Coherent oscillation between phonons and magnons

Abstract

Consider observing two different waves with the same frequency and wavelength. When these waves are coupled, the amplitude alternates between the two waves periodically, a phenomenon called coherent beating oscillation. Such phenomena can be seen in familiar coupled pendulums and, on a cosmic scale, neutrino oscillations: the oscillation between different types of neutrinos. In solids, on the other hand, there are various wave excitations responsible for their thermal and electromagnetic properties. Here we report the observation of coherent beating between different excitation species in a solid: phonons and magnons. By using time-resolved magneto-optical microscopy, magnons generated in Lu₂Bi₁Fe_3.4Ga_1.6O₁₂ gradually disappear by transforming to phonons, and after a while, they return to magnons. The period of the oscillation as a function of the field is consistent with the prediction of the magnon-phonon beating. The experimental results pave a way to coherent control of magnon-phonon systems in solids.

Introduction

Phonons refer to vibration waves of a crystal lattice in solids (Fig. 1a). They are responsible for the elastic and thermal properties of solids. On the other hand, magnons, or spin waves, refer to the wavy motion of magnetization in magnets, responsible for the magnetic and thermal properties (Fig. 1a)^1,2,3,4,5. Phonons and magnons can interact in solids via the magneto-elastic and magnetostatic couplings^6,7,8 (Fig. 1b).

Fig. 1: Concept of coherent oscillation between phonons and magnons, and time-resolved magneto-optical microscopy.

a A schematic illustration of phonons and magnons, b A schematic illustration of coherent oscillation between phonons and magnons. c The dispersion curves of phonon and magnon in lutetium iron garnet (LuIG). d A magnified view around A in Fig. 1c. The black curves represent the dispersion relation of hybridized magnon-phonon polaron, while the red and blue dashed curves represents dispersion relation of pure magnons and transverse acoustic phonons, respectively. e Optical setup for the time-resolved magneto-optical microscopy with the extended delay time. The excited magnetization dynamics is detected via the polarization rotation angle of the probe laser pulse induced by the magneto-optical Faraday effect in the sample. The detection is performed by an charge-coupled device (CCD) camera. f Magneto-optical image observed 3.5 ns after the pump pulse irradiation under the external magnetic field B = 11.5 mT parallel to the wavevector of the excited magnons. g, Wavenumber spectrum of the obtained magneto-optical images observed 3.5 ns after the excitation (B = 11.5 mT). The inset shows a magnified view.

The dynamics of phonons and magnons in each system is written in their dispersion curves, curves that show the relation between the wavenumber and frequency. Figure 1c shows dispersion curves of transverse acoustic (TA) phonons and magnons in a film of typical magnetic insulator, lutetium iron garnet (LuIG)^9,10. In the dispersion curves, the phonon dispersion and the magnon dispersion curves have an intersection, depicted as A in Fig. 1c, due to the magnetostatic gap in the magnon dispersion at k = 0. Around the intersection, a magnon-phonon hybridized state can be formed^11,12,13,14. This state, called a magnon polaron, was experimentally found to exhibit extremely long lifetime, much longer than pure magnons, attributed to the hybridization with phonons with long lifetime^15,16. In a lutetium iron garnet, the extended lifetime is confirmed by spin-heat conversion measurement even at room temperature¹⁷. As a result, when magnons are excited and sufficient time has passed, the pure magnons are attenuated, leaving magnon polarons. In magnon polarons, theoretically, the hybridization can cause level splitting, giving rise to an anti-crossing gap at the intersection in the dispersion curves, as shown in Fig. 1d. When the two states across the gap are excited simultaneously in a coherent way, coherent superposition of these two states, corresponding to beating oscillation between phonons and magnons, may be created (Fig. 1b).

We report the observation of coherent beating between phonons and magnons in lutetium iron garnet. By using time-resolved magneto-optical (TRMO) microscopy, we measured spatio-temporal magnetization dynamics, which couples with phonons in a thin film of lutetium iron garnet, LuIG. We found the coherent beating lasts up to tens of nanoseconds, which experimentally confirms strong coupling between magnons and phonons in the bare film of LuIG.

Results

Sample and measurement setup

To explore the beating oscillation in solids, we have developed TRMO microscopy. Figure 1e shows a schematic illustration of the experimental setup used in the present study. We used a thin film of Lu₂Bi₁Fe_3.4Ga_1.6O₁₂ (LuIG) with the thickness of 1.8 μm as a sample, exhibiting large magneto-optical effects^18,19 and small magnetization damping (See Supplementary Note 1). To excite magnetization dynamics, we focused a pulsed laser light (pump pulse) with the wavelength of 800 nm onto the sample, where the wavelength corresponds to almost half the energy of the bandgap of LuIG (~2.3 eV)^18,19. The duration and the energy of the pulse is 100 fs and 1.0 μJ per pulse, respectively. The pump pulse excites spin waves, or magnons, via the photo-induced demagnetization and the photo-induced expansion effects^{20,21,22,23,24}. We shaped the focus into a vertical line to selectively excite magnons with the wavevector k perpendicular to the vertical line (y axis) by using the Huygens-Fresnel interference²⁵. Next, we shined another weak light pulse (probe pulse) with the wavelength of 630 nm (100 fs, 50 nJ per pulse) on the sample from the normal direction. We measured the spatial distribution of the magneto-optical Faraday rotation of the probe pulse transmitted through the sample by using a CCD camera (See Method for details)²⁶. The Faraday rotation at each position on the sample θ_F(r, t) reflects the local magnetization precession amplitude projected along the propagation direction of the probe pulse. By sweeping the time delay between the pump and probe pulses, we obtained temporal evolution of the spatial images of the magnetization dynamics excited by the pump pulse. We extended the maximum delay time to 35 ns by using optical fibers, so that the frequency resolution of the obtained images reaches 28 MHz, which is enough to resolve the magnon-phonon gap frequency in the sample^7,8. All the measurements were performed at room temperature.

Observation of coherent oscillation between phonons and magnons

Figure 1f shows a spatial image of the polarization-rotation angle obtained at 3.5 ns after the pump-pulse irradiation. Vertical wave patterns appear in the vicinity of the focus of the pump pulse, demonstrating the magnon excitation by the pump pulse. By observing the sign change of the rotation caused by reversing the external magnetic field (See Supplementary Note 2), we confirmed that the polarization rotation is due mainly to the magneto-optical Faraday effect rather than the distortion-induced polarization rotation. Here, we define ${\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t)=\int {\theta }_{{{{{{{{\rm{F}}}}}}}}}({{{{{{{\bf{r}}}}}}}},t){e}^{i{{{{{{{\bf{k}}}}}}}}{{{{{{{\bf{r}}}}}}}}}d{{{{{{{\bf{k}}}}}}}}$ as the Fourier transform of the spatial image of the Faraday rotation angle with respect to the spatial coordinate. Figure 1g shows the $| {\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t){| }^{2}$ at t = 3.5 ns. In the k_x > 0 region [k = (k_x, k_y)], a region corresponding to waves propagating right in Fig. 1f, we see clear peaks of $| {\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t){| }^{2}$ at k_x = 2.93 × 10⁴ rad ⋅ cm⁻¹ ≡ k_TA (k_x = 1.38 × 10⁴ rad ⋅ cm⁻¹ ≡ k_LA) corresponding to the intersection point between magnons and TA (LA; longitudinal acoustic) phonons calculated from the parameters of LuIG. This shows that the magnon polarons are created at the intersections of the dispersion curves of magnons and phonons after the pump-pulse irradiation.

In Fig. 2a, we show the temporal evolution of the real part of ${\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t)$ at ∣k∣ = k_TA and k//B. Importantly, the envelope of the ${\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t)$ signal clearly oscillates; the envelope amplitude decreases from t = 15 ns to t = 20 ns, while it increases from t = 20 ns to t = 25 ns up to almost the same amplitude as at t = 15 ns. This oscillation is in contrast to the ordinary relaxation dynamics of magnons, which monotonically decreases, but rather implies oscillation of magnon density.

Fig. 2: Observation of magnon-phonon coherent oscillation.

a Temporal evolution of the real part of ${\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t)$ at k_x = k_TA under the magnetic field B = 11.5 mT parallel to k, where k_TA refers to the wavenumber of the intersection point between dispersion relations of transverse acoustic (TA) phonons and magnons. Red inverted triangles indicates t = 15 ns, 20 ns, and 25 ns after the pump pulse irradiation. b A frequency power spectrum of ${\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t)$ at k_x = k_TA. The blue filled circles represents experimentally obtained spectrum intensity, while the gray curve represents fitting curve. Inverted red triangle highlights peaks. Errors of the data are evaluated as a standard deviation, which is smaller than the data plot. c Theoretically calculated dispersion curves of magnon polarons around k_x = k_TA and k_y = 0, where we use the crystalline anisotropy energy K_c = 73.0 [J ⋅ m⁻³], uniaxial anisotropy energy K_u = −767.5 [J ⋅ m⁻³], saturation magnetization M_s = 14.8 [kA ⋅ m⁻¹], velocity of LA phonons v_LA = 6.51 [km ⋅ s⁻¹], velocity of TA phonons v_TA = 3.06 [km ⋅ s⁻¹] and magnon-phonon coupling constant b₂ = 1.8 × 10⁵ [J ⋅ m⁻³]. The black solid curves represent the dispersion curves of magnon polarons, while the blue and red dashed curves represent pure TA phonons and magnons, respectively. d Temporal evolution of the real part of ${\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t)$ at k_x = k_LA under the magnetic field B = 11.5 mT parallel to k, where k_LA refers to the wavenumber of the intersection point between dispersion relations of longitudinal acoustic (LA) phonons and magnons. e A frequency power spectrum of ${\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t)$ at k_x = k_LA. The black filled circles represents experimentally obtained spectrum intensity, while the gray curve represents fitting curve. Errors of the data are evaluated as a standard deviation, which is smaller than the data plot. f Theoretically calculated dispersion curves of magnon polarons around k_x = k_LA. The gray line and red curve represent the dispersion curves of LA phonons and magnons, respectively. g Temporal evolution of the real part of ${\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t)$ at k_x = k_TA under the magnetic field B = 11.5 mT perpendicular to k. h Temporal evolution of real part of ${\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t)$ at k_x = k_LA under the magnetic field B = 11.5 mT perpendicular to k. i, Magneto-optical images taken at different delay times.

In Fig. 2b, we show the frequency power spectrum of ${\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t)$, F_k(ω) = ∫θ_F(r, t)e^i(kr−ωt)dkdt, at ∣k∣ = k_TA, where k//B. The frequency spectrum exhibits two peaks, suggesting the level splitting at the intersection of the magnon and TA phonon dispersion curves. The splitting width 70 MHz coincides with the inverse of the period for the observed oscillation envelope shown in Fig. 2a (14 ns), implying that the splitting is related to the observed oscillation.

When we measure pure magnons directly with the TRMO technique, we should see the periodically oscillating signal as a function of time with the frequency of magnons. Since we measure magneto-optical Faraday rotation, the signal disappears when magnons are transformed into phonons. Therefore, the observed oscillation of the signal envelope shown in Fig. 2a implies the periodic coherent beating oscillation between magnon and phonon in the time domain.

To confirm that the observed envelope oscillation is magnon-phonon beating oscillation, we examine field-angular dependence; if the observed oscillation is the beating, it should disappear for LA phonons since the magnon-phonon coupling coefficient takes minimum for LA phonons when k//B. The effective magnetic field b = (b_y, b_z) acting on local magnetization due to the magnon-phonon coupling can be written as²⁷,

$$\left(\begin{array}{c}{b}_{y}\\ {b}_{z}\end{array}\right)=\left(\begin{array}{c}2{b}_{2}{\epsilon }_{xz}\cos {\theta }_{{{{{{{{\bf{k}}}}}}}}}\\ {b}_{1}{\epsilon }_{xx}\sin 2{\theta }_{{{{{{{{\bf{k}}}}}}}}}\end{array}\right),$$

(1)

where b₁ and b₂ are the magneto-elastic coupling constants, and θ_k is the relative angle between the wavevector of magnons and the saturation magnetization. ϵ_xx and ϵ_xz are the elastic strain tensor components induced by phonons. ϵ_xx (ϵ_xz) becomes nonzero for LA (TA) phonons. When θ_k = 0(k//B), LA phonons induce minimal effective fields, while TA phonons induce maximum effective fields. On the other hand, both LA and TA phonons induce minimal effective fields when θ_k = 90^∘(k⊥B). Therefore, the coherent oscillation should disappear when k_x = k_LA(k//B), k_x = k_LA(k⊥B), and k_x = k_TA(k⊥B). We show the temporal evolution of the real part of the experimentally obtained ${\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t)$ at k_x = k_LA(k//B) in Fig. 2d. When k//B, the envelope amplitude of the ${\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t)$ does not oscillate but monotonically decreases. The frequency spectrum of the ${\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t)$ exhibits a single peak at k_x = k_LA(k//B), as shown in Fig. 2e, consistent with the θ_k dependence of the effective field representing the magnon-phonon coupling [Fig. 2f]. Figure 2g and h show the temporal evolution of ${\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}$ at k_x = k_LA(k⊥B) and k_x = k_TA(k⊥B), respectively. The envelope of ${\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t)$ does not oscillate in both cases, consistent with the prediction that the effective field is minimized under these conditions. We also performed measurement at θ_k = 45°, where the magneto-elastic coupling is maximized for LA phonon, however, we could not see clear oscillation due to large damping of magnons (See Supplementary Note 3).

To demonstrate how coherent beating oscillation is observed in real space, we show the temporal change in the wave pattern excited by the pump pulse in Fig. 2i. At t = 15 ns, we see clear wave pattern with the characteristic wavelength of λ_TA  =  2π/k_TA = 2.1 μm, while at t = 20 ns, the period of the strong wave pattern changes to λ_LA = 2π/k_LA = 4.6 μm. This is because the magnon amplitude ${\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t)$ at k = k_TA decreases owing to coherent beating oscillation, while the ${\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t)$ at k = k_LA remains finite. After another 10 ns, the wave pattern of k_TA mode recovers its intensity, which are all consistent with the result shown in Fig. 2a and d.

Excitation spectra of magnons and coherent oscillation frequency

We now discuss the results in terms of the relation between magnon excitation spectra and coherent oscillation frequency. By using the effective field b, we calculate the coherent oscillation frequency as follows (See Method),

$${{\Delta }}f=\frac{1}{2\pi }\sqrt{{\left[{\omega }_{{{{{{{{\rm{m}}}}}}}}}(k;{{{{{{{\bf{B}}}}}}}})-{\omega }_{{{{{{{{\rm{p}}}}}}}}}(k)\right]}^{2}+{\sigma }_{k}^{2}},$$

(2)

where ${\sigma }_{k}={b}_{2}k\sqrt{\frac{2\gamma }{\rho {M}_{{{{{{{{\rm{s}}}}}}}}}{\omega }_{t}}}$ is the magnon-phonon gap frequency, γ = 2π × 2.8 × 10¹⁰Hz ⋅ T⁻¹ is the gyromagnetic ratio, ρ = 7.39 × 10³ kg ⋅ m⁻³ is the density of the sample, M_s = 1.48 × 10⁴ A ⋅ m⁻¹ is the saturation magnetization, and k = ∣k∣ is the wavenumber. ω_m(k; B) and ω_p(k) are the angular frequency of magnons and phonons, respectively. ω_t is the angular frequency at the intersection between magnon and phonon dispersion curves. In Fig. 3a, we show a magnified view of ∣F_k(ω)∣² near the intersection. The spectrum intensity is in good agreement with the theoretical calculation exhibiting anticrossing gap around the intersection of the dispersion curves. Figure 3b shows the oscillation frequency as a function of the wavenumber obtained from the time-domain analysis of the ${\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t)$. The obtained oscillation frequency exhibits a V-shaped curve which takes a minimum at the intersection point. The bottom frequency of the V-shaped curve remains finite within the error bar. The result demonstrates again the formation of the anti-crossing gap caused by the magnon-phonon coupling, which has yet to be observed in magnetic garnet films^28,29,30. When the external magnetic field is increased to 13.0 mT, the spectrum peak shifts towards higher wavenumbers (Fig. 3c). The magnetic field dependence confirms that the observed gap is not due to the phonons, which do not respond to external magnetic fields. The experimental result is well reproduced by Eq. (2), as shown in Fig. 3b and d, showing that the observed envelope oscillation of ${\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t)$ is attributed to the coherent oscillation between phonons and magnons.

Fig. 3: Wavenumber and field dependence of magnon-phonon coherent oscillation.

a Frequency spectrum F_k(ω) observed at B = 11.5 mT around the intersection of the magnon and transverse acoustic (TA) phonon dispersion curves. b Comparison between experimentally obtained gap between the upper branch and lower branch of the spectrum at B = 11.5 mT and the theoretical calculation of the gap frequency. Error bars represent standard deviation. c Frequency spectrum F_k(ω) observed at B = 13.0 mT around the intersection of the magnon and TA-phonon dispersion curves. d Comparison between experimentally obtained gap between the upper branch and lower branch of the frequency spectrum at B = 13.0 mT and the theoretical calculation of the gap frequency.

Discussion

We numerically calculated the temporal evolution of the magnon amplitude ${\tilde{a}}_{k}(t)$ by calculating the Fourier transform of the spectral magnon amplitude a_k(ω)³¹. We considered only the coupled dynamics between TA phonons and magnons, relevant to the observed oscillation. By considering the magnon-phonon interaction to the lowest order, Gilbert damping, and phonon relaxation, a_k(ω) can be written as follows (See Methods),

$${a}_{k}(\omega )=-\frac{{{{{{{{\rm{i}}}}}}}}{\sigma }_{k}{f}_{k}{g}_{{{{{{{{\rm{p}}}}}}}}}^{{{{{{{{\rm{ex}}}}}}}}}/2}{\left[{{{{{{{\rm{i}}}}}}}}({\omega }_{{{{{{{{\rm{m}}}}}}}}}-\omega )+{\kappa }_{{{{{{{{\rm{m}}}}}}}}}\right]\big[{{{{{{{\rm{i}}}}}}}}({\omega }_{{{{{{{{\rm{p}}}}}}}}}-\omega )+{\kappa }_{{{{{{{{\rm{p}}}}}}}}}\big]+{\sigma }_{k}^{2}/4},$$

(3)

where κ_m and κ_p are the relaxation constants of magnons and phonons, respectively. f_k is the external driving force for phonons, and ${g}_{{{{{{{{\rm{p}}}}}}}}}^{{{{{{{{\rm{ex}}}}}}}}}$ is the coupling between phonons and f_k (See Method). ω_m and ω_p are the angular frequency of magnon and phonon at a wavenumber k, respectively. In Fig. 4a and b, we compared the experimentally obtained temporal evolution of $| {\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t){| }^{2}$ as a function of the wavenumber with the calculated magnon amplitude $| {\tilde{a}}_{k}(t){| }^{2}$. The $| {\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t){| }^{2}$ and $| {\tilde{a}}_{k}(t){| }^{2}$ exhibit similar wavenumber-dependent oscillation. We estimated the magnon-phonon gap frequency σ_k and the relaxation constants κ_m and κ_p by fitting the experimentally obtained magnon amplitude, as shown in Fig. 4c. Table 1 shows the estimated magnon-phonon coupling strength and the relaxation constants of magnons and phonons at three different wavenumbers. The magnon-phonon coupling constant ${b}_{2}=\frac{{\sigma }_{k}}{k}\sqrt{\frac{\rho {M}_{{{{{{{{\rm{s}}}}}}}}}{\omega }_{t}}{2\gamma }}$ is estimated to be (1.8 ± 0.1) × 10⁵ J ⋅ m⁻³ for all the wavenumbers, which agrees with a previous magneto-striction study³². At k = 2.98 × 10⁴ rad ⋅ cm⁻¹, σ_k is as large as 53.0 rad ⋅ MHz, while the relaxation constants of magnons and phonons are κ_m = 2.10 rad ⋅ MHz and κ_p = 0.49 rad ⋅ MHz, corresponding to the values of lifetime τ_m = 2.99 μs and τ_p= 12.8 μs, respectively. For all wavenumbers, σ_k is greater than κ_m and κ_p. The result shows that the gap frequency σ_k is greater than the spectrum linewidth of both phonons and magnons in the vicinity of the anti-crossing, satisfying the condition of phonon-magnon strong coupling. We calculated the cooperativity $C={\sigma }_{k}^{2}/{\kappa }_{{{{{{{{\rm{m}}}}}}}}}{\kappa }_{{{{{{{{\rm{p}}}}}}}}}$ of the magnon-phonon coupling to evaluate the strength of the coupling in LuIG, which gives C = 2720 ± 2700 for k = 2.98 × 10⁴ rad ⋅ cm⁻¹, C = 125 ± 81 for k = 3.15 × 10⁴ rad ⋅ cm⁻¹, and C = 127 ± 53 for k = 3.30 × 10⁴ rad ⋅ cm⁻¹. The cooperativity value is much greater than previously reported magnon-phonon coupling in nanomagnet³⁰. The large cooperativity is attributed to the small intrinsic magnetic damping and high-quality factor of phonon in garnet crystals³³. Although the error of the cooperativity is still large owing to the limitation of pump-probe delay time, the result implies intrinsic magnon-phonon coupling in a plane film of LuIG can be comparable with strong coupling appears in magnon-photon coupling in a cavity^31,34. The magnon-phonon coupling in the film can be further enhanced by fabricating phononic or magnonic crystals out of the plane film³, which may aid in the control of magnons in magnonic circuits and devices. In addition, our demonstration of the magnon-phonon coherent oscillation provides a means of studying the dynamics of coherent superposition of coupled systems, which may pave a way to coherent control of magnetic and elastic properties in various magnetic materials.

Fig. 4: Parameter fitting of coherent oscillation.

a Experimentally obtained temporal evolution of $| {\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t){| }^{2}$ at B = 11.5 mT. b Calculated temporal evolution of magnon amplitude $| {\tilde{a}}_{{{{{{{{\rm{k}}}}}}}}}(t){| }^{2}$. c Temporal evolution of $| {\tilde{F}}_{{{{{{{{\bf{k}}}}}}}}}(t){| }^{2}$ at different wavenumbers. Gray curves represents fitting curves according to Eq. (3). Errors of the data are evaluated as a standard deviation, which is smaller than the data plot.

Table 1 Coupling constant σ_k, relaxation constant of magnons κ_m and phonons κ_p at different wavenumbers k in the Lu₂Bi₁Fe_3.4Ga_1.6O₁₂ film.

Method

Time-resolved magneto-optical microscopy

The time-resolved magneto-optical imaging method is realized by combining  the time-resolved optical spectroscopy and conventional magneto-optical imaging method, which enables the observation of magnetization dynamics in samples with the spatial resolution of 750 nm and temporal resolution of 1 ps. We used a 100 fs-duration Ti; Sapphire pulse laser system with the central wavelength of 800 nm and 1 kHz repetition frequency (Coherent Inc. Astrella). Pump pulses were prepared by separating a part of the emitted laser. To prepare probe pulses, the central wavelength of a part of the emitted laser was converted to 630 nm by an optical parametric amplifier. The power of the pump and probe pulses was 1.0 μJ and 50 nJ per pulse, respectively, which was controlled by the variable ND filter. The pump and probe pulses were linearly polarized along the y-axis (Fig. 1e) by using Glan-Taylor prisms. The pump pulse was shaped into a 2.3-μm wide and ~100-μm long vertical line by using a metallic slit. An objective lens collected the transmitted probe pulse with a magnification of 20 and was then introduced to an imaging setup. The imaging setup is composed of a half-waveplate mounted on a rotation stage, an analyzer, and a charge-coupled device (CCD) camera with another objective lens of which the magnification is two. The polarization of the probe pulses was measured using the rotation analyzer method. The detailed optical configuration and analysis are shown in the Supplementary Note 2. We switch on and off the pump beam with a mechanical shutter to measure the magneto-optical images with and without the spin-wave excitation by the pump beam, and take the difference between the images. By sweeping the time delay between the pump and probe pulses, we obtained the propagation dynamics of magnons in the sample. We used an optical fiber to extend the temporal delay between the pump and probe pulses up to 35.4 ns, of which the Fourier frequency resolution (28 MHz) corresponds to the typical energy scale of the magneto-elastic coupling³². We used Lu₂Bi₁Fe_3.4Ga_1.6O₁₂(LuIG) grown on a [001] plane of a gadolinium gallium garnet substrate by liquid phase epitaxy as a sample. LuIG is a substituted magnetic garnet with the same crystallographic structure as Y₃Fe₅O₁₂(YIG), which is known to exhibit small Gilbert damping. The detailed material characterization is described in Supplementary Note 1. LuIG exhibits a large magneto-optical effect (1.5 deg/μm at 630 nm) and small magnetization owing to the Ga substitution¹⁷, leading to the small ferromagnetic resonance (FMR) frequency. Owing to the small FMR frequency, the wavenumber of the dispersion intersection between magnons and phonons is small enough to be observed with the present TRMO microscopy. An external magnetic field was applied using a quadrupole electromagnet during the measurement. We analyzed the region in the magneto-optical image which is 10-μm away from the pump focus so as not to include the signal at the pump focus where strong nonlinear effect of phonon such as structural distortion may arise. Owing to the distance between pump focus and the analyzed region, the obtained time-resolved signal has a finite time delay depending on the group velocity of the magnon polaron.

Calculation of dispersion relation of magnon polaron

In this note, we calculate the dispersion relation of magnon polaron by considering the total Hamiltonian, which involves the spin Hamiltonian, lattice Hamiltonian, and magneto-elastic coupling Hamiltonian as follows⁸,

$${H}_{{{{{{{{\rm{m}}}}}}}}}=-\int {d}^{3}r\left({H}_{0}{M}_{z}+\frac{D}{2\gamma \hslash {M}_{{{{{{{{\rm{s}}}}}}}}}}\frac{\partial {M}_{i}}{\partial {x}_{j}}\frac{\partial {M}_{i}}{\partial {x}_{j}}\right)$$

(4)

$${H}_{{{{{{{{\rm{e}}}}}}}}}=\int {d}^{3}r\left(\frac{\rho }{2}\frac{\partial {u}_{i}}{\partial t}\frac{\partial {u}_{i}}{\partial t}+\frac{\alpha }{2}\frac{\partial {u}_{i}}{\partial {x}_{i}}\frac{\partial {u}_{j}}{\partial {x}_{j}}+\frac{\beta }{2}\frac{\partial {u}_{i}}{\partial {x}_{i}}\frac{\partial {u}_{j}}{\partial {x}_{j}}\right)$$

(5)

$${H}_{{{{{{{{\rm{me}}}}}}}}}=\int d{r}^{3}\frac{{b}_{2}}{2{M}_{{{{{{{{\rm{s}}}}}}}}}^{2}}{M}_{i}{M}_{j}\left(\frac{\partial {u}_{i}}{\partial {x}_{j}}+\frac{\partial {u}_{j}}{\partial {x}_{i}}\right),$$

(6)

where, the repetitive use of indices i, j indicates summation with i ≠ j, and H₀ is the external magnetic field, D = 2JSa²/γℏ is the exchange stiffness, γ is the gyromagnetic ratio, ℏ is the Dirac constant, S is total spin, J is the exchange constant, a is the lattice constant, ρ is the density of the material, α = c₁₂ + c₄₄ and β = c₄₄ are the elastic constant, b₂ is the shear magneto-elastic constant, M_i is the i -th component of magnetization, u_i is the i -th component of displacement, M_s is the saturation magnetization, ω_pμ(k) = v_μk is the dispersion relation of phonon with mode index μ, ${v}_{{{{{{{{\rm{L}}}}}}}}}=\sqrt{\alpha /\rho },{v}_{{{{{{{{\rm{T}}}}}}}}}=\sqrt{\beta /2\rho }$ is the velocity of phonon, where L and T represent TA phonon and LA phonon, respectively. ω_m(k) is the dispersion relation of magnons, derived as follows by using the long-wavelength limit of B. A. Kalinikos’s method⁹. In the long-wavelength limit, the dispersion relation of magnons is written as follows,

$${\omega }_{{{{{{{{\rm{m}}}}}}}}}(k)=\sqrt{\left({\omega }_{{{{{{{{\rm{H}}}}}}}}\beta }+{\omega }_{{{{{{{{\rm{M}}}}}}}}}{\lambda }_{{{{{{{{\rm{ex}}}}}}}}}{k}^{2}\right)\left[{\omega }_{{{{{{{{\rm{H}}}}}}}}\alpha }+{\omega }_{{{{{{{{\rm{M}}}}}}}}}{\lambda }_{{{{{{{{\rm{ex}}}}}}}}}{k}^{2}+{\omega }_{{{{{{{{\rm{M}}}}}}}}}\left(1+{g}_{k}\left({\sin }^{2}{\theta }_{{{{{{{{\bf{k}}}}}}}}}-1\right)+\frac{{\omega }_{{{{{{{{\rm{M}}}}}}}}}{g}_{k}\left(1-{g}_{k}\right){\sin }^{2}{\theta }_{{{{{{{{\bf{k}}}}}}}}}}{{\omega }_{{{{{{{{\rm{H}}}}}}}}\alpha }+{\omega }_{{{{{{{{\rm{M}}}}}}}}}{\lambda }_{{{{{{{{\rm{ex}}}}}}}}}{k}^{2}}\right)\right]},$$

(7)

where, ${\lambda }_{{{{{{{{\rm{ex}}}}}}}}}=2{A}_{{{{{{{{\rm{ex}}}}}}}}}/{\mu }_{0}{M}_{{{{{{{{\rm{s}}}}}}}}}^{2}\,\left[{{{{{{{{\rm{m}}}}}}}}}^{2}\right]$ is the exchange constant, ω_M = γμ₀M_s, ${g}_{k}=1-\left[1-\exp (-kd)\right]/(kd)$ with the sample thickness d. ω_Hα = γμ₀H_α and ω_Hβ = γμ₀H_β is the Larmor precession frequency defined by anisotropic effective field H_α, H_β. We ignore the term originating from compressive strain because it is zero when the relative angle between θ_k is either 0 or 90^∘. In this case, the total effective Hamiltonian is written as follows⁸,

$${H}_{{{{{{{{\rm{eff}}}}}}}}}=\mathop{\sum}\limits_{k\mu }\left[\hslash {\omega }_{{{{{{{{\rm{m}}}}}}}}}(k){\hat{a}}_{k}^{{{{\dagger}}} }{\hat{a}}_{k}+\hslash {\omega }_{{{{{{{{\rm{p}}}}}}}}\mu }(k){\hat{b}}_{k\mu }^{{{{\dagger}}} }{\hat{b}}_{k\mu }+\frac{{{{{{{{\rm{i}}}}}}}}}{2}\hslash {\sigma }_{k\mu }\left({\hat{a}}_{k}^{{{{\dagger}}} }{\hat{b}}_{k\mu }-{\hat{b}}_{k\mu }^{{{{\dagger}}} }{\hat{a}}_{k}\right)\right],$$

(8)

where ${\hat{a}}_{k}({\hat{a}}_{k}^{{{{\dagger}}} })$ is the annihilation (creation) operator of a magnon with a wavenumber k, and ${b}_{k\mu }({b}_{k\mu }^{{{{\dagger}}} })$ is the annihilation (creation) operator of a phonon with a wavenumber k and mode index μ = T, L. σ_kμ is the coupling constant between magnon and phonon which is written as follows,

$${\sigma }_{k\mu }={b}_{2}\sqrt{\frac{2\gamma }{\rho {\omega }_{{{{{{{{\rm{p}}}}}}}}\mu }(k){M}_{{{{{{{{\rm{s}}}}}}}}}}}.$$

(9)

By diagonalizing the Hamiltonian, we obtain the following condition for the frequency of the magnon polaron ω⁸,

$$({\omega }^{2}-{\omega }_{{{{{{{{\rm{p}}}}}}}}}^{2})(\omega -{\omega }_{{{{{{{{\rm{m}}}}}}}}})-\frac{1}{2}{\omega }_{p}{\sigma }_{k\mu }^{2}=0.$$

(10)

We used λ_ex = 4.2 × 10⁻¹³ [m²], v_L = 6.51[km ⋅ s⁻¹], v_T = 3.06 [km ⋅ s⁻¹], b₂ = 1.8 × 10⁵ [J ⋅ m⁻³], ρ = 7.39 × 10³ [kg ⋅ m⁻³].

Derivation of coherent oscillation frequency

In this note, we derive the coherent oscillation frequency from the total effective Hamiltonian defined in the last note. We consider the parallel magnetic field configuration where wavevector k and M are parallel. In this configuration, only the TA phonon couples with the magnons, thus we set mode index μ = T Here, we start from the effective total Hamiltonian as follows,

$${H}_{{{{{{{{\rm{eff}}}}}}}}}=\mathop{\sum}\limits_{k}\left[\hslash {\omega }_{{{{{{{{\rm{m}}}}}}}}}(k){\hat{a}}_{k}^{{{{\dagger}}} }{\hat{a}}_{k}+\hslash {\omega }_{{{{{{{{\rm{p}}}}}}}}\mu }(k){\hat{b}}_{k{{{{{{{\rm{T}}}}}}}}}^{{{{\dagger}}} }{\hat{b}}_{k{{{{{{{\rm{T}}}}}}}}}+\frac{{{{{{{{\rm{i}}}}}}}}}{2}\hslash {\sigma }_{k{{{{{{{\rm{T}}}}}}}}}\left({\hat{a}}_{k}^{{{{\dagger}}} }{\hat{b}}_{k{{{{{{{\rm{T}}}}}}}}}-{\hat{b}}_{k{{{{{{{\rm{T}}}}}}}}}^{{{{\dagger}}} }{\hat{a}}_{k}\right)\right].$$

(11)

The effective Hamiltonian can be diagonalized using Bogoliubov transformation defined as follows,

$${\hat{c}}_{k}={u}_{k}{\hat{a}}_{k}-{{{{{{{\rm{i}}}}}}}}{v}_{k}{\hat{b}}_{k{{{{{{{\rm{T}}}}}}}}},$$

(12)

$${\hat{d}}_{k}={u}_{k}{\hat{b}}_{k{{{{{{{\rm{T}}}}}}}}}-{{{{{{{\rm{i}}}}}}}}{v}_{k}{\hat{a}}_{k},$$

(13)

where,

$${u}_{k}=\sqrt{\frac{{\omega }_{{{{{{{{\rm{s}}}}}}}}}+{\omega }_{\delta }}{2{\omega }_{{{{{{{{\rm{s}}}}}}}}}}},\quad {v}_{k}=\sqrt{\frac{{\omega }_{{{{{{{{\rm{s}}}}}}}}}-{\omega }_{\delta }}{2{\omega }_{{{{{{{{\rm{s}}}}}}}}}}},\quad {u}_{k}^{2}+{v}_{k}^{2}=1$$

(14)

and

$${\omega }_{\delta }=({\omega }_{{{{{{{{\rm{p}}}}}}}}{{{{{{{\rm{T}}}}}}}}}-{\omega }_{{{{{{{{\rm{m}}}}}}}}})/2,\quad {\omega }_{{{{{{{{\rm{s}}}}}}}}}=\sqrt{{\omega }_{\delta }^{2}+{\sigma }_{k{{{{{{{\rm{T}}}}}}}}}^{2}/4}.$$

(15)

The Bogoliubov transformation leads to the diagonalized Hamiltonian as follows,

$$H=\mathop{\sum}\limits_{k}\left[\hslash {\omega }_{{{{{{{{\rm{c}}}}}}}}}(k){\hat{c}}_{k}^{{{{\dagger}}} }{\hat{c}}_{k}+\hslash {\omega }_{{{{{{{{\rm{d}}}}}}}}}(k){\hat{d}}_{k}^{{{{\dagger}}} }{\hat{d}}_{k}\right],$$

(16)

where ω_c(k) = (ω_pT + ω_m)/2 + ω_s, ω_c(k) = (ω_pT + ω_m)/2 − ω_s. Here, we define the coherent state of the hybridized wave as follows,

$${\hat{c}}_{k}\left|{\gamma }_{k}\right\rangle ={\gamma }_{k}\left|{\gamma }_{k}\right\rangle ,$$

(17)

$${\hat{d}}_{k}\left|{\delta }_{k}\right\rangle ={\delta }_{k}\left|{\delta }_{k}\right\rangle ,$$

(18)

where γ_k = u_kα_k − iv_kβ_kT (δ_k = u_kβ_kT − iv_kα_k) is the amplitude of upper (lower) branch of the magnon polaron, α_k (β_kT) is the amplitude of magnon (phonon), which are the eigenvalues of the coherent state $\left|{\gamma }_{k}\right\rangle ,\left|{\delta }_{k}\right\rangle ,\left|{\alpha }_{k}\right\rangle ,\left|{\beta }_{k{{{{{{{\rm{T}}}}}}}}}\right\rangle$. By using time-resolved magneto-optical microscopy, we can measure the amplitude of the magnon, which is proportional to the magnon number. The magnon number at an arbitrary time t is calculated by using the coherent state of magnon polaron, as follows,

$$\langle {n}_{{{{{{{{\rm{m}}}}}}}}}\rangle =\left\langle {\gamma }_{k},{\delta }_{k}\right|{\hat{a}}_{k}^{{{{\dagger}}} }(t){\hat{a}}_{k}(t)\left|{\gamma }_{k},{\delta }_{k}\right\rangle \kern16.7pc$$

(19)

$$\kern1pc= \, \left\langle {\delta }_{k},{\gamma }_{k}\right|\left({u}_{k}{\hat{d}}_{k}^{{{{\dagger}}} }(0){e}^{{{{{{{{\rm{i}}}}}}}}{\omega }_{d}t}+{{{{{{{\rm{i}}}}}}}}{v}_{k}{\hat{c}}_{k}^{{{{\dagger}}} }(0){e}^{{{{{{{{\rm{i}}}}}}}}{\omega }_{c}t}\right)\left({u}_{k}^{* }{\hat{d}}_{k}(0){e}^{-{{{{{{{\rm{i}}}}}}}}{\omega }_{d}t}-{{{{{{{\rm{i}}}}}}}}{v}_{k}^{* }{\hat{c}}_{k}(0){e}^{-{{{{{{{\rm{i}}}}}}}}{\omega }_{c}t}\right)\left|{\gamma }_{k},{\delta }_{k}\right\rangle \\ = \, | {u}_{k}{| }^{2}| {\gamma }_{k}{| }^{2}+| {v}_{k}{| }^{2}| \delta {| }^{2}+2{u}_{k}{v}_{k}| {\gamma }_{k}^{* }{\delta }_{k}| \sin (2{\omega }_{s}t+\phi ),$$

(20)

where, $\tan \phi ={{{{{{{\rm{Im}}}}}}}}({\delta_k }^{* }{\gamma }_{k})/{{{{{{{\rm{Re}}}}}}}}({\delta_k }^{* }{\gamma }_{k})$. For an initial state, we suppose α_k = 0, β_kT ≠ 0 as we excite phonon predominantly by the pump pulse. Then γ_k = − iv_kβ_kT, δ_k = u_kβ_kT and ϕ = π/2 holds, leading to the temporal evolution of magnon number as follows,

$$\left\langle {n}_{{{{{{{{\rm{m}}}}}}}}}\right\rangle =\frac{{\sigma }_{k}^{2}}{2{\omega }_{s}^{2}}| {\beta }_{kT}{| }^{2}\left[1-\cos (2{\omega }_{{{{{{{{\rm{s}}}}}}}}}t)\right].$$

(21)

Since the frequency difference between upper and lower branch of magnon polaron is expressed by Δf = ω_c − ω_d = 2ω_s, this derivation leads to the expression in Eq. (2). We plotted the Δf in Fig. 3b and d in the main part of our paper.

Numerical calculation of coherent oscillation

In this note, we derive the coherent oscillation amplitude as a function of wavenumber k and time t. We consider non-Hermitian Hamiltonian including loss of magnons and phonons, and external field under parallel configuration as follows,

$$H= \, \mathop{\sum}\limits_{k}\left[\hslash {\omega }_{{{{{{{{\rm{m}}}}}}}}}{\hat{a}}_{k}^{{{{\dagger}}} }{\hat{a}}_{k}+\hslash {\omega }_{{{{{{{{\rm{pT}}}}}}}}}{\hat{b}}_{k{{{{{{{\rm{T}}}}}}}}}{\hat{b}}_{k{{{{{{{\rm{T}}}}}}}}}+\frac{{{{{{{{\rm{i}}}}}}}}\hslash }{2}{\sigma }_{k{{{{{{{\rm{T}}}}}}}}}\left({\hat{a}}_{k}^{{{{\dagger}}} }{\hat{b}}_{k{{{{{{{\rm{T}}}}}}}}}-{\hat{b}}_{k{{{{{{{\rm{T}}}}}}}}}^{{{{\dagger}}} }{\hat{a}}_{k}\right)\right.\\ -\left.{{{{{{{\rm{i}}}}}}}}\hslash {\kappa }_{{{{{{{{\rm{m}}}}}}}}}{\hat{a}}_{k}^{{{{\dagger}}} }{\hat{a}}_{k}-{{{{{{{\rm{i}}}}}}}}\hslash {\kappa }_{{{{{{{{\rm{pT}}}}}}}}}{\hat{b}}_{k{{{{{{{\rm{T}}}}}}}}}^{{{{\dagger}}} }{\hat{b}}_{k{{{{{{{\rm{T}}}}}}}}}+\hslash {\kappa }^{{{{{{{{\rm{ex}}}}}}}}}\left({F}_{k}{(\omega )}^{{{{\dagger}}} }{\hat{b}}_{k{{{{{{{\rm{T}}}}}}}}}+{F}_{k}(\omega ){\hat{b}}_{k{{{{{{{\rm{T}}}}}}}}}^{{{{\dagger}}} }\right)\right],$$

(22)

where κ_m(κ_pT) is the damping constant of magnon (TA phonon), κ^ex is the coupling constant between phonon and external excitation force F_k(ω). The equation of motion of the operators leads to the following form.

$$-{{{{{{{\rm{i}}}}}}}}\omega {\hat{a}}_{k}=-{{{{{{{\rm{i}}}}}}}}{\omega }_{{{{{{{{\rm{m}}}}}}}}}{\hat{a}}_{k}-{\kappa }_{{{{{{{{\rm{m}}}}}}}}}{\hat{a}}_{k}+\frac{{\sigma }_{k{{{{{{{\rm{T}}}}}}}}}}{2}{\hat{b}}_{k{{{{{{{\rm{T}}}}}}}}}$$

(23)

$$-{{{{{{{\rm{i}}}}}}}}\omega {\hat{b}}_{k{{{{{{{\rm{T}}}}}}}}}=-{{{{{{{\rm{i}}}}}}}}{\omega }_{{{{{{{{\rm{pT}}}}}}}}}{\hat{b}}_{k{{{{{{{\rm{T}}}}}}}}}-{\kappa }_{{{{{{{{\rm{p}}}}}}}}}{\hat{b}}_{k{{{{{{{\rm{T}}}}}}}}}-{{{{{{{\rm{i}}}}}}}}{\kappa }^{{{{{{{{\rm{ex}}}}}}}}}{F}_{k}-\frac{{\sigma }_{k{{{{{{{\rm{T}}}}}}}}}}{2}{\hat{a}}_{k}.$$

(24)

The excitation spectrum of magnons can be calculated from the equation of motion as follows,

$$\left\langle {\hat{a}}_{k}(\omega )\right\rangle =-\frac{{{{{{{{\rm{i}}}}}}}}{\sigma }_{k}{F}_{k}{g}_{{{{{{{{\rm{p}}}}}}}}}^{{{{{{{{\rm{ex}}}}}}}}}/2}{\left[{{{{{{{\rm{i}}}}}}}}({\omega }_{{{{{{{{\rm{m}}}}}}}}}-\omega )+{\kappa }_{{{{{{{{\rm{m}}}}}}}}}\right]\left[{{{{{{{\rm{i}}}}}}}}({\omega }_{{{{{{{{\rm{p}}}}}}}}}-\omega )+{\kappa }_{{{{{{{{\rm{p}}}}}}}}}\right]+{\sigma }_{k{{{{{{{\rm{T}}}}}}}}}^{2}/4}.$$

(25)

Here, we suppose the spectrum intensity of the external force is determined by the pump focus shape in real space²⁵. Therefore the external force is expressed as follows,

$${F}_{{{{{{{{\bf{k}}}}}}}}}(\omega )=\int dtd{{{{{{{\bf{r}}}}}}}}\,{I}_{0}f(t)G({{{{{{{\bf{r}}}}}}}}){e}^{{{{{{{{\rm{i}}}}}}}}({{{{{{{\bf{k}}}}}}}}{{{{{{{\bf{r}}}}}}}}-\omega t)}$$

(26)

$$G({{{{{{{\bf{r}}}}}}}})={e}^{-\frac{{x}^{2}}{2{\sigma }_{x}^{2}}-\frac{{y}^{2}}{2{\sigma }_{y}^{2}}}$$

(27)

where σ_x(σ_y) is the length of the focus along x(y) − axis, r = (x, y). The spatial distribution of the excitation intensity is plotted in Fig. 5(a). f(t) is the temporal excitation function defined as follows,

$$f(t)=\tan {{{{{{{\rm{h}}}}}}}}\frac{t-{t}_{{{{{{{{\rm{s}}}}}}}}}}{{\sigma }_{t}}-\tan {{{{{{{\rm{h}}}}}}}}\frac{t-{t}_{{{{{{{{\rm{e}}}}}}}}}}{{\sigma }_{t}}+1,$$

(28)

where t_s (t_e) is the start (end) time of the square-wave type excitation in time, σ_t is the parameter to describe smoothness of the square wave. The temporal evolution of the f(t) is plotted in Fig. 5(b). In calculating coherent oscillation, we need only the component where k = ke_x, which is F_k(ω) since we consider parallel configuration. The calculated spectrum intensity is shown in Fig. 5(c). The intensity of the spectrum shows a peak at the crossing between dispersion relations of magnons and TA phonons as seen in the experimentally obtained excitation spectra in Fig. 3a and c.

Fig. 5: Numerical calculation of magnon excitation intensity.

a Heat map of G(r). σ_x and σ_y are set to realize plane-wave excitation of magnon polaron (σ_x = 40 nm, σ_y = 40 nm). b Time evolution of excitation intensity f(t). c Heat map of spectrum intensity calculated according to Eq. (25) (t_s = 1.5 ns, t_e = 1.6 ns, σ_t = 0.3 ns). The spectrum intensity takes peak at the dispersion crossing between transverse acoustic (TA) phonon and magnon, reproducing the experimental results.