# Melting of magnetic order in ${\mathrm{NaOsO}}_{3}$ by fs laser pulses

Flavio Giorgianni<EMAIL_ADDRESS>Paul Scherrer Institute, 5232
Villigen PSI, Switzerland  Max Burian  Paul Scherrer Institute, 5232 Villigen
PSI, Switzerland  Namrata Gurung  Paul Scherrer Institute, 5232 Villigen PSI,
Switzerland  Laboratory for Mesoscopic Systems, Department of Materials, ETH
Zurich, 8093 Zurich, Switzerland  Martin Kubli  Paul Scherrer Institute, 5232
Villigen PSI, Switzerland  Institute for Quantum Electronics, Physics
Department, ETH Zurich, 8093 Zurich, Switzerland  Vincent Esposito  Paul
Scherrer Institute, 5232 Villigen PSI, Switzerland  Urs Staub  Paul Scherrer
Institute, 5232 Villigen PSI, Switzerland  Paul Beaud  Paul Scherrer
Institute, 5232 Villigen PSI, Switzerland  Steven Lee Johnson  Paul Scherrer
Institute, 5232 Villigen PSI, Switzerland  Institute for Quantum Electronics,
Physics Department, ETH Zurich, 8093 Zurich, Switzerland  Yoav William Windsor
Paul Scherrer Institute, 5232 Villigen PSI, Switzerland  Fritz Haber Institut
od the Max Planck Society, Faradayweg 4-6, 14195 Berlin, Germany  Laurenz
Rettig  Paul Scherrer Institute, 5232 Villigen PSI, Switzerland  Fritz Haber
Institut od the Max Planck Society, Faradayweg 4-6, 14195 Berlin, Germany
Dmitry Ozerov  Paul Scherrer Institute, 5232 Villigen PSI, Switzerland  Henrik
Lemke  Paul Scherrer Institute, 5232 Villigen PSI, Switzerland  Susmita Saha
Paul Scherrer Institute, 5232 Villigen PSI, Switzerland  Laboratory for
Mesoscopic Systems, Department of Materials, ETH Zurich, 8093 Zurich,
Switzerland  Department of Physics and Astronomy, Uppsala University, 751 21
Uppsala, Sweden  Federico Pressacco  Physics Department and Centre for Free-
Electron Laser Science (CFEL), University of Hamburg, 22761 Hamburg, Germany
The Hamburg Centre for Ultrafast Imaging (CUI), 22761 Hamburg, Germany
Stephen Patrick Collins  Diamond Light Source, Harwell Science and Innovation
Campus, Didcot, Oxfordshire OX11 0DE, United Kingdom  Tadashi Togashi  Tetsuo
Katayama  Shigeki Owada  Makina Yabashi  Japan Synchrotron Radiation Research
Institute (JASRI), 1-1-1 Kouto, Sayo-cho, Sayo-gun, Hyogo 679-5198, Japan
RIKEN SPring-8 Center, 1-1-1 Kouto, Sayo-cho, Sayo-gun, Hyogo 679-5148, Japan
Kazunari Yamaura  International Center for Materials Nanoarchitectonics (WPI-
MANA), National Institute for Materials Science, Namiki 1-1, Tsukuba, Ibaraki
305-0044, Japan  Yoshikazu Tanaka  RIKEN SPring-8 Center, 1-1-1 Kouto, Sayo-
cho, Sayo-gun, Hyogo 679-5148, Japan  V. Scagnoli<EMAIL_ADDRESS>Paul Scherrer Institute, 5232 Villigen PSI, Switzerland  Laboratory for
Mesoscopic Systems, Department of Materials, ETH Zurich, 8093 Zurich,
Switzerland

###### Abstract

NaOsO3 has recently attracted significant attention for the strong coupling
between its electronic band structure and magnetic ordering. Here, we used
time-resolved magnetic X-ray diffraction to determine the timescale of the
photoinduced antiferromagnetic dynamics in NaOsO3. Our measurements are
consistent with a sub-100 fs melting of the antiferromagnetic long-range
order, that occurs significantly faster than the lattice dynamics as monitored
by the transient change in intensity of selected Bragg structural reflections,
which instead show a decrease of intensity on a timescale of several ps.

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††preprint: APS/123-QED

##  I Introduction

The discovery of sub-ps demagnetization in ferromagnets upon ultrafast laser
excitation Beaurepaire  _et al._ (1996) has triggered an intense wave of
research focusing on understanding the fundamental mechanisms involved in the
dissipation of the spin and orbital angular momenta Battiato  _et al._ (2010);
Mueller  _et al._ (2013); Turgut  _et al._ (2016); Hofherr  _et al._ (2017);
Ritzmann  _et al._ (2020); Dornes  _et al._ (2019). This fundamental research
interest in the ultrafast manipulation of magnetic order is complemented by
its potential applications for high-speed data storage and processing
technologies, as well as its relevance for faster spintronic architectures;
where, for example, one major goal of research on these systems is to develop
new, ultrafast methods of switching between metastable magnetic states.
Stanciu  _et al._ (2007); Lambert  _et al._ (2014); Schubert  _et al._
(2014a); Hadri  _et al._ (2017); Beens  _et al._ (2019); Ciuciulkaite  _et
al._ (2020)

Recent research efforts within ultrafast magnetism have been partly focused on
antiferromagnets Wadley  _et al._ (2016); Bodnar  _et al._ (2018); Meinert
_et al._ (2018); Meer  _et al._ (2021); Chen  _et al._ (2019) since they
possess resonant frequencies in the terahertz (THz) range, which is three
orders of magnitude higher than observed for ferromagnets (see Ref. Němec  _et
al._ , 2018 and reference therein). Furthermore, in these systems the angular
momentum can be directly exchanged between the spin up and spin down magnetic
sublattices. Having equivalent stable states with zero net angular momentum in
the spin system, antiferromagnets are expected to exhibit faster dynamics,
different from those observed in ferromagnets, where momentum transfer to the
lattice occurs. Given the ubiquity of antiferromagnetic materials, they offer
a rich playground for investigating ultrafast spin dynamics. Of particular
interest is the fact that in the majority of antiferromagnetic systems the
magnetic ordering is intimately related to the electronic structure of the
material. Therefore by manipulating the electronic structure, one also affects
the antiferromagnetic order parameter. Indeed, theoretical predictions Werner
_et al._ (2012); Mentink and Eckstein (2014); Eckstein and Werner (2016)
suggest that in antiferromagnetic materials falling in the weak and strong
electron coupling regime quenching of the antiferromagnetic order parameter
occurs concomitantly within the photo-excited dynamics of the electronic
system, which implies a few-fs or faster time scales are possible. Laser-
induced ultrafast reorientation or switching of the antiferromagnetic order
parameter has been demonstrated in several materials Kimel  _et al._ (2004,
2009); Kanda  _et al._ (2011); Satoh  _et al._ (2015); Kimel  _et al._ (2002);
Kampfrath  _et al._ (2011); Yamaguchi  _et al._ (2010); Baierl  _et al._
(2016); Němec  _et al._ (2018), including Mott insulators Kimel  _et al._
(2002); Kanda  _et al._ (2011); Kampfrath  _et al._ (2011). However, there
have been a limited amount of reports that discuss the ultimate time scale of
the antiferromagnetic quenching in strongly-correlated systems following an
ultrafast photo-excitation. Kimel  _et al._ (2002); Caviglia  _et al._ (2013);
Afanasiev  _et al._ (2019); Mazzone  _et al._ (2021); Stoica  _et al._ (2020)
In these reports, the time scales have been limited either by the experimental
time resolution or found to be in the 100-400 fs range, which is significantly
slower when compared with the values predicted theoretically Werner  _et al._
(2012); Mentink and Eckstein (2014); Eckstein and Werner (2016) or reported
for ferromagnetic materials and multilayer films Beaurepaire  _et al._ (1996);
Hohlfeld  _et al._ (1997); Scholl  _et al._ (1997); Stamm  _et al._ (2007);
Kirilyuk  _et al._ (2010); Rudolf  _et al._ (2012). It is therefore
interesting to ascertain on which timescales the electronic and magnetic
properties can be modified, in particular in systems in which
antiferromagnetic order occurs concomitantly with an abrupt change in the
electronic properties of the material, as occurring at an insulator-to-metal
transition.

In this work we report the ultrafast manipulation of the antiferromagnetic
order and the electronic structure in NaOsO3. The photoexcitation of NaOsO3 by
femtosecond laser pulses with photon energy above the insulating gap drives
simultaneously the dynamics of electrons and spins. To uniquely access the
laser-induced ultrafast spin dynamics we use time-resolved femtosecond X-ray
diffraction on a magnetic Bragg peak. Our results show that the manipulation
of the antiferromagnetic order parameter occurs on a sub-100 fs time scale.

NaOsO3 undergoes an insulator-to-metal transition concomitant with
antiferromagnetic ordering at $T_{\mathrm{IM}}=T_{\mathrm{N}}$= 410(1) K. Shi
_et al._ (2009) In this compound, the absence of crystallographic symmetry
breaking Calder  _et al._ (2012); Gurung  _et al._ (2018) is suggestive of a
magnetically driven insulator-to-metal transition. However, due to the
presence of energetically similar competing interactions, a consensus on the
nature of the metal-insulator mechanism operating in this perovskite is absent
Calder  _et al._ (2012); Middey  _et al._ (2014); Gurung  _et al._ (2018); Kim
_et al._ (2016a); Vale  _et al._ (2018a, b). In the insulating phase, with a
gap at low temperature of 102(3) meV Vecchio  _et al._ (2013), the magnetic
moment determined by neutron diffraction refinement is $1\,\mu_{B}$, and it
suggests a coexistence of localized and itinerant magnetism Calder  _et al._
(2012). Below $T_{\mathrm{N}}$, magnetic moments order almost parallel to the
$c$-axis in a G-type antiferromagnet with a very small
($<0.01~{}\mu_{\mathrm{B}}$) ferromagnetic component along the $b$-axis. The
strong enhancement of magnetic diffraction at the Os $L_{3}$ edge (10.787 keV)
makes this material very appealing for time resolved X-ray diffraction
measurements of the antiferromagnetic dynamics. The latter can be related to
the changes in the conductivity response of the material upon fs laser
excitation, which are expected to drive an insulator-to-metal transition in
the system.

This paper is organized as follows: in Sec. II we describe the sample
preparation and the experimental details of the pump-probe experiments. In
Sec. III, we present the experimental results describing the antiferromagnetic
dynamics in a time window of few ps subsequent to a sub 100 fs laser
excitation. These results are analysed with a time-dependent order parameter
model that was applied previously to manganite materials. Beaud  _et al._
(2014) We find that the melting of the antiferromagnetic order following a fs
laser excitation occurs on a timescale comparable with the experimental time
resolution and faster compared to other 5$d$ oxide materials. Lattice
dynamics, monitored via the transient change of the intensity of structural
Bragg peaks, occur on a time scale of several ps.

Figure 1: Time-resolved resonant X-ray diffraction experiment. a)
Antiferromagnetic order of NaOsO3 for T$<$TN in the $a$,$c-$ plane. The Os
magnetic moments (blue arrow) lying along the $c$-axis. The OsO6 octahedral
cages are colored in grey. b) Schematic setup of the optical-pump/X-ray-probe
experiment. The experiment was performed in grazing incidence geometry
$\alpha<1\deg$ and with the UV pump laser (400 nm) at an angle of 7 $\deg$
with the X-rays. The angle $\alpha$ ($\beta$) is the angle between the
incident (scattered) X-rays and the sample surface. The angle $\phi$ rotates
the sample about its surface normal.

##  II Experimental details

The resonant X-ray diffraction experiment was carried out at the EH2
endstation of the BL3 beamline at the SACLA X-ray free electron laser Ishikawa
_et al._ (2012), using a Huber four-circle ($2\theta$, $\theta$, $\chi$ and
$\phi$) diffractometer in horizontal scattering geometry equipped with a
single module MultiPort Charged Coupled Device (MPCCD) detector Kameshima  _et
al._ (2014). An illustration of the experimental setup is shown in Fig. 1. In
the experiment, a NaOsO3 single-crystal sample with a surface normal direction
close to [1 0 0]Pbnm and an area of $\sim$ 100 $\times$ 100 $\mu$m2 was
mounted on a goniometer with a nitrogen cryostream to stabilize the
temperature at approximately 293 K. A $\hat{\tau}_{L}=$65 fs full width at the
half maximum (FWHM) optical pulse with wavelength 400 nm excited the sample
with a repetition rate of 60 Hz. The laser spot size at the sample position
was measured to be 300 $\mu$m $\times$ 300 $\mu$m. The horizontal polarized
X-ray beam operating at 30 Hz and with a pulse duration of $\hat{\tau}_{X}=$10
fs (FWHM) was focused to a spot size of 10 $\mu$m $\times$ 10 $\mu$m. During
the experiment, the angle between the X-ray- and laser- beams was kept fixed
at 7 degrees. An X-ray grazing incidence geometry ($\alpha=0.5^{\circ}$) was
used to match the X-ray and laser penetration lengths, with the latter
estimated to be 50 nm from Ref. Vecchio  _et al._ , 2013 for a powder sample.
Independent shutters for the laser and X-rays beams were used to collect data
with and without laser excitation.

The X-ray beam energy was tuned within the vicinity of the Os L3 edge ($2p-5d$
transition) at around 10.787 keV ($\lambda=$0.115 nm). The temporal jitter
between X-ray and optical laser pulses was measured shot-by-shot using a
transmission grating based timing tool, which has an accuracy $\Delta
t_{TT}=$10 fs (FWHM) Katayama  _et al._ (2016). The temporal fingerprint of
each shot was then used to re-bin all data into $\Delta t_{bin}=50$ fs time-
segments Nakajima  _et al._ (2018). The effective time-resolution of the
experiment was estimated to be $\Delta
t_{\textrm{eff}}=\sqrt{\hat{\tau}_{L}^{2}+\hat{\tau}_{X}^{2}+\Delta
t_{TT}^{2}+\Delta t_{bin}^{2}}=80(5)$ fs (FWHM). As it is customary to model
time traces using an erf($t$/$\tau$) function Glownia  _et al._ (2010); Gierz
_et al._ (2015), which is characterized by a timescale $\tau$ which is
different from the timescale $\hat{\tau}$ expressed in terms of the FWHM, the
experimental time resolution $\tau_{eff}$ can also be expressed as
$\tau_{\textrm{eff}}$=$\Delta t_{\textrm{eff}}/\sqrt{4\ln(2)}$= 51(4) fs.

##  III Results and Modelling

In order to monitor the time evolution of the antiferromagnetic order
parameter in response to an ultrafast laser excitation, we have taken
advantage of the enhancement of the X-ray magnetic cross section in the
vicinity of the Os $L_{3}$ edge. Based on the experimental geometric
constraints, we have selected the (1 $\bar{5}$ 0) magnetic reflection as the
most suitable one with which to perform our pump-probe measurements.

Figure 2: Normalized diffraction intensity changes of the (1 $\bar{5}$ 0)
magnetic reflection at the Os L3 edge for several laser fluence at T=293 K.
Error bars reflect the standard deviation of the data within each bin. Solid
lines are fit to the data obtained with Eq. 1.

The time evolution of the (1 $\bar{5}$ 0) peak maximum intensity upon optical
laser excitation is illustrated in Fig. 2 for several laser excitation
fluences. In all of the time traces a sudden decrease of intensity is observed
within 100 fs after the optical laser excitation. Angular $\phi$-scans
(rocking curves) at fixed time delay (see Fig. 5) confirm that the magnetic
peak position in reciprocal space does not change within the first 10 ps after
excitation. Therefore, we observe a reduction of the diffracted intensity
related to the reduction of the sublattice magnetization up to 60% on a sub ps
timescale, followed by a recovery of the intensity on a several ps timescale.
The persistence of a sizable part of the magnetic diffraction intensity is
likely due to the imperfect matching of the laser and the X-ray penetration
depths in a crystal as observed, for example, in similar experiments Caviezel
_et al._ (2012); Windsor  _et al._ (2020); Mazzone  _et al._ (2021).

In order to extract an upper bound for the relevant timescales from these time
traces and to readily compare such values with other experimental results
Afanasiev  _et al._ (2019); Mazzone  _et al._ (2021); Caviglia  _et al._
(2013); Stoica  _et al._ (2020), we have fit the data using an error function,
which captures the fast decay time ($\tau$), multiplied by an exponential term
that captures the recovery time ($\tau_{rec}$). This model is appropriate when
the response of the material to the excitation is linear and significantly
faster than the time resolution of the experiment, which in this case is
determined by the pump and probe pulse duration as well as their relative
timing stability. The fitting function has the following analytical form:

$f(t)=\frac{1}{2}\Big{[}\text{erf}\Big{(}\frac{t-t_{0}}{\tau}\Big{)}+1\Big{]}\cdot\Big{[}A\,e^{-(t-t_{0})/\tau_{rec}}+c\Big{]}\,,$
(1)

where $A$, and $c$ are fit parameters that represent respectively the
amplitudes of the intensity reduction and the long-lived transient, which
lasts well beyond the 4 ps time window. Here, $t_{0}$ represents the time
where the X-rays and the optical laser impinge concomitantly on the sample. In
order to reproduce accurately the time traces, we have found that $t_{0}$ must
be treated as a fit parameter. The shift in $t_{0}$ we observe as a function
of the excitation laser fluence is possibly due to an uncontrolled drift of
$t_{0}$ during the experiment or to saturation effects. The latter would lead
also to a change in $\tau$ as a function of fluence, that we do not observe
outside experimental uncertainties.

Table 1: Parameters obtained by fitting the experimental data for the magnetic (1 $\bar{5}$ 0) reflection with Eq. (1). Numbers within brackets represent standard deviations. $F$ stands for fluence.  $F$ (mJ/cm2) |  $t_{0}$ (fs) |  $\tau$ (fs) | $A$ |  $\tau_{rec}$ (ps) | $c$
---|---|---|---|---|---
1.33 | 132(8) | 69(9) | -0.11(6) | 0.54(5) | -0.001(3)
3.07 | 133(4) | 69(7) | -0.13(3) | 0.71(6) | -0.034(4)
5.77 | 85(3) | 78(11) | -0.12(5) | 0.71(7) | -0.099(4)
11.88 | 74(3) | 75(4) | -0.10(5) | 0.47(5) | -0.180(5)
24.05 | 72(2) | 63(5) | -0.09(5) | 0.66(4) | -0.207(3)

Taking into account the caveats above, we observe that the fit of a time trace
returns a value of $\tau$ on the order of 65-75 fs for the fast decay constant
(see Table 1). Therefore, our experiments reveal the presence of a sub-100 fs
drop of the magnetic peak intensity, for all the laser fluences, with an
average decay time $\tau=\tau_{\text{AFM}}$ = 71 $\pm$6 fs. In addition, we
observe two distinct behaviours as a function of the laser fluence. For
fluences below 5 mJ/cm2 the antiferromagnetic ordering recovers almost
completely to the initial intensity with the first 2-3 ps, while for higher
laser fluences the intensity remains suppressed over the time window of our
measurements. A similar behaviour has been observed in other transition metal
based oxide materials Cavalleri  _et al._ (2001); Beaud  _et al._ (2009); de
Jong  _et al._ (2013); Esposito  _et al._ (2018a); Porer  _et al._ (2020);
Afanasiev  _et al._ (2019) and is often assumed to herald the occurrence of a
phase transition in the probed sample volume. In order to obtain a more
quantitative description of our measurements we apply a modified version of
the model presented in Refs. Beaud  _et al._ , 2014; Esposito  _et al._ ,
2018b; Burian  _et al._ , 2021 based on an effective time-dependent order-
parameter $\eta$, which in our case represents the staggered magnetization
associated with the antiferromagnetic ordering.

Within this model, there exists a threshold value $n_{c}$ of the absorbed
local energy density per volume $n$ above which the system undergoes a phase
transition where the order parameter vanishes. In our case this will
correspond to the melting of the long range antiferromagnetic order when the
ratio $n/n_{c}>1$, i.e. when the sublattice magnetization vanishes. The time-
dependent order-parameter model can take into account the effective time
resolution of the experiment $\Delta t_{\textrm{eff}}$ and the fact that the
optical laser pulses are absorbed as they propagate through the sample.

Specifically, we account for the depth-dependent excitation profile by
splitting the sample into N = 400 layers of thickness of $\Delta z=$1 nm. The
electronic excitation density $n_{i}$ of a layer at a depth $z_{i}$ is then
$n_{0i}=n_{0}\,e^{-z_{i}/z_{L}}$, where $z_{L}$ is the effective laser
penetration depth. For a given laser fluence $F$, $n_{0}=F/z_{L}$. As
explained in detail in Appendix A, we express the diffracted intensity
$I_{exc}(t)$ as

$I_{exc}(t)/I_{0}\propto\sum_{i=0}^{N}\left|\eta(z_{i},t)\right|^{2},$ (2)

where $\eta(z_{i},t)$ is the time-dependent order parameter normalized to
unity at time before excitation and for times during and after the excitation
given by

$\eta(z_{i},t)=\Big{(}1-\frac{\text{min}(n(z_{i},t),n_{c})}{n_{c}}\Big{)}^{\gamma_{0}}\,,$
(3)

where $n_{c}$ is a critical excitation density and $n(z_{i},t)$ is the
absorbed local energy density per volume, which depends on the incoming pump
fluence $F$ and $\gamma_{0}$ is analogous to a critical exponent of the
initial excitation. If $n(z_{i},t)>n_{c}$ the phase transition to the
paramagnetic state occurs and $\eta=0$. In order to account for the recovery
of the order parameter, an energy dissipation term is introduced in the
expression for $n(z_{i},t)$ Beaud  _et al._ (2014).

$n(z_{i},t)=\Big{(}n_{0}(z_{i},t)-\alpha
n_{c}\Big{)}e^{-\frac{t}{\tau_{rec}}}+\alpha n_{c},$ (4)

In using this form for $n(z_{i},t)$ we approximate the relaxation of the
electronic energy density as a two-step process. The time constant
$\tau_{\textrm{rec}}$ characterizes a fast relaxation via electron-phonon
interactions, which is followed by a much slower relaxation characterized by a
time constant much larger than the measurement time window. Here Beaud  _et
al._ (2014); Porer  _et al._ (2020)

$\alpha=1-\Big{(}1-\frac{n_{0}(z_{i},t)}{n_{c}}\Big{)}^{\frac{\gamma}{\gamma_{0}}},$
(5)

where $\gamma$ is an effective critical exponent of the quasi-thermalized
system after the initial relaxation process.

The quantity $\alpha$ is a function of $\gamma_{0}$ and $\gamma$, which can be
regarded as the critical exponents with respect to the initial excitation
($t\approx 0$) and after equilibration ( $t>>\tau_{rec}$). If
$\gamma_{0}=\gamma$, $\alpha=n_{0}(z_{i},t)/n_{c}$ and the variation in the
observed intensity will be the same at short and large time after the laser
excitation. If $\gamma_{0}>\gamma$, the change in the measured intensity will
be larger at the shorter time scales, signalling the presence of a mechanism
leading to a partial recovery of the order parameter with a time constant
$\tau_{rec}$. For $\gamma_{0}>>\gamma$, one could anticipate an almost
complete recovery of the order parameter, at least for the lowest excitation
fluences.

We assume for the initial electron energy density

$n_{0}(z_{i},t)dz=\frac{F}{2}\Big{(}1-e^{-\frac{dz}{z_{L}}}\Big{)}e^{-\frac{z_{i}}{z_{L}}}\Big{[}1+\text{erf}\Big{(}\frac{t}{\tau_{L}+\hat{\tau}}\Big{)}\Big{]},$
(6)

which describes the energy deposited by the pump laser pulse of duration
$\tau_{L}$ at a given depth $z_{i}$.

Figure 3: Fits to the time-dependent magnetic reflection data using the time
dependent order parameter model described in the text for different laser pump
fluences.

With such a model we are able to describe reasonably well all the observed
time traces, as shown in Fig. 3.  From the simultaneous fit of all the time
traces, we find the following values (and associated uncertainties) for the
fit parameters: the effective optical penetration depth $z_{L}=20.9(8)$ nm,
which is reasonable if we consider that the effective optical penetration
depth is reduced in our geometry when compared to the 50 nm value estimated
for a powder sample in normal incidence conditions Vecchio  _et al._ (2013).
The critical excitation density is $n_{c}=813(3)$ J/cm3 and the critical
exponent is $\gamma_{0}=1.28(7)$, which is considerably higher than the values
0.5 and 0.69(3) reported for PrCaMnO3 Beaud  _et al._ (2014) and for PrCaMnO4
Porer  _et al._ (2020), respectively. The estimated value for the $\gamma_{0}$
exponent in this experiment suggests an approximately quadratic relation
between the scattering intensity and the excitation energy density. The value
of $\gamma=0.08(1)$ is smaller than that reported for PrCaMnO3 Beaud  _et al._
(2014) and PrCaMnO4 Porer  _et al._ (2020), where it was found
$\gamma=0.20(1)$ and $\gamma=0.29(3)$, respectively. The differences between
the fitted values of both $\gamma$ and $\gamma_{0}$ for the present experiment
and these previous studies may be partially due to the fact that the previous
studies were on thin films, whereas the current study is on a bulk sample
where the laser excitation depth is smaller than the x-ray penetration depth.
Since the model does not explicitly treat energy transport effects, the fitted
values of the parameters may be influenced by these effects. Nevertheless, the
ratio of $\gamma$/$\gamma_{0}$ is significantly smaller for NaOsO3, making
plausible to conclude that the electronic energy dissipation term is more
effective in NaOsO3 as it is characterized by a relaxation constant
$\tau=0.48(3)$ ps, whereas in PrCaMnO3 it was $\tau=0.81$ ps. Such results
could possibly reflect the difference in the phase transition occurring it
those two classes of materials, with the manganites undergoing a change in the
structure of the material, while this should not be the case for NaOsO3 Calder
_et al._ (2012). In such a scenario, one would expect a recovery requiring a
lattice rearrangement to occur on a longer timescale than a reordering of
spins to their antiferromagnetic ground state ordering. The observed recovery
time-scale $\tau_{rec}$ points to the presence of an efficient thermalization
process of the transient free carrier population induced by the pump pulse,
consistent with the small gap and the partially delocalized nature of the
magnetism in NaOsO3.

Figure 4: Absorbed fluence dependence of the normalized intensity at 0.25 ps
and at 2.65 ps. Error bars represent the standard deviation of the measurement
points gathered in the vicinity of the nominal value for each fluence. The
dashed line specifies the fluence corresponding to the critical energy density
at the surface. Solid lines are obtained by using the model described in the
text.

Finally, the model, taking into account the experimental resolution, enables
us to extract the melting timescale of the antiferromagnetic order, estimated
in $\hat{\tau}=62(12)$ fs. The resulting timescale of the melting of the
antiferromagnetic order following a fs laser excitation is faster compared to
other 5$d$ oxide materials Dean  _et al._ (2016); Afanasiev  _et al._ (2019);
Mazzone  _et al._ (2021) (see Table 2) and, in agreement with the estimate
based on Eq. (1), and it is comparable with the experimental resolution. The
critical behaviour of the squared order parameter $\eta^{2}$ as a function of
the excitation fluence is illustrated in Fig. 4, which compares the measured
relative changes of the scattering intensity at 0.2-0.3 ps and 2.2-3 ps after
later laser excitation. The fluence $F_{c}$=2.5(2) mJ/cm2 corresponding to the
critical value of the energy density $n_{c}$ at the surface has a value
similar to that observed in the manganites Beaud  _et al._ (2014); Porer  _et
al._ (2020) and for the melting of the charge ordering in NdNiO3 Esposito  _et
al._ (2018b), which has also an insulator-to-metal transition coinciding with
antiferromagnetic ordering. The photoinduced magnetic response in the
nickelate shows as for NaOsO3 a prompt recovery of the magnetic ordering for
low laser fluences ($<$1mJ/cm2), suggesting that the critical fluence for this
nickelate is lower than for NaOsO3.  However, one should exercise caution in
comparing fluence values from different experiments as they strongly rely on
values of laser power and beam size measurements that can be affected by
significant uncertainties.

##  IV Discussion

There is a consistent body of evidence that impulsive laser excitations above
the band gap are able to drive an insulator-to-metal transition Fiebig  _et
al._ (2000); Ogasawara  _et al._ (2001, 2002); Ruello  _et al._ (2007) and,
for samples which concomitantly order antiferromagnetically, destroy the
antiferromagnetic long range ordering Caviglia  _et al._ (2013); Dean  _et
al._ (2016); Afanasiev  _et al._ (2019); Mazzone  _et al._ (2021); Stoica  _et
al._ (2020). In these materials, one expects to be able to use time-resolved
experiments to draw conclusions on the hierarchy of interactions leading to
the development of the insulator-to-metal transition. Specifically, it is
interesting to see if the insulator-to-metal transition in oxides based on
transition metal atoms could be categorized following the observed changes in
physical properties in response to impulsive laser excitation. For example,
NaOsO3 originally attracted interest as it was thought to be a rare example of
a Slater insulator Calder  _et al._ (2012); Gurung  _et al._ (2018), namely
showing a magnetically driven insulator-to-metal transition. While alternative
explanations of the nature of the insulator-to-metal transition have been
proposed Kim  _et al._ (2016a); Vale  _et al._ (2018b, a), it is clear that
there is a strong coupling between the magnetism and electronic structure.
Calder  _et al._ (2012); Middey  _et al._ (2014); Gurung  _et al._ (2018); Kim
_et al._ (2016a).

It would therefore be interesting to compare the sub-100 fs dynamics in NaOsO3
with those of NdNiO3 in which there is a large consensus on the fact that the
appearance of the antiferromagnetic phase is a byproduct of the concurrent
insulator-to-metal transition. NdNiO3 is a charge transfer insulator (a Mott
insulator has a gap between two bands of the same character, e.g. both 3d,
whereas for a charge transfer insulator the gap has a mixed character, e.g.
between the oxygen 2p and the transition metal 3d bands) and one would expect
the electronic response time $\tau{{}_{e}}$ to be faster than $\tau{{}_{m}}$,
the magnetic one. Vice versa, for NaOsO3 one would expect $\tau{{}_{m}}$ to be
comparable with $\tau{{}_{e}}$. In this respect, neither our data nor those
presented in Ref. Caviglia  _et al._ , 2013; Stoica  _et al._ , 2020 are
conclusive as they are limited by time resolution available in the time-
resolved X-ray measurements. Our experiments on NaOsO3 have shown that the
magnetic response to a fs optical excitation occurs within few tenths of fs
after the fs laser excitation. Modelling the time trace with Eq. (1) gives
$\tau{{}_{m}}\sim$71(6) fs, while the model described in Sec. III, which takes
into account the experimental time resolution gives $\tau{{}_{m}}\sim$62(12)
fs. These value are comparable with those of $\tau{{}_{e}}$ in NaOsO3 and
NdNiO3, which are also limited by the time resolution of the optical
experiments (see Table 2). While from our measurements at room temperature it
would be tempting to conclude that $\tau{{}_{m}}>\tau{{}_{e}}$, it must be
noted that the magnetization time response $\tau{{}_{m}}$ in antiferromagnetic
Sr2IrO4 can be halved when experiments are conducted at low temperatures Tm
sufficiently far from TN Afanasiev  _et al._ (2019).

Regrettably, drawing conclusions on the microscopic parameters governing the
timescale of the disappearance of the antiferromagnetic order parameter upon
impulsive laser excitation is difficult, due to the scarcity of experiments
reporting true timescales of the antiferromagnetic response. The fact that
this timescale seems to be faster in NaOsO3 than other Ir based oxides could
possibly point to the role played by the presence of itinerant magnetism
facilitating the electron mobility and therefore the further delocalization of
the magnetic moments after the fs laser excitation. However, recent
experiments on FeRh, a metallic antiferromagnetic at room temperature, suggest
that changes in the band structure occur only on the order of few hundreds of
fs Pressacco  _et al._ (2021). So it is clear that the full details of the
band structure should be taken into account in order to obtain a quantitative
description of the temporal evolution of the antiferromagnetic order
parameter. In this respect, it must be also mentioned that non-equilibrium
dynamic mean field theory predicts for $\tau{{}_{m}}$ timescales on the order
of a few fs. Afanasiev  _et al._ (2019) Accordingly, in order to gain a deeper
understanding of the interplay between the electronic structure and magnetism
in oxide materials upon laser excitation more experiments with better time
resolution are required.

Table 2: This table illustrates the decay constants $\tau_{m}$ and $\tau_{e}$, determined by Eq. (1), associated with the magnetic and electronic degree of freedom, respectively. Selected oxides compounds, whose timescale of the melting of the antiferromagnetic phase has been reported, are included. T${}_{m}\,$(K) indicates the temperature at which the pump-probe experiments on the antiferromagnetic order parameter were performed. The ratio Tm/TN is also reported, as it could affect the measured value of $\tau_{m}$.  Sample |  $\tau_{m}$ $(fs)$ |  $\tau_{e}$ $(fs)$ | T${}_{m}\,$(K) | Tm/TN | Reference
---|---|---|---|---|---
Sr3Ir2O7 |  $<$120111Limited by experimental time resolution. |  $<$9511footnotemark: 1 | 110 | 0.39 | [Mazzone  _et al._ , 2021, Chu  _et al._ , 2017]
Sr2IrO4 | 330 | 250 | 75 | 0.39222For the thin film sample used in Ref. [Afanasiev  _et al._ , 2019] T${}_{N}\sim$ 195 K. | [Afanasiev  _et al._ , 2019]
Sr2IrO4 | 560 | 250 | 190 | 0.9722footnotemark: 2 | [Afanasiev  _et al._ , 2019]
NdNiO3 |  $<$12511footnotemark: 1 |  $<$6011footnotemark: 1 | 40 | 0.27 | [Stoica  _et al._ , 2020, Liang  _et al._ , 2018]
NaOsO3 | 71(6) |  $<$9011footnotemark: 1 | 300 | 0.73 | this work

##  V Conclusions

In summary, we have presented the results of time-resolved X-ray experiments
on NaOsO3, which aimed to ascertain the time scales of the melting of the
antiferromagnetic structure. We have found that the magnetic order is quenched
on a timescale $\tau{{}_{m}}\sim 71(6)$ fs, which is faster than in other
perovskite compounds. These results may indicate the strong coupling between
the electronic and magnetic degrees of freedom in this material.  Our
measurements of changes in the structural Bragg reflections show evidence of
subsequent lattice dynamics extending over times of several picoseconds.

The raw data files that support this study are available via the Zenodo
repository zen .

## Acknowledgments

We thank D. Hsieh and D. Mazzone for sharing their data, which were used to
extract some of the timescales $\tau$ reported in Table 2. We thank I. Lo
Vecchio for sharing the NaOsO3 data from Ref. Vecchio  _et al._ , 2013. We are
also indebted with J. Mentink and M. Eckstein for stimulating discussion. This
research work was supported from funding provided by the Swiss National
Science Foundation, SNF Project No. 200021$\\_$162863\. Work in Japan was
partly supported by JSPS KAKENHI Grant Number JP20H05276, and Innovative
Science and Technology Initiative for Security (Grant Number JPJ004596) from
Acquisition, Technology, and Logistics Agency (ATLA), Japan. The experiment at
SACLA was performed with the approval of the Japan Synchrotron Radiation
Research Institute (JASRI; Proposal No. 2017A8021). S.S. acknowledges ETH
Zurich Post-Doctoral fellowship and Marie Curie actions for People COFUND
program and the Carl Tryggers foundation with project number (113705531).

## Appendix

###  V.1 Modelling of the antiferromagnetic X-ray diffracted intensity

In a static diffraction experiment the measured intensity of a selected
diffraction peak $I(q)$ is proportional to the square of the unit cell
structure factor $F(q)$, with $q$ being the momentum transfer. If the sample
is excited by an ultrashort laser pulse, the expression of the structure
factor needs to be modified to take into account the depth-dependent
excitation profile. Namely, close to the surface of the sample, the laser
excitation would suppress or modify the scattering cross section, while layers
further away from the surface could be unaffected, at least on very short time
scales. It is customary to account for this depth dependence by splitting the
sample into $N$ layers, each at a given depth zi. Each layer will scatter
X-rays differently and will contribute to the total diffracted intensity. To
calculate the total diffracted intensity two limiting cases are typically
considered. In the first case, the in-plane coherence length $\xi$ is much
larger than the X-ray effective penetration depth $\zeta$ (the X-ray
penetration depth $\zeta_{X}$ must be corrected by a geometrical factor that
takes into account the scattering geometry, $\zeta\sim 3\zeta_{X}$ in our
case). In this case the contributions from each layer are summed in amplitude
and the diffracted intensity is given by

$I(q,t)\propto\left|\sum_{i=0}^{N}F(q,t)\right|^{2}.$ (7)

In the opposite case, where the penetration depth $\zeta$ is much larger than
the in-plane coherence length $\xi$, the contributions of the different layers
should be added incoherently such that the diffracted intensity is expressed
as

$I(q,t)\propto\sum_{i=0}^{N}\left|F(q,t)\right|^{2}.$ (8)

In our experiment we are in an intermediate situation, where $\xi$ and $\zeta$
are of the same order of magnitude, so it is not evident which would be the
best approximation to describe the diffracted intensity. In order to model the
experimental results, we have accordingly empirically fitted the data first
with Eq. (7) and subsequently with Eq. (8). As shown by Fig. 3, the model
based on the expression of the intensity given in Eq. (8) reproduces well the
fluence dependence of the antiferromagnetic diffracted intensity. Conversely,
the model based on Eq. (7) does not provide a reasonable description of the
measured diffraction intensities. We have therefore concluded that, for our
specific experimental conditions, the model based on Eq. (8) is best suited to
describe the data we have gathered on the antiferromagnetic peak.

###  V.2 Lattice dynamics

Figure 5: Angular $\phi$-scans at different time delays at the maximum
excitation fluence 24.2 mJ/cm2 for the (1 $\bar{5}$ 0) magnetic and the
structural (1 $\bar{2}$ 2) Bragg peaks. The intensity for each time delay has
been normalized to maximum value within each scan for the ease of comparison
of the change of the peak position and peak FWHM. Figure 6: Time traces of the
change in intensity of the structural peak (2$\bar{4}$0) measured upon
photoexcitation for two different fluences.

To ascertain the presence of lattice deformation due to laser induced heating
or strain waves we have performed angular scans at selected time delays on a
magnetic and lattice peak. In order to maintain the X-ray grazing incident
angle fixed during the scan, we have scanned the diffractometer angle $\phi$.
The results, gathered with the laser fluence of 24.05 mJ/cm2 and normalized to
1 for the ease of comparison, are illustrated in Fig. 5a for the magnetic peak
(1 $\bar{5}$ 0) and in Fig. 5b for the structural peak (1 $\bar{2}$ 2). The
magnetic peak does not show an appreciable change of the peak position for
time delays up to 10 ps. Also at time delay of several tenths of ps, where
heat diffusion has occurred, the change in the peak position is tiny. Such a
behaviour is not observed for the structural peak. As shown in Fig. 5b, the
position and the shape of the (1 $\bar{2}$ 2) Bragg peak changes significantly
for time delays larger than 2.7 ps. Specifically, for time delay of $\sim$ 6
ps a second peak appears at $\phi=-11.76$, indicating the presence of a laser
induced heating effect. At 10 ps time delay the presence of the second peak is
more prominent and it is clearly visible as a distribution of intensity
between the two peaks, which we ascribe to the presence of a thermal gradient
in the sample. Finally at larger time delays, the heat has diffused from the
lattice planes close to the sample surface to all of the X-ray probed volume,
resulting in a sizable shift of the structural Bragg peak. A thermal gradient
is still present, as indicated by the larger full width at half maximum (FWHM)
and the asymmetric shape of the Bragg peak. The fact that the angular scans of
the two peaks show such a marked difference for time delays larger than 2.7 ps
can be understood as follows. The antiferromagnetic peak intensity decreases
strongly as the sample temperature approaches TN. Therefore, in the case of
the magnetic peak no second peak nor a distribution of intensity is to be
expected.

As a next step, in order to determine the laser induced lattice dynamics, two
structural lattice peaks (1 $\bar{2}~{}$2) (with no scattering contribution
from the Os ion) and (2 $\bar{4}$ 0) (with a scattering contribution from the
Os ion) were measured, using X-rays with an incident photon energy of 10.787
keV (0.115 nm), the same used for the antiferromagnetic peak. Typically,
phonon modes involving heavy ions are lower in frequency, and the laser
excitation energy is transferred from electronic systems first to high
energetic phonon modes. So one could expect a different time evolution for the
two reflections, subsequent to the laser stimulus. The timescales of the drop
in lattice peak intensities were found to be
$\tau_{\mathrm{O},\,\mathrm{Na}}\sim$11 ps (1 $\bar{2}$ 2) and
$\tau_{\mathrm{Os}}\sim$ 20 ps (see Fig. 6), so significantly slower than the
observed antiferromagnetic order parameter and in a time window where coherent
lattice expansion is expected to occur due to the laser heating effect in the
excited sample volume. We tentatively ascribe the different dynamics observed
to the different momentum transfer projection of the two reflections along the
surface normal direction. Unfortunately, due to the limited amount of
available measurement time, it was not possible to obtain more detailed
information on the lattice deformation and a complete fluence dependence for
such reflections. The observed dynamics is therefore ascribed to the expansion
of the crystal lattice due to the heat deposited by the excitation laser
pulses.

## References

  * Beaurepaire  _et al._ (1996) E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot, Phys. Rev. Lett. 76, 4250 (1996).
  * Battiato  _et al._ (2010) M. Battiato, K. Carva, and P. M. Oppeneer, Phys. Rev. Lett. 105, 027203 (2010).
  * Mueller  _et al._ (2013) B. Y. Mueller, A. Baral, S. Vollmar, M. Cinchetti, M. Aeschlimann, H. C. Schneider, and B. Rethfeld, Phys. Rev. Lett. 111, 167204 (2013).
  * Turgut  _et al._ (2016) E. Turgut, D. Zusin, D. Legut, K. Carva, R. Knut, J. M. Shaw, C. Chen, Z. Tao, H. T. Nembach, T. J. Silva, S. Mathias, M. Aeschlimann, P. M. Oppeneer, H. C. Kapteyn, M. M. Murnane, and P. Grychtol, Phys. Rev. B 94, 220408 (2016).
  * Hofherr  _et al._ (2017) M. Hofherr, P. Maldonado, O. Schmitt, M. Berritta, U. Bierbrauer, S. Sadashivaiah, A. J. Schellekens, B. Koopmans, D. Steil, M. Cinchetti, B. Stadtmüller, P. M. Oppeneer, S. Mathias, and M. Aeschlimann, Phys. Rev. B 96, 100403 (2017).
  * Ritzmann  _et al._ (2020) U. Ritzmann, P. Baláž, P. Maldonado, K. Carva, and P. M. Oppeneer, Phys. Rev. B 101, 174427 (2020).
  * Dornes  _et al._ (2019) C. Dornes, Y. Acremann, M. Savoini, M. Kubli, M. J. Neugebauer, E. Abreu, L. Huber, G. Lantz, C. A. F. Vaz, H. Lemke, E. M. Bothschafter, M. Porer, V. Esposito, L. Rettig, M. Buzzi, A. Alberca, Y. W. Windsor, P. Beaud, U. Staub, D. Zhu, S. Song, J. M. Glownia, and S. L. Johnson, Nature 565, 209 (2019).
  * Stanciu  _et al._ (2007) C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, Phys. Rev. Lett. 99, 047601 (2007).
  * Lambert  _et al._ (2014) C.-H. Lambert, S. Mangin, B. S. D. C. S. Varaprasad, Y. K. Takahashi, M. Hehn, M. Cinchetti, G. Malinowski, K. Hono, Y. Fainman, M. Aeschlimann, and E. E. Fullerton, Science 345, 1337 (2014), https://science.sciencemag.org/content/345/6202/1337.full.pdf .
  * Schubert  _et al._ (2014a) C. Schubert, A. Hassdenteufel, P. Matthes, J. Schmidt, M. Helm, R. Bratschitsch, and M. Albrecht, Applied Physics Letters 104, 082406 (2014a), https://doi.org/10.1063/1.4866803 .
  * Hadri  _et al._ (2017) M. S. E. Hadri, M. Hehn, G. Malinowski, and S. Mangin, Journal of Physics D: Applied Physics 50, 133002 (2017).
  * Beens  _et al._ (2019) M. Beens, M. L. M. Lalieu, A. J. M. Deenen, R. A. Duine, and B. Koopmans, Phys. Rev. B 100, 220409 (2019).
  * Ciuciulkaite  _et al._ (2020) A. Ciuciulkaite, K. Mishra, M. V. Moro, I.-A. Chioar, R. M. Rowan-Robinson, S. Parchenko, A. Kleibert, B. Lindgren, G. Andersson, C. S. Davies, A. Kimel, M. Berritta, P. M. Oppeneer, A. Kirilyuk, and V. Kapaklis, Phys. Rev. Materials 4, 104418 (2020).
  * Wadley  _et al._ (2016) P. Wadley, B. Howells, J. Železný, C. Andrews, V. Hills, R. P. Campion, V. Novák, K. Olejník, F. Maccherozzi, S. S. Dhesi, S. Y. Martin, T. Wagner, J. Wunderlich, F. Freimuth, Y. Mokrousov, J. Kuneš, J. S. Chauhan, M. J. Grzybowski, A. W. Rushforth, K. W. Edmonds, B. L. Gallagher, and T. Jungwirth, Science 351, 587 (2016).
  * Bodnar  _et al._ (2018) S. Y. Bodnar, L. Šmejkal, I. Turek, T. Jungwirth, O. Gomonay, J. Sinova, A. A. Sapozhnik, H.-J. Elmers, M. Kläui, and M. Jourdan, Nature Communications 9, 348 (2018).
  * Meinert  _et al._ (2018) M. Meinert, D. Graulich, and T. Matalla-Wagner, Phys. Rev. Applied 9, 064040 (2018).
  * Meer  _et al._ (2021) H. Meer, F. Schreiber, C. Schmitt, R. Ramos, E. Saitoh, O. Gomonay, J. Sinova, L. Baldrati, and M. Kläui, Nano Letters 21, 114 (2021).
  * Chen  _et al._ (2019) X. Chen, X. Zhou, R. Cheng, C. Song, J. Zhang, Y. Wu, Y. Ba, H. Li, Y. Sun, Y. You, Y. Zhao, and F. Pan, Nature Materials 18, 931 (2019).
  * Němec  _et al._ (2018) P. Němec, M. Fiebig, T. Kampfrath, and A. V. Kimel, Nature Physics 14, 229 (2018).
  * Werner  _et al._ (2012) P. Werner, N. Tsuji, and M. Eckstein, Phys. Rev. B 86, 205101 (2012).
  * Mentink and Eckstein (2014) J. H. Mentink and M. Eckstein, Phys. Rev. Lett. 113, 057201 (2014).
  * Eckstein and Werner (2016) M. Eckstein and P. Werner, Scientific Reports 6, 21235 (2016).
  * Kimel  _et al._ (2004) A. V. Kimel, A. Kirilyuk, A. Tsvetkov, R. V. Pisarev, and T. Rasing, Nature 429, 850 (2004).
  * Kimel  _et al._ (2009) A. V. Kimel, B. A. Ivanov, R. V. Pisarev, P. A. Usachev, A. Kirilyuk, and T. Rasing, Nature Physics 5, 727 (2009).
  * Kanda  _et al._ (2011) N. Kanda, T. Higuchi, H. Shimizu, K. Konishi, K. Yoshioka, and M. Kuwata-Gonokami, Nature Communications 2, 362 (2011).
  * Satoh  _et al._ (2015) T. Satoh, R. Iida, T. Higuchi, M. Fiebig, and T. Shimura, Nature Photonics 9, 25 (2015).
  * Kimel  _et al._ (2002) A. V. Kimel, R. V. Pisarev, J. Hohlfeld, and T. Rasing, Phys. Rev. Lett. 89, 287401 (2002).
  * Kampfrath  _et al._ (2011) T. Kampfrath, A. Sell, G. Klatt, A. Pashkin, S. Mährlein, T. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and R. Huber, Nature Photonics 5, 31 (2011).
  * Yamaguchi  _et al._ (2010) K. Yamaguchi, M. Nakajima, and T. Suemoto, Phys. Rev. Lett. 105, 237201 (2010).
  * Baierl  _et al._ (2016) S. Baierl, M. Hohenleutner, T. Kampfrath, A. K. Zvezdin, A. V. Kimel, R. Huber, and R. V. Mikhaylovskiy, Nature Photonics 10, 715 (2016).
  * Caviglia  _et al._ (2013) A. D. Caviglia, M. Först, R. Scherwitzl, V. Khanna, H. Bromberger, R. Mankowsky, R. Singla, Y.-D. Chuang, W. S. Lee, O. Krupin, W. F. Schlotter, J. J. Turner, G. L. Dakovski, M. P. Minitti, J. Robinson, V. Scagnoli, S. B. Wilkins, S. A. Cavill, M. Gibert, S. Gariglio, P. Zubko, J.-M. Triscone, J. P. Hill, S. S. Dhesi, and A. Cavalleri, Phys. Rev. B 88, 220401 (2013).
  * Afanasiev  _et al._ (2019) D. Afanasiev, A. Gatilova, D. J. Groenendijk, B. A. Ivanov, M. Gibert, S. Gariglio, J. Mentink, J. Li, N. Dasari, M. Eckstein, T. Rasing, A. D. Caviglia, and A. V. Kimel, Phys. Rev. X 9, 021020 (2019).
  * Mazzone  _et al._ (2021) D. G. Mazzone, D. Meyers, Y. Cao, J. G. Vale, C. D. Dashwood, Y. Shi, A. J. A. James, N. J. Robinson, J. Lin, V. Thampy, Y. Tanaka, A. S. Johnson, H. Miao, R. Wang, T. A. Assefa, J. Kim, D. Casa, R. Mankowsky, D. Zhu, R. Alonso-Mori, S. Song, H. Yavas, T. Katayama, M. Yabashi, Y. Kubota, S. Owada, J. Liu, J. Yang, R. M. Konik, I. K. Robinson, J. P. Hill, D. F. McMorrow, M. Först, S. Wall, X. Liu, and M. P. M. Dean, Proceedings of the National Academy of Sciences 118, e2103696118 (2021).
  * Stoica  _et al._ (2020) V. Stoica, D. Puggioni, J. Zhang, R. Singla, G. Dakovski, G. Coslovich, M. Seaberg, M. Kareev, S. Middey, P. Kissin, R. Averitt, J. Chakhalian, H. Wen, J. Rondinelli, and J. W. Freeland, arXiv , 2004.03694 (2020).
  * Hohlfeld  _et al._ (1997) J. Hohlfeld, E. Matthias, R. Knorren, and K. H. Bennemann, Phys. Rev. Lett. 78, 4861 (1997).
  * Scholl  _et al._ (1997) A. Scholl, L. Baumgarten, R. Jacquemin, and W. Eberhardt, Phys. Rev. Lett. 79, 5146 (1997).
  * Stamm  _et al._ (2007) C. Stamm, T. Kachel, N. Pontius, R. Mitzner, T. Quast, K. Holldack, S. Khan, C. Lupulescu, E. F. Aziz, M. Wietstruk, H. A. Dürr, and W. Eberhardt, Nature Materials 6, 740 (2007).
  * Kirilyuk  _et al._ (2010) A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82, 2731 (2010).
  * Rudolf  _et al._ (2012) D. Rudolf, C. La-O-Vorakiat, M. Battiato, R. Adam, J. M. Shaw, E. Turgut, P. Maldonado, S. Mathias, P. Grychtol, H. T. Nembach, T. J. Silva, M. Aeschlimann, H. C. Kapteyn, M. M. Murnane, C. M. Schneider, and P. M. Oppeneer, Nature Communications 3, 1037 (2012).
  * Shi  _et al._ (2009) Y. G. Shi, Y. F. Guo, S. Yu, M. Arai, A. A. Belik, A. Sato, K. Yamaura, E. Takayama-Muromachi, H. F. Tian, H. X. Yang, J. Q. Li, T. Varga, J. F. Mitchell, and S. Okamoto, Phys. Rev. B 80, 161104 (2009).
  * Calder  _et al._ (2012) S. Calder, V. O. Garlea, D. F. McMorrow, M. D. Lumsden, M. B. Stone, J. C. Lang, J.-W. Kim, J. A. Schlueter, Y. G. Shi, K. Yamaura, Y. S. Sun, Y. Tsujimoto, and A. D. Christianson, Phys. Rev. Lett. 108, 257209 (2012).
  * Gurung  _et al._ (2018) N. Gurung, N. Leo, S. P. Collins, G. Nisbet, G. Smolentsev, M. García-Fernández, K. Yamaura, L. J. Heyderman, U. Staub, Y. Joly, D. D. Khalyavin, S. W. Lovesey, and V. Scagnoli, Phys. Rev. B 98, 115116 (2018).
  * Middey  _et al._ (2014) S. Middey, S. Debnath, P. Mahadevan, and D. D. Sarma, Phys. Rev. B 89, 134416 (2014).
  * Kim  _et al._ (2016a) B. Kim, P. Liu, Z. Ergönenc, A. Toschi, S. Khmelevskyi, and C. Franchini, Phys. Rev. B 94, 241113 (2016a).
  * Vale  _et al._ (2018a) J. G. Vale, S. Calder, C. Donnerer, D. Pincini, Y. G. Shi, Y. Tsujimoto, K. Yamaura, M. M. Sala, J. van den Brink, A. D. Christianson, and D. F. McMorrow, Phys. Rev. Lett. 120, 227203 (2018a).
  * Vale  _et al._ (2018b) J. G. Vale, S. Calder, C. Donnerer, D. Pincini, Y. G. Shi, Y. Tsujimoto, K. Yamaura, M. Moretti Sala, J. van den Brink, A. D. Christianson, and D. F. McMorrow, Phys. Rev. B 97, 184429 (2018b).
  * Vecchio  _et al._ (2013) I. L. Vecchio, A. Perucchi, P. Di Pietro, O. Limaj, U. Schade, Y. Sun, M. Arai, K. Yamaura, and S. Lupi, Scientific Reports 3, 2990 (2013).
  * Beaud  _et al._ (2014) P. Beaud, A. Caviezel, S. O. Mariager, L. Rettig, G. Ingold, C. Dornes, S.-W. Huang, J. A. Johnson, M. Radovic, T. Huber, T. Kubacka, A. Ferrer, H. T. Lemke, M. Chollet, D. Zhu, J. M. Glownia, M. Sikorski, A. Robert, H. Wadati, M. Nakamura, M. Kawasaki, Y. Tokura, S. L. Johnson, and U. Staub, Nature Materials 13, 923 (2014).
  * Ishikawa  _et al._ (2012) T. Ishikawa, H. Aoyagi, T. Asaka, Y. Asano, N. Azumi, T. Bizen, H. Ego, K. Fukami, T. Fukui, Y. Furukawa, S. Goto, H. Hanaki, T. Hara, T. Hasegawa, T. Hatsui, A. Higashiya, T. Hirono, N. Hosoda, M. Ishii, T. Inagaki, Y. Inubushi, T. Itoga, Y. Joti, M. Kago, T. Kameshima, H. Kimura, Y. Kirihara, A. Kiyomichi, T. Kobayashi, C. Kondo, T. Kudo, H. Maesaka, X. M. Maréchal, T. Masuda, S. Matsubara, T. Matsumoto, T. Matsushita, S. Matsui, M. Nagasono, N. Nariyama, H. Ohashi, T. Ohata, T. Ohshima, S. Ono, Y. Otake, C. Saji, T. Sakurai, T. Sato, K. Sawada, T. Seike, K. Shirasawa, T. Sugimoto, S. Suzuki, S. Takahashi, H. Takebe, K. Takeshita, K. Tamasaku, H. Tanaka, R. Tanaka, T. Tanaka, T. Togashi, K. Togawa, A. Tokuhisa, H. Tomizawa, K. Tono, S. Wu, M. Yabashi, M. Yamaga, A. Yamashita, K. Yanagida, C. Zhang, T. Shintake, H. Kitamura, and N. Kumagai, Nature Photon 6, 540 (2012).
  * Kameshima  _et al._ (2014) T. Kameshima, S. Ono, T. Kudo, K. Ozaki, Y. Kirihara, K. Kobayashi, Y. Inubushi, M. Yabashi, T. Horigome, A. Holland, K. Holland, D. Burt, H. Murao, and T. Hatsui, Review of Scientific Instruments 85, 033110 (2014), https://doi.org/10.1063/1.4867668 .
  * Katayama  _et al._ (2016) T. Katayama, S. Owada, T. Togashi, K. Ogawa, P. Karvinen, I. Vartiainen, A. Eronen, C. David, T. Sato, K. Nakajima, Y. Joti, H. Yumoto, H. Ohashi, and M. Yabashi, Structural Dynamics 3, 034301 (2016).
  * Nakajima  _et al._ (2018) K. Nakajima, Y. Joti, T. Katayama, S. Owada, T. Togashi, T. Abe, T. Kameshima, K. Okada, T. Sugimoto, M. Yamaga, T. Hatsui, and M. Yabashi, Journal of Synchrotron Radiation 25, 592 (2018).
  * Glownia  _et al._ (2010) J. M. Glownia, J. Cryan, J. Andreasson, A. Belkacem, N. Berrah, C. I. Blaga, C. Bostedt, J. Bozek, L. F. DiMauro, L. Fang, J. Frisch, O. Gessner, M. Gühr, J. Hajdu, M. P. Hertlein, M. Hoener, G. Huang, O. Kornilov, J. P. Marangos, A. M. March, B. K. McFarland, H. Merdji, V. S. Petrovic, C. Raman, D. Ray, D. A. Reis, M. Trigo, J. L. White, W. White, R. Wilcox, L. Young, R. N. Coffee, and P. H. Bucksbaum, Opt. Express 18, 17620 (2010).
  * Gierz  _et al._ (2015) I. Gierz, M. Mitrano, H. Bromberger, C. Cacho, R. Chapman, E. Springate, S. Link, U. Starke, B. Sachs, M. Eckstein, T. O. Wehling, M. I. Katsnelson, A. Lichtenstein, and A. Cavalleri, Phys. Rev. Lett. 114, 125503 (2015).
  * Caviezel  _et al._ (2012) A. Caviezel, U. Staub, S. L. Johnson, S. O. Mariager, E. Möhr-Vorobeva, G. Ingold, C. J. Milne, M. Garganourakis, V. Scagnoli, S. W. Huang, Q. X. Jia, S.-W. Cheong, and P. Beaud, Phys. Rev. B 86, 174105 (2012).
  * Windsor  _et al._ (2020) Y. W. Windsor, A. Ernst, K. Kummer, K. Kliemt, C. Schüßler-Langeheine, N. Pontius, U. Staub, E. V. Chulkov, C. Krellner, D. V. Vyalikh, and L. Rettig, Communications Physics 3, 139 (2020).
  * Cavalleri  _et al._ (2001) A. Cavalleri, C. Tóth, C. W. Siders, J. A. Squier, F. Ráksi, P. Forget, and J. C. Kieffer, Phys. Rev. Lett. 87, 237401 (2001).
  * Beaud  _et al._ (2009) P. Beaud, S. L. Johnson, E. Vorobeva, U. Staub, R. A. D. Souza, C. J. Milne, Q. X. Jia, and G. Ingold, Phys. Rev. Lett. 103, 155702 (2009).
  * de Jong  _et al._ (2013) S. de Jong, R. Kukreja, C. Trabant, N. Pontius, C. F. Chang, T. Kachel, M. Beye, F. Sorgenfrei, C. H. Back, B. Bräuer, W. F. Schlotter, J. J. Turner, O. Krupin, M. Doehler, D. Zhu, M. A. Hossain, A. O. Scherz, D. Fausti, F. Novelli, M. Esposito, W. S. Lee, Y. D. Chuang, D. H. Lu, R. G. Moore, M. Yi, M. Trigo, P. Kirchmann, L. Pathey, M. S. Golden, M. Buchholz, P. Metcalf, F. Parmigiani, W. Wurth, A. Föhlisch, C. Schüßler-Langeheine, and H. A. Dürr, Nature Materials 12, 882 (2013).
  * Esposito  _et al._ (2018a) V. Esposito, L. Rettig, E. Abreu, E. M. Bothschafter, G. Ingold, M. Kawasaki, M. Kubli, G. Lantz, M. Nakamura, J. Rittman, M. Savoini, Y. Tokura, U. Staub, S. L. Johnson, and P. Beaud, Phys. Rev. B 97, 014312 (2018a).
  * Porer  _et al._ (2020) M. Porer, L. Rettig, E. M. Bothschafter, V. Esposito, R. B. Versteeg, P. H. M. van Loosdrecht, M. Savoini, J. Rittmann, M. Kubli, G. Lantz, O. J. Schumann, A. A. Nugroho, M. Braden, G. Ingold, S. L. Johnson, P. Beaud, and U. Staub, Phys. Rev. B 101, 075119 (2020).
  * Esposito  _et al._ (2018b) V. Esposito, L. Rettig, E. M. Bothschafter, Y. Deng, C. Dornes, L. Huber, T. Huber, G. Ingold, Y. Inubushi, T. Katayama, T. Kawaguchi, H. Lemke, K. Ogawa, S. Owada, M. Radovic, M. Ramakrishnan, Z. Ristic, V. Scagnoli, Y. Tanaka, T. Togashi, K. Tono, I. Usov, Y. W. Windsor, M. Yabashi, S. L. Johnson, P. Beaud, and U. Staub, Structural Dynamics 5, 064501 (2018b), https://doi.org/10.1063/1.5063530 .
  * Burian  _et al._ (2021) M. Burian, M. Porer, J. R. L. Mardegan, V. Esposito, S. Parchenko, B. Burganov, N. Gurung, M. Ramakrishnan, V. Scagnoli, H. Ueda, S. Francoual, F. Fabrizi, Y. Tanaka, T. Togashi, Y. Kubota, M. Yabashi, K. Rossnagel, S. L. Johnson, and U. Staub, Phys. Rev. Research 3, 013128 (2021).
  * Dean  _et al._ (2016) M. P. M. Dean, Y. Cao, X. Liu, S. Wall, D. Zhu, R. Mankowsky, V. Thampy, X. M. Chen, J. G. Vale, D. Casa, J. Kim, A. H. Said, P. Juhas, R. Alonso-Mori, J. M. Glownia, A. Robert, J. Robinson, M. Sikorski, S. Song, M. Kozina, H. Lemke, L. Patthey, S. Owada, T. Katayama, M. Yabashi, Y. Tanaka, T. Togashi, J. Liu, C. Rayan Serrao, B. J. Kim, L. Huber, C.-L. Chang, D. F. McMorrow, M. Först, and J. P. Hill, Nature Materials 15, 601 (2016).
  * Fiebig  _et al._ (2000) M. Fiebig, K. Miyano, Y. Tomioka, and Y. Tokura, Applied Physics B 71, 211 (2000).
  * Ogasawara  _et al._ (2001) T. Ogasawara, T. Kimura, T. Ishikawa, M. Kuwata-Gonokami, and Y. Tokura, Phys. Rev. B 63, 113105 (2001).
  * Ogasawara  _et al._ (2002) T. Ogasawara, K. Tobe, T. Kimura, H. Okamoto, and Y. Tokura, Journal of the Physical Society of Japan 71, 2380 (2002).
  * Ruello  _et al._ (2007) P. Ruello, S. Zhang, P. Laffez, B. Perrin, and V. Gusev, Phys. Rev. B 76, 165107 (2007).
  * Pressacco  _et al._ (2021) F. Pressacco, D. Sangalli, V. Uhlíř, D. Kutnyakhov, J. A. Arregi, S. Y. Agustsson, G. Brenner, H. Redlin, M. Heber, D. Vasilyev, J. Demsar, G. Schönhense, M. Gatti, A. Marini, W. Wurth, and F. Sirotti, Nature Communications 12, 5088 (2021).
  * Chu  _et al._ (2017) H. Chu, L. Zhao, A. de la Torre, T. Hogan, S. D. Wilson, and D. Hsieh, Nature Materials 16, 200 (2017).
  * Liang  _et al._ (2018) W. Liang, H. Hou, Y. Lin, and S.-N. Luo, Journal of Physics D: Applied Physics 52, 075303 (2018).
  * (72) Https://zenodo.org/record/4452113.
  * (73) A more involved expression as used for Pr${}_{0}.5$Ca0.5MnO3 is not needed here.
  * Gray  _et al._ (2018) A. X. Gray, M. C. Hoffmann, J. Jeong, N. P. Aetukuri, D. Zhu, H. Y. Hwang, N. C. Brandt, H. Wen, A. J. Sternbach, S. Bonetti, A. H. Reid, R. Kukreja, C. Graves, T. Wang, P. Granitzka, Z. Chen, D. J. Higley, T. Chase, E. Jal, E. Abreu, M. K. Liu, T.-C. Weng, D. Sokaras, D. Nordlund, M. Chollet, R. Alonso-Mori, H. Lemke, J. M. Glownia, M. Trigo, Y. Zhu, H. Ohldag, J. W. Freeland, M. G. Samant, J. Berakdar, R. D. Averitt, K. A. Nelson, S. S. P. Parkin, and H. A. Dürr, Phys. Rev. B 98, 045104 (2018).
  * Liu  _et al._ (2012) M. Liu, H. Y. Hwang, H. Tao, A. C. Strikwerda, K. Fan, G. R. Keiser, A. J. Sternbach, K. G. West, S. Kittiwatanakul, J. Lu, S. A. Wolf, F. G. Omenetto, X. Zhang, K. A. Nelson, and R. D. Averitt, Nature 487, 345 (2012).
  * Mayer  _et al._ (2015) B. Mayer, C. Schmidt, A. Grupp, J. Bühler, J. Oelmann, R. E. Marvel, R. F. Haglund, T. Oka, D. Brida, A. Leitenstorfer, and A. Pashkin, Phys. Rev. B 91, 235113 (2015).
  * Yamakawa  _et al._ (2017) H. Yamakawa, T. Miyamoto, T. Morimoto, T. Terashige, H. Yada, N. Kida, M. Suda, H. M. Yamamoto, R. Kato, K. Miyagawa, K. Kanoda, and H. Okamoto, Nature Materials 16, 1100 (2017).
  * Giorgianni  _et al._ (2019) F. Giorgianni, J. Sakai, and S. Lupi, Nature Communications 10, 1159 (2019).
  * Lange  _et al._ (2014) C. Lange, T. Maag, M. Hohenleutner, S. Baierl, O. Schubert, E. R. J. Edwards, D. Bougeard, G. Woltersdorf, and R. Huber, Phys. Rev. Lett. 113, 227401 (2014).
  * Schubert  _et al._ (2014b) O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, Nature Photonics 8, 119 (2014b).
  * Tarekegne  _et al._ (2017) A. T. Tarekegne, H. Hirori, K. Tanaka, K. Iwaszczuk, and P. U. Jepsen, New Journal of Physics 19, 123018 (2017).
  * Hubmann  _et al._ (2020) S. Hubmann, G. V. Budkin, M. Urban, V. V. Bel’kov, A. P. Dmitriev, J. Ziegler, D. A. Kozlov, N. N. Mikhailov, S. A. Dvoretsky, Z. D. Kvon, D. Weiss, and S. D. Ganichev, Journal of Infrared, Millimeter, and Terahertz Waves 41, 1155 (2020).
  * Houver  _et al._ (2019) S. Houver, L. Huber, M. Savoini, E. Abreu, and S. L. Johnson, Opt. Express 27, 10854 (2019).
  * Verwey (1939) E. J. W. Verwey, Nature 144, 327 (1939).
  * Birell and Davies (1982) N. D. Birell and P. C. W. Davies,  _Quantum Fields in Curved Space_ (Cambridge University Press, 1982).
  * (86) W. K. C. (2004),  _The Electrical Engineering Handbook_ (Elsevier, 2004).
  * Slater (1951) J. C. Slater, Phys. Rev. 82, 538 (1951).
  * Salén  _et al._ (2019) P. Salén, M. Basini, S. Bonetti, J. Hebling, M. Krasilnikov, A. Y. Nikitin, G. Shamuilov, Z. Tibai, V. Zhaunerchyk, and V. Goryashko, Physics Reports 836-837, 1 (2019), matter manipulation with extreme terahertz light: Progress in the enabling THz technology.
  * Calder  _et al._ (2015) S. Calder, J. H. Lee, M. B. Stone, M. D. Lumsden, J. C. Lang, M. Feygenson, Z. Zhao, J.-Q. Yan, Y. G. Shi, Y. S. Sun, Y. Tsujimoto, K. Yamaura, and A. D. Christianson, Nature Communications 6, 8916 (2015).
  * Kim  _et al._ (2016b) B. Kim, P. Liu, Z. Ergönenc, A. Toschi, S. Khmelevskyi, and C. Franchini, Phys. Rev. B 94, 241113 (2016b).
  * Kim  _et al._ (2008) B. J. Kim, H. Jin, S. J. Moon, J.-Y. Kim, B.-G. Park, C. S. Leem, J. Yu, T. W. Noh, C. Kim, S.-J. Oh, J.-H. Park, V. Durairaj, G. Cao, and E. Rotenberg, Phys. Rev. Lett. 101, 076402 (2008).
  * Moretti Sala  _et al._ (2014) M. Moretti Sala, S. Boseggia, D. F. McMorrow, and G. Monaco, Phys. Rev. Lett. 112, 026403 (2014).
  * Chapon and Lovesey (2011) L. Chapon and S. Lovesey, Journal of Physics: Condensed Matter 23, 252201 (2011).
  * Roskos  _et al._ (2007) H. Roskos, M. Thomson, M. Kreß, and T. Löffler, Laser & Photonics Reviews 1, 349 (2007).