# Ultrafast lattice disordering can be accelerated by electronic collisional
forces

Gilberto A. de la Peña Muñoz  Stanford PULSE Institute, SLAC National
Accelerator Laboratory, Menlo Park, CA 94025, USA.  Stanford Institute for
Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo
Park, CA 94025, USA.  Alfredo A. Correa<EMAIL_ADDRESS>Quantum Simulations
Group, Lawrence Livermore National Laboratory, Livermore, California 94551,
USA.  Shan Yang  Department of Mechanical Engineering and Materials Science,
Duke University, Durham, NC 27708, USA.  Olivier Delaire  Department of
Mechanical Engineering and Materials Science, Duke University, Durham, NC
27708, USA.  Yijing Huang  Stanford PULSE Institute, SLAC National Accelerator
Laboratory, Menlo Park, CA 94025, USA.  Stanford Institute for Materials and
Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, CA 94025,
USA.  Allan S. Johnson  ICFO-Institut de Ciencies Fotoniques, The Barcelona
Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860
Castelldefels (Barcelona), Spain.  IMDEA Nanoscience, Calle Faraday 9, 28049,
Madrid, Spain  Tetsuo Katayama  Japan Synchrotron Radiation Research
Institute, 1-1-1 Kouto, Sayo-cho, Sayo-gun, Hyogo 679-5198, Japan.  RIKEN
SPring-8 Center, 1-1-1 Kouto, Sayo, Hyogo 679-5148, Japan.  Viktor Krapivin
Stanford PULSE Institute, SLAC National Accelerator Laboratory, Menlo Park, CA
94025, USA.  Stanford Institute for Materials and Energy Sciences, SLAC
National Accelerator Laboratory, Menlo Park, CA 94025, USA.  Ernest Pastor
ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and
Technology, Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona),
Spain.  IPR–Institut de Physique de Rennes, CNRS-Centre national de la
recherche scientifique, UMR 6251 Université de Rennes, 35000 Rennes, France
David A. Reis  Stanford PULSE Institute, SLAC National Accelerator Laboratory,
Menlo Park, CA 94025, USA.  Stanford Institute for Materials and Energy
Sciences, SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA.
Samuel Teitelbaum  Physics Department, Arizona State University, Phoenix,
Arizona, USA.  Luciana Vidas  ICFO-Institut de Ciencies Fotoniques, The
Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3,
08860 Castelldefels (Barcelona), Spain.  Simon Wall  Department of Physics and
Astronomy, Aarhus University, Aarhus, Denmark.  Mariano Trigo
<EMAIL_ADDRESS>Stanford PULSE Institute, SLAC National Accelerator
Laboratory, Menlo Park, CA 94025, USA.  Stanford Institute for Materials and
Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, CA 94025,
USA.

###### Abstract

In the prevalent picture of ultrafast structural phase transitions, the atomic
motion occurs in a slowly varying potential energy surface determined
adiabatically by the fast electrons. However, this ignores non-conservative
forces caused by electron-lattice collisions, which can significantly
influence atomic motion. Most ultrafast techniques only probe the average
structure and are less sensitive to random displacements, and therefore do not
detect the role played by non-conservative forces in phase transitions. Here
we show that the lattice dynamics of the prototypical insulator-to-metal
transition of $\mathrm{VO_{2}}$ cannot be described by a potential energy
alone. We use the sample temperature to control the preexisting lattice
disorder before ultrafast photoexcitation across the phase transition and our
ultrafast diffuse scattering experiments show that the fluctuations
characteristic of the rutile metal develop equally fast ($120~{}\mathrm{fs}$)
at initial temperatures of 100 K and 300 K. This indicates that additional
non-conservative forces are responsible for the increased lattice disorder.
These results highlight the need for more sophisticated descriptions of
ultrafast phenomena beyond the Born-Oppenheimer approximation as well as
ultrafast probes of spatial fluctuations beyond the average unit cell measured
by diffraction.

##  I Main text

Ultrafast structural transformations are generally described in terms of atoms
moving in potential energy surfaces that are determined adiabatically by the
motion of the electrons on much faster timescales. This well-known Born-
Oppenheimer approximation is the usual description of nuclear motion in
chemical dynamics, coherent lattice vibrations [1, 2], and ultrafast
structural transformations in solids [3, 4, 5, 6, 7, 8, 9, 10, 11]. This
description ignores the bi-directional, non-adiabatic [12] exchange of energy
between the ions and the electrons [13, 14, 15]. The approximation breaks down
when there is no clear separation between the time- (or energy-) scale of the
atomic and electronic motions such as in metals or across IMTs. In solids,
this breakdown could manifest as enhanced lattice fluctuations caused by
random electron-lattice collisions and an effective electronic friction [12].
However, it is difficult to observe these lattice fluctuations by ultrafast
diffraction because it is primarily sensitive to _average_ structural changes.
The current framework of time-dependent potentials [16, 9, 10] seems to
describe the motion of the average structure captured by ultrafast diffraction
experiments, but new experiments are required to probe the fluctuations caused
by the non-conservative forces.

Figure 1:  Effect of non-conservative electronic forces on structural
dynamics. (a) Schematic structure of $\mathrm{VO_{2}}$. Light-blue (orange)
spheres represent vanadium atoms in the monoclinic M1 (rutile R) phase and
green spheres are the oxygen atoms. Some equivalent sets of vanadium atoms are
omitted for visual clarity. The M1 and rutile unit cells are shown with blue
and red lines, respectively. (b) Schematic of the potential surface at $t<0$
(gray lines) and distribution of vanadium displacements around the M1 order
parameter $\vec{\xi}$ for two initial temperatures $T_{1}>T_{2}$. At $t>0$
(c-d) photoexcitation changes the potential energy and the distribution shifts
towards $\vec{\xi}=0$ and broadens, a signature of increased disorder. (c) and
(d) correspond to scenario without, and with, electron-lattice collisions,
respectively. (e) and (f) show a schematic of the time evolution of the mean
(solid curves) and width (shaded area) of the displacement distributions
without (e) and with (f) collisions for the two temperatures. The solid lines
are slightly displaced vertically for clarity. Electron-lattice collisions
increase the disorder (variance) in (d) and (f) faster than in (c) and (e).

Here we overcome this limitation by simultaneously measuring the ultrafast
x-ray diffraction and diffuse scattering using an x-ray free electron laser.
Using these measurements we demonstrate the effect of the non-adiabatic forces
on an insulator-metal transition on a single-crystal of $\mathrm{VO_{2}}$.
Femtosecond x-ray diffraction following above-gap photoexcitation tracks the
evolution of the Bragg peaks of the equilibrium M1 phase, which reflect the
long-range-order (LRO) of the vanadium dimers that double the unit cell [17].
Importantly, their response directly reflects the shape of the photoexcited
potential[10]. More importantly, we simultaneously record the diffuse
intensity between Bragg peaks, a signature of spatial lattice fluctuations and
disorder, particularly in the vanadium dimers [18]. We change the equilibrium
sample temperature between $300$ and $100~{}\mathrm{K}$ to control the atomic
velocities prior to photo-excitation. We observe that both the peaks and
especially the diffuse intensity respond with virtually the same timescale
between the two sample temperatures. This is surprising because without
additional forces, the soft, flat potential surface[10] cannot accelerate the
disorder and the diffuse intensity cannot increase faster than dictated by the
initial velocities. Without additional sources of disorder the intensity onset
should slow down at lower temperature, as predicted by ab-initio simulations
of adiabatic potentials (Extended Data Fig. 1) and as observed in other
photoinduced order-disorder transitions [19]. We conclude that non-adiabatic
forces from electronic collisions with hot electrons [20, 21] are as important
as the potential [10] in the dynamics of the IMT in $\mathrm{VO_{2}}$. The
observed timescale of $120~{}\mathrm{fs}$, fast compared with other
metals[21], is indicative of an unusually strong electron-lattice coupling in
$\mathrm{VO_{2}}$. While these qualitative conclusions are drawn from general
arguments, we illustrate the behavior of the fluctuations semi-quantitatively
with a simple model that includes disorder.

On cooling, $\mathrm{VO_{2}}$ transforms at $T_{\text{c}}=340~{}\mathrm{K}$
from a metal [22, 23, 24] with a tetragonal rutile (R) structure [25] to a
monoclinic (M1) insulating phase (Fig. 1a). At $T<T_{\text{c}}$ the system
exhibits a frozen distortion represented by $\vec{\xi}$ in Fig. 1a and new
Bragg peaks at wavevectors $(0,1/2,1/2)_{\text{R}}$ in the rutile Brillouin
zone [10]. Ultrafast experiments have provided valuable information on the
transformation pathway between the M1 and R phases. The salient observations
are the fast $100~{}\mathrm{fs}$ loss of LRO in the vanadium dimers observed
by ultrafast diffraction, concomitant with a sharp change in optical
conductivity hallmark of a transient metal [26, 27, 3, 7, 28, 11]. However,
diffraction primarily probes the average order parameter $\vec{\xi}$, which
lacks sensitivity to spatial fluctuations caused by the non-conservative
forces.

Recent diffuse and inelastic x-ray scattering measurements in equilibrium
showed a direct connection between the soft transverse acoustic (TA) phonons
and the enhanced lattice fluctuations responsible for the high entropy of the
metallic phase [18]. Previous work using ultrafast diffuse scattering [10]
found unexpectedly fast lattice disordering upon photoexcitation, a
consequence of the order-disorder nature of this transition. Furthermore, ab-
initio calculations [18, 10, 11] show that the rutile and the photoexcited
states are characterized by a nearly flat, anharmonic potential along the
coordinate $\vec{\xi}$, which enables the large variance of displacements and
the high lattice entropy. Furthermore, this potential describes the diffuse
intensity[18] and the dynamics of the average structure extremely well[10].
These results underscore the importance of lattice fluctuations in both the
thermodynamic equilibrium and the transformation pathway as well as the need
for probes of fluctuations such as diffuse scattering.

Fig. 1b-f shows schematically the generic behavior of the atomic displacement
distribution across a photoinduced structural phase transition. We consider
two idealized but well distinguished scenarios: In one case the dynamics are
primarily governed by a slowly varying adiabatic potential (c, e), and in the
other scenario there are additional non-conservative forces due to electron-
lattice collisions (d, f). The distinction is that the width of the
distributions, visible as diffuse intensity, increases faster in (d) and (f)
due to additional non-adiabatic forces. In the context of VO2 the
distributions in Fig. 1 represent the vanadium positions relative to the
rutile structure (distributed over multiple unit cells) and their changes due
to photoexcitation. Initially at $t<0$ the system is dimerized (gray potential
curve in Fig. 1b) with a displacements distribution centered at $\vec{\xi}$
and initial width dictated by the initial temperature, $T_{1}$ and $T_{2}$,
respectively. The wide photoinduced potential at $t>0$ enables the
distributions to broaden resulting in enhanced spatial fluctuations (Figs. 1c
and 1d). The solid line and shaded areas in Figs. 1e and 1f depict the
dynamics of the mean and variance for the two scenarios, respectively. In the
scenario without additional collisions (e), the potential cannot accelerate
the disorder and the increase in the variance and diffuse intensity are
primarily determined by the initial temperature and become slower at lower
temperature. Such is the case of the ultrafast melting in InSb [5, 29, 6]
where the loss of LRO is slower for a lower initial temperature [19]. However,
when the electron-lattice collisions are strong (Fig. 1f) they can dominate
the increase in the disorder, which becomes nearly unaffected by the initial
temperature. Importantly, diffraction is only sensitive to the mean (solid
lines in Figs 1e-f) and insensitive to disorder, thus these two scenarios are
indistinguishable from a diffraction experiment alone. Here we use x-ray
diffuse scattering to show that $\mathrm{VO_{2}}$ behaves as in the case in
Fig. 1f and Fig. 1d, indicating that the photoexcited electrons cannot be
ignored in describing the dynamics.

Fig. 2a shows a representative static x-ray scattering image of the M1 phase
at $100~{}\mathrm{K}$ taken with $12~{}\mathrm{keV}$ x-ray photons at the
SACLA x-ray free electron (XFEL) facility in Japan [30] (see Methods for
further experimental details). The bright spots correspond to points near the
zone-center of the M1 Brillouin zone (BZ) where the intensity is primarily
from the structure factor of the nearest M1 Bragg point. In this geometry, we
observe contributions from two domains of the M1 structure which we label M1
and M1’ (see Extended Data Fig. 2 for BZs for M1’), where M1’ is rotated from
M1 by $90^{\circ}$ around the rutile c-axis. The red lines in Fig. 2a
represent the boundaries of the BZs for the M1 domain. In Fig. 2b-c we show
representative snapshots of the relative intensity, $I(t)/I(t<0)$, where
$I(t)$ is the intensity at time $t$ and $I(t<0)$ is the equilibrium intensity
before the pump pulse. There are appreciable changes in the diffuse intensity
at $t=44~{}\mathrm{fs}$ which develop almost fully at $110~{}\mathrm{fs}$,
with little change up to several tens of $\mathrm{ps}$ [10]. The sample
returns to equilibrium after several nanoseconds, shorter than the
$40~{}\mathrm{ms}$ repetition period of the pump pulses. Fig. 2d shows the
evolution of the $(\bar{1}\,2\,2)_{\text{M1}}$ and
$(\bar{1}\,1\,3)_{\text{M1}}$ structure factors at $T=100~{}\mathrm{K}$ (lower
panel) together with the diffuse intensity integrated over the regions labeled
$Q_{1}$ and $Q_{2}$ in Fig. 2c (top panel). The diffuse intensity is
characteristic of the disordered vanadium dimers and strongly resembles the
equilibrium pattern of the rutile phase [10].

Figure 2: Ultrafast x-ray diffraction and diffuse scattering from
$\mathrm{VO_{2}}$. (a) Representative static x-ray intensity captured with
$12~{}\mathrm{keV}$ photons at a base temperature of $100~{}\mathrm{K}$. The
bright spots originate from wavevectors near Bragg points, which are labeled
by the corresponding Miller indices for domains M1 and M1’ (see text for
details). Solid red lines represent the Brillouin zone (BZ) edges for the M1
structure; BZs for M1’ are omitted for clarity. (b-c) Relative diffuse
intensity at representative delays of $44$ and $110~{}\mathrm{fs}$ after
photoexcitation with a near-infrared laser pulse centered at
$800~{}\mathrm{nm}$ with $50~{}\mathrm{fs}$ pulse duration. The regions of
reciprocal space labeled $Q_{1}$ and $Q_{2}$ are characteristic of the broken
dimers [10]. (d) Integrated intensity as a function of time for representative
Bragg peaks and for diffuse regions $Q_{1}$ and $Q_{2}$.

Fig. 3 (a) shows the evolution of the relative intensity of the
$(\bar{1}\,2\,2)_{\text{M1}}$ Bragg peak (lower panel) at initial temperatures
of $100~{}\mathrm{K}$ and $300~{}\mathrm{K}$. The timescale for the intensity
drop of $100~{}\mathrm{fs}$ is indistinguishable between the two temperatures
(see Extended Data Fig. 3 for timescale definition and fits). We also show the
integrated dynamics of the relative intensity in the $Q_{1}$ region at the two
temperatures (top panel). Here too, the intensity increases with virtually
identical timescale of $120~{}\mathrm{fs}$ between the two temperatures. There
is also no appreciable difference in the timescale of the $Q_{2}$ region
between the two temperatures (middle panel). These timescales are well above
the experimental resolution $50~{}\mathrm{fs}$, and are also consistent with
prior measurements at a different XFEL facility [10]. Fig. 3 (b) shows the
fluence dependence of the relative intensity at $4~{}\mathrm{ps}$ for
$300~{}\mathrm{K}$ and $100~{}\mathrm{K}$. This shows that the material can be
transformed at both initial temperatures, and is consistent with ultrafast
reflectivity measurements[31]. The results of the $Q_{1}$ and $Q_{2}$ regions
in Fig. 3 (a) imply that the preexisting fluctuations at $t<0$, controlled by
the initial temperature, do not affect the disordering initiated by
photoexcitation and that the system quickly loses memory of the initial
thermal distribution. This contrasts sharply with other order-disorder
transitions such as ultrafast melting of InSb where the initial melting
dynamics are inertial and temperature dependent [19]. It also contrasts
strongly with the slower disorder at lower temperature predicted by ab-initio
molecular dynamics (AIMD) (see Extended Data Fig. 1), which only considers
forces in the Born-Oppenheimer approximation. The fast disordering at low
temperatures observed here implies that additional fluctuating forces not
captured by the potential, which otherwise describes the quasi-equilibrium
diffuse intensity, phonon dispersion[18] and Bragg peaks dynamics quite well
[10, 11], must be affecting the lattice.

Figure 3: Influence of the initial temperature on the dynamics. (a) Comparison
of the dynamics of the relative intensity of the $(\bar{1}\,2\,2)$ peak and
the diffuse intensity in regions $Q_{1}$ and $Q_{2}$ between $100$ and
$300~{}\mathrm{K}$ at incident fluences of $46.5~{}\mathrm{mJ/cm^{2}}$ and
$28~{}\mathrm{mJ/cm^{2}}$, respectively, indicated by the green circles in
(b). (b) Incident fluence dependence of the relative intensity of
$(\bar{1}22)$ at $t=4~{}\mathrm{ps}$ for $100$ and $300~{}\mathrm{K}$.

To build intuition of how the potential and nonconservative forces affect the
dynamics of disorder we develop a simple model based on Langevin dynamics. We
focus on the motion of the vanadium atoms relative to the high-symmetry rutile
positions on the plane perpendicular to $[1,-1,0]_{\text{R}}$ shown as blue
and orange spheres in Fig. 1a. For simplicity, we consider only the vanadium
pairs that move primarily in this plane [17] parameterized as
$\mathbf{r}_{j}=(x_{j},y_{j})$, with $x_{j}=y_{j}=0$ being the average rutile
positions and $j$ a spatial index (see Methods for details). We define a local
potential centered on each high-symmetry site

$U(x,y,t)=r(t)(x^{2}+y^{2})+\frac{1}{2}(x^{4}+y^{4}),$ (1)

where $r(t<0)=-1$ and $r(t>0)=0$. The energy units are such that $U(\pm 1,\pm
1,t<0)=-1$ with minima at $x=\pm 1$, $y=\pm 1$ corresponding to the M1
positions in normalized coordinates, as shown by the contour plots in Fig. 4a.
The filled circles represent the vanadium atoms in equilibrium at
$T(t<0)=T_{\text{i}}=0.005$ in units relative to the depth of the potential
well. Here $T_{\text{i}}$ is the initial temperature. Photoexcitation makes
$r(t>0)=0$, such that the potential becomes wide with minimum at $x=y=0$ as
suggested by observations and ab-initio calculations [10, 11]. The contours in
Figs. 4b-e show $U(x,y,t>0)$. The sudden change in the potential causes the
distribution to broaden quickly, as shown by the purple dots in Fig 4b-e at
$t=0.2~{}\mathrm{ps}$. This broadening is associated with larger spatial
fluctuations and stronger diffuse scattering [10, 18]. As we show below, this
model illustrates that without additional forces the broadening is slower for
lower $T_{\text{i}}$.

Figure 4:  Model of the disorder with non-conservative forces. (a) Contour
plot of the potential $U(x,y,t<0)$ and $U(x,y,t>0)$ (b - e), respectively. The
dots in (a) represent the vanadium atoms in the initial thermal distribution
at $T=0.005$ and out-of-equilibrium in (b-e). Top and bottom rows represent
two idealized scenarios from Eq. (2): 1) constant bath temperature (b-c) and
(f), and 2) bath temperature jump (d-e) and (g). All trajectories in (b-e)
start with the thermal distribution in (a) at $T=0.005$. (f-g) Relative
intensity of 2D Bragg peaks $(2.5,2.5)$ (in reciprocal lattice units of the 2D
lattice) representative of the dimerized doubled unit cell (bottom panels) and
the change in diffuse intensity, $\Delta I(t)$ integrated along wavevectors
$(0.1,0.1)-(0.45,0.45)$ (top panels). For easier comparison between
temperatures, $\Delta I(t)$ curves in (f, g) are shown as percentage of their
respective saturation value at $t>2$ ps. The legends in (f-g) show the
temperature of the initial distribution $T(t<0)$ in units of the potential
well depth. In (f) the bath temperature is constant, in (g) it changes as
$T(t)=T_{\text{i}}+0.1\times\theta(t)$ (See text for details).

The fast motion of the electrons in the transient metal causes additional
forces not captured by the time-dependent potential in Eq. (1). We treat the
electrons as an effective thermal bath characterized by a time-dependent
temperature $T(t)$ that causes stochastic and damping forces on the vanadium
lattice [12, 13, 14, 15, 32]. The equations of motion for the $\mathbf{r}_{j}$
in dimensionless units are

$\frac{1}{a}\ddot{\mathbf{r}}_{j}=-\boldsymbol{\nabla}U(\mathbf{r}_{j},t)-\gamma\dot{\mathbf{r}}_{j}+\boldsymbol{\eta}_{j}(t)$
(2)

where $U(\mathbf{r},t)$ is defined in Eq. (1), $\gamma=5.3~{}\mathrm{ps}^{-1}$
is a phenomenological damping that must be strong enough to attenuate coherent
oscillations in the Bragg peaks in Fig. 2d (see also [10]) and
$a=(2\pi\,6\,\mathrm{THz})^{2}/2$ defines the units of time such that the
small amplitude oscillations in the initial potential have a frequency of
$6~{}\mathrm{THz}$ [33]. The term $\boldsymbol{\eta}_{j}(t)$ represents
uncorrelated random forces due to the electron bath and satisfies the
fluctuation-dissipation theorem, with zero mean and variance
$\langle\eta_{j}^{2}\rangle=2\gamma T(t)$, where $T$ is given relative to the
potential well depth. As we explain below, this stochastic term is crucial to
accelerate the disorder and increase in diffuse intensity. Importantly, the
only temperature in the model, $T(t)$, is the temperature of the electron
bath. The vanadium lattice does not have a well-defined temperature at early
times. In all cases $T(t<0)=T_{\text{i}}$ and the lattice is initially in
equilibrium. In spite of its simplicity, as we show next, Eq. (2) captures the
behavior of the fluctuations, diffraction and diffuse intensity reasonably
well.

The timescale $\gamma^{-1}=0.18~{}\mathrm{ps}$ is associated with electronic
friction where the damping constant $\gamma$ is related to the electron-phonon
coupling constant $g$ [12, 13] (see in particular Eq. (14) in Ref. [13]).
Assuming that the dominant contribution to the damping $\gamma$ is from
electronic collisions we obtain $g=3.7\times 10^{18}~{}\mathrm{Wm^{-3}K^{-1}}$
which is an order of magnitude larger than that of typical metals [34]. This
attests to the large electron-lattice interaction of $\mathrm{VO_{2}}$ and the
importance of the nonconservative forces the electrons exert on the lattice.

In the following simulations we take $T(t<0)=T_{\text{i}}$ with $T_{\text{i}}$
the initial temperature (Fig. 4a) and $r(t>0)=0$, which changes the potential
energy as depicted in Fig. 4b-e. We consider two scenarios with different
behavior in the electron bath: 1) constant temperature, $T(t>0)=T_{\text{i}}$,
shown in Fig. 4b-c and 4f; and 2) sudden temperature jump,
$T(t>0)=T_{\text{i}}+0.1$, shown in Fig. 4d-e and 4g. We also present in
Extended Data Fig. 1 an AIMD simulation which shows that the disorder slows
down at low temperature in the absence of collisions, even for this complex
potential derived from first principles. Clearly, the fast disorder observed
experimentally does not result from peculiarities in the potential, but from
collisions with hot electrons [phenomenologically, the third term in Eq. (2)]
that quickly spread the displacement distribution.

Figures 4f and 4g show the simulated dynamics of the $(2.5,2.5)$ Bragg peak
(bottom panels) and diffuse intensity (top panels) integrated over the
wavevector range $(0.1,0.1)-(0.45,0.45)$ for several initial temperatures (see
Methods for details on the structure factor calculation). In Fig. 4f the bath
temperature is constant $T(t>0)=T_{\text{i}}$, and in 4g it changes as
$T(t>0)=T_{\text{i}}+0.1$. Note that $T_{\text{i}}=0.005$ corresponds to
$100~{}\mathrm{K}$ and $T(t>0)=T_{\text{i}}+0.1$ corresponds to a bath at 2000
K, in agreement with estimates of the photoexcited electronic temperature
[10]. The Bragg peaks are normalized to the initial intensity and the change
in diffuse intensities $\Delta I(t)$ are shown as percentage of their
respective saturation value at $t>2$ ps. Fig. 4f shows that while
$T_{\text{i}}$ does not affect the Bragg peaks significantly, in scenario 1)
the diffuse intensity increases more slowly at lower $T_{\text{i}}$. This
behavior agrees qualitatively with the observed slowdown of the Debye-Waller
factor (a measure of disorder) with the initial temperature[19] during
ultrafast melting of InSb [5, 29, 6], but contrasts with our observations of
disorder in $\mathrm{VO_{2}}$. Note that because of the average change in
symmetry between monoclinic and rutile and associated structure factor change,
the Debye-Waller factor on Bragg peaks is not a good measure of disorder here.

The situation for a step in $T(t)$ for scenario 2) in Fig. 4g is quite
different. Here, $\Delta I(t)$ (top panel) changes with the same timescale
independently of $T_{\text{i}}$, indicating that the system quickly loses
memory of the initial lattice fluctuations because of the additional kicks
from the last term in Eq. (2). Note that in this case too, the Bragg peak
intensity (lower panel) is dominated by the average distortion and cannot
distinguish the two scenarios (compare lower panels in Fig. 4f and 4g).

These results indicate that electronic degrees of freedom can exert additional
forces on the lattice that are not captured by the potential energy surfaces
often used in phenomenological[16], or first-principles-based [10, 11, 35]
descriptions of ultrafast structural dynamics. We showed that these forces
have a strong influence in the lattice dynamics of $\mathrm{VO_{2}}$ and they
most strongly affect the fluctuations rather than the average motion and thus
their characterization requires measurements beyond those of diffraction [10,
36, 20, 21]. Materials control by light pulses will require not only more
sophisticated measurements of structural dynamics, but also a fundamental
description of the coupled electron and lattice degrees of freedom beyond the
Born-Oppenheimer approximation [37].

##  II  Acknowledgements

G.A.P.M., Y.H., V.K., D.A.R., S.T. and M.T. were supported by the US
Department of Energy, Office of Science, Office of Basic Energy Sciences
through the Division of Materials Sciences and Engineering under Contract No.
DE-AC02-76SF00515. A. A. C. work was supported by the Center for Non-
Perturbative Studies of Functional Materials Under Non-Equilibrium Conditions
(NPNEQ) funded by the Computational Materials Sciences Program of the U.S.
Department of Energy, Office of Science, Basic Energy Sciences, Materials
Sciences and Engineering Division and performed under the auspices of the U.S.
Department of Energy by Lawrence Livermore National Laboratory under Contract
DE-AC52-07NA27344. OD was supported by the US Department of Energy (DOE),
Office of Science, Basic Energy Sciences, Materials Sciences and Engineering
Division, under Award No. DE-SC0019978. ASJ, EP, LV and SW were funded through
the European Research Council (ERC) under the European Union’s Horizon 2020
Research and Innovation Programme (Grant Agreement No. 758461) and
PGC2018-097027-B-I00 project funded by MCIN/ AEI /10.13039/501100011033/ FEDER
“A way to make Europe” and CEX2019-000910-S [MCIN/ AEI/10.13039/501100011033],
Fundació Cellex, Fundació Mir-Puig, and Generalitat de Catalunya (AGAUR Grant
No. 2017 SGR 1341, CERCA program). ASJ and EP acknowledge support from the
Marie Skłodowska-Curie grant agreement no. 754510 (PROBIST). ASJ acknowledges
support of a fellowship from ”la Caixa” Foundation (ID 100010434), fellowship
code LCF/BQ/PR21/11840013 and the Agencia Estatal de Investigacion (the R&D
project CEX2019-000910-S, funded by MCIN/ AEI/10.13039/501100011033, Plan
National FIDEUA PID2019-106901GB-I00, FPI). TK acknowledges support from JSPS
KAKENHI (Grant Numbers JP19H05782, JP21H04974, and JP21K18944). Ultrafast
x-ray measurements were performed at BL3 of SACLA with the approval of the
Japan Synchrotron Radiation Research Institute (JASRI) (Proposal Nos.
2019A8038 and 2019B8075). Preliminary x-ray characterization was performed at
the Stanford Synchrotron Radiation Lightsource (SSRL). Use of the SSRL is
supported by the US Department of Energy, Office of Science, Office of Basic
Energy Sciences under Contract No. DE-AC02-76SF00515.

##  III  Authors contributions

M.T., S.W. and O.D. conceived the initial experiment. M.T. supervised the
project. G.A.P.M., Y.H., A.J., T.K, V.K., E.P., D.A.R., S.T., L.V., S.W. and
M.T. performed the experiment at SACLA. G.A.P.M. analyzed the data. M.T. and
A.A.C. developed the model, and M.T., G.A.P.M. and S.Y. performed the
simulations. M.T. and A.A.C. wrote the initial draft with feedback from all
the authors.

##  IV Competing Interests Statement

The authors declare no competing interests.

## References

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##  V  Methods

###  V.1 Experimental details

The ultrafast x-ray scattering measurements were performed at the Spring-8
Angstrom Compact free electron LAser (SACLA) facility in Japan using x-ray
pulses with $10~{}\mathrm{fs}$ duration and $12~{}\mathrm{keV}$ photon energy
(probe). The sample consisted of a single-crystal of $\mathrm{VO_{2}}$ with
the sample normal along the rutile $[1,1,0]_{R}$ direction [10]. The sample
was photo-excited with laser pulses (pump) centered at a wavelength of
$800~{}\mathrm{nm}$, with duration of $40~{}\mathrm{fs}$. The arrival time of
pump and probe were measured and corrected on a shot-by-shot basis giving an
overall time resolution of $50~{}\mathrm{fs}$ full width at half maximum
(FWHM). We used a grazing geometry with the incident x-ray beam making an
angle of $0.5~{}\mathrm{deg}$ with respect to the sample surface to match the
penetration depth of the optical pulse. The pump was nearly co-linear with the
probe ($5~{}\mathrm{deg}$). Scattered x-ray photons were captured by the
MultiPort Charged Coupled Device (MPCCD) detector, which we placed
$96~{}\mathrm{mm}$ away from the sample to cover a large section of reciprocal
space. A nitrogen cryo-stream was used to cool the sample to
$100~{}\mathrm{K}$.

###  V.2 Model of disorder dynamics

The two-dimensional model is based on the dynamics of particles coupled to a
Langevin bath. The vanadium atoms evolve according to Eq. (2) in the main
text. The coordinates are normalized such that the positions at the minima of
the potential are $\mathbf{r}=(\pm 1,\pm 1)$. Starting with a staggered
initial condition $\mathbf{r}_{0}=(1,1)(-1)^{i+j}$ with $i$ and $j$ the
indices of the vanadium in the plane, we run the simulation using an Euler-
Maruyama scheme for $4~{}\mathrm{ps}$ to obtain an initial thermal ensemble.
In Fig. 4a we show a snapshot of this distribution thermalized at a
temperature $T=0.005$ in units of the potential well depth. The time dependent
excitation is modeled as a change in the coefficient from $r(t<0)=-1$ to
$r(t\geq 0)=\exp(-i/x_{0})\theta(t)-1$, which accounts for a spatially
decaying excitation along the sample depth (index $i$), necessary to reproduce
the Bragg peak contrast observed experimentally due to a mismatch of the x-ray
penetration depth. We find that $x_{0}=1/3$ of the x-ray penetration depth
reproduced the observed $0.5$ contrast in the Bragg peaks, however this does
not affect the resulting time-scales. For simplicity we assume a square
lattice with lattice constant $a_{\text{R}}=5.78~{}\mathrm{\text{\AA}}$.

We compute the structure factor as
$S(\mathbf{Q},t)\propto\langle|\sum_{j}e^{i\mathbf{Q}\cdot(\mathbf{R}_{j}+\mathbf{r}_{j}(t))}|^{2}\rangle$,
where $\mathbf{R}_{j}$ are the equilibrium positions in the high-symmetry
structure, $\mathbf{Q}$ is the momentum transfer in reciprocal lattice units
(rlu), and the average is over realizations of the random forces. Wavevectors
in rlu are given in the reciprocal lattice of the high-symmetry structure such
that half-integers represent Bragg peaks of the distorted 2D structure
analogous to the M1 structure.

##  VI  Data Availability Statement

Raw data were generated at the SACLA large-scale facility. Derived data
supporting the findings of this study are available from the corresponding
author upon request.

##  VII  Code Availability Statement

Computer codes used in this study are available from the corresponding author
upon request.