Cavity magnomechanics

Xufeng Zhang,1 Chang-Ling Zou,1, 2 Liang Jiang,2 and Hong X. Tang1, 2, ∗

1Department of Electrical Engineering, Yale University, New Haven, Connecticut 06511, USA2Department of Applied Physics, Yale University, New Haven, Connecticut 06511, USA

(Dated: November 17, 2015)

A dielectric body couples with electromagneticfields through radiation pressure and electrostric-tive forces, which mediate phonon-photon cou-pling in cavity optomechanics [1]. In a mag-netic medium, according to Korteweg-Helmholtzformula [2], magnetostrictive forces should ariseand lead to phonon-magnon interaction. Herewe report such a coupled phonon-magnon systembased on ferrimagnetic spheres, which we termas cavity magnomechanics, by analogy to cav-ity optomechanics. Coherent phonon-magnon in-teractions, including electromagnetically inducedtransparency and absorption, are demonstrated.Excitingly, due to strong hybridization of magnonand microwave photon modes and their high tun-ability, our platform exhibits new features in-cluding parametric amplification of magnons andphonons, triply resonant photon-magnon-phononcoupling and phonon lasing. Our work demon-strates the fundamental principle of cavity mag-nomechanics and its application as a new in-formation transduction platform based on co-herent coupling between photons, phonons andmagnons.

Mechanical oscillators have been recently widely stud-ied as a transducer mediating the coherent signal conver-sion between different systems [1]. Particularly, radiationforce [3–7], electrostatic force [8–10] and piezoelectricforce [11, 12] have been utilized for coupling phonon withoptical or microwave photons. Such interaction mecha-nisms lead to the fast development of a variety of cavityelectro- and opto-mechanical systems [1], but they allintrinsically lack good tunability. The magnetostrictiveforce [2] provides an alternative mechanism to allow adifferent information carrier – magnon – to couple withphonon. Magnon is a collective excitation of magnetiza-tion, whose frequency can be tuned at will by adjustingbias magnetic field [13–15]. The magnetostrictive inter-action has been long overlooked for information process-ing as it is negligibly weak in commonly used dielectricor metallic materials. However, in magnetic materialssuch as yttrium iron garnet (YIG, Y3Fe5O12), the mag-netostrictive force becomes dominant, which provides agreat opportunity to establish an highly tunable hybridsystem for coherent information processing. Thanks to

∗ To whom correspondence should be addressed. E-mail:hong.tang@yale.edu

the excellent material property of YIG, the magnome-chanical system can be further integrated with opto- orelectro-mechanical elements, providing an excellent plat-form for quantum state transfer among different physicalsystems.

Here, we demonstrate an intriguing cavity magnome-chanical system in which magnon couples with phononthrough magnetostrictive interaction, resulting in hall-mark coherent phenomena such as magnomechanicallyinduced transparency/absorption (MMIT/MMIA) andmagnomechanical parametric amplification (MMPA).During such processes, magnons are in the hybridizedstate with cavity microwave photons as they are stronglycoupled to each other [16–20]. Therefore coherent sig-nal conversions among these three different informationcarriers are realized in a single device. The magneticfield dependence of magnon provides our system with un-precedented tunability compared with opto- or electro-mechanical systems. Moreover, the great flexibility ofthis system allows us to achieve triple-resonance amongmagnon, phonon and photon, which drastically enhancesthe magnomechanical interaction. The principles demon-strated in our room temperature experiments can bereadily applied to the quantum regime at millikelvin tem-perature, opening up great opportunities in various ap-plications, such as tunable microwave filter and amplifier[21], long-lifetime quantum memories [22], microwave-to-optics conversion [9].

Magnetostrictive interactionThe device used in our experiments is schematicallyshown in Fig. 1A. The key component is a highly polishedsingle crystal YIG sphere glued to the end of a silica fiberfor supporting purpose (Fig. 1B). With an external mag-netic field H biased along z direction, a uniform magnonmode resonates in the YIG sphere at frequency ωm = γH,where γ is the gyromagnetic ratio. The YIG sphere isalso an excellent mechanical resonator (Fig. 1C) thanksto its superior material and geometrical properties. Thevarying magnetization induced by the magnon excitationinside the YIG sphere causes deformation of its sphericalgeometry (and vise versa), introducing the coupling be-tween magnon and phonon modes (Fig. 1D). Consideringthe large frequency mismatch between the magnon andthe phonon modes (gigahertz v.s. megahertz) with ourexperiment parameters, a strong parametric drive is usedto compensate their frequency difference. In this case,the system is described by an radiation pressure-like, dis-

persive interaction Hamiltonian H = ~gmbm†m(b + b†),

where ~ is the reduced Planck’s constant, b (m) is the

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Spheroidal phonon modesThe magnetostrictive coupling strength is determined bythe mode overlap between the uniform magnon mode andthe phonon modes. In a YIG sphere, there exist variousphonon modes, each with a different displacement pro-file and consequently a different coupling strength withthe magnon mode. Figure 2A plots the typical profilesof the lowest order spheroidal phonon modes S1,l,ma

(land ma are the angular and azimuthal mode numbers,respectively), among which the S1,2,2 mode shows thehighest coupling strength when the bias field is along thedirection of maximum displacement (Fig. 2B). Thereforein our experiments we focus only on the S1,2,2 mode.Although a YIG sphere with a smaller diameter is favor-able for achieving larger coupling strengths (Fig. 2B), italso results in a higher frequency for the phonon mode(Fig. 2C), which in turn leads to lower responsivity tothe parametric drive, so a trade-off has to be made whenchoosing the sphere size. In our experiments, a 250-µm-diameter YIG sphere is used, corresponding to a phononfrequency ωb/2π = 11.42 MHz and a coupling strengthgmb/2π ≤ 9.9 mHz. With an external drive of 0 dBm,the linear magnon-phonon coupling can be enhanced toaround 30 kHz, which is two orders of magnitude largerthan the phonon dissipation rate κb.

Magnetostriction mediates the coupling betweenmagnons and photons. However, in order to achieve co-herent magnon-phonon coupling, it is further requiredthat phonon mode should have relatively long lifetime.Single crystal YIG has a garnet structure that is known toexhibit very low mechanical damping and therefore sup-ports a material-limited phonon lifetime over a millisec-ond [23]. The supporting fiber that is glued to the YIGsphere reduces the phonon lifetime (Fig. 2D). In our ex-periments, the measured linewidth of S1,2,2 phonon modewith a 125-µm-diameter supporting fiber is 2κb/2π = 300Hz, which is sufficiently small for observing coherentmagnon-phonon coupling phenomena.

Coherent magnomechanical interactionFigure 1E plots the schematics of our measurement setupat room temperature ambient condition. The YIG sphereis placed inside a three-dimensional microwave cavity(Fig. 1A). A weak probe signal is sent into the cavitythrough a coaxial probe, and by sweeping its frequency ωs

and measuring the reflection, we can infer the interactionamong photon, magnon and phonon inside the cavity.The YIG sphere is positioned at the maximum microwavemagnetic field of the cavity TE011 mode, which resonatesat ωa/2π = 7.86 GHz. By controlling the bias magneticfield, we tune the magnon close to resonance with thecavity photon mode. This leads to the hybridization be-tween magnon and photon [16–19], which shows up inthe reflection spectrum as a pair of split normal modes(Fig. 1F). Because each of the two hybrid modes con-

tains magnon components, it coherently couples with thephonon modes when the cavity is parametrically drivenby a strong microwave signal at ωd.

We first study the coherent magnomechanical cou-pling for each individual hybrid mode by applying anoff-resonance microwave drive. In this case, the cavitymagnomechanical system can be described by

Hmb = ~gmb(b+ b†)(cos2 θA†+A+ + sin2 θA†−A−), (1)

where the two hybrid modes interact with the phononmode independently. Here, A+ = cos θa + sin θm and

A− = − sin θa + cos θm are quantized boson operatorsfor hybridized excitations constituted by magnon andmicrowave photon (a), with θ = 1

2 arctan 2gma

∆mavaries

with photon-magnon coupling strength gma and photon-magnon detuning ∆ma = ωm − ωa. In our system,both the magnon and the cavity photon modes have arelatively narrow linewidth (2κm/2π = 1.12 MHz and2κa/2π = 3.35 MHz). As a result, the hybrid modelinewidth is well below the phonon frequency, leadingour system deep inside the resolved sideband regime, byanalogy with optomechanical systems [1]. In this case,the nonlinear interaction can be converted either intothe linear beam splitter model ~(G±A

†±b + G∗±A±b

†) or

the parametric oscillator model ~(G±A†±b† + G∗±A±b)

with the presence of an external drive, where G± =A±,ssgmb(1∓cos2θ)/2 is the enhanced coupling strength.Here, A±,ss is the steady state amplitude of the hybridmode, corresponding to the effective pumping of the mi-crowave drive on magnon due to the magnon-photon hy-bridization.

Figures 3A and B plot the measured reflection spec-tra for a series of bias magnetic fields with a microwavedrive at a fixed frequency ωd. To avoid the influence ofthe other hybrid mode, the driving signal is red (blue)detuned for the lower (upper) hybrid mode, as illus-trated by the top insets. For the red-detuned drive,the power is held constant at 26 dBm. In the spec-tra, the broad Lorentzian-shaped resonance dip corre-sponds to the hybrid mode A−, while the very sharpmodification of the spectra at the two-photon detuning∆sd = ωb is evidence of coherent magnomechanical in-teraction. The zoomed-in spectra in Fig. 3A show thatthese phonon-induced resonances have a Fano-like shapethat varies with bias magnetic field. When the drive-resonance detuning ∆d− = ωd−ω− = −ωb, the Fano-likeresonance changes into a symmetric Lorentzian-shapedtransparency peak (MMIT). In contrast, the Fano-likeresonances in the spectra for the blue-detuned drive(with a constant power of 22 dBm) show an oppositesymmetry (Fig. 3B). When the drive is blue detuned to∆d+ = ωd − ω+ = ωb, such a resonance becomes aLorentzian-shaped absorption dip (MMIA).

One distinct advantage of magnon is that its frequencyis determined by the external bias magnetic field andtherefore can be conveniently tuned in a broad range.By varying ∆ma, the percentage of magnon componentin the hybrid mode changes. Therefore, the hybrid mode

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experiences different effective dissipation rate, externalcoupling rate, as well as effective coupling strength withthe phonon mode. As a result, the coherent magnome-chanical interaction is magnetically controllable, whichcan be quantified by the dependence of the cooperativityC = G2

±/κ±κb on the bias magnetic field. The measuredC-H relation is plotted in Fig. 3D. For each measurementunder a specific bias condition, the drive frequency is de-tuned from the hybrid mode by ∆d± = ±ωb, as indicatedby the crosses in Fig. 3C, while the driving power is fixedconstant at 30 dBm. We can see there exists an optimalcondition for a maximum C, as a the result of the compe-tition between the magnon and photon components in thehybrid mode: more magnon component yields strongermagnetostrictive coupling, while more photon componentleads to a higher driving efficiency. From these measure-ment results we can extract the magnon-phonon couplingstrength gmb/2π = 4.1 mHz, in accordance with our the-oretical prediction (Fig. 2B).

Triply resonant cavity magnomechanicsThe great flexibility of our system leads to tremendousadvantages. For instance, it allows us to work in theinteresting triple-resonance condition, where both max-imum hybridized modes simultaneously couple with thephonon mode, as described by

Hmb =1

2~gmb(b+ b†)(A†+A− + A†−A+). (2)

By adjusting the direction of bias field or the positionof the YIG sphere inside the cavity, we can tune the hy-brid mode splitting to match the phonon frequency ωb.In this case, both the drive and probe photons can beapplied on-resonance with the hybrid modes (top insetof Figs. 4A and B), resulting in a drastically enhancedmagnomechanical coupling. For the red-detuned drive,the transparency windows at various driving powers areplotted in Fig. 4A. In addition to the red shift of the cen-ter frequency, the linewidth of the transparency windowsexhibits a clear linear dependence on the driving power(Fig. 4C, red squares). With a driving power of only8.0 dBm, the linewidth increases from its intrinsic value0.62 kHz to 2.12 kHz, corresponding to a cooperativityC = 2.4. As a comparison, a driving power of 34 dBmis used to achieve the same cooperativity when the driveis applied off-resonance, indicating the drastic enhance-ment of the magnomechanical interaction induced by thetriple-resonance condition. The reflection signal for theblue detuning situation is plotted in Fig. 4B at variousdriving powers. As the driving power increases, the cen-ter frequency of the small phonon-induced resonance in-side the hybrid mode is blue shifted, and its linewidthlinearly decreases (Fig. 4C, blue circles).

A direct comparison of Figs. 4A and B reveals dis-tinctly different spectral lineshapes of the phonon-induced resonances. The same as in the case of off-resonance drive, we observed MMIT for the red-detuneddrive in the triply resonant system, with the peak height

and linewidth of the transparency window increasingwith the driving power. While for the blue-detuned drive,we observed the transition from MMIA to MMIT, andthen to MMPA and eventually self-sustained oscillationas we increase the driving power. These observationslead to a unified explanation about the modified spec-tral lineshape (which is not limited to the triple reso-nance situation): the coupling with phonon introducesadditional dissipation and phase shift to the hybridizedmodes and therefore changes their lineshapes. With thepresence of a parametric drive, the effective dissipationrate of the hybrid mode is modified from κ to κ(1± C),which increases for the red-detuned drive while decreasesfor blue-detuned drive. Given a fixed external couplingrate κe, the on-resonance reflectivity of the cavity is

r =1± C − 2κe

κ

1± C. (3)

Therefore, depending on the external coupling conditionand the driving power, the reflection spectra lineshapevaries among MMIT, MMIA or MMPA.

The measured on-resonance reflectivity for an under-coupled hybrid mode agrees well with our theoreti-cal model (Fig. 4D). For the red-detuned drive, the in-creasing linewidth with elevated driving power causesthe mode further under-coupled and therefore the on-resonance reflectivity increases. On the contrary, forthe blue-detuned situation, the decreasing linewidth firstleads to critical coupling and then over coupling condi-tion, yielding a dip in the reflectivity followed by a rapidincrease which diverges as C ≈ 1 at a driving power of6.2 dBm. The deviation of the measured reflectivity fromthe theoretical prediction can be attributed to thermalinstability or gain-bandwidth-product limitation, whichalso limit the highest measurable parametric gain to 3dB. When the hybrid mode is tuned to over coupled, theincrease of the parametric gain with the driving poweris more gradual, and therefore a much higher paramet-ric gain up to 25 dB is achieved before reaching insta-bility (Fig. 4E). The observed MMPA is similar to theelectromechanical parametric amplifiers [24] but with un-precedented tunability. Further increasing the drivingpower leads the system into the instable regime where thephonon mode experiences self-sustained oscillation. Thethreshold behavior of the measured emission power fromthe Stokes sideband, as shown by the inset of Fig. 4D,indicates the onset of the phonon lasing [25].

ConclusionThe demonstration of the coherent magnon-phonon inter-action, including the MMIT (MMIA) and MMPA, pro-vides a versatile platform for the coherent informationprocessing. Besides, as YIG also possesses great opti-cal properties such as low optical loss and optomagneticnonreciprocity, our study shows great potential for inte-grating different systems, including microwave, optical,mechanical and magnonic systems, in a single device andrealizing information inter-conversion among these dif-

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ferent information carriers. Distinguished from opto- orelectro-mechanical systems, our cavity magnomechanicalsystem shows high level of tunability which allows theresonance be externally controlled in a wide frequencyrange. Moreover, such a complex system is compatible

with superconducting quantum circuits [26]. All of theseare not only crucial for realizing long desired functionssuch as microwave-to-optical conversion [9–11, 27, 28],but also provide a flexible platform that intrigues thefundamental study of exotic magnetic excitations.

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ACKNOWLEDGEMENTS

We thank N. Zhu for gluing the YIG sphere to thesilica fiber. This work was supported by DARPAMTO/MESO program (N66001-11-1-4114). C.Z., L.J.and H.X.T. acknowledge support from LPS through anARO grant (W911NF-14-1-0563) and Packard Founda-tion. L.J. also acknowledges support from the Alfred P.Sloan Foundation.

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Figure 1. Device schematic and measurement setup. (A) Schematic of the device that consists of a three-dimensionalcopper cavity (only bottom half is shown) and a YIG sphere. The YIG sphere is placed near the maximum microwave magneticfield (along y direction) of the cavity TE011 mode. A uniform external magnetic field (H) is applied along z direction to biasthe YIG sphere for magnon-photon coupling. (B) Optical image of the highly polished 250-µm-diameter YIG sphere thatis glued to a 125-µm-diameter supporting silica fiber. The gluing area is minimized to reduce the contact damping to thephonon mode. Scale bar: 100 µm. (C) Simulated mechanical displacement (u) of the S1,2,2 phonon mode in the YIG spherewhich has the strongest magnomechanical interaction with the uniform magnon mode. (D) An intuitive illustration of themagnomechanical coupling. Top panel shows the uniform magnon excitation in the YIG sphere. Bottom panel shows thatthe magnon-induced magnetization (vertical yellow arrows) causes the deformation (compression along y direction) of the YIGsphere (and vise versa). (E) Schematic illustration of the measurement setup. VNA: vector network analyzer; SRC: microwavesource for driving; AMP: microwave amplifier; BPF: bandpass filter; PWR: microwave power meter; CIR: circulator; BSF:bandstop filter; DUT: device-under-test. (F) Black curve: cavity reflection spectrum when magnon is on-resonance with the

cavity photon mode. The two dips represent the maximum hybridized modes A± = 1√2(a ± m). Red and blue vertical lines

indicate the applied drive and probe, respectively. The probe is swept across the hybrid mode resonance. ∆sd: two photon(probe-drive) detuning; ∆d−: drive-resonance detuning.

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Figure 2. Analysis of the phonon modes and magnetostrictive coupling strengths. (A) Simulated displacementprofiles of the low order phonon modes in the YIG sphere (with a small supporting fiber). S1,l,ma represents the spheroidalmode with a radial mode number of 1, an angular mode number of l, and an azimuthal mode number of ma. Only one ofthe 2l + 1 degenerate modes is plotted for each l. (B) Theoretical prediction of the magnomechanical coupling strength asa function of YIG sphere diameter for the S1,2,0 (black) and S1,2,2 modes (red and blue, corresponding to different bias fielddirections). Solid lines are numerical calculations while symbols are analytical fittings. (C) Phonon mode frequency as afunction of the YIG sphere diameter. Solid lines are the theoretical calculations, showing an inverse proportional dependence,while red circles are the measurement results. (D) Simulated phonon linewidth due to clamping loss as a function of thesupporting fiber diameter for the S1,2,ma modes. Black dot indicates the experiment parameter, showing an anchor-loss-limitedlinewidth of 20 Hz.

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lower hybrid mode A− as a function of the two-photon detuning ∆sd for a series of different bias magnetic fields. The broaddip corresponds to the lower hybrid mode resonance, whose lineshape changes with bias magnetic field because of the changein the ratio between magnon and photon components. A strong (26 dBm) microwave drive is red-detuned with a fixed drivingfrequency ωd. A Fano-like narrow resonance line shows up inside the hybrid mode, which turns into a Lorentzian transparencypeak when ∆d− = −ωb. Zoom-in shows detailed spectra of the magnomechanically induced resonances (shaded area in (A)).

(B) Measured reflection spectra near the upper hybrid mode A+ with a blue-detuned strong drive (22 dBm) for various biasmagnetic fields. When ∆d+ = ωb, the magnomechanically induced narrow resonance shows up as a Lorentzian absorptiondip. Zoom-in shows detailed spectra of the shaded area in (B). (C) Reflection spectra of the hybrid magnon-photon modesat various bias magnetic fields. The crosses indicate the drive frequency and bias magnetic field used for each data point in(D). (D) The magnomechanical cooperativity as a function of bias magnetic field. For each measurement, the microwave driveis detuned from the hybrid mode by ∆d± = ±ωb, while the probe is swept across the hybrid mode resonance. Red squares(blue circles) are for the red (blue) detuning situation. Solid lines in (D) and in the zoom-in of (A) and (B) are theoreticalcalculations using only a single fitting parameter gmb.

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Figure 4. Enhanced magnomechanical coupling in the triply resonant system. (A) Magnomechanically inducedtransparency (MMIT) signal for a red-detuned drive at various driving powers. (B) Magnomechanically induced absorption(MMIA) and magnomechanical parametric gain (MMPA) signal for a blue-detuned drive at various driving powers. (C) Thelinewidth of the magnomechanically induced resonance as a function of the drive power. (D) Magnomechanical-interaction-modified on-resonance reflectivity of the hybrid mode as a function of the drive power. Shaded area indicates the instableregime. Inset: Measured power of the Stokes sideband of the driving signal. The threshold behavior indicates the onset ofphonon lasing. (E) Magnomechanical parametric gain as a function of the drive power in an over-coupled hybrid system. Inset:Measured reflection spectrum that shows a 25-dB gain. In the main panels of (C)–(E), blue circles (red squares) are for theblue (red) detuning, and solid lines are theoretical calculations.