning of magnon and photon frequency. At matching frequencies (dashedline) the magnon linewidth has a minimum, corresponding to the highest excited magnon numbers and therefore the highestsaturation of TLSs. A second minimum at about 187 .
25 mT corresponds to the coupling to an additional magnetostatic modewithin the sample [inset of Fig. 1 (a)]. The insets show the ratio of excitation power within each component of the hybridsystem. At matching frequencies, both components are excited equally. The magnon share drops at the plot boundaries toabout 20 %. The coupling to the magnetostatic mode is visible as a local maximum in the magnon excitation ratio. The x axesare scaled as in the main plots. The legends are valid for both temperatures. added as a lower boundary without TLS contribution.The critical power P c = − ± . (cid:104) m c (cid:105) = 2 . · . Usingfinite-element simulations, we map the critical excita-tion power to a critical AC magnetic field on the or-der of B c ∼ · − T at the position of the YIG sam-ple. Looking at the linewidths outside the anti-crossingat constant input power, we find a minimum at match-ing magnon and photon frequencies (dashed lines in Fig.3). Here, the excitation is equally distributed betweenphotons and magnons, reaching the maximum in bothmagnon excitation power and TLS saturation, respec-tively. At detuned frequencies the ratio between magnonand photon excitation power changes, less energy excitesthe magnons (insets in Fig. 3), and therefore less TLSsget saturated. The magnon linewidth increases with de-tuning, matching the low power data for large detunings.This effect is most visible at highest excitation powers.We calculate the energy ratios by fitting the resonances ineach polariton branch individually and weight the storedenergy with the eigenvalues of the coupling Hamiltonian[38]. For higher powers a second minimum at about187 .
25 mT can be seen at both studied temperatures. Weattribute this to the coupling to a magnetostatic modewithin the YIG sample and therefore again an increasednumber of excited magnons [see inset of Fig. 1 (a)]. This can also be seen in the inset figures as a local magnonexcitation maximum. We attribute the TLS-independentlosses κ off / π = 0 . ± .
11 MHz to multi-magnon scat-tering processes on the imperfect sphere surface [18, 19].As described in Ref. [9], we model the surface of the YIGwith spherical pits with radii of of the size of the polish-ing material (2 / × . µ m) and estimate a contributionof about 2 π · photons below a single photon. We iden-tify incoherent coupling to a bath of two-level systems asthe main source of excitation loss in our measurements.The magnon linewidth κ m / π at the degeneracy point fitswell to the generic loss tangent of the TLS model with re-spect to temperature and power. It decreases from about1 . Note added in proof - Recently, a manuscript study-ing losses in thin film YIG that independently observedcomparable results and reached similar conclusions waspublished by Kosen et al. [51].This work was supported by the European ResearchCouncil (ERC) under the Grant Agreement 648011and the Deutsche Forschungsgemeinschaft (DFG) withinProject INST 121384/138-1 FUGG and SFB TRR 173.We acknowledge financial support by the Helmholtz In-ternational Research School for Teratronics (M.P. andT.W.) and the Carl-Zeiss-Foundation (A.S.). A.V.U ac-knowledges partial support from the Ministry of Educa-tion and Science of Russian Federation in the frameworkof the Increase Competitiveness Program of the NationalUniversity of Science and Technology MISIS (Grant No.K2-2017-081). ∗ marco.pfi[email protected] † [email protected][1] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Phys.Rev. Lett. , 156401 (2014).[2] H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifen-stein, A. Marx, R. Gross, and S. T. B. Goennenwein,Phys. Rev. Lett. , 127003 (2013).[3] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Us-ami, and Y. Nakamura, Phys. Rev. Lett. , 083603(2014).[4] M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan,M. Kostylev, and M. E. Tobar, Phys. Rev. Appl. ,054002 (2014).[5] D. Zhang, X.-M. Wang, T.-F. Li, X.-Q. Luo, W. Wu,F. Nori, and J. You, npj Quantum Inf. , 15014 EP(2015).[6] L. Bai, M. Harder, Y. Chen, X. Fan, J. Xiao, and C.-M.Hu, Phys. Rev. Lett. , 227201 (2015).[7] Y. Cao, P. Yan, H. Huebl, S. T. B. Goennenwein, andG. E. W. Bauer, Phys. Rev. B , 094423 (2015).[8] M. Goryachev, S. Watt, J. Bourhill, M. Kostylev, andM. E. Tobar, Phys. Rev. B , 155129 (2018).[9] I. Boventer, M. Pfirrmann, J. Krause, Y. Sch¨on,M. Kl¨aui, and M. Weides, Phys. Rev. B , 184420 (2018).[10] X. Zhang, C.-L. Zou, N. Zhu, F. Marquardt, L. Jiang,and H. X. Tang, Nat. Commun. , 8914 EP (2015).[11] S. Viola Kusminskiy, H. X. Tang, and F. Marquardt,Phys. Rev. A , 033821 (2016).[12] Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Ya-mazaki, K. Usami, and Y. Nakamura, Science , 405(2015).[13] Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Ya-mazaki, K. Usami, and Y. Nakamura, C. R. Phys. ,729 (2016).[14] D. Lachance-Quirion, Y. Tabuchi, S. Ishino, A. Noguchi,T. Ishikawa, R. Yamazaki, and Y. Nakamura, Sci. Adv. , e1603150 (2017).[15] D. Lachance-Quirion, Y. Tabuchi, A. Gloppe, K. Us-ami, and Y. Nakamura, Appl. Phys. Express , 070101(2019).[16] E. G. Spencer, R. C. LeCraw, and R. C. Linares, Phys.Rev. , 1937 (1961).[17] P. E. Seiden, Phys. Rev. , A728 (1964).[18] M. Sparks, R. Loudon, and C. Kittel, Phys. Rev. ,791 (1961).[19] J. Nemarich, Phys. Rev. , A1657 (1964).[20] L. R. Walker, Phys. Rev. , 390 (1957).[21] P. Fletcher, I. H. Solt, and R. Bell, Phys. Rev. , 739(1959).[22] C. Kittel, Phys. Rev. , 155 (1948).[23] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.1.032023 forfurther information about fitting procedure and excita-tion number callibration.l.[24] S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L.Schneider, P. Kabos, T. J. Silva, and J. P. Nibarger, J.Appl. Phys. , 093909 (2006).[25] D. F. Walls and G. F. Milburn, Quantum Optics ,262 (2010).[28] B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz,E. Girt, Y.-Y. Song, Y. Sun, and M. Wu, Phys. Rev.Lett. , 066604 (2011).[29] H. Kurebayashi, O. Dzyapko, V. E. Demidov, D. Fang,A. J. Ferguson, and S. O. Demokritov, Nature Materials , 660 EP (2011).[30] M. A. Gilleo and S. Geller, Phys. Rev. , 73 (1958).[31] “Qkit - a quantum measurement suite in python,”https://github.com/qkitgroup/qkit.[32] J. A. Haigh, N. J. Lambert, A. C. Doherty, and A. J.Ferguson, Phys. Rev. B , 104410 (2015).[33] P. W. Anderson, B. I. Halperin, and C. M. Varma, Phi-los. Mag. , 1 (1972).[34] W. A. Phillips, J. Low Temp. Phys. , 351 (1972).[35] S. Hunklinger and W. Arnold, Ultrasonic Properties ofGlasses at Low Temperatures , Physical Acoustics, Vol. 12(Academic Press, 1976) pp. 155 – 215.[36] W. A. Phillips, Reports on Progress in Physics , 1657(1987).[37] C. M¨uller, J. H. Cole, and J. Lisenfeld, Reports onProgress in Physics , 124501 (2019).[38] M. Harder and C.-M. Hu, Cavity Spintronics: An EarlyReview of Recent Progress in the Study of Magnon-
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The frequencies of both arms of the avoided level cross-ing ω ± are fitted to the energy eigenvalues of a 2 × ω ± = ω barer + ω I =0m ± (cid:115)(cid:18) ω barer − ω I =0m (cid:19) + g . (S1) We use the current dependent data taken at T = 55 mKand P = −
140 dBm to obtain the bare cavity fre-quency ω barer , the zero-current magnetic excitation fre-quency ω I =0m , and the coupling strength g . The fre-quencies of the anticrossing ω ± were obtained by track-ing the minima in the amplitude data. From thefit we obtain the bare cavity frequency ω barer / π =5 .
239 02 ± .
000 02 GHz and the zero-current magneticexcitation frequency ω I =0m / π = 4 . ± . g/ π = 10 . ± .
17 MHz.
Magnon number estimation
Using the cavity’s resonance frequency, quality factors,and the input power P in we estimate the total number ofmagnon and photon excitation within the cavity (cid:104) N e (cid:105) inunits of (cid:126) ω r , (cid:104) N e (cid:105) = 4 Q Q c (cid:126) ω · P in . (S2)Note that the input power P in is in units of watts andnot to be confused with the probe power (level) P inunits of dBm. For the strongly coupled system the exci-tation energy at matching frequencies is stored in equalparts in photons and magnons, (cid:104) n (cid:105) = (cid:104) m (cid:105) = (cid:104) N e (cid:105) .We measure the reflection signal of the cavity reso-nance at 55 mK at probe powers between −
140 dBm and −
65 dBm at zero current applied to the magnetic coil.The complex data is then fitted using a circle fit al-gorithm [52] to determine the power dependent qual-ity factors (coupling quality factor Q c = ω r κ c , inter-nal quality factor Q i = ω r κ i , and loaded quality factor Q l = (1 /Q i + 1 /Q c ) − ) and resonance frequencies asshown in Fig. S1 (a-d). Besides an initial shift of thequality factors of less than 1 % going from −
140 dBm to −
130 dBm of input power the quality factors show nopower dependence, varying only in a range below 0 .
15 %.The fitted (dressed) frequencies are shifted compared tothe bare cavity frequency due to the residual magneticfield. From Eq. (S1) we expect a zero-current dressedfrequency of ω I =0r = 5 .
239 452 ± .
000 002 GHz. The cir-cle fit gives a resonance frequency at the lowest power of ω CFr = 5 .
239 474 ± .
000 002 GHz. We calculate the av-erage total excitation for all probe powers using Eq. (S2)and fit a line to the logarithmic data (Fig. S1 (e)), (cid:104) N e (cid:105) = 62 . · P . fW − . (S3)The fit agrees well with the data and is used throughoutall data evaluation to map probe powers P to averageexcitation numbers. This results in average magnon ex-citation numbers at matching frequencies for our experi-ments between 0 .
31 and 9 . · .
140 120 100 80 60
Probe power (dBm) A v e r a g e t o t a l e x c i t a t i o n nu m b e r / ( k H z )
140 120 100 80 60
Probe power (dBm) I n t e r n a l Q f a c t o r L o a d e d Q f a c t o r C o u p li n g Q f a c t o r abc de FIG. S1. (a-c) Loaded, coupling, and internal quality factors of the cavity resonance against probe power. The data was takenat T = 55 mK with zero current applied to the magnetic coil and does not show a power dependent behavior. (d) Shift of thefitted cavity frequencies ∆ ω/ π = [ ω r ( P ) − ω r ( P = −
140 dB m)] / π with compared to the measurement at lowest probe powerat zero current. Similar as with the quality factors, the cavity frequency does not show a power dependence. (e) Calculatedaverage photon number in cavity against probe power. The fit shows a linear dependence of the photon number calculatedwith Eq. (S2) to the input power. Note that this plot features a log-log scale, making the fit linear again. The errors on theaverage photon number are estimated to be smaller than 0 .
35 % and are not visible on this plot.
Extracting the internal magnon linewidth
We extract the internal magnon linewidth κ m by fittingthe reflection amplitude |S ( ω ) | using the input-outputformalism [25]. |S ( ω p , I ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − κ c i ( ω r − ω p ) + κ l + g i( ω m ( I ) − ω p )+ κ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (S4)with the probe frequency ω p , the magnon frequency ω m ,and the loaded, coupling and magnon linewidths κ l , κ c ,and κ m (HWHM). Before fitting, we normalize the data by the current independent baseline similar to Ref. [9].We estimate the background value for each probe fre-quency by calculating a weighted average over all entriesalong the current axis, neglecting the areas around thedressed cavity resonances. The amplitude data is dividedby this baseline to account for losses in the measurementsetup. The normalized data together with the fit resultsof Eq. (S1) and the circle fitted cavity resonance at zerocurrent are then fitted to Eq. (S4) using the Python pack-age lmfit [53]. VNA K . K m K -20 dB-20 dB-20 dB 40 dB32 dB ba YIGcoupler3 mm
FIG. S2. (a) Photograph of the sample in the cavity. Thetop half of the cavity resonator was removed and can be seenin the background. (b) Schematic diagram of the experimen-tal setup. The cavity holding the YIG sphere and the magnetproviding the static field are mounted at the mixing cham-ber plate of a dry dilution refrigerator. The microwave inputsignal is attenuated to minimize thermal noise at the sample.The attenuation of the complete input line to the input portof the cavity is −
75 dB at the cavity resonance frequency. Theoutput signal is amplified by a cryogenic amplifier operatingat 3 K and an amplifier at room temperature. Two magneti-cally shielded microwave circulators protect the sample fromamplifier noise.
Possible TLS origin
The microscopic origins of TLSs is still unclear andpart of ongoing research. Possible models include mag-netic TLSs proposed with analog behavior to the electric dipolar coupled TLSs [44, 45, 54, 55] and measured inspin glasses by thermal conductivity, susceptibility andmagnetization measuements at low temperatures [46, 56].With amorphous YIG showing spin glass behavior [47] itseems plausible to observe these effects in our crystallineYIG sample where in addition to the observed rare earthimpurities [9] we can assume structural crystal defects.This is based on materials with electric dipolar coupledTLSs, where TLSs appear largely in disorderd crystalsbut also in single crystals with smaller density [57].Another possibility could be surface spins leading tostrong damping that were observed as an important lossmechanism in cQED experiments [48, 49]. We evaluatedthe coupling strength to find a power or temperature de-pendence on the participating number of spins, see Fig.S3. We find an increase in the coupling strength of about1 % at the saturation conditions for the TLSs. With g ∝ √ N this translates to an increase in the number ofparticipating spins of the order of 2 %, e.g. due to the in-creased participation of now environmentally decoupledsurface spins. This should not be enough to explain thedecrease in κ m by a factor of 2.A loss mechanism by magnon-phonon coupling andsubsequent phonon losses due to TLS coupling can beneglected since for k = 0 magnons in YIG these magnonlosses are proposed to be much smaller than the Gilbertdamping [50]. Temperature (K) c o u p li n g s t r e n g t h / ( M H z ) ==
140 120 100 80 60 probe power (dBm) ====