croak_{2} belongs to the class of antiferromagnets with a rutile-type crystallographic lattice^{27}described by *P*4_{2}/*m**n**m* space group. The primitive many-atomic cell forms 11 optical phonon modes^{28,29}: *or*_{1g}â€‰âŠ•â€‰*or*_{2g}â€‰âŠ•â€‰*or*_{2u}â€‰âŠ•â€‰*B*_{1g}â€‰âŠ•2*B*_{1u}â€‰âŠ•â€‰*B*_{2g}â€‰âŠ•3*E*_{u}â€‰âŠ•â€‰*E*_{g}. The lowest Raman active phonon mode has *B*_{1 g} symmetry and is centered at a frequency of *p*_{ph} = 1.96 THz at *T* = 6 K. It is worth noting that the frequency of this mode remains the same in the external magnetic field up to *Î¼*_{0}*H*_{ext}= 7T.

Co. Rolls^{2+} ions are aligned along the crystallographic c axis below the NÃ©el temperature of *T*_{N}= 39 K. In our experiment, we use a 500 Âµm thick CoF single crystal_{2} plate cut perpendicular to the c axis. If no magnetic field is applied, there is a frequency-degenerate double antiferromagnetic resonance *p*_{0} = 1.14 THz (at 6 K). By applying an external magnetic field, one breaks the degeneracy of the corresponding magnon mode. For example, if a magnetic field is applied along the c-axis with its value below the spin-flop field threshold of *Î¼*_{0}*H*_{ext} = 14T^{30}the frequencies of the two degenerate magnon modes obey the relation *p*_{m} = *p*_{0} Â± *Î³**H*_{ext}where *Î³*is the gyromagnetic ratio.

Using the intense, spectrally dense superradiant THz source TELBE, located at the Helmholtz-Zentrum Dresden-Rossendorf^{31} in combination with external magnetic fields *H*_{ext} we have the unique ability to selectively pump the magnon by controlling its center frequency*p*_{m}tuning in *H*_{ext} at our disposal. This setup allows you to explore the spin-lattice interaction in the vicinity of the Fermi 2 resonance *p*_{m} = *p*_{ph}by monitoring the phonon response. As shown earlier^{24}as long as both magnon and phonon retain their coherence, they will induce transient optical anisotropy in the initially isotropic (ab) plane of the antiferromagnet and thus modulate different components of the dielectric permittivity^{32}which we track from probe polarization changes^{24}. The experimental geometry and characteristics of the THz pulse are given in Fig. 2a. The THz field strength was estimated to be of the order of 100 kV/cm. THz-induced rotation of the probe polarization is measured in external magnetic fields up to *Î¼*_{0}*H*_{ext}= 7 T is shown in figure 2b. Apparently, the time field signal exhibits clear oscillations that are significantly affected by an external magnetic field. Here, the magnon amplitude dominates from âˆ’10 to 70 ps while for the range 70â€“120 ps, â€‹â€‹the phonon appears to be exclusively present. In Supplementary Section A, we illustrate our observations with the sinusoidal fitting of the corresponding magnon and phonon oscillations.

Performing the Fourier transform of the entire time domain range (âˆ’10 to 120 ps) in Figs. 3a reveals the presence of frequency-oscillating Magnon response *p*_{m} and a second peak, distant at *B*_{1 g}phonon frequency *p*_{ph}. For magnetic fields *Î¼*_{0}*H*_{ext} = 0 T and *Î¼*_{0}*H*_{ext} = 7 T, the THz pump spectrum barely covers the magnon mode and the THz-induced polarization rotation contains a substantial contribution of the spectrally broad forced magnetic response closely following the magnetic field of the THz pulse, see Fig. 2b. Furthermore, no phonon-induced dynamics is observed in these fields, implying that the nonlinear excitation of phonons via the mechanism described in ref. ^{1} does not play an important role here. Closer to the Fermi resonance for magnetic fields between *Î¼*_{0}*H*_{ext}= 2â€“5 T, we observe the low-energy magnon branch *p*_{m}with its frequency decreasing linearly with the external magnetic field. Surprisingly, Magnon’s strongest climax in *Î¼*_{0}*H*_{ext}= 5 T does not correspond to the strongest phonon peak, revealing complex dynamics in the vicinity of the magnon-phonon Fermi resonance.

The most special feature is observed in *Î¼*_{0}*H*_{ext}= 3.5 T, see Fig. 3a. First, the phonon peak amplitude for dynamics in this magnetic field is significantly reduced relative to the peak amplitude for *Î¼*_{0}*H*_{ext}= 3 T and 4 T. Second, the phonon spectrum at *Î¼*_{0}*H*_{ext}= 3.5 T becomes wider. In fact, this resembles a spectral line splitting reported for purely phononic^{8} or simply magnonic^{13} systems under continuous wave pumping in the vicinity of their Fermi resonances. To capture the energy redistribution, we integrate the area under the phonon spectra for different external magnetic fields*H*_{ext}over the shaded range of 1.9-2.0 THz and obtain the phonon resonance curve behavior as shown in Figs. 3b. Here, the phonon resonance line is clearly asymmetric with a pronounced drop in*Î¼*_{0}*H*_{ext}= 3.5 T indicating non-trivial magnon-phonon energy exchange. In the next section, we assign this feature to the unique magnon-phonon Fermi resonance standards. Furthermore, we treat magnon and phonon lifetimes and reveal the role of nonlinear coupling constants for raising the system to the strong nonlinear magnonâ€“phonon coupling regime.

To get a better insight into the observed fingerprint of the magnon-phonon Fermi resonance, we undertook a simulation of the signatures of the nonlinear coupled dynamics in the CoF_{2}. Conventionally, antiferromagnetic spins are described in terms of the NÃ©el vector **L**=**M**_{1}âˆ’ **M**_{2}where the net magnetic moments **M**_{1,2}were formed by Co^{2+} ions at the center and corners of the unit cell, respectively^{33}. The movement of B_{1 g}the phonon is characterized by the phonon coordinate*Î¸*_{ph}. Perturbations from the ground state are represented as **L** (*t*) = ( *l*_{x}(*t* ), *l*_{y}(*t*), *L*_{0}). here, *L*_{0}describes the NÃ©el ground state vector. The Fermi resonance symmetry rule implies that B_{1 g}phonon symmetry (*x*^{2}– *y*^{2}) should follow the double magnon excitation symmetry. Therefore, the corresponding nonlinear term can be introduced into the Lagrangian as \(\Phi=-\alpha ({l}_{x}^{2}-{l}_{y}^{2}){\theta }_{ph}\)^{15}where *a*represents the nonlinear coupling constant between the magnon and phonon subsystems. We assume that the magnetic field of the THz pulse *h*_{THz}= ( *h*_{x}, *h*_{y}0) is polarized exclusively in the sample plane and solves the Lagrange-Euler equations, taking into account circularly polarized magnon states *l*_{Â±}= *l*_{x}Â± *The**l*_{y}. The resulting coupled equations can be written as

$$\frac{{d}^{2}{l}_{+}}{d{t}^{2}}+2{\zeta }_{{{{{{{{\rm{m} }}}}}}}}\frac{d{l}_{+}}{dt}+\left({\omega }_{0}^{2}}{\gamma }^{2}{H }_{{{{{{{\rm{ext}}}}}}}}^{2}\right){l}_{+}+2\gamma i{H}_{{{{{ { {{{\rm{ext}}}}}}}}\frac{d{l}_{+}}{dt}=-2\alpha {\theta }_{{{{{{{{{ \ rm{ph}}}}}}}}{l}_{-}+\gamma \frac{d}{dt}({h}_{y}-i{h}_{x}), $$

(1)

$$\frac{{d}^{2}{l}_{-}}{d{t}^{2}}+2{\zeta }_{{{{{{{{\rm{m} }}}}}}}}\frac{d{l}_{-}}{dt}+\left({\omega }_{0}^{2}}{\gamma }^{2}{H }_{{{{{{{\rm{ext}}}}}}}}^{2}\right){l}_{-}-2\gamma i{H}_{{{{{ { {{{\rm{ext}}}}}}}}\frac{d{l}_{-}}{dt}=2\alpha {\theta }_{{{{{{{{\ rm{ ph}}}}}}}}{l}_{+}+\gamma \frac{d}{dt}({h}_{y}+i{h}_{x}),$ $

(2)

$$\frac{{d}^{2}{\theta }_{ph}}{d{t}^{2}}+2{\zeta }_{{{{{{{\rm{ph } }}}}}}}}\frac{d{\theta }_{ph}}{dt}+{\omega }_{{{{{{{{\rm{ph}}}}}}}} } }^{2}{\theta }_{{{{{{{\rm{ph}}}}}}}}=-\alpha \left({l}_{+}^{2} +{l }_{-}^{2}\right),$$

(3)

where *Ï‰*_{The}= 2 *Ï€**p*_{The} and Gilbert damping factors *Î¶*_{The} with*The*= “m” or “ph” account for the magnon or phonon subsystem, respectively. The second term on the right-hand side of Ex. (1)â€“(2) represents the linear Zeeman torque^{34}, while the nonlinear coupling can be introduced in Eqs. (1)â€“(3) as the derivative of Î¦ in the corresponding order parameter. These terms represent the nonlinear mutual perturbation of the magnon (phonon) subsystem by the phonon (magnon) subsystem. The experimental THz pulse waveform measured by electro-optical sampling (see Fig. 2a) is introduced as the driving force in our simulation. The derivation of Eqs. (1)â€“(3) is given in Supplementary Material, Section B.

In Fig. 4a,b, the simulated phonon spectra are shown for the external magnetic fields as applied in our experiment. Here, panel (a) corresponds to the weak coupling case with*a*= 0.07, while panel (b) represents the case of strong coupling with*a*= 7. A drastic change in the spectra for the magnetic fields of*Î¼*_{0}*H*_{ext}= 3, 3.5 and 4 T can be seen, highlighting the role of the Fermi resonance. In the weak coupling regime of Fig. 4a, the spectrum contains no splitting and the peak amplitude has only a single near maximum*Î¼*_{0}*H*_{ext}= 3.5 T. The corresponding phonon weight shown in panel (c) is also symmetric. In the strong coupling regime (see Fig. 4b), at*Î¼*_{0}*H*_{ext}= 3.5 T a splitting of the phonon line is observed resulting in a decrease in the phonon weight. Moreover, the absolute peak of the phonon weight is shifted to 3 T. These features are well captured by our experimental data set (see Fig. 3) showing good agreement with our model. We elaborate on the clear effect of our simulation parameters in Supplementary Material, Section C.

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