Theoretical description
Derivation of the nonlinear torque
We developed a model for the generation of the propagating magnons in the nonlinear conversion regime, and derive a model for the magneto-optical detection of the spin waves. We start by writing the Landau–Lifshitz equations for antiferromagnetic spin dynamics42
$$\begin{array}{c}\frac{{\rm{d}}{\boldsymbol{M}}}{{\rm{d}}t}=\gamma \left({\bf{M}}\times \frac{\delta W}{\delta {\bf{M}}}\right)+\gamma \left({\bf{L}}\times \frac{\delta W}{\delta {\bf{L}}}\right),\\ \frac{{\rm{d}}{\bf{L}}}{{\rm{d}}t}=\gamma \left({\bf{M}}\times \frac{\delta W}{\delta {\bf{L}}}\right)+\gamma \left({\bf{L}}\times \frac{\delta W}{\delta {\bf{M}}}\right).\end{array}$$
(4)
In equation (4), M is the ferromagnetic component M1 + M2 and L is the antiferromagnetic component M1 − M2. The terms \(\frac{\delta W}{\delta {\bf{M}}}\) and \(\frac{\delta W}{\delta {\bf{L}}}\) represent the internal effective fields in the spin system. W denotes the free energy of the spin system and for the orthoferrite is given by
$$W=\frac{1}{2}\,J{M}^{2}+D[{M}_{x}{L}_{z}-{M}_{z}{L}_{x}]+\frac{1}{2}({K}_{y}-{K}_{x}){L}_{y}^{2}+\frac{1}{2}({K}_{z}-{K}_{x}){L}_{z}^{2}+\frac{1}{4}{K}_{4}{L}^{4}+{q}^{{\prime} }{(\nabla {\bf{M}})}^{2}+q{(\nabla {\boldsymbol{L}})}^{2}-{\bf{M}}\cdot {\bf{h}}(t),$$
(5)
in which J is the exchange constant, D is the Dzyaloshinskii–Moriya constant, Kx,y,z,4 are the magnetic anisotropy constants, q′ and q are the exchange stiffness constants, and h(t) is the effective field that drives the spin dynamics. Here \(J < < D < < {K}_{x,y,z} < < {K}_{4}\). Note that in equation (4) the damping is neglected for simplicity. We add phenomenological damping to our final equations of motion.
We write the equations of motion for each separate component of the M and L vectors in the Γ2 phase \(({{\bf{L}}}_{{\bf{0}}}=(\mathrm{0,0},{L}_{0}),{{\bf{M}}}_{{\bf{0}}}=({M}_{0},\mathrm{0,0}))\), and carry out the linearization procedure, writing the spin deflections as the sum of the static and dynamic magnetizations, and assume that the dynamic component is small:
$$\begin{array}{c}{\bf{M}}(t)={{\bf{M}}}_{0}+{\bf{m}}(t),\\ {\bf{L}}(t)={{\bf{L}}}_{{\bf{0}}}+{\bf{l}}(t),\\ \left|{\bf{m}}\left(t\right)\right|\ll \left|{{\bf{M}}}_{{\bf{0}}}\right|,\left|{\bf{l}}\left(t\right)\right|\ll \left|{{\bf{L}}}_{{\bf{0}}}\right|.\end{array}$$
(6)
Also, we describe the spin deflection in terms of angular coordinates.
$$\begin{array}{l}{l}_{y}(t)={L}_{y}(t)-{L}_{y,0}={L}_{0}\sin (\varphi (t))\approx {L}_{0}\varphi (t),\\ {l}_{z}(t)={L}_{z}(t)-{L}_{0}^{z}={L}_{0}\cos (\varphi (t))\approx {L}_{0}(1-\varphi {(t)}^{2}/2)-{L}_{0}\approx 0.\end{array}$$
(7)
From the assumption discussed above, we can neglect all terms containing products φ2. Typically, the product of the spin deflection and effective field that excites the spin deflection is also neglected, as the spin deflection is φ = 0 before the excitation. This is, however, not the case at the arrival of the second pump in our experiment, so we retain the terms containing \(\varphi (t)h(t).\)
As a result, we come to the Klein–Gordon equation with an additional nonlinear torque.
$$\frac{{{\rm{\partial }}}^{2}{\varphi }_{2}(z,t)}{{\rm{\partial }}{t}^{2}}+({\omega }_{0}^{2}-{v}_{{\rm{s}}{\rm{w}}}^{2}{{\rm{\nabla }}}^{2}){\varphi }_{2}(z,t)+2\alpha \frac{{\rm{\partial }}{\varphi }_{2}(z,t)}{{\rm{\partial }}t}=-{\omega }_{h}\frac{{\rm{\partial }}{h}_{x}(z,t)}{{\rm{\partial }}t}+{\omega }_{D}{\omega }_{h}{\varphi }_{1}(z,t){\mathop{h}\limits^{ \sim }}_{2}(z,t).$$
(8)
Here vsw is the spin-wave propagation velocity and α is the damping parameter. We obtained this simplified expression by introducing the following parameters:
$$\begin{array}{c}\,\,{\omega }_{0}=\sqrt{{\omega }_{{\rm{E}}}{\omega }_{{\rm{A}}}},\\ \,\,{\omega }_{{\rm{E}}}=\gamma {L}_{0}\,J,\\ \,\,{\omega }_{{\rm{A}}}=\gamma {L}_{0}({K}_{{\rm{z}}}-{K}_{{\rm{x}}}),\\ \,\,{\omega }_{D}=\gamma {L}_{0}D,\\ \,\,{\omega }_{h}=\gamma {h}_{0},\\ \,\,{v}_{{\rm{s}}{\rm{w}}}^{2}={\gamma }^{2}{L}_{0}^{2}\,Jq,\\ \mathop{h}\limits^{ \sim }(z,t)=\frac{h(z,t)}{{h}_{0}}.\end{array}$$
(9)
The first term on the right-hand side in equation (8) represents the linear torque, and the second term on the right-hand side represents the newly derived nonlinear torque induced by the interaction of the spin deflection driven by the first pump pulse and the light-induced effective field of the second pulse. We see that this nonlinear torque is proportional to the Dzyaloshinskii–Moriya interaction D, the effective field and the spin deflection. Thus, such a nonlinear torque acts only when the external excitation couples with magnons in an out-of-equilibrium altermagnetic system.
As the laser pulse duration is much shorter than the spin precession period, we may approximate the pulses to act as a Dirac delta function27,43, arriving at time t1 and t2.
We assume that the effective fields of both pulses h1 and h2 are confined to the same distance d near the boundary:
$${h}_{1,2}(z,t)={h}_{0}{{\rm{e}}}^{-z/d}\delta (t-{t}_{1,2}).$$
(10)
We have denoted the spin deflections induced by pump 1 and 2 as φ1 and φ2, respectively. Note that in between the arrival of the first and second pump and changing the indices 2 to 1, we retrieve the linear Klein–Gordon equation, as φ(t < t1) = 0.
Detection of the spin dynamics
The spin dynamics is probed in a similar manner to the experiment of ref. 26, so here we expand on this detection formalism to explain the observed modulation frequencies.
First, we remind ourselves of the general expression of the magneto-optical polarization rotation of the reflected light induced by the magnetization near the surface44,45.
$${\theta }_{K}=i\frac{a{k}_{0}^{2}}{2k}\frac{{t}_{0}\widetilde{{t}_{0}}}{{r}_{0}}{\int }_{0}^{\infty }{\rm{d}}z{\prime} {e}^{2ik{z}^{{\prime} }}M(z,t).$$
(11)
When we neglect the nonlinear torque, we know that the spin deflections induced by the first pump pulse are described by27:
$${\varphi }_{1}(z,t)={\int }_{-\infty }^{\infty }[\,{f}_{1}(\omega ){{\rm{e}}}^{-i{k}_{{\rm{sw}}}(\omega )z}+{p}_{1}(\omega ){{\rm{e}}}^{-z/d}]{{\rm{e}}}^{i\omega t}{\rm{d}}\omega .$$
(12)
The first term in equation (12) corresponds to the freely propagating solution, and the second term refers to pump-driven uniform precession. Note that this spin-wave solution is obtained analytically in the Fourier domain, and the integral represents the inverse Fourier transformation to the time domain. From the analytical solution in the Fourier domain and by using the exchange boundary conditions46, we found the expressions for \({f}_{1}(\omega )\) and \({p}_{1}(\omega )\) for the case of the impulsive excitation:
$$\begin{array}{l}{p}_{1}(\omega )=\frac{-i\omega {\omega }_{h}\widetilde{h}(\omega )}{-{\omega }^{2}+{\omega }_{0}^{2}+2i\alpha \omega -{v}_{\mathrm{sw}}^{2}/{d}^{2}},\\ {f}_{1}(\omega )=\frac{1/d-\xi }{\xi -i{k}_{\mathrm{sw}}(\omega )}{p}_{1}(\omega ).\end{array}$$
(13)
To calculate the magneto-optically detected spin deflection, it is first necessary to obtain the solution to the nonlinear Klein–Gordon equation for the spin deflections (equation (8)). The solution to the linear equation is known, and will simply add another propagating wave starting at t2 through interference, yielding only a phase factor \({{\rm{e}}}^{i\omega {t}_{2}}\). Therefore, we focus on solving the equation for only the nonlinear torque, which can be found analytically by transforming the equation to the Fourier domain.
$$\left(-{\omega }^{2}+{\omega }_{0}^{2}+2i\alpha \omega -{v}_{\mathrm{sw}}^{2}{\nabla }^{2}\right){\varphi }_{2}\left(\omega ,z\right)={\omega }_{D}{\omega }_{h}{\varphi }_{1}\left(z,\tau \right){{\rm{e}}}^{-z/d}{{\rm{e}}}^{-i\omega \tau }.$$
(14)
We substitute the known linear solution for \({\varphi }_{1}(z,\tau )\) as the inverse Fourier transformation in equation (12) and the spatiotemporal profile of the effective field \(h(z,t).\)
The solution to equation (14) will have a similar form to the solution for the linear case, but the amplitudes are modified, and the effective field of this nonlinear torque will effectively be confined in a region d/2 from the material surface.
$${\varphi }_{2}\left(\omega \right)={f}_{2}\left(\omega \right){{\rm{e}}}^{-i{k}_{\mathrm{sw}}\left(\omega \right)z}+{p}_{2}\left(\omega \right){{\rm{e}}}^{-2z/d}.$$
(15)
The amplitude of the solution driven by the nonlinear torque \({p}_{2}(\omega )\) can be found and the amplitude of the freely propagating solution is obtained by defining a pinning parameter ξ that describes the restrictions of spin precession at the boundary, and applying the exchange boundary condition27.
Note that in the limit ξ → ∞, spin precession at the boundary is forbidden. After applying the exchange boundary conditions, we obtain the amplitudes for both components of the solution:
$$\begin{array}{c}{p}_{2}(\omega )=\frac{{\omega }_{D}{\omega }_{h}{\int }_{-\infty }^{+\infty }{p}_{1}(\varOmega ){{\rm{e}}}^{-i(\omega -\varOmega )\tau }{\rm{d}}\varOmega }{-{\omega }^{2}+{\omega }_{0}^{2}+2i\alpha \omega -4{v}_{{\rm{s}}{\rm{w}}}^{2}/{d}^{2}},\\ {f}_{2}(\omega )=\frac{2/d-\xi }{\xi -i{k}_{{\rm{s}}{\rm{w}}}(\omega )}{p}_{2}(\omega ).\end{array}$$
(16)
Note that the integral represents the inverse Fourier transformation. Finally, as our results are sensitive to the mx mode in our experiment, we convert the derived deflection of the ly component to the mx component, using the expression:
$${m}_{x}(\omega )=\frac{1}{i\omega }\left({\omega }_{A}-\frac{{v}_{{\rm{sw}}}^{2}}{{\omega }_{E}}{\nabla }^{2}\right){l}_{y}(\omega ).$$
(17)
We remind ourselves that \({m}_{x}(z,t)\approx {M}_{0}\varphi (z,t).\) Now we substitute the solutions (15) and (16) into equation (11), to obtain the 2D spectrum of the magneto-optical detection experiment, as a function of Ω and ω:
$${\theta }_{K}^{p}(\omega ,\varOmega )=i\frac{a{k}_{0}^{2}}{2k}\frac{\mathop{{t}_{0}}\limits^{ \sim }{t}_{0}}{{r}_{0}}\frac{1}{i\omega }\left({\omega }_{A}-\frac{4{c}^{2}}{{d}^{2}{\omega }_{E}}\right)\left(\frac{1}{2k+2i/d}\right)\frac{{\omega }_{D}{\omega }_{h}{p}_{1}(\varOmega ){{\rm{e}}}^{-i\omega \tau }}{-{\omega }^{2}+{\omega }_{0}^{2}+2i\alpha \omega -4{v}_{{\rm{s}}{\rm{w}}}^{2}/{d}^{2}}$$
(18)
and
$$\begin{array}{c}{\theta }_{K}^{f}(\omega ,\varOmega )=i\frac{a{k}_{0}^{2}}{2k}\frac{\mathop{{t}_{0}}\limits^{ \sim }{t}_{0}}{{r}_{0}}\frac{1}{i\omega }\left({\omega }_{A}+\frac{{c}^{2}{k}_{{\rm{s}}{\rm{w}}}{(\omega )}^{2}}{{\omega }_{E}}\right)\left(\frac{2/d-\xi }{\xi -i{k}_{{\rm{s}}{\rm{w}}}(\omega )}\right)\\ \times \left(\frac{1}{2k-{k}_{{\rm{s}}{\rm{w}}}(\omega )}\right)\frac{{\omega }_{D}{\omega }_{h}{p}_{1}(\varOmega ){e}^{-i\omega \tau }}{-{\omega }^{2}+{\omega }_{0}^{2}+2i\alpha \omega -4{v}_{{\rm{s}}{\rm{w}}}^{2}/{d}^{2}}.\end{array}$$
(19)
This expression is plotted in Fig. 4, and shows the peaks at the diagonal frequencies as a result of interference and the peak at the off-diagonal as a result of the magnon conversion by the nonlinear torque. In our calculation, we used the parameters shown in Extended Data Table 1.
Note that the light-induced effective field used in our simulations is relatively small compared to that of previous reports47, owing to the low pump fluence used in our experiment. For simplicity, we have derived all of the above equations assuming the Brillouin conditions for normal incidence. Although the parameters of n and β are not explicitly specified in our equations, they are used for the proper projection of the probe wave vector on the magnon wave vector to use the Brillouin condition for oblique incidence:
$$2kn\,\cos (\beta )={k}_{{\rm{sw}}}.$$
(20)
Altermagnetic symmetry of orthoferrites
Here we show that the orthoferrites satisfy the criteria to be classified as altermagnets. The criteria formulated in ref. 10 are as follows: “there is an even number of magnetic atoms in the unit cell”, “there is no inversion centre between the sites occupied by the magnetic atoms” and “the two opposite-spin sublattices are connected by crystallographic rotation transformation (may be combined with translation or inversion transformation)”.
The orthoferrites have four Fe atoms in the unit cell, thus satisfying the first criterion. The Fe ions occupy inversion centres and there is no inversion centre between them; hence, the second criterion is satisfied as well. Finally, the antiferromagnetic structure of HoFeO3 in the Γ2 phase, relevant for the present experiment, is invariant with respect to the screw axis transformation (that is, rotation around the x axis plus translation), satisfying the third condition. More details on the symmetry of the orthoferrites can be found in ref. 48. Moreover, altermagnets may be classified on the basis of their strong magneto-optical responses13. One of the characteristic features of the orthoferrites is their strong magneto-optical and optomagnetic responses47. For instance, the Faraday effect in the orthoferrites has been shown to scale linearly with the magnetic order parameter L (ref. 49), which is a manifestation of altermagnetism.
HoFeO3 material properties and sample information
Holmium orthoferrite (HoFeO3) is a weak ferromagnet, with antiferromagnetically ordered spins below the Neel temperature of approximately 650 K (ref. 50). The non-vanishing Dzyaloshinskii–Moriya interaction slightly cants the otherwise antiparallel spins, thus resulting in a weak net ferromagnetic moment. The canting angle in the weak ferromagnets is proportional to the Dzyaloshinskii–Moriya interaction constant D (ref. 51). Similar to the other orthoferrites, HoFeO3 is an insulator, with a bandgap energy, Eg, of about 3 eV (ref. 52). As a result, the orthoferrite exhibits a strong absorption of photons with an energy higher than this bandgap. The absorption enables the nanoscale confinement of the optical excitation of spins next to the sample facet and is essential for generating a propagating wave packet of magnons26.
HoFeO3 is a unique orthoferrite in the sense that its magnetic phase structure is complex, having more phases of spin orientation than the other orthoferrites. Here we summarize its magnetic properties. The magnetic structure is described by the antiferromagnetic vector L = M1 − M2 and the ferromagnetic vector M = M1 + M2. The magnetic phases are defined by the three temperatures T1 ≈ 38 K, T2 ≈ 52 K and T3 ≈ 58 K (ref. 53). At low temperatures T < T1, the Fe spins are in the Γ2 phase, in which the ferromagnetic moment M aligns along the crystallographic a axis, and the antiferromagnetic moment L aligns along the c axis. At high temperatures T > T3, the spin system enters the Γ4 phase, in which M aligns along the c axis and L aligns along the a axis. In between these temperatures, L gradually rotates from the c axis to the a axis, first through the b–c plane (Γ12 phase, T1 < T < T2) and then through the a–c plane (Γ24 phase, T2 < T < T3). HoFeO3 features extremely low damping of magnon modes54 and strong linear magneto-optics49, which is one of the signatures of altermagnetism55.
The HoFeO3 sample measured in our experiments is c-cut and has a thickness of about 60 μm. Our experiments are typically carried out at temperatures close to the temperature T1, such that the equilibrium weak ferromagnetic moment is aligned along the a axis. The sample’s a axis is oriented horizontally, along with a small magnetic field to saturate the domains.
Experimental setup
A detailed layout of the experimental setup is depicted in Extended Data Fig. 1.
The HoFeO3 sample is placed in an open-cycle cryostat, which is cooled with liquid helium, down to temperatures of 5 K. The temperature is regulated by controlling the helium flow and the heat applied using a temperature controller. We use an electromagnet to apply a small magnetic field of 25 mT to saturate the magnetic domains. The laser pulses are generated by a Spectra-Physics Ti:sapphire laser amplifier, which outputs photons with a wavelength of 800 nm, with a repetition rate of 1 kHz. Most of the generated 800-nm light is guided into an optical parametric amplifier, which converts the 800-nm light into photons with other wavelengths. The optical parametric amplifier allows us to tune the photon energy in the UV–Vis–NIR range. The remainder of the 800-nm light is attenuated and guided through a delay line, and after attenuation with neutral density filters illuminates a BBO crystal that converts the 800-nm photons to 400-nm photons through second-harmonic generation. The residue of the 800-nm light is removed with a Schott BG39 filter. The intensity of the pump pulses can be tuned with a combination of polarizer and half-wave plate. The linear polarization of the light is rotated with a half-wave plate (λ/2). A lens is placed such that the sample is nearly in the focal plane of the lens. The 400-nm light excites the spin dynamics in the HoFeO3.
We probe the spin dynamics magneto-optically by measuring the polarization rotation of the probe light reflected from the sample. For our probe pulse, we use the output of the optical parametric amplifier. Typically, in our experiments we choose 660 nm for the probe wavelength. This light is tightly focused in the area illuminated by the pump pulse. We confirmed with a knife edge measurement that the focal spot of the probe pulse (≈100 μm) is much smaller than the focal spot of the pump pulse (≈1 mm). Typically, we pump with a pulse energy of approximately 0.8 μJ, which results in a fluence of approximately 0.1 mJ cm−2. The reflection from the surface of the sample is captured and collimated by a second lens. The light is guided and focused into a homemade pair of balanced photodetectors. A Wollaston prism is used to separate the orthogonal polarizations of the light.
The dynamical Kerr rotation is obtained by tracking the amplified difference of the signals in the photodiodes as a function of the time delays. The signals are analysed with a lock-in amplifier. The lock-in reference is coupled to the modulation frequency of the pump (500 Hz) and allows our results to be sensitive only to the pump-induced changes of the signal.
For our double-pump experiment, we use a beam splitter to split the pump pulse into two pump pulses, in a Michelson interferometer arm. The distance between the mirrors and beam splitters is adjustable, thereby allowing variation of the time delay between the two pumps. For convenience, we refer to the two orthogonal interferometer arms as stage 1 and stage 2. We carry out our double-pump experiment in two distinct configurations. In the first configuration, the chopper is placed after the interferometer arms (chopper 2). As a result, both pumps are modulated at 500 Hz, and our results are sensitive to the signal induced by both pumps.
In the other configuration, the chopper is placed in one of the arms of the Michelson interferometer (stage 1, chopper 1). In this configuration, only the pump pulses travelling through this arm are modulated at the 500-Hz lock-in reference frequency. Hence, in our measurements, we directly observe only the signal induced by the pump from stage 1. The pump from stage 2 affects only the signal from stage 1, but we cannot directly detect the pump-induced signal from this stage. Hence, our results are sensitive only to the nonlinear modulations of this second pump, and linear effects such as the coherent superposition of the spin waves launched by the separate pumps are not observable in this configuration.
The concept of such a 2D spectroscopy experiment is illustrated in Extended Data Fig. 2. In the left-hand side of the figure, we consider the case of only interference, or no interaction between the modes. We see that the amplitude of the spin wave will be modulated only at the frequencies of the modes themselves, resulting in the emergence of the diagonal peaks (f0, f0) and (fk, fk). On the other hand, if a peak appears at the off-diagonal (f0, fk) as in our experiments, this indicates that the amplitude of the modes oscillating at frequency fk is modulated with frequency f0. Such an effect can be explained only as being due to the presence of a nonlinear torque due to the coupling between the photon and the magnon. As compared to a single-pump experiment, the second pump pulse exerts an additional torque on the already deflected spin that allows the generation of oscillations at frequencies fk from the frequency f0, thus up-conversion. We note that this up-conversion is impossible in the single-pump experiment, as the duration of the excitation is much shorter than the precession period.
Analysis procedure
We measure 2D scans by first setting the pump delay stage to a fixed time delay, and measuring the spin dynamics by varying the probe delay stage. In our experimental setup (Extended Data Fig. 1), the light used as the probe has a fixed arrival time. The pumps pass through a delay line and are split in a Michelson interferometer, in which one of the arms is static and the other is moved to vary the time delay between the pumps. In this configuration, varying the time delay between the two pumps will also affect the temporal overlap between pump 2 and the probe. Hence, care should be taken when creating the 2D Fourier spectra.
As the measured signals are commonly associated with a step after the arrival of the second pump (due to light-induced phase transition), this offset distorts the Fourier analysis by adding zero-frequency components. We remove this offset by fitting data after the step with a polynomial, and subtract this fit from the data. We shift our starting point of the fit according to the delay between the two pumps, such that only the data after arrival of the second pump are fitted. In the main text, these subtracted data are shown in Fig. 3a. Although our focus is on the modulation after the arrival of the second pump, in the following section we also compare the amplitudes of the spin waves before and after the arrival of the second pump to illustrate the induced modulation by the second pump.
We checked for the occurrence of any artefacts in our analysis. For the case when both pumps are modulated, we checked this by taking the two separate reference scans obtained by measuring with the single pumps from both arms of the Michelson interferometer. We fit the data a few picoseconds after the pump–probe overlap, to remove low-frequency artefacts occurring due to the step. We temporally shift one of these reference scans according to the experimental pump delays while keeping the other scan static, and add both the scans for each experimentally used time delay between the pumps, thus creating a temporal 2D map. We carry out the 2D Fourier transform of these data, to obtain the 2D spectrum with respect to pump and probe delay, as shown in Fig. 3d. From this analysis procedure we obtained a single diagonal peak corresponding to the interference of the two spin waves, indicating that the off-diagonal peaks we found are not a result of artefacts in our analysis procedure.
For the case when the single static pump is modulated, we again created a temporal 2D map. As the signals from the time-shifted pump are not visible in this configuration, we have the signal for each time delay. We carry out the 2D Fourier transform and obtain the spectrum in Extended Data Fig. 7d. As expected, we observe no modulation in this reference scan, resulting in only zero-frequency features along the pump delay. The absence of any diagonal and off-diagonal peaks indicates that our off-diagonal peak is explained by a physical nonlinearity. It also highlights that the observed diagonal peak in Extended Data Fig. 7c may be explained by a similar nonlinearity.
