# Causality violations in realistic simulations of heavy-ion collisions

Christopher Plumberg  Illinois Center for Advanced Studies of the Universe,
Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL
61801, USA  Dekrayat Almaalol  Department of Physics, Kent State University,
Kent, OH 44242, USA  Travis Dore  Illinois Center for Advanced Studies of the
Universe, Department of Physics, University of Illinois at Urbana-Champaign,
Urbana, IL 61801, USA  Jorge Noronha  Illinois Center for Advanced Studies of
the Universe, Department of Physics, University of Illinois at Urbana-
Champaign, Urbana, IL 61801, USA  Jacquelyn Noronha-Hostler  Illinois Center
for Advanced Studies of the Universe, Department of Physics, University of
Illinois at Urbana-Champaign, Urbana, IL 61801, USA

###### Abstract

We show that causality is violated in the early stages of state-of-the-art
heavy-ion hydrodynamic simulations. Up to 30% of the initial fluid cells
violate nonlinear causality constraints while the rest are uncertain. Only
after 2-3 fm/c of evolution, do we find that $\sim 50\%$ of the fluid cells
are definitely causal. Inclusion of pre-equilibrium evolution significantly
reduces the number of acausal cells, but it does not eliminate them. Our
findings suggest that relativistic causality places stringent constraints on
the available model parameter space of heavy-ion collision simulations.

Introduction: Relativistic viscous hydrodynamics is a vital tool for the
phenomenological modeling of ultrarelativistic heavy ion-collisions [1, 2, 3].
The robust success of its predictions at the LHC [4, 5, 6] at the percent
level, and the ability to fit standard observables [7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20], provide strong evidence for the formation of a
fluid-like state of deconfined matter known as the quark-gluon plasma (QGP) in
nuclear collisions [21].

Comparisons to experimental data require modeling all the stages of a heavy-
ion collision in a fully relativistic language, i.e., the initial [22, 23, 11]
and the pre-equilibrium stages [24, 25, 26, 27, 28], the relativistic
hydrodynamic evolution [3], and the post-hydrodynamic phase [29, 30, 31, 32,
33]. The relativistic viscous fluid evolution in heavy-ion collisions is
determined by equations of motion [34, 35, 36] for an extended set of
dynamical variables that include the standard hydrodynamic fields
(temperature, chemical potentials, and flow velocity) and also the non-
equilibrium currents, such as the shear-stress tensor, $\pi_{\mu\nu}$, and the
bulk scalar, $\Pi$. This multi-stage description of heavy-ion collisions,
rooted in relativistic hydrodynamics, has provided key insight into the fluid-
like properties of the QGP such as, for instance, the temperature dependence
of its transport coefficients [14, 37, 38, 39, 40, 41].

Despite its success, the applicability of hydrodynamics to systems as small
and short-lived as nuclear collisions is far from trivial. In heavy-ion
collisions very large initial spatial gradients are expected to occur [42, 22,
43, 44], which drive the system to a far-from-equilibrium state that defies
our current understanding of relativistic hydrodynamics. While there has been
progress towards properly defining relativistic hydrodynamics far from
equilibrium [45] (see also the reviews [46, 47]), traditionally [48]
hydrodynamics is expected to accurately describe the long time, long
wavelength behavior of many-body systems close to equilibrium. The issue
concerning the applicability of hydrodynamics has become even more pressing
after the experimental observation that collective behavior compatible with
hydrodynamics is present even in small systems (e.g. proton-nucleus
collisions) [49, 50, 51, 52].

As a matter of fact, while a strong connection has been established between
the spatial anisotropy of the initial state energy density and the final flow
harmonics in nucleus-nucleus collisions [53, 54, 55, 56, 57, 58, 59, 60], this
begins to break down in small systems [44, 61, 62, 63] due to the strong
influence of initial out-of-equilibrium contributions [64]. Therefore,
conceptual issues concerning the emergence of relativistic hydrodynamics and
its domain of applicability have direct relevance to the phenomenological
modeling of the QGP.

In the far-from-equilibrium domain, dissipative contributions to the energy-
momentum tensor of the system can become comparable to the equilibrium
pressure $P$. In this regime, viscous terms contribute significantly to the
nonlinear evolution of the fluid [65] and the well-known constraints on the
transport coefficients derived in [66, 67, 68] using linearized perturbations
around equilibrium are not sufficient to ensure a well-defined causal
evolution. Thus, while in all the current heavy-ion simulations the transport
coefficients are chosen to satisfy these linear constraints, it is not known
whether causality actually holds in such simulations in the nonlinear far-
from-equilibrium regime probed by heavy-ion collisions.

This question can now be answered using the new constraints derived in [69]
involving the magnitude of the viscous currents and transport coefficients,
which ensure that relativistic causality [70] holds in the nonlinear regime of
the general class of hydrodynamic equations of motion [34, 35, 36] used in
heavy-ion collision simulations. These constraints define the physically
allowable space of out-of-equilibrium corrections to the initial state,
providing new theoretical guidance for the hydrodynamic description of heavy-
ion collisions.

In this work we explore the consequences of these causality constraints for
the most well-behaved scenario for heavy-ion simulations: central Pb+Pb
collisions at the LHC. This is done using two state-of-the-art open-source
hydrodynamic frameworks that we label the “VISHNU framework” that couples
TRENTo+free-streaming+VISHNU [71, 37, 38, 40] and the “MUSIC framework” [72,
73, 22, 74, 75, 28, 27], which couples IP-Glasma+(K${\o}$MP${\o}$ST)+MUSIC. We
find that both frameworks generically yield causality violations throughout a
significant portion of the early time evolution, for typical parameter
settings determined via comparisons to experimental data. For the MUSIC
framework, we study if variations in the pre-hydrodynamic phase [28, 27] can
ameliorate these violations. The inclusion of pre-equilibrium evolution
significantly reduces acausal behavior, though specific details concerning the
nature of the interactions in this phase do not play a significant role. Our
analysis suggests that the nonlinear constraints imposed by relativistic
causality should be taken into account in the assessment of viable regions of
hydrodynamic parameter space and, ultimately, in the quantitative extraction
of the properties of the QGP formed in nuclear collisions.

Modeling: We have specifically chosen the VISHNU and MUSIC frameworks because
they both are open-source, incorporate a fully initialized energy-momentum
tensor $T^{\mu\nu}$, have constrained their parameters through a Bayesian
analysis [76, 37, 38], and have been extensively compared to experimental
data. An alternative approach also exists that only initializes the energy
density profiles [16, 11], which has produced relatively equivalent results to
experimental data except for a handful of observables such as momentum-flow
correlations [77, 78, 79, 80, 81]. It has previously been shown that TRENTo
and IP-Glasma have fairly comparable results in terms of energy density
eccentricities [23, 14, 82]. However, subtle differences remain that are most
likely due to the scaling of the initial energy density distribution with the
thickness functions [83, 84].

Figure 1: From top to bottom: the VISHNU scenario and the three MUSIC
scenarios (no K${\o}$MP${\o}$ST, free-streaming K${\o}$MP${\o}$ST, EKT
K${\o}$MP${\o}$ST). All systems are characterized by strong causality
violations at early times and particularly at the edge of the system (only the
cells that have not frozen out are shown). The approach to completely causal
evolution depends on the details of each model implementation.

For the VISHNU framework, we adopt the Bayesian tune to p+Pb and Pb+Pb data at
the LHC [37, 38]. This combines initial conditions generated from TRENTo [23]
with a conformal, pre-hydrodynamic free-streaming phase [25, 26], a boost-
invariant hydrodynamic phase [85, 71], and finally a post-hydrodynamic
rescattering phase implemented with UrQMD [29, 30]. For the hydrodynamic
transport coefficients, we use the maximum-likelihood parameters presented in
[38]. A single, central Pb+Pb event at $\sqrt{s_{NN}}=2.76A$ TeV is generated
and the random seed is set to 1 in order to ensure reproducibility of our
results. The freeze-out criterion is imposed automatically at an energy
density of $e_{FO}\approx 0.265$ GeV/fm3.

In the MUSIC framework, the initial conditions are generated using IP-Glasma,
coupled to classical Yang-Mills evolution [73, 22], followed by a boost-
invariant hydrodynamic evolution performed with MUSIC starting at $\tau=0.4$
fm$/c$ [74]. We also consider scenarios with an intervening pre-hydrodynamic
phase starting at $\tau=0.1$ fm$/c$ and propagated until $\tau=0.8$ fm$/c$
implemented using K${\o}$MP${\o}$ST [28, 27]: “FS” evolves the initial
conditions using free-streaming and “EKT” implements the full effective
kinetic theory. Therefore, we consider three different scenarios in the MUSIC
framework: (i) IP-Glasma + MUSIC; (ii) IP-Glasma + K${\o}$MP${\o}$ST (FS) +
MUSIC; (iii) IP-Glasma + K${\o}$MP${\o}$ST (EKT) + MUSIC. A single, central
Pb+Pb event at $\sqrt{s_{NN}}=2.76A$ TeV is generated with a random seed value
of 1615404198. All scenarios use a fixed $\eta/s=0.12$ and used the
$[\zeta/s](T)$ parameterization from [86], and freeze out occurs at
$e_{FO}=0.18$ GeV/fm3 or at $T_{FO}=145$ MeV. The equilibrium pressure $P$ is
defined using an interpolation of hadron resonance gas and lattice QCD-based
equations of state in both frameworks [87, 88].

Figure 2: Fractions of the total number of hydrodynamic cells ($e\geq e_{FO}$)
in the causal (left), indeterminate (center), or acausal (right) categories,
plotted as functions of the rescaled time evolution in each framework. Figure
3: The momentum anisotropy $\epsilon_{2,p}$ (left) and spatial eccentricity
$\epsilon_{2,x}$ (right) as a function of time in the different frameworks
considered. The black dots represent the points at which exactly half of the
hydrodynamic cells in each scenario (with $e\geq e_{FO}$) have become
explicitly causal. Whereas most of the final $\epsilon_{2,p}$ in the VISHNU
framework is built up after the majority of the system has become causal, a
significant amount of the system’s flow in the MUSIC framework originates when
less than half the system can be conclusively identified as causal. Similar
conclusions apply to the time evolution of the spatial eccentricity
$\epsilon_{2,x}$ in each scenario.

The constraints derived in Ref. [69] apply to the set of Israel-Stewart-like
[34, 35, 36] equations of motion used in both the VISHNU and MUSIC frameworks.
The conditions were found by determining the characteristics of the
corresponding nonlinear system of PDEs, which were used to obtain a set of
_necessary_ conditions for causality, i.e., if the system fails to satisfy
these necessary conditions causality is violated. _Sufficient_ conditions for
causality were also derived, i.e., if the sufficient conditions are satisfied
then causality is guaranteed to hold. Both sets of conditions correspond to
simple inequalities involving the transport coefficients and the viscous
currents, i.e. $\Pi$ and the four eigenvalues $\\{0,\Lambda_{i}\\}$ of
$\pi^{\mu}_{\nu}$ (with $i=1,2,3$ and $\sum_{i=1}^{3}\Lambda_{i}=0$), which
can be evaluated at each time step. For the explicit expressions of the
constraints, we refer the reader to [69].

The causality analysis of [69] is used here to sort grid points in the
simulations into three different categories, identified by different colors:

  * •

Blue: points at which the sufficient conditions (and consequently the
necessary conditions) hold, implying that causality is respected.

  * •

Red: points at which one or more necessary conditions (and consequently
sufficient conditions) are violated, implying that causality is unquestionably
violated.

  * •

Purple: points at which all necessary conditions are satisfied, but one or
more sufficient conditions fail. Thus, the analysis cannot determine if
causality is violated.

On very rare occasions, points occur where the pre-conditions for the
applicability of the causality analysis fail to hold. In our simulations, this
is typically due, e.g., to values of $\Lambda_{i}$ which lead to the
positivity of $e+P+\Pi+\Lambda_{i}$ failing to hold. We color these points
orange in our plots below. We also color those points green at which the
diagonalization of $\pi^{\mu}_{\nu}$ fails to produce a completely transverse
flow vector $u^{\mu}$. However, because both orange and green points occur so
infrequently, they are barely visible in the plots and will be neglected in
the remainder of this study.

Results: The time evolution of the causality analysis for a typical Pb+Pb
event is shown for VISHNU and all scenarios for the MUSIC framework in Fig. 1.
Note that only the fluid cells that have not yet frozen out are plotted. The
hydrodynamic simulations are all characterized by pervasive violations of
relativistic causality, particularly in the first 1-2 fm$/c$ of the collision.
For VISHNU, most of the severe causality violation occurs near the edge of the
system where Knudsen and inverse Reynolds numbers [43, 44] become large,
though still above the freeze out condition. For the MUSIC framework, without
K${\o}$MP${\o}$ST, approximately $75\%$ of cells in the initial state violate
causality. However, the inclusion of K${\o}$MP${\o}$ST pre-equilibrium
evolution significantly reduces the causality violation present in the IP-
Glasma initial state, bringing it down to approximately 1/3 of fluid cells.
EKT has a slight improvement over FS but the difference is small.

These plots demonstrate some qualitative features that are likely due to
different choices in the transport coefficients in the MUSIC and VISHNU
frameworks. For instance, VISHNU appears to switch the regions at the edge
from acausal to causal first and work its way inwards (with a small region of
indeterminable cells at the center at late times). In contrast, MUSIC has
acausal and indeterminate regions at the edges throughout the expansion but
appears to have more causal regions at the center. This may be due to the
significantly larger bulk viscosity currently used in the MUSIC framework [86,
89] or to the smoother initial condition from TRENTO.

In Fig. 2 we show the time evolution of the fraction of fluid cells (with
$e\geq e_{FO}$) in each category identified by the causality analysis, plotted
as a function of the rescaled time
$\Delta\tau\equiv\tau-\tau_{\mathrm{hydro}}$ (where $\tau_{\mathrm{hydro}}$ is
the time at which hydrodynamics begins). For the earliest stages of each
scenario considered, up to 75% of the fluid cells violated causality. During
roughly the first 20% of the evolution most of the system’s fluid cells are
either acausal (red) or indeterminate (purple/green). All simulations
considered do eventually converge to a regime where the hydrodynamic evolution
is completely causal everywhere. These observations hold quite generally for
the different events, centralities, and collision systems that we have
considered so far. Our results appear to be consistent with similar
conclusions drawn in a recent work [90], which has also addressed the issue of
causality in nuclear collisions. Although we find that the fraction of
causality-violating cells at a given time can include up to $\sim 75\%$ of the
system, we note that the acausal cells account for only about $\sim 1-2\%$ of
the total number of cells in the simulation throughout the evolution of the
system (because the acausal cells turn causal at larger times), which appears
to be consistent with Ref. [90].

We have focused primarily on the hydrodynamic phase and have not yet explored
the consequences of causality violations for experimental observables such as
anisotropic flow or the HBT radii [91, 92]. In lieu of these standard probes
of the nuclear collisions’ momentum-space and coordinate-space geometries, we
consider instead the momentum anisotropy $\epsilon_{2,p}$ and the spatial
eccentricity $\epsilon_{2,x}$, which we define by

$\displaystyle\epsilon_{2,p}$
$\displaystyle=\sqrt{\left(\left<T^{xx}-T^{yy}\right>_{1}^{2}+\left<2T^{xy}\right>_{1}^{2}\right)/\left<T^{xx}+T^{yy}\right>_{1}^{2}}$
(1) $\displaystyle\epsilon_{2,x}$
$\displaystyle=\sqrt{\left(\left<x^{2}-y^{2}\right>_{e\gamma}^{2}+\left<2xy\right>_{e\gamma}^{2}\right)/\left<x^{2}+y^{2}\right>_{e\gamma}^{2}},$
(2)

where $\gamma=\sqrt{1-u_{x}^{2}-u_{y}^{2}}$ is the Lorentz factor for the
local flow velocity, and we have introduced the shorthand

$\displaystyle\left<f(x,y)\right>_{w}$ $\displaystyle=\left.\int
dx\,dy\,w(x,y)f(x,y)\right/\int dx\,dy\,w(x,y)$ (3)

to represent the weighted average of $f$ with respect to $w$. We study in Fig.
3 how these quantities evolve with time in different scenarios considered. The
black dots indicate the point in time for each scenario when half of the fluid
cells are certainly causal. We observe in the MUSIC scenarios that the
majority of the $\epsilon_{2,p}$ anisotropy is built up at early times (up to
$20-30\%$ of evolution time) and that nearly half of the final anisotropy is
built up when most of the system either explicitly violates causality or does
not meet the sufficient conditions for causality. This indicates that
enforcing causality criteria may lead to measurable effects for final state
observables [90], which should be useful additions to Bayesian analyses which
seek to realistically extract quantitative properties of the QGP.

| Initial | Final | Initial | Final
---|---|---|---|---
|  $\epsilon_{2,x}$ [all] |  $\epsilon_{2,x}$ [causal] |  $\epsilon_{2,x}$ [all] |  $\epsilon_{2,x}$ [causal] |  $\epsilon_{2,p}$ [all] |  $\epsilon_{2,p}$ [causal] |  $\epsilon_{2,p}$ [all] |  $\epsilon_{2,p}$ [causal]
VISHNU | 0.0396 | 0.0515 | 0.0218 | 0.0347 | 0.00281 | 0.0238 | 0.0277 | 0.0348
MUSIC (EKT pre-eq.) | 0.100 | 0.135 | 0.0425 | 0.0426 | 0.0182 | 0.0332 | 0.0679 | 0.0680
MUSIC (FS pre-eq.) | 0.101 | 0.147 | 0.0442 | 0.0442 | 0.0151 | 0.0223 | 0.0640 | 0.0638
MUSIC (no pre-eq.) | 0.0995 | 0.113 | 0.0332 | 0.0332 | 0.0072 | 0.0276 | 0.0538 | 0.0539
Table 1: A summary of the effects of acausal/indeterminate cells on the
initial and final values of $\epsilon_{2,p}$ and $\epsilon_{2,x}$. The initial
values are averaged over the early stages of each system until at least half
of the fluid cells are causal; the final values are taken from the latest
freeze-out time, using either all cells at the final timestep, or using only
cells which are causal at the final timestep.

In Table 1, we show the effects of including or excluding
acausal/indeterminate fluid cells in the evaluation of the initial and final
$\epsilon_{2,p}$ and $\epsilon_{2,x}$ for each considered scenario. Note that,
as is standard practice, the anisotropies represent integrals over the full
transverse plane, implying that cells with $e\leq e_{FO}$ are also included
when evaluating Eqs. (1) - (3). In order to minimize numerical instabilities,
we evaluate the initial anisotropies by first averaging the numerator and
denominator of Eq. (3) appearing in Eqs. (1) and (2) separately over all times
up to the point at which exactly half of the fluid cells are causal.

Omitting the acausal and indeterminate cells from the calculation leads to
significantly different estimates for the initial and final values of
$\epsilon_{2,p}$ and $\epsilon_{2,x}$ and, thus, substantially different
interpretations of the underlying physics. Generally, the eccentricities are
larger when only causal cells are considered (and, conversely, their radii are
smaller). Although we cannot at this stage remove the effects of causality
violations entirely from our simulations, these results suggest that both
collective dynamics and spatial geometry will be affected once causality
constraints are taken into account.

Conclusion: Holding simulations to the criterion of relativistic causality
places powerful constraints on the space of parameters currently available to
state-of-the-art hydrodynamic simulations of nuclear collisions. We find that
the standard open-source VISHNU and MUSIC frameworks used in this work, with
parameters constrained by experimental data, both yield acausal evolution
throughout the earliest stages of central Pb+Pb collisions at the LHC, with up
to 75% of fluid cells explicitly violating causality at the beginning of
hydrodynamics. Although we do not consider small systems here, our preliminary
results suggest that the situation there is even more problematic (see also
[90]), with almost the entire system evolving acausally at early times (since
it is known already that Knudsen and Reynolds numbers remain large throughout
the entire evolution even for intermediate systems [93]). An extension of our
causality study to small systems will be presented elsewhere.

In the case of the MUSIC framework, we have shown that the inclusion of a pre-
equilibrium phase prior to the hydrodynamic evolution significantly reduces
the amount of causality violation found in this work, though it does not
eliminate it. We note that much of this analysis depends on our understanding
of the pre-equilibrium phase, which is typically modeled in a conformally
invariant manner, whereas it is known that the equation of state of quantum
chromodynamics [94, 87, 95] used in the hydrodynamic evolution is far from
being conformal even at the temperatures probed at top LHC energies at early
times (see [77] for a discussion of the consequences of matching a pre-
equilibrium conformal phase to a nonconformal hydrodynamic evolution). Thus,
further improvements in the pre-equilibrium phase [96, 97, 98, 41] will be
relevant to fix this acausal behavior found in hydrodynamic simulations of the
QGP formed in heavy-ion collisions.

A possible solution to the issues found in this paper would be the systematic
implementation of causality constraints into Bayesian analyses (e.g., [38]),
which would allow the causality requirements to dictate which regions of
parameter space are most viable. Such an analysis, performed taking into
account both nucleus-nucleus and also small systems, would be crucial to
determine the range of values of transport coefficients and the initial
viscous currents that are physical and compatible with experimental data.
Alternatively, in situations where causality violation is concentrated at the
edge of the system, a core-corona approach wherein only fluid cells that are
causal are run through hydrodynamics (the core) and all other fluid cells (the
corona) are hadronized [99, 100, 101, 102] may be more applicable.

Acknowledgements: We thank M. Disconzi for discussions and N. Cruz Camacho, G.
S. Denicol, M. Luzum, A. Mazeliauskas, B. Schenke, and C. Shen for helpful
comments concerning the hydrodynamic simulations. J.N.H, T.D., and C.P. are
supported by the US-DOE Nuclear Science Grant No. DE-SC0020633. J.N. is
partially supported by the U.S. Department of Energy, Office of Science,
Office for Nuclear Physics under Award No. DE-SC0021301. D.A. is supported by
the U.S. Department of Energy, Office of Science, Office for Nuclear Physics
under Award No. DE-SC0013470.

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# First-order kinetics bottleneck during photoinduced ultrafast insulator-
metal transition in 3D orbitally-driven Peierls insulator CuIr2S4

M. Naseska1, P. Sutar1, Y. Vaskivskyi1, I. Vaskivskyi1, D. Vengust1, D.
Svetin1, V. V. Kabanov1, D. Mihailovic1,2, T. Mertelj1,2 1Complex Matter
Department, Jozef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia
2Center of Excellence on Nanoscience and Nanotechnology Nanocenter (CENN
Nanocenter), Jamova 39, 1000 Ljubljana, Slovenia  *<EMAIL_ADDRESS>
###### Abstract

The spinel-structure CuIr2S4 compound displays a rather unusual orbitally-
driven three-dimensional Peierls-like insulator-metal transition. The low-$T$
symmetry-broken insulating state is especially interesting due to the
existence of a metastable irradiation-induced disordered weakly conducting
state. Here we study intense femtosecond optical pulse irradiation effects by
means of the all-optical ultrafast multi-pulse time-resolved spectroscopy. We
show that the structural coherence of the low-$T$ broken symmetry state is
strongly suppressed on a sub-picosecond timescale above a threshold excitation
fluence resulting in a structurally inhomogeneous transient state which
persists for several-tens of picoseconds before reverting to the low-$T$
disordered weakly conducting state. The electronic order shows a transient gap
filling at a significantly lower fluence threshold. The data suggest that the
photoinduced-transition dynamics to the high-$T$ metallic phase is governed by
first-order-transition nucleation kinetics that prevents the complete
ultrafast structural transition even when the absorbed energy significantly
exceeds the equilibrium enthalpy difference to the high-$T$ metallic phase. In
contrast, the dynamically-decoupled electronic order is transiently suppressed
on a sub-picosecond timescale rather independently due to a photoinduced Mott
transition.

Keywords: ultrafast metal-insulator transition, ultrafast optical
spectroscopy, transient reflectivity, coherent phonons, CuIr2S4, spinel

##  I Introduction

Kinetics of first order phase transitions are important both from the point of
view of applications as well as fundamental science. For example, the control
of kinetics during the transformation of various forms of steel (Papon  _et
al._ , 2002) is used to tailor its microstructural and mechanical properties.
An example of interesting fundamental physics related to first order phase
transition kinetics is inflation of the early universe, where a supercooled
metastable system might been transformed via droplet nucleation into the
Higgs-field broken-symmetry state (Linde, 1982).

In solids, ultrafast first-order insulator-metal (IM) phase transitions could
be instrumental for ultrafast sensor and nonvolatile memory applications.
Their ultrafast kinetics therefore attracted a great deal of attention (Becker
_et al._ , 1994; Fiebig  _et al._ , 2000; Cavalleri  _et al._ , 2001; Perfetti
_et al._ , 2006; Baum  _et al._ , 2007; Wall  _et al._ , 2012; De Jong  _et
al._ , 2013; Fukazawa  _et al._ , 2013; Stojchevska  _et al._ , 2014; Morrison
_et al._ , 2014; Wegkamp  _et al._ , 2014; Abreu  _et al._ , 2015; O’Callahan
_et al._ , 2015; Haupt  _et al._ , 2016; Zhang  _et al._ , 2016; Lantz  _et
al._ , 2017; Jager  _et al._ , 2017; Laulhé  _et al._ , 2017; Singer  _et al._
, 2018; Ligges  _et al._ , 2018; Ronchi  _et al._ , 2019; Wall  _et al._ ,
2018; Otto  _et al._ , 2019; Vidas  _et al._ , 2020) with strong focus on VO2
(Becker  _et al._ , 1994; Cavalleri  _et al._ , 2001; Baum  _et al._ , 2007;
Wall  _et al._ , 2012; Wegkamp  _et al._ , 2014; Morrison  _et al._ , 2014;
O’Callahan  _et al._ , 2015; Jager  _et al._ , 2017; Wall  _et al._ , 2018;
Otto  _et al._ , 2019; Vidas  _et al._ , 2020), V2O3 (Abreu  _et al._ , 2015;
Lantz  _et al._ , 2017; Singer  _et al._ , 2018; Ronchi  _et al._ , 2019) and
1 _T_ -TaS2 (Perfetti  _et al._ , 2006; Stojchevska  _et al._ , 2014; Haupt
_et al._ , 2016; Laulhé  _et al._ , 2017; Ligges  _et al._ , 2018). There are
however still open questions how the inherent first-order meta-stability
manifests itself when phases with concurrent electronic and lattice orders are
driven across the first-order phase boundary on ultrafast time scales.

In VO2, for example, the change of the low-$T$ electronic-order-induced
monoclinic lattice potential is believed to be rather abrupt (Baum  _et al._ ,
2007; Wall  _et al._ , 2012; Jager  _et al._ , 2017; Wall  _et al._ , 2018)
with a sub-picosecond V-V dimerization suppression suggesting that the
high-$T$ metallic rutile phase symmetry is restored on a $\sim 100$-fs
timescale. This is contrasted by recent observation of an additional transient
metastable monoclinic metallic phase in a significant fraction of the
polycrystaline sample grains (Otto  _et al._ , 2019). While the presence of
the metastable monoclinic metallic phase is still rather controversial (Vidas
_et al._ , 2020) the electronic and lattice orders might be transiently
decoupled with the emergence of the metallic rutile lattice structure somehow
inhibited.

In V2O3 the inherent first-order meta stability plays a determining role with
the nucleation and growth kinetics of the metallic nanodroplets governing the
dynamics across the first-order IM phase transition (Abreu  _et al._ , 2015;
Ronchi  _et al._ , 2019). The importance of the nucleation and growth kinetics
was noted also in a first-order ultrafast charge-density-wave phase
transformation in $1T$-TaS2 (Haupt  _et al._ , 2016; Laulhé  _et al._ , 2017).

Figure 1: CuIr2S4 structure. (a) Schematics (Momma and Izumi, 2011) of the
CuIr2S4 structure in the high-$T$ metallic phase. The different sets of Ir-ion
rows alternating along the $\left\langle 110\right\rangle$ or
$\left\langle-110\right\rangle$ cubic directions are indicated by the
differently colored bonds. (b) In the low-$T$ insulating phase one set of rows
undergoes a Peierls distortion with the Ir4+ dimers (red dumbbells)
interleaved with Ir3+-ion pairs (green). The compasses indicate the basis
orientations while the black lines indicate the high-$T$ cubic unit cell. The
Cu ions are omitted for clarity. (c) A schematic diagram of the
Ir-t2g-orbitals derived quasi-1D electronic bands density of states in the
low-$T$ phase (Khomskii and Mizokawa, 2005). The shaded regions correspond to
the occupied states and the dominant optical transitions at the experimental
photon energy(Sarkar  _et al._ , 2009) are indicated by the red arrow.

An interesting opportunity for studying the ultrafast first-order IM phase
transition (Nagata  _et al._ , 1994) kinetics is offered in the spinel-
structure CuIr2S4 compound that shows a rather unusual orbitally-driven
(Khomskii and Mizokawa, 2005) three-dimensional Peierls-like IM transition.
Upon cooling, the metallic cubic (MC) phase (Figure 1 (a)) undergoes a first
order transition to a triclinic (Ishibashi  _et al._ , 2001; Radaelli  _et
al._ , 2002) insulating phase (Figure 1 (b)) at $T_{\mathrm{IM}}\sim$233 K.
The triclinic insulating phase is characterized by Ir3+:Ir4+ charge
disproportionation accompanied with Peierls-like dimerization of Ir4+-ion
pairs along a set of cubic $\left\langle 110\right\rangle$ and
$\left\langle-110\right\rangle$ Ir-ion rows alternating along the
corresponding $\left\langle 001\right\rangle$ direction.

The transition can be understood in terms of quasi-1D Ir-t2g-orbitals derived
hybridized bands that are split due to the band Jahn-Teller effect into two
narrower fully occupied $xz$\- and $yz$-derived bands and a broader 3/4
occupied $xy$-derived band that simultaneously becomes gaped due to the
Peierls tetramerization (see Fig 1 (c)) (Khomskii and Mizokawa, 2005). The
Ir-$5d$ orbitals appear broad enough that the correlation effects do not play
a major role in the transition (Sasaki  _et al._ , 2004; Khomskii and
Mizokawa, 2005; Sarkar  _et al._ , 2009) contrary to V2O3 and VO2.

The CuIr2S4 is interesting also due to the low temperature X-ray- and visible-
light-induced metastable disordered weakly conducting (DWC) phase (Ishibashi
_et al._ , 2002; Furubayashi  _et al._ , 2003; Takubo  _et al._ , 2005;
Kiryukhin  _et al._ , 2006; Takubo  _et al._ , 2008; Božin  _et al._ , 2011).
The presence of short-range incommensurate structural correlations observed
(Kiryukhin  _et al._ , 2006) in the DWC phase suggests presence of _competing
instabilities_ that appear after suppression of the insulating phase. The
formation of the DWC phase was studied mostly with weak continuous
excitations. Competing instabilities, however, might lead to formation of even
more conducting metastable state under strongly nonequilibrium conditions as
in the case of 1$T$-TaS2, where a strong ultrafast excitation leads to a
metastable hidden (H) metallic phase (Stojchevska  _et al._ , 2014). In order
to search for a possible hidden metallic state, different from the DWC phase,
and to study kinetics during ultrafast first-order IM phase transition in
CuIr2S4 we therefore performed a systematic femtosecond multi-pulse all-
optical investigation of the single-crystal transient reflectivity relaxation
dynamics in CuIr2S4 as a function of excitation fluence.

The low-$T$ quasi-equilibrium transient reflectivity in CuIr2S4 is dominated
by the low-frequency broken-symmetry-induced coherent-phonons response that
discontinuously vanishes across the equilibrium IM transition. (Naseska  _et
al._ , 2020) The coherent response can therefore be used as a time-resolved
probe for the broken-lattice-symmetry dynamics in the strongly excited low-$T$
phases. (Yusupov  _et al._ , 2010; Wall  _et al._ , 2012) This can be done by
means of our advanced multi-pulse method (Yusupov  _et al._ , 2010), where we
use a strong driving laser pulse to _initiate_ a photoinduced transition and
_probe_ the subsequent evolution by means of a weaker pump- (P) probe (Pr)
pulses sequence.

##  II Methods

### Sample growth and characterization

Single crystals of CuIr2S4 were grown from Bi solution and characterized as
described in Ref. (Naseska  _et al._ , 2020). Here we present ultrafast
optical data from two cleaved crystals designated S1 and S2. The orientation
of the S1 cleaved surface was determined from electron back scatter
diffraction Kikuchi patterns to be close to the $\left\langle
221\right\rangle$ plane while the orientation of the S2 cleaved surface was
inferred from the Raman selection rules to be close to the $\left\langle
001\right\rangle$ plane. (Naseska  _et al._ , 2020)

### Multi-pulse transient reflectivity measurements

The multi-pulse transient reflectivity measurements. (Yusupov  _et al._ ,
2010; Naseska  _et al._ , 2018) were performed using 50-fs linearly polarized
laser pulses at 800 nm wavelength and the $200-250$ kHz repetition rate. In
addition to the pump (P) and probe (Pr) pulses at $\hbar\omega=1.55$ eV we
used another intense driving (D) pulse (also at $\hbar\omega=1.55$ eV) with a
variable delay with respect to the pump (P) pulse (see Figure 2).

Figure 2: Schematics of the multi-pulse experiment with notation of the delays
adapted from Ref. (Madan  _et al._ , 2017).

The multi-pulse transient reflectivity $\Delta R_{3}/R$ was measured by
monitoring the intensity of the Pr beam. The direct contribution of the
unchopped D beam to the total transient reflectivity, $\Delta R$, was rejected
by means of a lock-in synchronized to the chopper that modulated the intensity
of the P beam only. Due to the chopping scheme the measured quantity in the
multi-pulse experiments is the difference between the transient reflectivity
in the presence of P and D pulses, $\Delta
R_{\mathrm{DP}}(t_{\mathrm{Pr}},t_{\mathrm{P}},t_{\mathrm{D}})$, and the
transient reflectivity in the presence of the D pulse only, $\Delta
R_{\mathrm{D}}(t_{\mathrm{Pr}},t_{\mathrm{D}})$:

$\Delta R_{3}(t_{\mathrm{Pr}},t_{\mathrm{P}},t_{\mathrm{D}})=\Delta
R_{\mathrm{DP}}(t_{\mathrm{Pr}},t_{\mathrm{P}},t_{\mathrm{D}})-\Delta
R_{\mathrm{D}}(t_{\mathrm{Pr}},t_{\mathrm{D}}),$ (1)

where $t_{\mathrm{Pr}}$, $t_{\mathrm{P}}$ and $t_{\mathrm{D}}$ correspond to
the Pr, P and D pulse arrival times, respectively. In the limit of vanishing D
pulse fluence $\Delta R_{3}/R$ reduces to the standard two-pulse transient
reflectivity $\Delta R/R$.

The P/D and Pr beam diameters were in the ranges of 40-70 and 18-30 $\mu$m,
respectively. The probe fluence was $\sim 5$0 $\mu$J/cm2. For the multi-pulse
measurements the fluence of the P pulse, $\mathcal{F_{\mathrm{P}}}\lesssim
100$ $\mu$J/cm2, was kept in the linear response region. The polarizations of
the P and D beams were perpendicular to the probe beam polarization with a
random orientation with respect to the crystal axes.

##  III Results

The DWC phase emergence depends on the absorbed dose (Furubayashi  _et al._ ,
2003; Koshimizu  _et al._ , 2009), but the threshold dose was determined only
for the case of high-energy ion irradiation(Koshimizu  _et al._ , 2009). Since
the exact optical DWC-phase-creation conditions are not known we measured also
the low-$T$ DC photoconductivity (see Supplemental Material (SM) (Naseska  _et
al._ , 2021)) and found, as suggested previously,(Naseska  _et al._ , 2020)
that even at the lowest feasible excitation fluences, $F\sim$100 $\mu$J/cm2,
the threshold dose is exceeded on a timescale faster than a single transient
reflectivity scan acquisition time of $\sim$100 s. The pristine insulating
triclinic phase is therefore inaccessible in the pump-probe experiments at low
$T$ and the previously reported(Naseska  _et al._ , 2020) equilibrium low-$T$
transient reflectivity refers to the DWC phase. Despite the absence of a long
range Ir4+-dimer order in the DWC phase(Kiryukhin  _et al._ , 2006) the
previously characterized coherent oscillations (CO) due to the displacive
excitation of coherent phonons(Zeiger  _et al._ , 1992; Naseska  _et al._ ,
2020, 2021) (DECP) show well defined peaks with the low-$T$ dephasing times
exceeding $\sim 100$ ps.

Figure 3: Transient reflectivity pump-fluence dependence. (a) Fluence-
normalized transient reflectivity as a function of $F_{\mathrm{P}}$ in the
low-$T$ phase. A high-$T$ MC-phase trace ($T=245$ K) is shown on top for
comparison. Due to the normalization the absence of changes at low
$F_{\mathrm{P}}$ indicates linear scaling with $F_{\mathrm{P}}$. The traces
are vertically offset for clarity and the thin lines correspond to the DECP
model fit. (b) The low-$F_{\mathrm{P}}$ transient reflectivity before and
after exposure at the highest $F_{\mathrm{P}}$. (c) Transient reflectivity at
elevated $F_{\mathrm{P}}$ at longer delay. Please note the logarithmic scale
after the break.

In Figure 3 (a) we show the pump fluence, $F_{\mathrm{P}}$, dependence of the
standard two-pulse transient reflectivity, $\Delta R/R$, at $T=9$ K. The CO
scale almost linearly with increasing $F_{\mathrm{P}}$ up to $\sim 2$ mJ/cm2.
With increasing $F_{\mathrm{P}}$ the CO appear suppressed in the
$F_{\mathrm{P}}$-normalized scans and the signal becomes dominated by the sub-
ps relaxation component. The detailed DECP-model(Naseska  _et al._ , 2020,
2021) component analysis (see. Figure 4) reveals that the sub-ps exponential
component (SEC) amplitude, $A\mathrm{{}_{e}}$, departs from the
low-$F_{\mathrm{P}}$ linear dependence and starts to increase more steeply
above $F_{\mathrm{ce}}\sim 0.6$ mJ/cm2 already (see inset to Figure 4 (e)).
Concurrently, the amplitudes of the three weaker, high-frequency modes,
designated O3, O4 and O5, show an onset of saturation while the amplitudes of
the two stronger, low-frequency modes, O1 and O2, show an increased slope.
Above $F_{\mathrm{c}}\sim 3$ mJ/cm2 the amplitudes of all modes drop and
vanish at the highest $F_{\mathrm{P}}$. The SEC does not show any anomalies
around $F_{\mathrm{c}}$, but shows saturation of $A_{\mathrm{e}}$ and the
relaxation time, $\tau_{\mathrm{e}}$, above $F\mathrm{{}_{P}\sim 5}$ mJ/cm2.

Figure 4: Pump fluence dependence of the DECP fit parameters. (a) The
frequency, (b) damping and (c) amplitude of the coherent oscillations as a
function of $F_{\mathrm{P}}$. In the shaded region, above $F_{\mathrm{P}}\sim
3.5$ mJ/cm2, the coherent oscillations cannot be completely described by the
DECP theory. (d) The effective temperature (see text) obtained from the
frequency shifts of the two strongest coherent modes as a function of
$F_{\mathrm{P}}$. (e) The relaxation time and (f) amplitude of the exponential
components as a function of $F_{\mathrm{P}}$. The black dashed line is the
extrapolation of the low $F_{\mathrm{P}}$ behavior. The inset to (e) shows (f)
in expanded scale. The vertical and horizontal dashed lines in correspond to
$F_{\mathrm{ce}}$, $F_{\mathrm{c}}$ and $T_{\mathrm{IM}}$, respectively.

The high-$F$ transient reflectivity shows an initial fast,
$\tau_{\mathrm{e}}=0.5$ ps, relaxation followed by slower $F$-dependent
dynamics (see Figure 3 (c)). The slow dynamics sets in above
$F_{\mathrm{P}}\sim 5$ mJ and consists of a plateau emerging into a broad dip
at $\sim 60$ ps at the highest $F_{\mathrm{P}}$ followed by relaxation
extending to a $\sim 1$ ns timescale.

The high-$F_{\mathrm{P}}$ slow dynamics was investigated further by means of
the multi-pulse experiments. The multi-pulse transient reflectivity, $\Delta
R_{3}/R$, well above $F_{\mathrm{c}}$ is shown in Figure 5 for different D-P
delays, $t_{\mathrm{DP}}$. At short $t_{\mathrm{DP}}$, the multi-pulse
transient reflectivity resembles the 2-pulse transient reflectivity of the
low-$T$ state around $F_{\mathrm{P}}\sim F_{\mathrm{c}}$, but, with _much
strongly damped_ CO. After $t_{\mathrm{DP}}\sim 5$ ps the CO start to
gradually reemerge also on longer P-Pr delays, $t_{\mathrm{PPr}}$. The
multipulse transients are _completely different_ from the high-$T$ MC phase
transients indicating that the highly excited transient state cannot be
associated with the high-$T$ MC phase.

The DECP fit component analysis shown in Figure 6 indicates that the initial
amplitudes of the strongest two modes (see Figure 6 (c)) _do not appear
suppressed_ even immediately after the D pulse while the damping factors (see
Figure 6 (b)) exceed the equilibrium value (Naseska  _et al._ , 2020) of $\sim
0.01$ THz by up to two orders of magnitude recovering by an order of magnitude
on a $\sim 100$ ps $t_{\mathrm{DP}}$ timescale. Just before the arrival of the
subsequent D pulse, after 5 $\mu$s, the damping is reduced close to the
equilibrium-state value, together with exponential components parameters
(Figure 6 (e) and (f)). The low-$T$ state is therefore completely recovered
after each D pulse, albeit, at a higher $T$ than the cryostat base temperature
due to the heat buildup. The behavior is similar also at an increased
$F_{\mathrm{D}}=8.1$ mJ/cm2 with slower recovery. (Naseska  _et al._ , 2021)

The suppression of the CO is delayed with respect to the D pulse for $\sim
0.6$ ps (see inset to Figure 5) and does not depend on the small changes of
the $t_{\mathrm{DP}}$ delay (Naseska  _et al._ , 2021). The delay is
consistent with the duration of the initial sub-picosecond transient observed
for the high-$F$ transient reflectivity in Figure 3 (a).

No irreversibility of the transient reflectivity was observed after exposing
the sample to long trains of pump pulses at the highest $F_{\mathrm{P}}$ as
shown in Figure 3b. To check for any heat build up effects single D-pulse
experiments were also performed. No long-lived changes of the ultrafast
transient reflectivity were found.(Naseska  _et al._ , 2021)

Figure 5: Multi-pulse transient reflectivity, $\Delta R_{3}/R$, as a function
of the delay between the D and P pulses, $t_{\mathrm{DP}}$. The arrows at the
short-$t_{\mathrm{DP}}$ scans emphasize the CO while the arrow at the
$t_{\mathrm{DP}}=5$-$\mu$s scan indicates the arrival of the subsequent D
pulse. The two top-most traces, shown for comparison, correspond to the
thermal equilibrium MC- and DWC-phase transient reflectivity. The traces are
vertically offset for clarity and the thin lines in both panels correspond to
the DECP model (Zeiger  _et al._ , 1992) fit (see SM (Naseska  _et al._ ,
2021)).

##  IV Discussion

The onset of the nonlinearities with increasing $F_{\mathrm{P}}$, at
$F_{\mathrm{ce}}\sim 0.6$ mJ/cm2, is mostly expressed in the SEC that can be
associated with the electronic degrees of freedom only. The SEC total duration
(including the risetime) of $\sim 500$ fs significantly exceeds the time
resolution of the setup ($\sim 80$ fs) so the nonlinearity cannot be
associated with the pump optical-transition nonlinearity, but is associated
with the low-energy electronic degrees of freedom.

Focusing on the lattice degrees of freedom, which show strong $F_{\mathrm{P}}$
nonlinearity on the larger fluence scale above $F_{\mathrm{c}}$, we first
estimate the transient heating taking the $T$-dependent frequencies of the
strongest two coherent modes,(Naseska  _et al._ , 2020) O1 and O2 at 1.86 THz
and 2.34 THz, respectively, as proxies for the lattice temperature. The
obtained effective temperatures (see Figure 4 (d)) reach $T_{\mathrm{IM}}$ at
a fluence that is close to $F_{\mathrm{c}}$. This indicates that the symmetry-
breaking lattice distortion is significantly affected only after the lattice
degrees of freedom are heated close to $T_{\mathrm{IM}}$. This is consistent
with the absorbed energy at $F_{\mathrm{c}}$ that is comparable to the
equilibrium enthalpy difference(Naseska  _et al._ , 2021) between the high-$T$
MC phase at $T\sim T_{\mathrm{IM}}$ and the low-$T$ phase at $T\sim 100$ K.
Here the initial state $T$ at $t_{\mathrm{DP}}\sim 5$ $\mu$s (see Figure 6
(d)) is also estimated from the coherent mode frequencies.

The suppression of the CO with increasing $F$ above $F\mathrm{{}_{c}}\sim 3$
mJ/cm2 is not abrupt as in the equilibrium case. This can be attributed to the
finite-optical-penetration-depth induced excitation inhomogeneity. The
thickens of a thin surface layer, within which the absorbed energy threshold
is exceeded, reaches the optical penetration depth thickness only at the
external fluence of $e\times F_{\mathrm{c}}\sim 8$ mJ/cm2, consistent with the
fluence where the CO are completely suppressed (see Figure 4 (c)). The
excitation inhomogeneity results in a rather large error bar of
$F\mathrm{{}_{c}}$, of the order of $\pm 0.5$ mJ/cm2, but does not hinder a
clear separation between $F\mathrm{{}_{ce}}$ and $F\mathrm{{}_{c}}$.

The electronic ordering is therefore affected at lower excitation densities
than the lattice order. The drop of $\tau_{\mathrm{e}}$ from the equilibrium
value of a few picoseconds to $\sim 200$ fs indicates that the gap-induced
relaxation bottleneck (Naseska  _et al._ , 2020) is transiently suppressed
above $F_{\mathrm{ce}}$ already. The estimated photoexcited carriers density
at $F_{\mathrm{ce}}$ is of the order(Naseska  _et al._ , 2021) of $10^{20}$
cm-3 resulting in the plasma frequency of $\sim 0.4$ eV that is larger than
the insulating gap(Wang  _et al._ , 2004) of $\Delta_{\mathrm{I}}\sim 0.15$
eV. The bottleneck suppression can therefore be attributed to a sub-picosecond
transient washout of the gap due to the (screening-induced) Mott transition
that lasts at least few hundred fs. On this timescale a significant amount of
the photoexcited carrier energy can be incoherently damped to phonons since
the inelastic electron-phonon scattering time can be as fast as $\sim 10$
fs(Lundstrom, 2000). However, the low frequency phonons corresponding to Ir-
ions displacements (Naseska  _et al._ , 2020) that set the lattice order
parameter dynamics appear coherent until the gap is restored. According to
Ref. [Khomskii and Mizokawa, 2005] the equilibrium IM transition is a
combination of orbital ordering and Peierls charge density wave transition.
The Ir-t2g-orbitals derived bands are split (band Jahn-Teller effect) into two
fully occupied $xz$\- and $yz$-derived bands and a broader 3/4 occupied
$xy$-derived band that simultaneously becomes gaped due to the Peierls
tetramerization (see Fig 1 (c)). While the photoexcited Mott transition
transiently suppresses the $xy$-derived band charge density wave, the band
Jahn-Teller effect remains effective stabilizing the broken symmetry lattice
deformation until the gap is restored. A similar decoupling of the electronic
and lattice orders on a short timescale has been previously observed in TiSe2.
(Porer  _et al._ , 2014)

Figure 6: D-P delay dependence of the DECP fit parameters. (a) A density plot
of the Fourier-transform spectra from data in Figure 5. The horizontal lines
correspond to the equilibrium CO frequencies. (b) The CO dampings and (c) the
corresponding amplitudes as a function of $t_{\mathrm{DP}}$. (d) The effective
temperature obtained from the frequency shifts of the two strongest coherent
modes as a function of $t_{\mathrm{DP}}$. (e) The relaxation time and (f)
amplitude of the exponential components as a function of $t_{\mathrm{DP}}$.
Open diamonds in (b), (c), (e) and (f) correspond to the experiment at
$F_{\mathrm{D}}=8.1$ mJ/cm2.

Turning to the highly excited transient state (HETS), the multi-pulse
experiments clearly confirm that even at $F_{\mathrm{D}}\sim 8$ mJ/cm2 the
excited volume does not switch into the high-$T$ MC state despite the large
absorbed energy. In the most excited part of the volume, near the surface, the
absorbed energy exceeds the equilibrium enthalpy difference between the
high-$T$ MC state just above $T_{\mathrm{IM}}$ and the initial low-$T$ state
by _more than two times_. The presence of the strongly damped broken-symmetry
(Naseska  _et al._ , 2020) CO with the unsuppressed initial amplitude (see
Figure 6 (c)) indicates that the local lattice symmetry remains broken in most
of the excited volume, albeit with strong dephasing due to fluctuations and/or
increased disorder.

The broken symmetry state could be stabilized by the pressure exerted by the
surrounding unexcited bulk low-$T$ phase. Application of pressure is known to
stabilize the triclinic insulating phase (Oomi  _et al._ , 1995), which has
$\sim 0.7$ % larger density (Furubayashi  _et al._ , 1994) than the MC phase.
Taking the compressibility(Garg  _et al._ , 2007), $\beta\sim 6\cdot 10^{-3}$
GPa-1, one obtains the pressure of $\sim 1.1$ GPa leading to an increase(Oomi
_et al._ , 1995) of $T\mathrm{{}_{IM}}$ to $T\mathrm{{}_{IM}}^{\prime}\sim
260$K.

In addition to the $T\mathrm{{}_{IM}}$ increase, the pump optical penetration
depth could increase due to the optical-transition bleaching at high-$F$
resulting in saturation of the absorbed energy density.(Naseska  _et al._ ,
2018) Such saturation cannot be entirely excluded and the maximum experimental
$T_{\mathrm{eff}}$ indeed does not appear to significantly exceed
$T\mathrm{{}_{IM}}^{\prime}$ (Figures 4 (d) and 6 (d)). We should note,
however, that at the high excitation $T_{\mathrm{eff}}$ does not directly
correspond to the most excited volume region since the CO are inhomogeneously
suppressed so $T_{\mathrm{eff}}$ is somewhat weighted towards the less-excited
volume region.

While the above effects can contribute to stabilization of the broken symmetry
state, they _cannot account_ for the observed large CO dephasing.
Extrapolating the equilibrium $\gamma(T)$(Naseska  _et al._ , 2020) to the
estimated(Naseska  _et al._ , 2021) maximum possible transient $T\sim 400$ K
we obtain $\gamma\sim 0.08$ and $\sim 0.16$ for the O1 and O2 mode,
respectively, which are at least$\sim 5$ times smaller than the observed
$\gamma$ below $t_{\mathrm{DP}}\sim 20$ ps (Figure 6 (b)). The picosecond
transient state therefore cannot be characterized as an ordinary super-heated
equilibrium pressure-stabilized low-$T$ phase.

According to the classical kinetic theory of first order phase transitions
(Lifshitz  _et al._ , 1995) the formation of a stable phase from a metastable
one proceeds through formation of macroscopic droplets that, due to the
surface energy terms, grow deterministically only if their size exceeds a
critical size, $R_{\mathrm{c}}$. The nucleation of the droplets with
$R>R_{\mathrm{c}}$ is a stochastic process driven either by fluctuations or
tunneling, the later being proposed for the broken-symmetry-state droplet
formation in the supercooled early universe (Linde, 1982). In the present case
thermal fluctuations are likely large enough that the tunneling should not
play any role.

The observed HETS could, therefore, be tentatively understood in this context
where the fluctuating droplets of the high-$T$ MC phase introduce strong
dynamical disorder into the triclinic phase matrix, but fail to reach the
critical size or their growth is too slow to reach the MC phase before the
heat diffusion spreads the absorbed energy across a larger volume, recovering
the stability of the triclinic phase. Since there was no observable variation
of the pump scattering with $t_{\mathrm{DP}}$ in the multipulse experiments
the droplets are presumably much smaller than the pump wavelength of 800 nm
and are most likely size-limited by the characteristic low-$T$-phase twin
lamella thickness (Sun  _et al._ , 2001) of $\sim 20$ nm.

Increasing the excitation fluence beyond $F\sim 8$ mJ/cm2 the standard 2-pulse
$\Delta R/R$ (see Fig. 3 (c)) shows evolution of a broad dip between $\sim 10$
and $\sim 200$ ps. The dip might signify an onset of MC-phase droplets growth.
Unfortunately, this excitation region was not investigated in multipulse
experiments in order to avoid sample cracking111Observed in earlier
experiments upon high laser induced average thermal load. due to the high
D-beam average thermal load222The average thermal load in the multipulse
experiments is doubled with respect to the standard 2-pulse experiments since
D-beam is unchopped.. Further experiments with lower repetition laser source
are therefore necessary to reach a more definitive conclusion regarding the
behavior beyond $F\sim 8$ mJ/cm2.

The HETS is expected to be quite metallic and different from the DWC state
since the sub-picosecond component behavior indicates the suppression of the
gap at lower $F$ already. However, the evidence is indirect so more direct
experimental evidence from time-resolved low-energy probes is necessary to
clearly establish the degree of metallicity.

Since the HETS state appears after strong ultrafast excitation on a sub-
picosecond timescale, in a manner similar to the metastable metallic H phase
(Stojchevska  _et al._ , 2014; Ravnik, 2019) in 1$T$-TaS2, it is necessary to
compare the phenomena. 1$T$-TaS2 is, unlike CuIr2S4, a layered compound with
the equilibrium phase diagram that is significantly more complicated than the
phase diagram of CuIr2S4. In particular, the low-$T$ insulating commensurate
(C) polaron ordered phase is separated from the symmetric high-$T$ metallic
phase ($T>543$ K) by intermediate incommensurate polaron ordered phases that
are quite conducting.

The H phase is created after the C phase exposure to a single ultrafast
optical pulse above a critical fluence, $F_{\mathrm{H}}\sim 1$
mJ/cm2.(Stojchevska  _et al._ , 2014) On the long timescales the H phase
appears as a disordered texture of the frozen C-phase polaron
domains(Gerasimenko  _et al._ , 2019a). An additional metastable amorphous (A)
polaron glass phase has also been observed recently that is created at higher
excitation fluences than the H phase, $F\gtrsim F_{\mathrm{A}}\sim 3.5$
mJ/cm2.(Gerasimenko  _et al._ , 2019b)

The spatial textures of the H and A phases and their evolution on the
ultrafast timescales have not been yet determined and the microscopic pathway
from the C to either H or A phase is still unclear. Since different C-phase
domains in the H phase have different phase shifts relative to the underlying
atomic lattice(Gerasimenko  _et al._ , 2019a) a global reconfiguration of the
polaron lattice must take place during the transition already above
$F_{\mathrm{H}}$. This implies that the C polaron lattice “melts” into some
transient state before condensing into the H (A) phase. This transient state
could be, similarly to the present case on the picoseconds timescales,
composed from droplets of a “melted” phase that are embedded into the C phase.
The droplet phase could correspond to either some incommensurate phase,
polaron liquid or even the high-$T$ metallic phase without polarons.

The CO in 1$T$-TaS2 are not so strongly suppressed during the creation of the
H phase and become suppressed only for $F>F_{\mathrm{A}}.$(Ravnik  _et al._ ,
2018, 2020) The H phase creation conditions in 1$T$-TaS2 are therefore more
similar to the excitation above $F_{\mathrm{ce}}$ in CuIr2S4, where the
lattice $T$ remains below $T\mathrm{{}_{IM}}$, while the fluences above
$F_{\mathrm{c}}$, where also the lattice $T$ transiently exceeds
$T\mathrm{{}_{IM}}$, correspond better to the A-phase creation conditions in
1$T$-TaS2. Owing to the similar penetration depths of $\sim 30$ nm and $\sim
40$ nm in 1$T$-TaS2(Gerasimenko  _et al._ , 2019b) and CuIr2S4(Naseska  _et
al._ , 2021), respectively, and the similar equilibrium transition
temperatures it is not surprising that $F_{\mathrm{c}}$ and $F_{\mathrm{A}}$
are rather similar.

Contrary to 1$T$-TaS2 in CuIr2S4 the same DWC state forms on the long
timescales irrespective of the excitation fluence and the excitation type. It
was shown(Kiryukhin  _et al._ , 2006; Božin  _et al._ , 2011) that in the DWC
state the long range low-$T$ order is replaced by a short range ($\xi\sim 2$
nm) incommensurate modulation with preserved _local_ Ir4+ dimerization. The
DWC state is, therefore, structurally more comparable to the A phase than the
H phase of 1$T$-TaS2. However, the electronic gap in the DWC state is not
suppressed(Takubo  _et al._ , 2008) like in the A state of
1$T$-TaS2(Gerasimenko  _et al._ , 2019b) and the resistivity is larger,
$\rho\sim 65$ $\Omega$cm, than in the A state of 1$T$-TaS2 with $\rho$ in the
$\sim 10$ m$\Omega$ range(Gerasimenko  _et al._ , 2019b). The DWC and A state
(as well as H state) are therefore quite different. This could be connected to
the absence of strong correlations and weak neighboring Ir-ion chain coupling
in CuIr2S4 making the electronic gap less sensitive to the disorder-induced
doping.

Comparing CuIr2S4 with VO2 and V2O3 we observe similarities, but also
differences. All three compounds show transient intermediate states with
suppressed gaps (Morrison  _et al._ , 2014; Wegkamp  _et al._ , 2014; Lantz
_et al._ , 2017; Singer  _et al._ , 2018) that are structurally distinct from
the high-$T$ metallic phases suggesting a transient decoupling of the
structural and electronic orders. The decoupling in both oxides has been
attributed to the correlation effects (Wegkamp  _et al._ , 2014; Lantz  _et
al._ , 2017). In our case, however, the effect can be attributed to a
photoinduced Mott transition.

The timescales of the structural transitions are quite different. In VO2 the
transition is abrupt (Jager  _et al._ , 2017; Wall  _et al._ , 2018; Otto  _et
al._ , 2019) while V2O3 (Singer  _et al._ , 2018; Ronchi  _et al._ , 2019) and
CuIr2S4 show first-order kinetics bottlenecks which slow down the transition.
In this context it is worth noting that the lattice volume change at the
equilibrium IM transition is the smallest, 0.1 % in VO2 (Marezio  _et al._ ,
1970) in comparison to -1.3 % in V2O3 (Rozier  _et al._ , 2002) and 0.7 % in
CuIr2S4 (Furubayashi  _et al._ , 1994), suggesting a smaller inter-phase-
boundary energy cost enabling easier nucleation of the high-$T$ metallic phase
droplets.

##  V Conclusions

We showed that in the strongly nonequilibrium regime upon a femtosecond
photoexcitation the low-temperature ultrafast structural dynamics in CuIr2S4
is dominated by the first-order-transition nucleation kinetics that prevents
the complete ultrafast photoinduced structural transition into the high-$T$
phase at unexpectedly large excitation densities. Concurrently, the
dynamically-decoupled electronic order is suppressed rather independently on a
sub-picosecond timescale at much weaker excitation density. While the
electronic order suppression likely results in a transient metallization of
the highly nonequilibrium broken lattice-symmetry state no evidence was found
that femtosecond excitation would lead to a long lived metastable state more
conducting than the slowly formed disordered weakly conducting phase.

## Acknowledgments

The authors acknowledge the financial support of Slovenian Research Agency
(research core funding No-P1-0040 and young researcher funding No. 50504) for
financial support. We would also like to thank V. Nasretdinova and E.
Goreshnik for the help at the sample characterization, M. Aničin and A. Bavec
for help at transient reflectivity measurements and E. Bozin for fruitful
discussions.

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