Nonreciprocal Multi-mode and Indirect Couplings in Cavity Magnonics
Chi Zhang, Chenglong Jia, Yongzhang Shi, Changjun Jiang, Desheng Xue, C. K. Ong, Guozhi Chai
NNonreciprocal Multi-mode and Indirect Couplings in Cavity Magnonics
Chi Zhang, Chenglong Jia, Yongzhang Shi, Changjun Jiang, Desheng Xue , C. K. Ong , , and Guozhi Chai ∗ Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education,Lanzhou University, Lanzhou, 730000, People’s Republic of China, Department of Physics, Xiamen University Malaysia,Jalan Sunsuria, Bandar Sunsuria, 43900, Sepang, Selangor, Malaysia. (Dated: January 1, 2021)We investigate the magnon-photon couplings by employing a small magnet within an irregularresonant cavity, which leads to a desirable nonreciprocity with a big isolation ratio. Moreover, thehigher-order couplings between the spin wave modes with the polarized photon modes also exhibitthe nonreciprocity. These couplings between polarized photon and spin waves could be regardedas an indirect multi-modes coupling between the ferromagnetic resonance (FMR) mode and spinwave mode magnons mediated by the cavity mode photons. We also derive a coupling matrix topredict the characteristics of this kind of indirect coupling. The existence of the indirect couplingsbroaden the field range of the nonreciprocity of the system. The achieved nonreciprocal multi-mode magnon-photon couplings in a single system offer a feasible method to improve the signaltransmission quality. It may also benefit for the unidirectional processing of quantum informationand the developing of cavity magnonic devices.
I. INTRODUCTION
Cavity magnonics, which connect the magnons (thequanta of spin waves [1]) and photons (the quanta ofelectromagnetic waves [2]) through magnon-photon cou-pling, have been developed as one of the central goalsin magnon-based cavity quantum electrodynamics [3–6].The magnon-photon coupling is first proposed by Soykaland Flatt´e in 2010 [7, 8], and then realized in many ex-periments since 2013 [9, 10]. So far the magnon-photoncouplings were found to be pronounced not only withFMR mode but also with the spin wave resonance mode[11–16].Among the researches of magnon-photon coupling,some experimental results reveal the indirect couplingsin cavity magnonics [17–22]. Hyde et al. experimen-tally observed an indirect coupling between the two mi-crowave cavity mode photons through a yttrium iron gar-net (YIG) sphere and the indirect coupling modes havea higher transmission rate than the two uncoupled cav-ity modes in 2016 [17]. Then, a kind of indirect cou-pling between two magnons mediated by cavity photonswas reported detailedly [18, 19]. After that, The indi-rect interaction between a magnon mode and a cavityphoton mode mediated by traveling photons was system-atically studied [21, 22]. These ideas pave a way to inves-tigate two or more unrelated modes in coupling systems.Meanwhile, the nonreciprocal magnon-photon couplings,as well as the nonreciprocal propagation of photons andmagnons [23–30], might also be benefit for applying theMagnon-photon coupling in future quantum coherent in-formation processing. Very recently, the nonreciprocalmicrowave transmission achieved in an open cavity spin-tronic system with a considerably large isolation ratio ∗ Correspondence email address: [email protected] and flexible controllability [31]. On this basis, the broad-band nonreciprocity can also be realized by locally con-trolling the magnonic radiation [32, 33].In this work, we designed a system which provided twotypes of magnon-photon interactions, one is the couplingbetween FMR mode and cavity-mode photon, the otheris multi-mode high-order coupling which is identified asthe indirect coupling between the FMR mode and for-ward volume magnetostatic spin wave (FVMSW) modemediated by the cavity mode. Beyond the multi-modecoupling, we achieved a desirable nonreciprocity in notonly the coupling between FMR mode and cavity-modephoton but also other high order indirect couplings. Thismeans that we established a relationship between FMRmode and spin-wave-mode magnons.
II. EXPERIMENTS
As shown in Fig. 1, the experimental system consistsof a resonant cavity, which is made up of the copper,and a piece of single crystal yttrium iron garnet (YIG)wafer, which has a low microwave magnetic-loss param-eter and high-spin-density [34]. The shape of the cav-ity is designed as a quadrant stadium so that we canbreak the time-reversal symmetry (TRS) by introducinga small magnetic material into the cavity [35, 36]. Thequadrant stadium is combined with a quadrant with ra-dius as 40 mm and a square with side length of 40 mm.The height of the cavity is 5mm. We define the inter-section of axes x , y and z as the origin of coordinateswith unit as millimeter. Therefore, we can provide thecoordinate of the antenna 1, antenna 2 and the centerof YIG wafer with coordinates as (10mm, 20mm, 5mm),(70mm, 20m, 5mm) and (15mm, 20mm, 2.5mm) respec-tively. In addition, with a wafer shape, more spin wavemodes can be excited by applying magnetic fields [14].The single crystal YIG wafer has a diameter of 5.64 mm, a r X i v : . [ phy s i c s . a pp - ph ] D ec a thickness of 0.48mm, a saturation magnetization M s = 1753 G, and a gyromagnetic ratio γ = 2.62 MHz/Oe.The Gilbert damping of single crystaline YIG is as lowas α = 3 × − [37]. An electromagnet, whose magneticfield can be varied by changing the magnitude of theapplied current, is employed to apply a static magneticfield along z direction. Meanwhile, the input and outputmicrowaves were fed and measured by a vector networkanalyzer (VNA, Agilent E8363B), which was connectedwith two microwave cables and two antennas (boughtfrom Allwin Technology Inc.). In our experiments, wechose the seventh eigenmode of 10.05 GHz as the cav-ity mode and the microwave magnetic field distributionis shown in Fig. 1(b). The black circle stands for theYIG wafer. The arrows describe the directions of themicrowave magnetic vectors. The strength of microwavemagnetic field at the position of YIG wafer is strong thatcan induce the strong coupling more easily [38–40]. Thedamping of this mode is described as β = 2 . × − . Figure 1. (a) Sketch of our experimental setup. A vectornetwork analyzer is employed to provide the microwaves signalinto the cavity load with a YIG wafer (YIG-cavity) throughtwo microwave cables with two antennas. An electromagnetis employed to apply a static magnetic field along z directionwhich is perpendicular to the YIG wafer. (b) Simulation ofthe microwave magnetic Field vectors distribution at 10.05GHz in the YIG-cavity system. The black circle stands forthe YIG wafer. The arrows describe the directions of themicrowave magnetic field. III. RESULTS AND DISCUSSIONA. Nonreciprocal couplings
In this paper, we will make use of the microwave trans-mission spectra to characterize the experimental phe-
Figure 2. Microwave transmission coefficient S and S rawdata of the YIG-cavity as a function of frequency at the mag-netic field of 0 (a) and 5220 Oe (b). Black hollow circlesand red hollow squares expresses the microwave transmis-sion coefficient S and S , respectively. The orientationsof the peaks represent the transmission (upward peaks) andloss (downward peaks) of the microwave. nomena discussed subsequently. Figure 2(a) shows themicrowave transmission coefficients S raw data (blackhollow circles) and S raw data (red hollow squares) ofthe cavity loaded with a YIG wafer (YIG-cavity) as afunction of frequency without magnetic field. The fre-quency range in the figure is set from 9 GHz to 11 GHzand the two experimental curves are almost overlapped.Then, we applied static magnetic fields in order to ob-serve the variation between transmission coefficients S and S . We picked up the most typical results with theapplied magnetic field H = 5220 Oe, which is shown inFig. 2(b). Black hollow circles and red hollow squaresexpresses the microwave transmission coefficient S and S , respectively. Several peaks could be found fromtwo curves. Especially, The orientation of the peaks ofthe microwave transmission coefficient S and S near9.84 GHz and 10.10 GHz (marked by blue rectangle) aretotally opposite. It suggests that this system leads toa nonreciprocal microwave transmission at such a mag-netic field. The orientations of the peaks represent thetransmission (upward peaks) and loss (downward peaks)of the microwave. Microwaves could not transmit as themicrowave transmission coefficients are close to -80 dB.In order to understand this phenomenon clearly, weswept the applied static magnetic fields with a smallerfrequency range between 9.70 GHz to 10.30 GHz. Thedensity mapping image of the magnitude of transmissioncoefficients are shown in Fig. 3. Figures 3(a) and 3(b)show the density mapping images of the magnitude of S and S transmission coefficients as the static field direc- Figure 3. The density mapping image of the amplitude of the transmission coefficients through the cavity as a function offrequency and the applied static magnetic field. The deeper colour of image express the larger microwave transmission loss. (a) S mapping at the magnetic field added along direction + z ( H + ). (b) S mapping at the magnetic field added along direction+ z ( H + ). (c) S mapping at the magnetic field added along direction - z ( H − ). (d) S mapping at the magnetic field addedalong direction - z ( H − ). Blue dashed line indicates the FMR mode fitting with the Kittel equation [Eq. (1)] and black, yellow,red and green short dashed lines indicate the four FVMSW modes fitting with the FVMSW dispersion relation [Eq. (2)]. tion is added along + z , respectively. Correspondingly,Figs. 3(c) and 3(d) show the density mapping images ofthe magnitude of S and S transmission coefficients asthe static field direction is added along - z . The mappingspectra are measured by varying the static magnetic field H from 4870 to 5370 Oe with a step size of 6.8 Oe. TheFMR mode f K (blue dashed line) is calculated by theKittel equation [41] f K = γ (cid:113) ( H + ( N x − N z ) M s )( H + ( N y − N z ) M s ) . (1)For the YIG wafer used in this work, the saturation mag-netization M s is 1750 G and the gyromagnetic ratio γ is2.62 MHz/Oe. As the demagnetizing field is not uniformin cylindrical shape magnetized bodies, the demagnetiz-ing factors can not be calculated analytically. We usethe experimental results of the demagnetizing factors ofthe oblate ellipsoid with same dimensional ratio, which ispresented in the textbook. N x and N y are the demagne-tizing factors as 0.07, N z is the demagnetizing factor as0.86 [42]. There are other magnon modes called spin wavemodes besides the FMR mode in the system. We canconfirm that these spin wave modes are FVMSW modesbecause the direction of the applied magnetic field is per-pendicular to the YIG wafer [14, 34, 43, 44]. Therefore,we plot the FVMSW mode frequency f F versus appliedfield dependence by the calculation formula [34]. f F = γ (cid:112) f H ( f H + f M (1 − (1 − e − µ mn ) / µ mn )) , (2) where f H = γ f eff , with f eff as the effective magnetic field.In our experiment, the effective magnetic field is the sumof the applied magnetic field and the demagnetizing field. f M = γM s , and µ mn stands for the m-th eigenvalue ofthe Bessel function of the first kind of order n. The valueof µ mn could be found from the textbook: µ = 2.405, µ = 3.832, µ = 5.136, µ = 6.38 [45]. Then we putthese values into Eq. (2) and obtained four FVMSWmodes as shown in Fig. 3, marked as black, yellow, redand green dashed lines, respectively. Obviously, threeanti-crossing phenomena can be found in every mappingimage, indicating the normal mode splitting of the hy-brid magnon mode and the photon mode. It is worthpaying attention that only modes with odd mode num-ber m can induce the couplings. It’s because that themodes with even mode number m would cancel couplingstrength with the uniform cavity field [12]. In addition,we find that the couplings shown in the density mappingimages of S and S transmission coefficients are oppo-site to each other with the same applied magnetic field,and the TRS is broken completely. The deeper color ofimage indicates the larger microwave transmission loss.As the direction of the applied magnetic field changedfrom + z into - z , the S and S transmission coefficientsare exchanged also. It implies a nonreciprocity which ischiral symmetric in this system.We calculated the isolation ratio (ISO) which is de-fined as S − S so as to express the nonreciprocity Figure 4. ISO of the amplitude of the transmission coefficients S at the magnetic field along direction + z (a) and - z (b). clearly. Figures. 4(a) and 4(b) represent the ISO of thetransmission coefficients at the magnetic field along di-rection + z and - z , respectively. The absolute values ofISO come up to 40 dB so that we could get three strongnonreciprocal magnon-phonon couplings. The ISO of thecoupling which induced by the FMR mode with a cavitymode is stronger than other two couplings. It is becausethe photon modes of these two high-order couplings aredominated by a single linear polarization [33]. It con-firms that such a microwave nonreciprocal system with alarge isolation ratio could be obtained from an special res-onator cavity within a small YIG wafer. The strong reci-procity would occurs in different mode in a much broadrange while changing the strength of the magnetic field.The unidirectional invisibility for microwave propagationcould be found from the chiral symmetry. B. Indirect couplings
In order to comprehend the experiment phenomenamore clearly, we fitted the results by the proper equa-tions as shown in fig. 5. Figures 5(a) show the couplingsin the frequency versus applied static magnetic field withthe fitting curves. Blue line stands for the FMR mode f K calculated by Eq. (1). Gray and green lines fitting withthe FVMSW dispersion relation Eq. (2) indicate twoFVMSW modes which induce the couplings. The cav-ity mode frequency f p is located at 10.05 GHz (magentaline). The dispersion of the frequencies and the couplingstrength g / π induced by FMR mode and cavity photonmode can be described as [9] f ± = 12 ( f p + f K ) ± (cid:113) ( f p − f K ) + ( g / π ) , (3) Figure 5. The couplings in the frequency versus applied staticmagnetic field with the fitting curves. The experiment dataare described as black triangles. Blue line indicates the FMRmode fitting with the Kittel equation [Eq. (1)]; gray and greenlines fitting with Eq. (2) describe two FVMSW modes whichinduce the couplings; magenta line indicates the cavity modefrequency located at 10.05 GHz; orange curves are fitting withthe coupling equation [Eq. (3)]. (a) Purple curves are fittingwith the traditional two-mode model described as [Eq. (4)]and so are the red curves. (b) Olive and cyan curves indicatethe polarized cavity modes fitting with f +0 in [Eq. (3)] and f (cid:48) +1 in [Eq. (6)], respectively. By fitting the experimental results as shown in Fig. 5with orange dash lines, g / π = 0.23 GHz. Then, wetried to fit the high-order couplings by the traditionaltwo-mode model described as [9, 14] f ± = 12 ( f p + f F ) ± (cid:113) ( f p − f F ) + ( g / π ) , (4)where f ± are frequencies of the coupled resonances, g / π is the coupling strength. The fitting curves areshown in fig. 5(a). It is clearly that the fitting curvesof the high-order couplings don’t agree well with the ex-perimental results. It indicates that high order magnon-photon couplings may not be directly interacted by theFVMSW modes with the fixed cavity modes base on thetraditional two-mode model. A new coupling model needto be put forward. As shown in experimental results, dif-ferent magnon modes can not coupled directly. In thisway the two magnon modes might be indirectly coupledthrough the cavity mode in this system. We propose outan matrix equation as Eq. (5) to predict the charac-teristics of indirect coupling between magnon modes viacavity photon mode in this system. Here the diagonalterms are the uncoupled resonance conditions of the twomagnon modes and the cavity mode. The off-diagonalterms are the coupling strengths. Two zeros indicatethat there is no direct coupling between the two magnonmodes. Here the cavity mode is modelled as a mechanicaloscillator with amplitude h f , and the FMR and FVMSWmode in YIG are described by the mechanical oscillatorwith amplitude m K and m F , respectively. Γ denotes the impedance matching parameter for the cavity mode. h is the microwave field used to drive resonance in the cav-ity and is eliminated by normalization in the microwavetransmission. f − f + i αf K f g f g f f − f + 2 iβf P f g f g f f − f + 2 iαf F f m k h f m F = f h (5)Comparing with the FMR mode, there is an obviousattenuation with the coupling strength induced by theFVMSW magnon mode. Hence the coupling strengthinduced by the FVMSW magnon mode doesn’t have agreat influence on the cavity photon mode. To furtherunderstand the nature of this indirect interaction be-tween the two magnon modes, a coupled harmonic os-cillator system was used to model the f -H dispersion,damping evolution, and amplitudes during indirect cou-pling. The coupling strength g / π and the frequenciesof the coupled resonances f (cid:48) ± f (cid:48) ± = 12 ( f +0 + f F ) ± (cid:112) ( f +0 − f F ) + ( g / π ) , (6)where f (cid:48) ± are described by the purple curves. g / π is the coupling strength of this coupling with a value of0.11 GHz. The fitting curves are shown in fig. 5(b).The olive curve is used to describe the polarized cavityphoton mode. Similarly, we are able to fit the couplinginduced by the other FVMSW mode (green lines) withanother polarized mode, where f (cid:48) ± are described by thered dashed curves. The coupling strength g / π is 0.08GHz. The cyan curve is used to describe the second-polarization cavity photon mode. Essentially, this kind ofcoupling is induced by the FMR mode with the FVMSWmodes indirectly and the cavity mode is the intermediary.The indirect coupling induced by two magnon modes arenot analogous to some other works based on the acousti-cal mode and optical mode [13, 19, 21].The coupling strengths changing with increasing modenumber are shown in Fig. 6. It shows all of the cou-pling strengths discussed above. Dashed line and blackcircle describe the coupling strength of the coupling in-duced by the FMR mode with the cavity mode. Redand blue circles describe the coupling strengths of thehigh-order couplings with odd and even mode numberm, respectively. Besides the results that already been dis-cussed, the relationship between these coupling strengthsalso arouse our interest and we analyzed it combining thequantum electrodynamic system. The coupling strength g obeys the √ N scaling law, where N is the atom num-ber. Accordingly, the coupling strength in the magnon- Figure 6. Coupling strengths. Dashed and black circle indi-cates the strength of the coupling induced by the FMR modewith the cavity mode. Red circles indicate the strengths ofthe high-order couplings with odd mode number m. Blue cir-cles indicate the strengths of the high-order couplings witheven mode number m. photon coupling systems could be regarded as g = g √ N ,where N is the net spin number of the magnet and g is the coupling strength of a single Bohr magneton tothe cavity [10, 40, 46, 47]. The net spin density of thepure YIG is 2 . × µ B / cm , the total Bohr mag-neton number of our YIG wafer can be calculated as2 . × . As we calculated the single spin couplingstrength of FMR-cavity mode, g / π is calculated to be15.5 mHz. Comparing with the theoretical value calcu-lated by g / π = γ (cid:112) µ (cid:126) ω/V = 38 mHz [46], where µ is the permeability of vacuum, the coupling efficiency ofour system is 40.8%. In the same way, we calculated thesingle spin coupling strength of other higher-order modesas g / π = 7.4 mHz and g / π = 5.4 mHz. The to-tal single spin coupling strength of these three couplingsis about 28.4 mHz, with the total coupling efficiency as74.7%. IV. SUMMARY
In summary, we obtained two types of magnon-photoninteractions by designing a cavity which can be used tobroken the TRS easily with a YIG wafer. Not only theusual coupling between FMR mode and cavity mode butalso the indirect couplings between the FMR mode andFVMSW modes mediated by the cavity mode show anadmirable nonreciprocity with a large isolation ratio. Al-though the single spin coupling strength of the couplingbetween FMR mode with the cavity mode is about 60%smaller than the theoretical value, the total single spincoupling strength of these three coupling is graduallyapproaching to the theoretical value. Compared withthe former researches about the nonreciprocal magnon-photon couplings, appearance of such indirect couplings widened the range of the nonreciprocity in a single sys-tem, which may be beneficial for quantum unidirectionalof processing information and the unidirectional deviceswhich can be used in a wide magnetic field range. It alsopaves a way to build a relationship between two unre-lated magnon modes which can exert a positive influenceon the development of cavity magnonics devices.
ACKNOWLEDGEMENTS
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