# Coherent THz Spin Dynamics in Antiferromagnets
Beyond the Approximation of the Néel vector

F. Formisano  0000-0001-8789-4518 Institute for Molecules and Materials,
Radboud University, 6525 AJ Nijmegen, The Netherlands  T. T. Gareev
0000-0002-1678-2444 Institute for Molecules and Materials, Radboud University,
6525 AJ Nijmegen, The Netherlands  D. I. Khusyainov  0000-0003-1332-4146
Institute for Molecules and Materials, Radboud University, 6525 AJ Nijmegen,
The Netherlands  A. E. Fedianin  0000-0002-5093-6830 Ioffe Institute, Russian
Academy of Sciences, 194021 St. Petersburg, Russia  R. M. Dubrovin
0000-0002-7235-7805 Ioffe Institute, Russian Academy of Sciences, 194021 St.
Petersburg, Russia  P. P. Syrnikov  Ioffe Institute, Russian Academy of
Sciences, 194021 St. Petersburg, Russia  D. Afanasiev  0000-0001-6726-3479
Institute for Molecules and Materials, Radboud University, 6525 AJ Nijmegen,
The Netherlands  R. V. Pisarev  0000-0002-2008-9335 Ioffe Institute, Russian
Academy of Sciences, 194021 St. Petersburg, Russia  A. M. Kalashnikova
0000-0001-5635-6186 Ioffe Institute, Russian Academy of Sciences, 194021 St.
Petersburg, Russia  J. H. Mentink  0000-0002-4082-1984 Institute for Molecules
and Materials, Radboud University, 6525 AJ Nijmegen, The Netherlands  A. V.
Kimel  0000-0002-0709-042X<EMAIL_ADDRESS>Institute for Molecules and
Materials, Radboud University, 6525 AJ Nijmegen, The Netherlands

###### Abstract

Controlled generation of coherent spin waves with highest possible frequencies
and the shortest possible wavelengths is a cornerstone of spintronics and
magnonics. Here, using the Heisenberg antiferromagnet $\mathrm{RbMnF}_{3}$, we
demonstrate that laser-induced THz spin dynamics corresponding to pairs of
mutually coherent counter propagating spin waves with the wavevectors up to
the edge of the Brillouin zone cannot be understood in terms of magnetization
and antiferromagnetic (Néel) vectors, conventionally used to describe spin
waves. Instead, we propose to model such spin dynamics using the spin
correlation function. We derive a quantum-mechanical equation of motion for
the latter and emphasize that, unlike the magnetization and antiferromagnetic
vectors the spin correlations in antiferromagnets do not exhibit inertia.

Exploring efficient routes for excitation of coherent spin waves with the
shortest possible wavelength and the highest achievable frequency is one of
the major challenges of today’s spintronics, magnonics, and magnetic data
storage [1; 2; 3; 4; 5]. Among magnetic materials, antiferromagnets have the
highest frequencies of spin dynamics in the THz range, which is a significant
advantage for becoming a crucial ingredient of ultra-high speed spintronic
devices [6; 7; 8; 9; 10; 11]. It has been demonstrated that ultrashort laser
pulses can be employed for the generation of coherent spin waves in
practically all classes of magnetically ordered materials [12; 13; 14; 15].
The wavelength of the optically excited spin wave is, in principle, defined by
the size of the illuminated volume, and can be further decreased by
introducing inhomogeneities [16; 17]. However, the spin waves thus generated
will still have frequencies and wavevectors close to the center of the
Brillouin zone. An appealing alternative approach for the generation of spin
waves with larger k-vectors is based on concomitant excitation of the two
counter-propagating waves [18; 5]. In antiferromagnets, such mutually coupled
pairs of spin waves excited over the whole Brillouin zone form a so-called
two-magnon mode, which couples to light via Raman scattering [19], and can be
triggered by ultrashort laser pulses [20; 21].

Until now, the approximation, which neglects individual strongly coupled
quantum mechanical spins and treats a magnet as a continuous medium, has been
fundamental to understanding spin dynamics, including that at ultrafast
timescales [22]. Using such an approximation, the simplest two-sublattice
antiferromagnet can be modeled as two antiferromagnetically coupled
ferromagnets with oppositely directed magnetizations $|\mathbf{M}_{1}|$ =
$|\mathbf{M}_{2}|$ = $M_{0}$, the dimensionless antiferromagnetic vector
$\mathbf{L}$ = ($\mathbf{M}_{1}$ \- $\mathbf{M}_{2}$)/$2M_{0}$ and the net
magnetization $\mathbf{M}$ = ($\mathbf{M}_{1}$ \+ $\mathbf{M}_{2}$)/$2M_{0}$.
The propagation of an antiferromagnetic spin wave with the wavevector
$\mathbf{k}$ and the angular frequency $\Omega_{\mathbf{k}}$ in this
approximation is described as spatio-temporal variations of the orientations
and lengths of L and M. It is clear that upon decreasing the wavelength and
approaching the edge of the Brillouin zone, the approximation based on such
macroscopic vectors should eventually fail since it is substantiated only if
the characteristic length scale of spin wave is much larger than the interspin
distance. Despite this fact, the breakdown has not been examined for
experimentally observed laser-induced spin dynamics of antiferromagnets.

Here, using the example of the Heisenberg antiferromagnet
$\mathrm{RbMnF}_{3}$, we demonstrate that the spin dynamics corresponding to
mutually coherent spin waves with large opposite wavevectors cannot be
understood if the light-spin interaction is modeled in conventional terms of
macroscopic magnetization $\mathbf{M}$ and antiferromagnetic vector
$\mathbf{L}$. Instead, we propose to model such spin dynamics using spin
correlations. We derive the equation of motion for the spin correlation
function and demonstrate that, unlike $\mathbf{M}$ and $\mathbf{L}$, the spin
correlations in antiferromagnets do not exhibit inertia. It means that there
is an optimal pulse duration for excitation of the spin correlation dynamics,
while the macrospin dynamics in the antiferromagnet can be excited impulsively
even by an infinitesimally short pulse.

The excitation of the two-magnon mode relies on the perturbation of exchange
coupling, and the symmetry of the latter is intrinsically linked to the
crystal structure, while for the models in terms of M and L an orientation of
magnetic moments is important. Therefore, it was crucial to choose a material
in which the orientation of the magnetic moments and the resulting
antiferromagnetic vector $\mathbf{L}$ are distinctly different from the main
crystallographic axes. Here we select the cubic fluoroperovskite insulator
$\mathrm{RbMnF}_{3}$ (point group $m\overline{3}m$), which is very close to
the isotropic 3D Heisenberg antiferromagnet below the Néel temperature
$T_{N}=83$ K [23; 24]. The spins ($S=5/2$) of the $\mathrm{Mn}^{2+}$ ions form
two equivalent magnetic sublattices coupled antiferromagnetically. Very weak
magnetic anisotropy aligns the antiferromagnetic vector $\mathbf{L}$ along one
of the $\langle 111\rangle$ directions [25], as shown in Fig. 1(b). The two-
magnon mode in RbMnF3 was observed in spontaneous Raman scattering experiments
[Fig. 1(a)]; has the $E_{g}$ symmetry and frequency 4 THz at low temperature
[26; 27; 28; 29].

To excite coherent spin dynamics corresponding to the two-magnon mode, we
perform a femtosecond two-color magneto-optical pump-probe experiment in the
transmission geometry, as shown in Fig. 1(c). The laser pulses with duration
45 fs and central photon energy 1.55 eV were emitted by the Ti:Sapphire
regenerative amplifier at a repetition rate of 1 kHz and were split into pump
and probe ($\hbar\omega_{pr}=1.55$ eV) beams with an energy per pulse ratio of
10:1. The pump pulses were tuned by an optical parametric amplifier (OPA) to a
central photon energy $\hbar\omega_{p}=1.03$ eV and a fluence of 1.2 mJ/cm2
and then were focused normally onto the sample surface into a spot with a
diameter of 500 $\mu$m. The pump and probe pulses were linearly polarized at
angles $\alpha$ and $\beta$ with respect to the $x\|[100]$ axis, respectively.
The angles were controlled by half-wave plates. The 620 $\mu$m-thick slab
sample with the optical quality of the surfaces was cut perpendicular to the
cubic $z\|[001]$ axis from a single crystal of $\mathrm{RbMnF}_{3}$ grown
using the Czochralski method. The projection of the antiferromagnetic vector
$\mathbf{L}$ on the sample $xy$ plane makes an angle of 45∘ with the $x-$axis,
as shown in Fig. 1(c). The sample was placed in the continuous-flow liquid
helium cryostat. The spin dynamics triggered by an intense pump pulse induces
in the sample a certain type of optical anisotropy [14]. The latter is
detected using a balance detection scheme to measure the induced ellipticity
of the polarization of the probe pulse as a function of the time delay
$\Delta{t}$ between pump and probe pulses. Note that in this detection scheme,
only modulation of the medium parameters which is nearly homogeneous within
the probed area is detected.

Figure 1:  (a) Coupling of two photons with frequencies differing by
$2\Omega_{\mathbf{k}}=\omega_{1}-\omega_{2}$ with a pair of the coherent spin
waves with frequencies $\Omega_{\mathbf{k}}$ and opposite wavevectors
$\mathbf{k}$ throughout the Brillouin zone. The dominant contribution to the
process from the spin waves at the edge of the Brillouin zone is shown. (b)
Crystallographic and magnetic structure of RbMnF3. The arrows indicate
antiferromagnetically ordered spins aligned along the $\langle 111\rangle$
direction. (c) Experimental geometry. Linearly polarized optical pump pulse
excites spin dynamics which is detected via transinet ellipticity of the
optical probe pulse time-delayed by $\Delta t$. The projection of the
antiferromagnetic vector $\mathbf{L}$ on the $xy$ sample plane is shown.

Figure 2(a) shows a temporal evolution of the pump-induced ellipticity in the
antiferromagnetic phase of $\mathrm{RbMnF}_{3}$. The pump and probe pulses
polarizations were $\alpha=0^{\circ}$ and $\beta=45^{\circ}$. One clearly
distinguishes oscillations damped within a few ps. The Fourier spectra of the
time trace for $T=5$ K reveals a clear resonance at 4 THz. The frequency and
amplitude of these oscillations decrease upon heating up to $T_{N}$ [Figs.
2(b,c)]. The frequency and its temperature behavior are in fair agreement with
those of the two-magnon mode obtained in spontaneous Raman scattering
experiments [26; 27; 28; 29]. The detected two-magnon mode corresponds to the
simultaneous excitation of pairs of mutually coherent spin waves with opposite
wavevectors throughout the Brillouin zone dominated by the waves with the
shortest wavelengths and the highest frequencies $\Omega_{\pi/a}$, as shown in
Fig. 1(a) [19; 26; 30; 31; 32]. Thus, we can confidently conclude that the
experimentally detected oscillations correspond to the spin dynamics
originating from the pairs of spin waves with opposite wavevectors at the
edges of the Brillouin zone.

To comprehend the observed spin dynamics, we performed measurements of the
pump-induced ellipticity at various angles of the pump $\alpha$ and probe
$\beta$ polarizations at $T=5$ K [Fig. 3]. The most pronounced oscillations of
the probe ellipticity are observed when the pump polarization is along the
crystallographic $x$ and $y$ axes ($\alpha=0,~{}90^{\circ}$) and the probe
polarization is at an angle $\beta=45^{\circ},~{}135^{\circ}$. Moreover, it is
clearly seen that the phase of the oscillations is shifted by $\pi$ when
either pump or probe polarization is rotated by $90^{\circ}$ [Figs. 3(a,b)].
To gain further insight, we performed two series of experiments for fixed pump
($\alpha=0^{\circ}$) and rotated probe ($\beta=0^{\circ}\ldots 360^{\circ}$)
polarization and for fixed probe ($\beta=45^{\circ}$) and rotated pump
($\alpha=0^{\circ}\ldots 360^{\circ}$) polarizations. The variations of the
signed Fourier amplitudes of the obtained waveforms are fairly well described
by $\cos{2\alpha}$ and $\sin{2\beta}$ [Fig. 3(d)].

Figure 2:  (a) Transient probe ellipticity at different temperatures below
$T_{N}=83$ K. Pump and probe pulses are linearly polarized at angles
$\alpha=0^{\circ}$ and $\beta=45^{\circ}$, respectively. Temperature
dependences of the (b) frequency and (c) amplitude of the oscillations seen in
(a). (d) Dependence of the amplitude of the oscillations on the pump pulse
fluence at $T=5$ K. The solid lines in (b-d) are guides for the eye.

In contrast to earlier reported results of ultrafast spin dynamics [22], the
observed polarization dependencies cannot be explained in terms of the
macroscopic magnetization $\mathbf{M}$ and antiferromagnetic vector
$\mathbf{L}$. In particular, it was argued that excitation of pairs of
mutually coherent spin waves with opposite wavevectors in isostructural
antiferromagnetic fluoroperovskite $\mathrm{KNiF}_{3}$ [33] modulate the
length of the antiferromagnetic vector $\mathbf{L}$ at the two-magnon
frequency homogeneously across the excitation area. The latter is detected in
experiments as oscillations of magnetic linear birefringence at the same
frequency. In $\mathrm{KNiF}_{3}$ $\mathbf{L}$ is along one of the main
crystallographic axes, e.g., the $x$ axis, while in the case of
$\mathrm{RbMnF}_{3}$, the projection of $\mathbf{L}$ on the $xy$ sample plane
is rotated over $45^{\circ}$ from the $x$ and $y$ axes. Nevertheless, the pump
polarization dependences are exactly the same in both antiferromagnets and
possess maxima when the pump is polarized along one of the main
crystallographic axes. Moreover, as the maximum signal in $\mathrm{RbMnF}_{3}$
is observed when the probe is initially polarized along or perpendicular to
the projection of $\mathbf{L}$ on the sample plane, the corresponding optical
anisotropy does not originate from the oscillations of the length of
$\mathbf{L}$ and cannot be understood in terms of magnetic linear
birefringence. The probing geometry could agree with the transverse dynamics
of $\mathbf{L}$ detected via such effect, which, however, yields the
paradoxical conclusion that $\mathbf{L}$ homogeneously oscillates at the
frequency twice as high as the maximal spin wave frequency (Suppl. Mater.
[34]).

To describe the observed polarization dependences, we have to go beyond the
conventional approach in terms of dynamics of L. We assume the
antiferromagnetic Heisenberg exchange interaction with Hamiltonian
$\hat{H}_{0}=J\sum_{\bm{\delta}}\hat{C}(\bm{\delta})$, where $J$ is the
exchange coupling constant between the nearest neighbor magnetic ions,
$\bm{\delta}$ is the vector between them. For a cubic crystal,
$\hat{C}(\bm{\delta})=2\sum_{i}\hat{\mathbf{S}}(\mathbf{r}_{i})\hat{\mathbf{S}}(\mathbf{r}_{i}+\bm{\delta})$,
where $\mathbf{r}_{i}$ is a vector to site $i$. The expectation value of
$\hat{C}(\bm{\delta})$ gives the equal time spin correlation
$\langle\hat{S}(\mathbf{r}_{i},t)\hat{S}(\mathbf{r}_{i}+\bm{\delta},t)\rangle$
along $\bm{\delta}$, where $\langle\ldots\rangle$ denotes averaging [35]. For
a cubic Mott insulator, the electric field
$\mathbf{E}=\left(E^{x},E^{y},E^{z}\right)$ of light propagating along the
$z-$axis and polarized in the $xy-$plane perturbs the Hamiltonian according to
the phenomenological expression [4]

$\displaystyle\Delta\hat{H}$
$\displaystyle=A(\omega_{p})\left[\left(E^{x}\delta^{x}\right)^{2}\hat{C}(\delta^{x})+\left(E^{y}\delta^{y}\right)^{2}\hat{C}(\delta^{y})\right],$
(1)

where $A(\omega)\delta^{\nu}\delta^{\nu}\hat{C}(\delta^{\nu})$ is proportional
to the dielectric permittivity at frequency $\omega$ with spectral dependence
$A(\omega)$ defined by exchange coupling, optical bandgap, and crystal
structure. We note that Eq. (1) agrees with the expression for the
perturbation of the Hamiltonian in the case of two-magnon Raman scattering in
an antiferromagnet [27].

Figure 3:  (a,b) Experimental and (c) calculated transient probe ellipticity
as excited and detected at (a) fixed pump polarization $\alpha=0$ and the
probe polarizations $\beta=45,135^{\circ}$, (b,c) fixed probe polarization
$\beta=45^{\circ}$ and the pump polarizations $\alpha=0,90^{\circ}$. (d) The
signed Fourier amplitude of 4 THz oscillations as a function of pump
polarization angle $\alpha$ at the probe polarization angle $\beta=45^{\circ}$
(purple symbols), and probe polarization angle $\beta$ at the pump
polarization angle $\alpha=0^{\circ}$ (red symbols). The sign change
corresponds to the waveform phase shift by $\pi$. The purple and red lines
show the fit of the experimental data by $\cos 2\alpha$ and $\sin 2\beta$
functions, respectively. The experimental data are obtained at $T=5$ K.

To derive the equation for the dynamics of spin correlations, we consider
small transverse spin-deflections with respect to the collinear
antiferromagnetic ground state. It is convenient to represent these spin
deflections in reciprocal space, since spin correlations for different bonds
are not independent:
$[\hat{C}_{i}(\delta^{\nu}),\hat{C}_{j}(\delta^{\mu})]\neq\delta_{ij}\delta_{\nu\mu}$,
with $\delta_{ij}$ being a Kronecker delta. Indeed, spin correlations can
share the same spin operator, for example,
$\hat{\mathbf{S}}(\mathbf{r}_{i}+\bm{\delta}^{\nu})=\hat{\mathbf{S}}(\mathbf{r}_{j}+\bm{\delta}^{\mu})$.
This means that perturbation of exchange bonds along the $x$ axis also
triggers dynamics of spin correlations along the $y$ and $z$ axes, etc.
Although the spin correlation itself is a scalar, not all components of the
spin correlation feature the same dynamics. Analogously to the
antiferromagnetic vector $\mathbf{L}$, we therefore write the different
components of the spin correlation as a vector
$\hat{\mathbf{C}}_{\mathbf{k}}=\left(\hat{C}_{\mathbf{k}}^{(1)},\hat{C}_{\mathbf{k}}^{(2)},\hat{C}_{\mathbf{k}}^{(3)}\right)$,
where

$\displaystyle\hat{C}^{(1)}_{\mathbf{k}}$
$\displaystyle=\frac{1}{2S}\left(\hat{S}^{X}_{\mathbf{k}}\hat{S}^{X}_{\mathbf{-k}}+\hat{S}^{Y}_{\mathbf{k}}\hat{S}^{Y}_{\mathbf{-k}}\right),$
$\displaystyle\hat{C}^{(2)}_{\mathbf{k}}$
$\displaystyle=\frac{1}{2S}\left(\hat{S}^{X}_{\mathbf{k}}\hat{S}^{Y}_{\mathbf{-k}}-\hat{S}^{Y}_{\mathbf{k}}\hat{S}^{X}_{\mathbf{-k}}\right),$
(2) $\displaystyle\hat{C}^{(3)}_{\mathbf{k}}$
$\displaystyle=\frac{1}{4S}\left((\hat{S}^{X}_{\mathbf{k}})^{2}+(\hat{S}^{Y}_{\mathbf{k}})^{2}+(\hat{S}^{X}_{\mathbf{-k}})^{2}+(\hat{S}^{Y}_{\mathbf{-k}})^{2}\right),$

with $X,Y$ being mutually orthogonal components perpendicular to the spin
quantization axis $Z$. Here we kept only the leading order, i.e., quadratic
spin deflections [34] and static terms $\propto-S^{2}$ are omitted, which is
justified by the linear dependence of the observed effect on the pump fluence
[Fig. 2(d)]. Further, although temperature is not included in the theory, the
results described below do not rely on explicit evaluation of the equations in
the $T=0$ ground state. Note that in equilibrium,
$\langle\hat{C}^{(2)}_{\mathbf{k}}\rangle=0$. However, it can be nonzero
during dynamics. In terms of the vector components, the leading order
deflections to the full correlation are given by

$\displaystyle\hat{C}_{\mathbf{k}}(\delta^{\nu})=4S\left(\hat{C}^{(3)}_{\mathbf{k}}+\cos(k^{\nu}\delta^{\nu})\hat{C}^{(1)}_{\mathbf{k}}\right),$
(3)

which shows that the correlation function features a term
$\propto\hat{C}^{(1)}$ not present in the dynamics of $\Delta
L^{Z}\propto\sum_{\mathbf{k}}\langle\hat{C}^{(3)}_{\mathbf{k}}\rangle$ [33].
The dependence of the correlation on $\bm{\delta}$ stems entirely from the
$\propto\hat{C}^{(1)}$ term, that was ignored before. From the commutations
relations for the spin operators, it follows that
$\hat{\mathbf{C}}_{\mathbf{k}}$ satisfies the cross product in hyperbolic
space,
$\hat{\mathbf{C}}_{\mathbf{k}}\times\hat{\mathbf{C}}_{\mathbf{k}}=i\hbar\hat{\mathbf{C}}_{\mathbf{k}}$,
featuring a minus sign for the terms in the $\hat{C}^{(3)}_{\mathbf{k}}$
component as compared to the ordinary cross product [36]. Hence, from
Heisenberg equations of motion, the dynamics of the correlations in reciprocal
space is determined by

$\displaystyle\hbar\frac{\partial\hat{\mathbf{C}}_{\mathbf{k}}}{\partial
t}-\hat{\mathbf{C}}_{\mathbf{k}}\times\left(-\frac{\partial\hat{H}_{\mathbf{k}}}{\partial\hat{\mathbf{C}}_{\mathbf{k}}}\right)=0,$
(4)

where

$\displaystyle\hat{H}_{\mathbf{k}}=\left(J+A(\omega_{p})\left(E^{x}\delta^{x}\right)^{2}\right)\hat{C}_{\mathbf{k}}(\delta^{x})$
$\displaystyle\quad\quad+\left(J+A(\omega_{p})\left(E^{y}\delta^{y}\right)^{2}\right)\hat{C}_{\mathbf{k}}(\delta^{y})+J\hat{C}_{\mathbf{k}}(\delta^{z}).$
(5)

Solving this equation of motion for the impulsive perturbation $\Delta\hat{H}$
gives precession of $\hat{\mathbf{C}}_{\mathbf{k}}$ with dominating frequency
$\approx 2zJS/\hbar$ ($z=6$ is the number of nearest neighbors), which
corresponds to the frequency of the two-magnon mode [see Fig. 3(c) and Eqs.
(19,21) in Suppl. Mater. [34]]. Thus, we derived the equation of motion for
spin correlations which describes the dynamics of this parameter, in agreement
with our and previously reported experiments. We note, that, unlike M and L in
antiferromagnets [37], the dynamics of spin correlation is described by the
differential equation of the first order and hence the spin correlations do
not show inertia.

From Eq. (4), we see that the polarization dependence of the pump-induced spin
correlations is dictated by the light-induced perturbations of the effective
fields
$-{\partial\hat{H}_{\mathbf{k}}}/{\partial\hat{\mathbf{C}}_{\mathbf{k}}}$. In
Suppl. Mater. [34] we demonstrate that for the experimental geometry
considered one then obtains a polarization dependence $\propto\cos{2\alpha}$,
in agreement with what is observed experimentally [Fig. 3(d)].
Phenomenologically, this polarization dependence can be understood as follows.
From Eq. (1) one sees that an inequality $E^{\nu}\neq 0$ gives
$\partial\Delta\hat{H}/\partial\hat{C}(\delta^{\nu})\neq 0$. It means that
pumping the antiferromagnet by a laser pulse polarized along one of the main
crystallographic axes brings the system out of the thermodynamic equilibrium,
and finding the new equilibrium will result in a change of the spin
correlations. Such changes in the spin correlation functions contribute to the
dielectric permittivity as
$\Delta\varepsilon^{\mu\nu}=\partial^{2}\langle\Delta\hat{H}\rangle/\partial
E^{\nu}\partial E^{\mu}$, which may lead to a pump-induced anisotropy
$\varepsilon^{xx}-\varepsilon^{yy}\neq 0$ in the initially optically isotropic
cubic system. For the $m\overline{3}m$ point group, we obtain the pump-induced
change of the dielectric permittivity
$\varepsilon^{xx}-\varepsilon^{yy}=2gE^{2}\cos{2\alpha}$, where
$g=\xi^{xxxx}-\xi^{yyxx}$, and $\bm{\xi}$ is the phenomenological polar tensor
[38]. In the range of low absorption, such pump-induced change of the
dielectric permittivity results in the ellipticity of the probe pulse which
obeys $\sin 2\beta$ dependence on the incoming probe polarization, observed in
the experiment [Fig. 3(d)]. Hence, the model based on spin correlation
reproduces the experimental observation in terms of symmetry of excitation and
detection.

Furthermore, the model correctly describes the linear dependence of the effect
on the pump pulse fluence [Fig. 2(d)], since the perturbation of the effective
field in Eq. (4) is linear in the pump pulse fluence $\propto(E^{\nu})^{2}$.
This contrasts with the result one obtains considering dynamics of L. Indeed,
the spin wave frequencies $\Omega_{\mathbf{k}}$ are much lower than those of
the pump photons $\omega$, and the spin wave excitation involves a nonlinear
process of down conversion $\omega_{1}-\omega_{2}=2\Omega_{k}$ [see Fig.
1(a)]. Hence, in the lowest order, a light-induced torque acting on
$\mathbf{M}$ or $\mathbf{L}$ and the amplitude of the excited spin wave are
linear with respect to the pump fluence, as observed for the optically-driven
spin waves with low k [12; 37]. As the spin wave wavevectors $k\approx\pi/a$
are much larger than the wavevector of light $q\approx 0$, they can be
detected only in pairs such that $\mathbf{k}-\mathbf{k}=\mathbf{q}\approx 0$.
Therefore, the signal detected by the probe pulse can only originate from a
product of the spin-wave amplitudes. Hence, the detected signal must be
quadratic with respect to the pump fluence (see Suppl. Mater. [34]).

In our model [Eqs. (1,4)], when the electric field of the pump pulse is along
one of the main crystallographic axes, it off-resonantly excites virtual
charge transfer transitions between $\mathrm{Mn}^{2+}$ and $\mathrm{F}^{1-}$
ions and, thus, effectively changes the exchange interaction between
$\mathrm{Mn}^{2+}$ spins [39]. As a result, the spin correlation function
$\hat{C}(\bm{\delta})$ is impulsively altered along the corresponding
crystallographic axis. In the following, the spin correlation function along
this axis will oscillate at the frequency of two-magnon mode which is defined
by the energy of the exchange interaction taking twice. The correlations along
the two other axes are affected as well, since different correlations can
comprise the same spins. Obeying the energy conservation law, spin
correlations along the two other crystallographic axes oscillate with the
opposite phase, and the total amplitude along all three axes equals zero.
Naturally, such kind of dynamics cannot be comprehended in terms of macrospins
$\mathbf{L}$ and $\mathbf{M}$, which do not distinguish anisotropic changes of
spin-spin coupling [Suppl. Mater. [34]]. Despite this fact, the dynamics
manifests itself in the modulation of the symmetric part of the dielectric
permittivity which distinguishes the spin correlations along different axes.
Thus, the spin dynamics is probed by measuring dynamic linear birefringence
(for details see Suppl. Mater. [34]). We stress that the considered scenario
for excitation and detection of the two-magnon mode holds for the isotructural
KNiF3 [21; 33] as well. The model suggested in those papers remains valid but
represents the next-order effect involving further lowering of the symmetry or
strong magnetic anisotropy [35].

In conclusion, we have demonstrated a regime of ultrafast spin dynamics in
antiferromagnetic $\mathrm{RbMnF}_{3}$, where the models considering the
antiferromagnetic vector $\mathbf{L}$ and the net magnetization $\mathbf{M}$,
fail. Instead, we propose to describe the regime in terms of the spin
correlation function, derive the corresponding equations of motion, and reveal
that, unlike the macrospins, the spin correlation function in antiferromagnets
does not possess inertia. As a consequence, a pump pulse with a duration equal
to about 1/8th period of the spin wave at the edge of the Brillouin zone, i.e.
$\approx$30 fs for $\mathrm{RbMnF}_{3}$, excites oscillations of spin
correlations with the largest amplitude (see Ref. [35] for details). This
contrasts with the optimal excitation of spin waves in an antiferromagnet at
the center of the Brillouin zone, which simply requires the shortest laser
pulses [37]. Contrary to models in terms of $\mathbf{M}$ and $\mathbf{L}$, the
spin correlation function is capable to describe adequately response to an
anisotropic perturbation of short-scale spin-spin exchange coupling by a
polarized laser pulse and also reveals the effect of the latter on the optical
properties of the medium. An important outcome of the developed model is the
intuitive analytical formula for time-dependent spin dynamics as well as
polarization dependences of optical signals in pump-probe experiments,
unavailable in earlier reported considerations of spin dynamics at the edge of
Brilloun zone [20; 33; 35]. We note that regimes of ultrafast spin dynamics
beyond precession of L and M are being also studied by atomistic [40; 41] and
multiscale simulation [42], which, however, focus on strongly-dissipative
light-matter interactions and still yield longitudinal and transverse dynamics
of M and L. Our experimental and theoretical findings open up novel
perspectives for ultrafast antiferromagnetic spintronics and magnonics
governed by laws different from those established for low-energy spin waves.

Supplementary Material. We present a detailed derivation of the equation of
motion for spin correlations and the equation for
$\varepsilon^{xx}-\varepsilon^{yy}$, and derive the polarization dependences
directly from these equations. We also discuss the key problems one faces when
describing the observed dynamics using a Néel vecotr L.

Acknowledgements. We are grateful to A.K. Zvezdin, V.N. Gridnev, A.I.
Nikitchenko, D. Bossini, H. Hedayat, and P.H.M. van Loosdrecht for fruitful
discussions, S. Semin, and C. Berkhout for technical support. J. H. M.
acknowledges funding by the European Research Council under ERC Grant
Agreement No. 856538 (3D-MAGiC), and Horizon Europe project No. 101070290
(NIMFEIA). The work of T. T. G. was funded by the European Union’s Horizon
2020 research and innovation program under the Marie Skłodowska-Curie Grant
Agreement No. 861300 (COMRAD). R. M. D. acknowledges the support of RSF (Grant
No. 22-72-00025). A. M. K. acknowledges support from RFBR (Grant No.
19-52-12065).

Conflict of Interest. The authors have no conflicts of interest to disclose.

The authors declare that this work has been published as a result of peer-to-
peer scientific collaboration between researchers. The provided affiliations
represent the actual addresses of the authors in agreement with their digital
identifier (ORCID) and cannot be considered as a formal collaboration between
the aforementioned institutions.

Author contributions. $\mathbb{F.~{}Formisano}$: Investigation (equal);
Writing - review & editing (supporting). $\mathbb{T.~{}T.~{}Gareev}$: Data
curation (equal); Formal analysis (equal); Investigation (equal); Methodology
(equal); Visualization (equal); Writing - review & editing (equal).
$\mathbb{D.~{}I.~{}Khusyainov}$: Data curation (equal); Formal analysis
(equal); Investigation (equal); Methodology (equal); Visualization
(supporting); Writing - review & editing (supporting).
$\mathbb{A.~{}E.~{}Fedianin}$: Formal analysis (equal); Investigation (equal);
Methodology (equal); Writing - original draft (equal); Writing - review &
editing (equal). $\mathbb{R.~{}M.~{}Dubrovin}$: Data curation (equal); Formal
analysis (equal); Funding acquisition (lead); Investigation (equal);
Methodology (equal); Visualization (equal); Writing - original draft (equal);
Writing - review & editing (equal). $\mathbb{P.~{}P.~{}Syrnikov}$: Resources
(equal). $\mathbb{D.~{}Afanasiev}$: Methodology (equal); Supervision (lead);
Writing - review & editing (equal). $\mathbb{R.~{}V.~{}Pisarev}$: Resources
(lead); Writing - review & editing (equal). $\mathbb{A.~{}M.~{}Kalashnikova}$:
Conceptualization (lead); Funding acquisition (lead); Writing - original draft
(lead); Writing - review & editing (lead). $\mathbb{J.~{}H.~{}Mentink}$:
Conceptualization (lead); Funding acquisition (lead); Methodology (lead);
Supervision (lead); Writing - original draft (lead); Writing - review &
editing (lead). $\mathbb{A.~{}V.~{}Kimel}$: Conceptualization (lead); Funding
acquisition (lead); Resources (lead); Supervision (lead); Writing - original
draft (lead); Writing - review & editing (lead).

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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