Bright and dark states of two distant macrospins strongly coupled by phonons

We study the collective dynamics of two distant magnets coherently coupled by acoustic phonons that are transmitted through an intercalated crystal. By tuning the ferromagnetic resonances of the two magnets to an acoustic resonance of the crystal, we control a coherent three levels system. We show that the parity of the phonon mode governs the nature of the indirect coupling between the magnets: the resonances with odd / even phonon modes correspond to out-of-phase / in-phase lattice displacements at the magnets, leading to bright / dark states in response to uniform microwave magnetic fields, respectively. The sample is a tri-layer garnet consisting of two thin magnetic layers epitaxially grown on both sides of a half-millimeter thick non-magnetic single crystal. In spite of the relatively weak magneto-elastic interaction, the long lifetimes of the magnon and phonon modes in the sample are the key to unveil this long range strong coupling. This demonstrates that garnets are a great platform to study multi-partite hybridization process between magnon and phonons at microwave frequencies.


I. INTRODUCTION
The coherent transfer of information between different wave forms is an important ingredient of quantum information processing as it allows interfacing, storage, and transport of quantum states [1][2][3][4]. The coupling between two distant quantum states, (1) and (2) can be coherently mediated by an intermediate third level (i), which may establish non-local entanglement [5]. Such a tripartite hybridization, generates two collective modes under a uniform excitation field that are labeled bright or dark [6][7][8]. Here we demonstrate control of the collective dynamics of two magnets by the polarity of the mutual coupling mediated by phonons through a high quality single crystal.
Strong coupling is possible only when the coupling rate between wave forms is larger than the geometric average of their relaxation rates [9,10], which corresponds to cooperativity parameters C > 1. Strong coupling between a microwave cavity mode and two superconducting qubits has long been realized [11,12]. The discovery of also strong coupling between a microwave circuit and ferrimagnetic objects [13][14][15][16][17] has encouraged the study of coherent multilevel systems of two or more magnetic objects indirectly coupled by cavity photons, by both theory [18,19] and experiments [20][21][22]. Studies focus on the dispersive regime, in which the two systems of interest are detuned and the coupling is mediated by virtual excitations of the intermediate level [23][24][25][26]. Moreover, until today, relatively little effort has been put to attempt to control the polarity of the coupling, which can permute the binding and anti-binding states.
Acoustic phonons in materials with low ultrasonic attenuation are excellent candidates to establish strong coupling with magnets as well [27,28]. Acoustic phonon pumping via the magneto-elastic and -rotational couplings can lead to increased damping of the magnetization dynamics [29][30][31]. Unidirectional phononic currents generated by magnetization dynamics have been predicted [32] and nonreciprocal propagation of acoustic phonons due to magneto-elastic coupling has been observed [33,34]. Coherent acoustic phonons can be also exploited to transfer incoherent spins [35]. Magnetoelastic effects have been reported in various systems, such as metallic magnetic layers [36][37][38][39], nano-sized ferromagnets [40], dilute magnetic semiconductors [41], and layered structures [42] as well as magnetic insulators [43][44][45]. In applications, the magneto-elastic coupling has also been used to design a low phase-noise magnetoacoustic resonator consisting of a thin yttrium iron garnet (YIG) film epitaxially grown on gallium-gadolinium garnet (GGG) [46] or high-speed oscillating magnetic field sensors [47]. We reported coherent long-range transfer of angular momentum in YIG|GGG|YIG structures via circularly polarized acoustic waves propagating through a mm thick GGG spacer [43]. However, the magnetic resonances were detuned by more than the magnetoelastic coupling strength, thereby falling short of a full hybridization of the distant magnets.
In the following, we demonstrate hybridization of two macrospins, which establishes our system as a promising solid state platform for quantum information processing, allowing coherent communication between distant magnon states via phonons in a monolithic device. We achieve full hybridization by careful tuning of both Kittel frequencies into a joint resonance with a phonon mode. At the degeneracy point, the magnon-phonon coupling induces a bonding-antibonding splitting that can be larger than the combined broadening, leading to bright and dark states of maximal and minimal coupling to an external readout antenna. We control the hybridization by either tilting the sample relative to an applied magnetic field, H, or by applying a vertical temperature gradient. Being able to select the index of the intermediate phonon mode offers an additional knob to control the parity of the coupling. We also detect the emerging bright and dark states electrically by Pt contacts on the YIG layers by spin pumping and the inverse spin Hall effect. The bright magnetic states emerge by the coupling to odd phonon modes. We observe a more than factor of 2 enhanced spin pumping signal, which is caused by the constructive interference of the amplitudes of the distant magnetic layers. On the other hand, by tuning the system to a dark state via an even index phonon mode, the microwave absorption and therefore the spin pumping signal are strongly suppressed.

II. MODEL
We consider two macrospins with oscillating magnetic components m 1 and m 2 with eigenfrequencies ω 1 and ω 2 , respectively. They are coupled by a standing acoustic wave with amplitude u i and frequency ω i /(2π) = v/λ i , where the integer i is the mode index and v is the sound velocity. λ i /2 = (2d + s)/i is half of the phonon wavelength that fits into the total crystal thickness, with d and s being the film thicknesses of the YIG and GGG films. The tripartite hybridization of the dimensionless state vector |Ψ = (m 1 , u, m 2 ) is governed by the Hamiltonian (omitting the weak dipolar coupling): .
The phase shift of 0 • /180 • at the free boundary conditions at the outer surfaces for phonon modes with even/odd index i alternates the sign of the mutual coupling between the top and bottom magnetic layers. We account it in Eq. (1) by Ω 1 = −Ω and Ω 2 = (−1) i Ω 1 [43].
At the degeneracy point (ω 1 = ω i = ω 2 ) the eigenfrequencies relative to ω i and corresponding normalized eigenvectors read: − Ω √ 2 : respectively for even and odd indices (upper and lower sign).
In a bracket notation of the state vectors, the linear response to an excitation |ε e jωt oscillating at frequency ω/(2π) defines the dynamic susceptibility: where I is the identity matrix and |ε = µ 0 h (1, 0, 1)/ √ 2 represents an external microwave magnetic field h coupled inductively to the two macrospins, with µ 0 being the vacuum permeability and γ the gyromagnetic ratio.
Imaginary terms jη s/a (j = √ −1) added to the diagonal elements of the Hamiltonian broaden the energy levels, representing the magnetic and acoustic relaxation rates respectively. Fig. 1(d,f) and Fig. 1(c,e) show with a bi-variate color code the absorbed power under stationary conditions in the second layer and acoustic system under the microwave excitation, P 2 = κ Im m 2 | χ |ε and P a = κ Im u i | χ |ε , respectively, with κ = µ 0 M 2 ω. In these plots, the spectral integral of P at fixed detuning are a definite-positive conserved quantity. A negative P 2 [43] (here in green) implies that the top layer returns power to the electromagnetic field when driven by an acoustic mode with phase shifted by 90 • in advance of the microwave drive. The "negative-power" acoustic modes change from even to odd (or vice versa) when crossing an acoustic resonance, because magnetic oscillations acquire a phase shift of 180 • that is added to the phase of the acoustic drive. The total power absorbed by the magnetic system vanishes for the dark state at the triple resonance, but does not become negative.
In Fig. 1(c-f), we plot the result as a function of detuning δ ≡ (ω 2 − ω 1 )/2 between the magnetic layers under the constraint ω i = (ω 1 + ω 2 )/2. When an even phonon mediates the interaction, we find an anticrossing. We point out, that the amplitude of the gap created by indirect coupling is √ 2 Ω, which is enhanced by √ 2 from the value produced by direct coupling. On the other hand, coupling by the odd phonon mode leads an additional central mode that does not depend on δ. The maximum value of m 2 is enhanced (compare the line cuts in Fig. 1(g) and (h)) by a constructive interference of the direct and indirect excitation amplitudes.
We can then compute P 2 as a function of the cooperativity, C = Ω 2 /(2η s η a ), and ρ ≡ h 1 /h 2 , an asymmetry in the amplitudes of the external microwave magnetic fields at each layers, where |ε = µ 0 h 2 (ρ, 0, 1)/ √ 2. At δ = 0, the contrast between bright and dark states follows the analytical expression: This ratio diverges for ρ = 2/C + 1, indicating that for any value of C , there is an optimum coupling asymmetry ρ to obtain the maximum contrast. In the asymptotic limit of C 1, we recover that the optimum is simply the symmetric system (ρ ≈ 1).

III. EXPERIMENTS
We test and confirm the above model by room temperature spin pumping experiments with two nominally identical d = 200 nm thick YIG films grown by liquid phase epitaxy on both sides of an s = 0.5 mm thick GGG substrate as shown schematically in Fig. 2(a). We denote the bottom and top YIG layers as YIG1 and YIG2, respectively. We monitor the YIG2 dynamics via the inverse spin Hall voltage (V ISHE ) [49] across a 7 nm thick, 240 µm wide Pt, and 950 µm long Pt stripe as shown in Fig. 2(a). [50] V ISHE peaks at H K2 as shown in Fig. 2(b). We also measure the absorption power by a microwave antenna close to YIG1 as shown in Fig. 2(c) for ω/(2π) = 5.6 GHz and θ H = +17 • [51]. At the resonance field H K1 the YIG1 Kittel mode dominates absorption by its proximity to the antenna, but we also see YIG2 by the weak line at H K2 that is shifted by 1.5 mT. The asymmetric coupling ratio ρ can be decreased by a larger separation between the microwave strip and the sample. The splitting between H K1 and H K2 betrays a small unintended asymmetry in the growth process that may be caused by a small difference in the saturation magnetization or magnetic anisotropies. The YIG2 linewidth is narrower in the V ISHE than in the microwave absorption spectra, which we attribute to a higher degree of homogeneity on the scale of the area covered by the Pt contact (see Appendix). We use below two different schemes to accomplish the triple resonance condition with H K1 = H K2 .
Both the angle θ H and a temperature gradient in the sample can tune the two Kittel modes.  Fig. 2(d)) . The small deviations of less than ±0.3% plotted in Fig.2(e) show a crossover angle θ c ≈ 24 • as indicated by an arrow. The magnetic resonance conditions depend on temperature, so a temperature gradient over the stack can tune H K1 − H K2 , with the advantage that each of H K 's varies much less than for the angular scan. We can heat the system simply by the Ohmic dissipation of a dc current I in the Pt contact. Since the dc source increases the noise level, we tune to the cavity resonance frequency, which has a quality factor of about 30. The higher power enhances the signal to noise ratio of the spin pumping signal. We detect the spin pumping voltage by lock-in at a frequency of about 3.1 kHz, thereby removing parasitic dc signals. The spin Seebeck voltage induced by microwave heating is negligibly small (see Appendix). At the magnetic field tilt angle θ H = 15 • the two resonances are so close that they cross already at small temperature gradients. In Fig. 2(f), the resonance fields from both the top and bottom layers are plotted as a function of the Joule heating power I 2 R. Since the magnetization decreases linearly with temperature, the accompanied reduction of the resonance field can be used as a thermome-ter [53]. From the shift of the YIG2 and YIG1 resonance fields, we estimate a temperature increase of 61 K and 31 K for I = 26 mA (I 2 R ∼ 70 mW), respectively [54]. The crossover current I c is at ∼13.6 mA, corresponding to Joule heating of I 2 c R ≈ 19 mW. Higher temperature expands the crystal and reduces the sound velocity, so the phonon frequencies shift as a function of T .
Tuning by temperature gradients In Fig. 3(a-f), we plot V ISHE as a function of I 2 R and magnetic field at six different frequencies f in the grey scale indicated in the bottom panel. We average the data for positive and negative currents to remove effects of the current direction such as spin-orbit torques or bolometric voltages [55]. We overlay V ISHE by blue solid lines that are the maxima of the microwave absorption spectra as a function of magnetic field, i.e. the resonances of the bottom layer. Small kinks emphasized by the green and orange arrows are evidence for crossings of the magnon layer as a function of Joule heating and magnetic field observed at different frequency tuning relative to acoustic resonances (microwave power is set at 0 dBm and sample orientation is set at θH = 15 • ). The blue solid lines are the maxima of the microwave absorption spectra that are dominated by YIG1. The right panel compares the modulation of the spin pumping signal measured either at zero detuning between the two Kittel resonances (purple vertical dotted line) or at large detuning (cyan vertical dotted line). The green and yellow arrows indicate the positions of odd and even phonon resonances, observable as small cusps in the microwave spectra (blue lines). The circles in (c) and (f) emphasize the triple resonance at which all frequencies agree. (c) At the intersection with an even lattice mode the peak splits and becomes "dark". (f) At the intersection with an odd mode, the intensity gains a factor of 4, which confirms that the magnetization amplitudes of both YIG1 and YIG2 interfere constructively into a "bright" mode. mode with odd and even phonon modes, respectively [56]. The decrease of the ultrasound velocities v [57] causes a downward shift of the phonon frequencies with increasing temperature [58]. The distortions are expected for f = i v(1 − ζI 2 R)/[2(2d + s)], where i is the number of nodes in the standing wave acoustic phonon amplitude (i is of the order of 1400 here). We indeed observe abnormalities for different indices i and heating powers I 2 R, so several phonon modes contribute to the window opened by the (moderate) temperature gradient generated by the ≤ 70 mW heating power. The decrease in v is proportional to the heating power with coefficient ζ = 0.048 mW −1 as estimate from the five phonon lines that shift by 3.5×5 = 17.5 MHz.
While the peak position of V ISHE decreases linearly with temperature, similar to the microwave signal of the bottom layer, its amplitude changes dramatically when H K1 ≈ H K2 . The signals vary strongly in the power range of 0-40 mW, but saturate in the 40 -70 mW interval. In Fig. 3(f) we observe an enhanced peak by more than a factor of 2 when an odd phonon mode comes close to the crossing at I = I c indicated by the vertical purple-dotted line. On the other hand, according to Fig. 3(c) an even phonon mode at I = I c splits the peak. In our experiments, m 1 is excited stronger than m 2 by its proximity to the antenna, while a complete dark state emerges only when the amplitudes of m 1 and m 2 are the same, as shown in Fig. 1. At a current far from the crossover I c , indicated by the vertical dotted cyan line and the corresponding line cut in the right panels of Fig. 3, such features are absent. The V ISHE signal at the peak center of the purple curves is enhanced by a factor of 4 (attenuated by a factor of 2) when an odd (even) acoustic mode mediates the coupling.
These factors can be compared to the predictions by Eq. (5). While the value of C ≈ 6 (see below), which depends on the material parameters, is fixed, the value of ρ is adjusted depending on the sample mounting (spacer between the garnet sample and the microwave antenna) and the power used. According to Eq.(5), such ratio between bright and dark state corresponds to an asymmetric inductive coupling ρ between YIG1 and YIG2 close to 3, which is smaller than previous estimation [43]. We attribute this partially to the use of large power, which saturates the excitation of the bottom layer while the top layer is still in the linear regime at the chosen microwave input power.

Tuning by magnetic field angle
Next we interpret the V ISHE spectra as a function of the polar angle θ H without applying any current to the Pt. Figure 4(a) is a gray-scale density plot of V ISHE versus frequency and magnetic field close to the normal configuration (θ H = 5 • ), with large differences between the  resonance frequency of the two layers, indicated by the of blue and red arrows. Green and orange horizontal lines mark the odd and even phonon resonances, respectively, that are spaced by 3.5 MHz. At large detuning, as shown in Fig. 4(a), magnon-phonon hybridization just reduces the spin-pumping voltage. This can be interpreted as an increased broadening that indicates an enhanced magnetization damping by leakage into the lattice [43].
Around θ H = 24 • the magnetic resonance of both layers are very close, as indicated by the blue and red arrows in Fig. 4(c), which looks very different from panel (a). The minima at the green lines for θ H = 5 • evolve into maxima, while the minima at the orange lines be-come deeper. We highlight these features in the adjacent panels (b) and (d) in terms of the signals plotted along the resonance condition. ∆ 2 in panel (d) is the difference between the signals at two adjacent odd and even phonon crossings (blue points), while the difference between the voltages shifted by a quarter of a phonon wavelength from the crossing (red points) is ∆ 1 . The factor sin θ M eliminates the well-known angle dependence of the inverse spin Hall effect [59]. Furthermore, by subtracting V ISHE , i.e. the average over the frequency range at a fixed angle, we remove possible deviations from the sin θ M dependence caused by an elliptical precession [60]. Figure 4(f) emphasizes that the modulation either changes sign near the crossover angle θ c or develops a maximum, which agrees with the expectations from the 3-oscillator model. These shapes may be interpreted in terms of the real and imaginary parts of a dynamic susceptibility of the coupled magnetic layers. Figure 5 summarizes in more detail the numerical solutions of the 3 coupled equations for Ω/(2π) = 1.5 MHz, η s /(2π) = 0.5 MHz, η a /(2π) = 0.35 MHz, which lead to C ≈ 6. We use ρ ≈ 7, a value close to our low power estimation [43]. Even though our model is strictly linear, it captures the qualitative features well even at the high applied microwave powers [61]. Both Figs. 4(e,f) and Fig. 5(f)) find a double sign change of the signal when crossing the triple resonance. When the detuning between the two Kittel modes vanishes in Fig. 5(c), the bright and dark states appear at the triple crossing indicated by green and dark orange circles. In theory, the bright and dark states appear exactly at the phonon frequencies. The measured extrema in Fig. 4(e) slightly deviate from the predictions because at the degeneracy points the signals are sensitive to higher order corrections. In the fully hybridized regime, the pure bright and dark states emerge as shown in Fig. 1(g) and (h) only when ρ = 1, i.e. when the two layers are evenly excited by the microwaves. If ρ = 1, then the contrast in the dynamics of m 2 is reduced, leading to a deviation from the model presented in Fig.1. We do not observe any evidence for damping-mediated coupling that induces level attraction rather than repulsion [62][63][64]. This is an interesting subject for future studies.

IV. CONCLUSIONS
We demonstrate coherent coupling of two macrospins by phonon exchange over a millimeter distance in YIG|GGG |YIG phononic spin valves. The phenomenology is generic for coupled tri-partite systems that support bonding and anti-bonding states at the degeneracy points. The additional knob to switch the polarity of the mutual coupling through relatively small changes of the mechanical resonator (e.g. ∼ 0.1% thickness variations) drastically tunes the inductive coupling of a magnetic system by ultrasound, and allows an integrated solution for long-range communication between distant quantum states. Modulation at short time scales (shorter than the relaxation times but longer than the reciprocal coupling rates) should allow coherent permutation between bright and dark states by adiabatic fast passage [65][66][67]. This effect could protect quantum information by switching on and off the radiation damping to an external readout antenna. While our paper focuses essentially on the dark or bright states, where the coupled dynamics of the two macrospins are respectively quenched or enhanced by the coherent sum of the two contributions, a continuous control of the mutual phase between the two oscillators measured by a phase-resolved technique could be beneficial to better quantify the coherent coupling phenomena. The microwave heating can give rise to a spin Seebeck effect which induces spin currents and therefore an additional voltage in Pt may arise in the lock-in signal. We estimate a temperature rise of about 100 mK for a continuous microwave heating based on the increase of resistance as shown in Fig. 6, with ∆R = R 0 ζ∆T , where ζ = 0.0021 K −1 for Pt. Using the standard SSE coefficient of S = 0.08 µV/K for YIG|Pt system [68], the Pt length of L Pt = 0.95 mm, and the SSE voltage expression of V SSE = SL Pt ∆T /(s + 2d), we obtain the SSE voltage of about 15 nV, which is close to the uncertainty of the measured spin pumping voltage and may be safely disregarded.

FIG. 6. Measured
RPt under continuous microwave excitation ON (red) and OFF (black) at 6 dBm and 5.11 GHz. The background level increases by 25 mΩ due to the microwave heating. In addition to that, the magnetic resonance, represented by the foldover shape on the red curve due to the strong excitation power, also increases RPt.

Appendix B: Angular dependence of the Kittel formula
We fit the angular dependence of the resonance field by the following formula that includes the uniaxial H U and cubic H C anisotropy fields [52].
where θ H and θ M represent the directions of the magnetic field and magnetization, respectively. θ H and θ M are related according to: Our results agree well with the above formula. The fit for the top layer H K2 (θ H ) leads to the parameters µ 0 H C2 = 7.6 mT, µ 0 H U2 = 3.5 mT, the gyromagnetic ratio of γ 2 /(2π) = 28.5 MHz/mT, and magnetization µ 0 M 2 = 0.172 T (solid line in Fig. 2(d)).

Appendix C: Identification of phonon resonances
The frequencies of the standing phonon states decrease with increasing temperature due to Joule heating caused by increasing current. We trace their positions systematically by the integrated intensity of the microwave absorption signal. Fig. 7(a) shows the shift of the raw absorption spectra. Periodic dips in the integrated signals as a function of Joule heating reveal the phonon resonances in Fig. 7(b-g).
Appendix D: Detailed comparison of magnetic resonance spectra obtained from spin pumping and diode absorption methods For the sake of completeness, we provide in Fig. 8 a zoom around of the Kittel mode of the YIG2 layer of both the spin pumping and diode absorption spectra as shown in Fig.2 (b) and (c). It shows that indeed the YIG2 resonances positions are slightly shifted (by 0.3 mT) between the two methods. Also the linewidth measured from the spin pumping method is narrower than that from the diode absorption. We attribute this to the inhomogeneous broadening, where in one case the spin pumping is sensitive to the spins below the Pt surface, while in the other case the inductive signal is sensitive to the bulk magnetization dynamics.