Nonlinear magnetization dynamics driven by strong terahertz fields
Matthias Hudl, Massimiliano d'Aquino, Matteo Pancaldi, See-Hun Yang, Mahesh G. Samant, Stuart S. P. Parkin, Hermann A. Dürr, Claudio Serpico, Matthias C. Hoffmann, Stefano Bonetti
NNonlinear magnetization dynamics driven by strong terahertz fields
Matthias Hudl, Massimiliano d’Aquino, Matteo Pancaldi, See-Hun Yang, Mahesh G. Samant, StuartS. P. Parkin,
3, 4
Hermann A. D¨urr, Claudio Serpico, Matthias C. Hoffmann, and Stefano Bonetti
1, 8, ∗ Department of Physics, Stockholm University, 106 91 Stockholm, Sweden Department of Engineering, University of Naples “Parthenope”, 80143 Naples, Italy IBM Almaden Research Center, San Jose CA 95120, USA Max-Planck Institut f¨ur Mikrostrukturphysik, 06120 Halle, Germany Department of Physics and Astronomy, Uppsala University, 751 20 Uppsala, Sweden DIETI, University of Naples Federico II, 80125 Naples, Italy SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA Department of Molecular Sciences and Nanosystems,Ca’ Foscari University of Venice, 30172 Venezia Mestre, Italy (Dated: September 26, 2019)We present a comprehensive experimental and numerical study of magnetization dynamics in athin metallic film triggered by single-cycle terahertz pulses of ∼
20 MV/m electric field amplitudeand ∼ Since Faraday’s original experiment [1] and until twodecades ago, the interaction between magnetism andlight has been mostly considered in a unidirectional way,in which changes to the magnetic properties of a mate-rial cause a modification in some macroscopic observableof the electromagnetic radiation, such as polarizationstate or intensity. However, the pioneering experiment ofBeaurepaire et al. [2], where femtosecond optical pulseswere shown to quench the magnetization of a thin-filmferromagnet on the sub-picoseconds time scales, demon-strated that intense laser fields can conversely be usedto control magnetic properties, and the field of ultrafastmagnetism was born. Large research efforts are nowadaysdevoted to the attempt of achieving full and deterministiccontrol of magnetism using ultrafast laser pulses [3–10], afundamentally difficult problem that could greatly affectthe speed and efficiency of data storage [11].Recently, it has been shown that not only femtosecondlaser pulses, but also intense single-cycle terahertz (THz)pulses [12] can be used to manipulate the magnetic orderat ultrafast time scales in different classes of materials[13–18]. The main peculiarity of this type of radiation,compared with more conventional femtosecond infraredpulses, is that the interaction with the spins occurs notonly through the overall energy deposited by the radia-tion in the electronic system, but also through the Zee-man torque caused by the magnetic field component ofthe intense THz pulse. This is a more direct and efficientway of controlling the magnetization, and to achieve thefastest possible reversal [19, 20]. However, an accuratedescription of the magnetization dynamics triggered bystrong THz pulses is still missing. In this Letter, we present a combined experimental andnumerical study of the magnetization dynamics triggeredby linearly polarized single-cycle THz pulses with peakelectric (magnetic) fields up to 20 MV/m (67 mT). Weinvestigate not only the fast time scales that are com-parable to the THz pulse duration ( ∼ Experimental details.
Room temperature experi-mental data is obtained from a time-resolved pump-probe method utilizing the magneto-optical Kerr effect(MOKE), Refs. [23, 24]. A sketch of the experimentalsetup is presented in Fig. 1 (a). Strong THz radiation isgenerated via optical rectification of 4 mJ, 800 nm, 100fs pulses from a 1 kHz regenerative amplifier in a lithiumniobate (LiNbO ) crystal, utilizing the tilted-pulse-frontmethod [25]. In contrary to the indirect (thermal) cou-pling present in visible- and near-infrared light-matter in-teraction, THz radiation can directly couple to the spinsystem via magnetic dipole interaction (Zeeman inter-action) [26]. In this respect, a fundamental aspect isthe orientation of the THz polarization, which is con-trolled using a set of two wire grid polarizers, one vari-ably oriented at ± ◦ and a second one fixed to +90 ◦ (or -90 ◦ ) with respect to the original polarization direc-tion of E THz . As depicted in Fig. 1 (b), the magneticfield component of the THz pulse H THz is fixed along a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p (a)(e) (f) (b) +H THz -H Ext
OUTWPLNO OAPMEM+SP PMPumpProbe ±45° ±90° − − M OK E s i g n a l ( a . u . ) − − +H Ext -H Ext − − M OK E s i g n a l ( a . u . ) − − +H THz -H THz (a)(c)(e) (d)(f) (b) +H
THz -H Ext
OUTWPLNO OAPMEM+SP PMPumpProbe ±45° ±90°
FIG. 1. (Color online) (a) Schematic drawing of the exper-imental setup: BD - Balanced detection using two photodi-odes and a lock-in amplifier, WP - Wollaston prism, EM + S- Electromagnet with out-of-plane field and sample, OAPM- Off-axis parabolic mirror, P - Wire grid polarizer, LNO -Lithium niobate, and M - Mirror. (b) Sample geometry: Theelectric field component of the THz pulse at the sample po-sition is oriented parallel to the y -axis direction, E THz (cid:107) y ,and the magnetic field component parallel to the x -axis direc-tion, H THz (cid:107) x . A static magnetic field H Ext is applied alongthe z -axis direction. (c-f) Experimental MOKE data showingthe influence of reversing the external magnetic field ( ± H Ext , H THz = const.) on the THz-induced demagnetization (c) andon the FMR oscillations (d). The influence of reversing theTHz magnetic field pulse ( ± H THz , H Ext. = const.) on theTHz-induced demagnetization and on the FMR oscillations isshown in (e) and (f), respectively. (The shaded green area isa guide to the eye.) the x -axis direction and is therefore flipped by 180 ◦ byrotating the first polarizer. An amorphous CoFeB sample(Al O (1.8nm)/Co Fe B (5nm)/Al O (10nm)/Sisubstrate) is placed either in the gap of a ±
200 mT elec-tromagnet or on top of a 0.45 T permanent magnet. Inboth cases, the orientation of the externally applied field H Ext is along the z -axis direction, i.e. out of plane withrespect to the sample surface. However, a small compo-nent of this external bias field lying in the sample planeparallel to the y -axis direction has to be taken into ac- count, due to a systematic (but reproducible) small mis-alignment in positioning the sample. The THz pumpbeam, with a spot size (cid:31) ≈ (cid:31) ≈ µ m (FWHM),overlap spatially on the sample surface in the center ofthe electromagnet gap. Being close to normal incidence,the MOKE signal is proportional to the out-of-plane com-ponent M z of the magnetization, i.e. polar MOKE geom-etry [27]. The probe beam reflected from the sample sur-face is then analyzed using a Wollaston prism and twobalanced photo-diodes, following an all-optical detectionscheme [4]. Results and discussion.
The experimental data demon-strating THz-induced demagnetization and the magneticfield response of the spin dynamics is shown in Fig. 1(c-f) when µ H Ext = 185 mT. For short timescales onthe order of the THz pump pulse, τ ∼ τ ∼
100 fs af-ter time zero (t ∼ τ ∼ τ ∼
100 ps. During and after magnetizationrecovery, a relaxation precession (corresponding to theFMR) is superimposed, see Fig. 1 (d)+(f). The effect ofreversing the externally applied magnetic field H Ext onthe MOKE measurements is illustrated in Fig. 1 (c)+(d).This data shows that all the different processes just iden-tified (demagnetization, coherent magnetization responsein the range t < t (cid:46) t + 2 ps, and FMR response)are indeed magnetic, as they all reverse their sign uponreversal of the sign of the bias magnetic field. Fig. 1(e)+(f) instead depict the effect of reversing the THz fieldpolarity, while keeping the externally applied field con-stant. This data illustrate the symmetry of the magneticresponse with respect to the THz magnetic field. Thedemagnetization and FMR response, whose amplitudescales quadratically with the THz magnetic field H THz ,remain unchanged upon reversal of the terahertz mag-netic field, while the coherent magnetization response,linear in H THz , reverses its sign.The dependence of the THz-induced FMR response onthe magnitude of the magnetic bias field ( H Ext ) is shownin Fig. 2 (a). Oscillation amplitude and resonance fre-quency are summarized in Fig. 2 (b). The oscillationamplitude is obtained from fitting a damped sinusoidalfunction to the data shown in Fig. 2 (a), and the reso-nance frequency follows from subsequent Fourier trans-formation of that fitting function. The relationship be-tween magnetic resonance field H r and FMR frequency f FMR can be described by the phenomenological Kittel M OK E s i g n a l ( a . u . ) F M R a m p . ( a . u . ) F r e q u e n c y ( G H z ) (a) (b)(c) FIG. 2. (Color online) Experimental data showing the mag-netic field dependence of the FMR. (a) MOKE signal fordifferent out-of-plane fields µ H Ext - 20, 50, 75, 100, 125,150, 175 and 185 mT in ascending order. The solid line is adamped sine function fitted to the experimental data. Bothfit and data curve are displaced on the y axis by a constantoffset. (b) Fourier transformation of the data shown in (a)vs. FMR amplitude. (c) FMR frequency as a function of thein-plane component of H Ext . The solid line corresponds toKittel equation, Eq. (1). equation [28] f FMR = γµ π (cid:112) H r ( H r + M eff ) (1)with gyromagnetic ratio γ and effective magnetization M eff = M S − H ⊥ K , where H ⊥ K is the perpendicularanisotropy field. From room-temperature magnetizationmeasurements (see the Supplemental Material), a satu-ration magnetization µ M S = 1.84 T and an anisotropyfield µ H ⊥ K = 0.76 T are obtained. The value of H r rep-resents the small in-plane component of H Ext (i.e. alongthe y -axis direction), which can be inferred from the mea-surements in Fig. 2 (a) in order to get good agreementwith the experimental data, as shown in Fig. 2 (c). Numerical modelling.
A detailed picture of the THz-induced demagnetization and FMR response for an exter-nal magnetic field generated by a 0.45 T permanent mag-net and a THz pulse of E
THz = 18 MV/m ( µ H THz ∼ H Ext , correspondingto the right-most point in Fig. 2 (c).This complete set of good-quality data has been used (b) M z / M ( % ) (a) (b) FIG. 3. (Color online) Comparison of experimental data(open markers) and macrospin simulations (solid lines). (a)THz-induced demagnetization for an external applied field of µ H Ext ∼
450 mT and a THz pulse of E
THz = 18 MV/m,( µ H THz ∼
60 mT). The blue markers/lines correspond to+ H THz , and the orange markers/lines to − H THz . (b) Exper-imental data and macrospin simulations under similar condi-tions as (a) showing FMR on a longer timescale. for validating all the assumptions considered in ourmacrospin simulations, whose results are shown as solidlines in Fig. 3. Indeed, magnetization dynamics of a uni-form ferromagnetic material can be modeled and under-stood from macrospin simulations where the descriptionof the magnetic state is simplified by a single-domain ap-proximation. The phenomenological LLG equation [22]can be used to obtain a first approximate description ofthe magnetization continuum precession and relaxation.Since the LLG equation preserves the length of the mag-netization vector ( | M | = const.) [29], it does not accountfor laser-induced demagnetization and it ignores relevantphysical phenomena such as scattering effects [11]. Apossibility to account for optically- or thermally-induceddemagnetization is the use of the Landau-Lifshitz-Bloch(LLB) equation [30], which has been shown to describeultrafast demagnetization processes [31]. However, theparameters involved in the LLB equation depend ontemperature, and a description of their temporal evo-lution due to the interaction with ultrafast pulses isneeded. Such a description is often provided using a two-temperature model [8, 31], which then has to be coupledto the LLB equation. However, such a two-temperaturemodel relies on several phenomenological material pa-rameters, and little insight on the physics is gained bythis approach, in particular at longer time scales. Sincehere we are interested in these time scales, we simplifythe problem by just assuming a phenomenological LLGequation with non-constant magnetization magnitude, aphysical quantity which can be readily measured.Our version of the LLG equation reads (see the Sup-plemental Material) d M dt = − γ (cid:48) (cid:2) M × H e + α M S ( M × ( M × H e )) (cid:3) − c ( M , H e ) M , (2)with γ (cid:48) = γ/ (1 + α ), where the gyromagnetic ratio is | γ/ π | ≈ M = M m , where M( t )is the magnetization amplitude and m ( t ) the unit vec-tor. The effective magnetic field H e includes the exter-nal field H Ext , THz field H THz and demagnetization field H D . The Gilbert damping α = 0 .
01 was independentlymeasured with conventional FMR spectroscopy, as shownin the Supplemental Material. In Eq. (2), the first andsecond terms on the right-hand side describe coherentspin precession and the macroscopic spin relaxation, re-spectively, whereas the third term describes the fast de-magnetization along the m direction controlled by thefunction c ( M , H e ). A non-constant magnetization mag-nitude is also needed for reproducing the observed FMRoscillation, as discussed in the Supplemental Material.Hence, we propose a semi-empirical approach to iden-tify c ( M , H e ) and, consequently, the time evolution ofthe magnetization vector length M( t ) from experimentaldemagnetization data. For this purpose, the change inM( t ) as a function of time is calculated from the cumu-lative integral of the incident THz pulse [16]:∆M( t ) ∝ e − t/τ R (cid:90) t −∞ H THz ( ξ ) dξ, (3)where the exponential term describes the recovery of themagnetization on a timescale of ∼
100 ps for CoFeB. Theshape of the incident THz pulse (in the time domain)is proportional to E THz ( t ), which can be measured viaelectro-optical sampling in GaP [32]. We notice that theTHz magnetic field H THz ( t ) inside the material, to beused in Eq. (3), can be derived from the measured free-space E THz ( t ) after taking into account the whole samplestack, e.g. by means of the transfer matrix method [33].Eventually, the function c ( M , H e ) is obtained from Eq.(3) after re-scaling it with experimentally obtained de-magnetization values (see Ref. [16] for more details).After determining c ( M , H e ), all the terms in Eq. (2)are known, and the equation can be used to computethe dynamics of the magnetization orientation in termsof the spherical angles θ ( t ) , φ ( t ). Indeed, Eq. (2) is nu-merically solved in spherical coordinates (M , θ, φ ) (seethe Supplemental Material for details) using a custom-made Python solver based on the 4th-order Runge-Kuttamethod, which provides a simple implementation andgood accuracy of the numerical solution [29].It is now interesting to explore the expected response ofthe magnetization to THz fields with strength larger thanthe ones considered in this work, but nowadays accessi-ble with table-top sources. For simplicity and generality,only the free-space value of the THz peak electric field isreported in the below discussion. In case of THz fields ∆ M y , z ( % ) A m p lit ud e ( a . u . ) × − M z M y ∆ M y , z ( % ) A m p lit ud e ( a . u . ) × − M z M y (a)(c) (b)(d) FIG. 4. (Color online) Comparison of macrospin simulationdata for moderate (20 MV/m) and high (200 MV/m) THz-peak-field stimulus. The external magnetic field is set to µ H Ext = 450 mT, i.e. replicating the experimental condi-tions from Fig. 3. Panel (a) shows the relative change of theM y (blue line) and M z (light blue line) magnetization com-ponent for a E THz peak field of 20 MV/m and (b) shows thecorresponding Fourier spectrum. (c) and (d) show the simu-lation data and corresponding Fourier spectrum for an E
THz peak field of 200 MV/m, respectively. smaller than E
THz ∼
100 MV / m, the demagnetizationfollows the square of the amplitude of the THz field, seeRef. [16]. Lacking detailed experimental data, it is rea-sonable to assume that THz-induced demagnetization forhigher THz peak fields (E THz >
100 MV / m) can be de-scribed by the positive section of an error function, allow-ing for a quadratic behavior for small demagnetizationand a saturation for large demagnetization approaching100% [17]. From our experimental data, we derive a func-tional description of the demagnetization as a function ofthe THz field such as Demag = f(E) = erf( A · E ), withTHz peak field E and fitting parameter A ≈ . · − m V − . (See the Supplemental Material for further de-tails.)With this assumption, the macrospin simulation re-sults for THz fields E THz = 20 MV / m and E THz = 200MV / m are presented in Fig. 4 (a-b) and Fig. 4 (c-d),respectively. For E THz = 200 MV / m, a clear nonlin-ear response of the magnetization to the THz field isfound, illustrated by the second harmonic oscillation inthe M y component of the magnetization. The simulatedTHz-induced demagnetization for E THz = 200 MV / m ison the order of ∆M z ∼ y andM z components of the magnetization at THz pump peakfields of 20 MV/m and 200 MV/m are depicted. TheFourier data of the 200 MV/m simulation shown in Fig. 4(c) clearly shows a second harmonic peak at ∼
14 GHz,present for M y but not for M z . A similar behavior wasobserved recently by performing FMR spectroscopy ofthin films irradiated with femtosecond optical pulses in-ducing either ultrafast demagnetization [34], by excitingacoustic waves [35], and by two-dimensional THz mag-netic resonance spectroscopy of antiferromagnets [36]. Inour case, the high-harmonic generation process is solelydriven by the large amplitude of the terahertz magneticfield that is completely off-resonant with the uniform pre-cession mode. This would allow for exploring purely mag-netic dynamics in regimes that are not accessible withconventional FMR spectroscopic techniques, where high-amplitude dynamics are prevented by the occurrence ofso-called Suhl’s instabilities, i.e. non-uniform excitationsdegenerate in energy with the uniform mode. Such non-resonant, high THz magnetic fields are within the capa-bilities of recently developed table-top THz sources [37],and can also be generated in the near-field using meta-material structures as described by Refs. [38–40].In summary, we investigated magnetization dynam-ics induced by moderate THz electromagnetic fields inamorphous CoFeB, in particular the ferromagnetic reso-nance response as a function of applied bias and THzmagnetic fields. We demonstrate that semi-empiricalmacrospin simulations, i.e. solving the Landau-Lifshitz-Gilbert equation with a non-constant magnitude of themagnetization vector to incorporate THz-induced de-magnetization effect, are able to describe all the detailsof the experimental results to a good accuracy. Exist-ing models of terahertz spin dynamics and spin pumpingwould need to be extended to include the evidence pre-sented here [41, 42]. Starting from simulations describ-ing experimental data for THz-induced demagnetization,we extrapolate that THz fields one order of magnitudelarger drive the magnetization into a nonlinear regime.Indeed, macrospin simulations with THz fields on the or-der E THz ∼
200 MV / m ( µ H THz ∼
670 mT) predict a sig-nificant demagnetization of ∆M z ∼ ∗ [email protected][1] M. Faraday, Philosophical Transactions of the Royal So-ciety of London , 1 (1846).[2] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y.Bigot, Phys. Rev. Lett. , 4250 (1996).[3] B. Koopmans, M. Van Kampen, J. T. Kohlhepp, andW. J. M. de Jonge, Phys. Rev. Lett. , 844 (2000).[4] M. van Kampen, C. Jozsa, J. T. Kohlhepp, P. LeClair,L. Lagae, W. J. M. de Jonge, and B. Koopmans, Phys.Rev. Lett. , 227201 (2002).[5] B. Koopmans, J. J. M. Ruigrok, F. Dalla Longa, andW. J. M. de Jonge, Phys. Rev. Lett. , 267207 (2005).[6] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk,A. Tsukamoto, A. Itoh, and T. Rasing, Phys. Rev. Lett. , 047601 (2007).[7] G. Malinowski, F. Dalla Longa, J. H. H. Rietjens, P. V.Paluskar, R. Huijink, H. J. M. Swagten, and B. Koop-mans, Nature Physics , 855 (2008).[8] B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf,M. F¨ahnle, T. Roth, M. Cinchetti, and M. Aeschlimann,Nature Materials , 259 (2010).[9] T. Kampfrath, A. Sell, G. Klatt, A. Pashkin, S. M¨ahrlein,T. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, andR. Huber, Nature Photonics , 31 (2011).[10] T. Kubacka, J. A. Johnson, M. C. Hoffmann, C. Vicario,S. de Jong, P. Beaud, S. Gr¨ubel, S.-W. Huang, L. Huber,L. Patthey, Y.-D. Chuang, J. J. Turner, G. L. Dakovski,W.-S. Lee, M. P. Minitti, W. Schlotter, R. G. Moore,C. P. Hauri, S. M. Koohpayeh, V. Scagnoli, G. Ingold,S. L. Johnson, and U. Staub, Science , 1333 (2014).[11] J. Walowski and M. M¨unzenberg, Journal of AppliedPhysics , 140901 (2016).[12] M. C. Hoffmann, S. Schulz, S. Wesch, S. Wunderlich,A. Cavalleri, and B. Schmidt, Optics Letters , 4473(2011).[13] M. Nakajima, A. Namai, S. Ohkoshi, and T. Suemoto,Opt. Express , 18260 (2010).[14] K. Yamaguchi, M. Nakajima, and T. Suemoto, Phys.Rev. Lett. , 237201 (2010).[15] T. H. Kim, S. Y. Hamh, J. W. Han, C. Kang, C.-S. Kee,S. Jung, J. Park, Y. Tokunaga, Y. Tokura, and J. S. Lee,Applied Physics Express , 093007 (2014).[16] S. Bonetti, M. C. Hoffmann, M.-J. Sher, Z. Chen, S.-H.Yang, M. G. Samant, S. S. P. Parkin, and H. A. D¨urr,Phys. Rev. Lett. , 087205 (2016).[17] M. Shalaby, C. Vicario, and C. P. Hauri, Applied PhysicsLetters , 182903 (2016).[18] S. Schlauderer, C. Lange, S. Baierl, T. Ebnet, C. P.Schmid, D. C. Valovcin, A. K. Zvezdin, A. V. Kimel,R. V. Mikhaylovskiy, and R. Huber, Nature , 383(2019).[19] I. Tudosa, C. Stamm, A. B. Kashuba, F. King, H. C.Siegmann, J. St¨ohr, G. Ju, B. Lu, and D. Weller, Nature , 831 (2004).[20] S. J. Gamble, M. H. Burkhardt, A. Kashuba, R. Al-lenspach, S. S. P. Parkin, H. C. Siegmann, and J. St¨ohr,Phys. Rev. Lett. , 217201 (2009).[21] L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjet. , 153(1935).[22] T. L. Gilbert, IEEE Transactions on Magnetics , 3443(2004). [23] M. R. Freeman and J. F. Smyth, Journal of AppliedPhysics , 5898 (1996).[24] W. K. Hiebert, A. Stankiewicz, and M. R. Freeman,Phys. Rev. Lett. , 1134 (1997).[25] M. C. Hoffmann and J. A. F¨ul¨op, Journal of Physics D:Applied Physics , 083001 (2011).[26] H. Hirori and K. Tanaka, Journal of the Physical Societyof Japan , 082001 (2016).[27] Z. Q. Qiu and S. D. Bader, Review of Scientific Instru-ments , 1243 (2000).[28] C. Kittel, Phys. Rev. , 155 (1948).[29] P. Horley, V. Vieira, J. Gonz´alez-Hern´andez, V. Dugaev,and J. Barnas, Numerical Simulations of Nano-ScaleMagnetization Dynamics , edited by J. Awrejcewicz, Nu-merical Simulations of Physical and Engineering Pro-cesses: Chapter 6 (InTech, 2011).[30] D. A. Garanin, V. V. Ishchenko, and L. V. Panina, The-oretical and Mathematical Physics , 169 (1990).[31] U. Atxitia, D. Hinzke, and U. Nowak, Journal of PhysicsD: Applied Physics , 033003 (2017).[32] M. C. Hoffmann, S. Schulz, S. Wesch, S. Wunderlich,A. Cavalleri, and B. Schmidt, Opt. Lett. , 4473 (2011).[33] M. Born and E. Wolf, Principles of Optics: Electromag- netic Theory of Propagation, Interference and Diffractionof Light (Cambridge University Press, Cambridge, 1999).[34] A. Capua, C. Rettner, and S. S. P. Parkin, Phys. Rev.Lett. , 047204 (2016).[35] C. L. Chang, A. M. Lomonosov, J. Janusonis, V. S.Vlasov, V. V. Temnov, and R. I. Tobey, Phys. Rev. B , 060409(R) (2017).[36] J. Lu, X. Li, H. Y. Hwang, B. K. Ofori-Okai, T. Kurihara,T. Suemoto, and K. A. Nelson, Phys. Rev. Lett. ,207204 (2017).[37] M. Shalaby, A. Donges, K. Carva, R. Allenspach, P. M.Oppeneer, U. Nowak, and C. P. Hauri, Physical ReviewB , 014405 (2018).[38] Y. Mukai, H. Hirori, T. Yamamoto, H. Kageyama, andK. Tanaka, New Journal of Physics , 013045 (2016).[39] D. Polley, M. Pancaldi, M. Hudl, P. Vavassori,S. Urazhdin, and S. Bonetti, Journal of Physics D: Ap-plied Physics , 084001 (2018).[40] D. Polley, N. Z. Hagstr¨om, C. von Korff Schmising,S. Eisebitt, and S. Bonetti, Journal of Physics B: Atomic,Molecular and Optical Physics , 224001 (2018).[41] L. Bocklage, Scientific Reports , 22767 (2016).[42] L. Bocklage, Phys. Rev. Lett.118