# Ionization Dynamics in Intense Laser-Produced Plasmas

M. S. Cho<EMAIL_ADDRESS>Lawrence Livermore National Laboratory, 7000 East
Avenue, Livermore, California 94550, USA  A. L. Milder  Laboratory for Laser
Energetics, 250 East River Road, Rochester, New York 14623, USA  Department of
Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada  W. Rozmus
Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1,
Canada  H. P. Le  H. A. Scott  Lawrence Livermore National Laboratory, 7000
East Avenue, Livermore, California 94550, USA  D. T. Bishel  Laboratory for
Laser Energetics, 250 East River Road, Rochester, New York 14623, USA
Department of Physics and Astronomy, University of Rochester, Rochester, New
York 14627, USA  D. Turnbull  Laboratory for Laser Energetics, 250 East River
Road, Rochester, New York 14623, USA  S. B. Libby  Lawrence Livermore National
Laboratory, 7000 East Avenue, Livermore, California 94550, USA  M. E. Foord
Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore,
California 94550, USA

(August 28, 2024)

###### Abstract

The ionization dynamic of argon plasma irradiated by an intense laser is
investigated to understand transient physics in dynamic systems. This study
demonstrates that significant delayed ionization responses and stepwise
ionization processes are crucial factors in determining the ionization state
of such systems. When an intense laser begins to ionize an initially cold
argon plasma, the conditions change rapidly, leading to a delayed response in
ionization. Consequently, the dynamics do not reach a steady state, even if
the electron temperature and density appear unchanged, particularly when the
atomic transition process is not sufficiently rapid compared to the relevant
time scales. Furthermore, in this case, numerous highly excited states are
created primarily through collisional excitation. Thus, even low-energy
photons can predominantly ionize plasmas, challenging the conventional belief
that such photon energies insufficient to overcome the binding energy of bound
electrons typically contribute less to the ionization. These findings
underscore the necessity of incorporating these processes in ionization
modeling within radiation hydrodynamic simulations for various laser-plasma
experiments.

Ionization processes are crucial in determining a plasma’s properties. These
properties include, but are not limited to, electrical conductivity Krisch and
Kunze (1998), heat transport characteristics Farmer  _et al._ (2020); Milder
_et al._ (2020); Le  _et al._ (2019), and optical emission spectra De Giacomo
and Hermann (2017). The ionization state also affects the sound speed and
collisionality of the plasma. This impacts not only laser-plasma instabilities
but also diagnostic techniques such as Thomson scattering Sheffield  _et al._
(2010). In these techniques it is often difficult to distinguish between
changes in ionization state and temperature. The study of ionization dynamics
thus becomes indispensable for comprehending the behavior of plasmas under
laser irradiation. This knowledge is vital for applications ranging from
material processing to fusion research, where control and predictability of
plasma properties are essential Falcone (2024); Hu  _et al._ (2024); Hurricane
_et al._ (2024).

In this Letter, we elucidate two surprising phenomena in the ionization
dynamics of intense laser-induced plasmas: a long delayed ionization response
from rapidly changing conditions, and the dominance of a two-step ionization
mechanism that involves collisional excitation followed by photoionization.
Initially, as the plasma undergoes rapid changes in its conditions, the
ionization fails to achieve a steady state at each timestep, leading to a lag
in ionization. Notably, this effect can persist even when plasma conditions
appear to stabilize, resulting in an ionization level that is consistently
more than 15% lower than expected under steady-state conditions. Additionally,
despite photon energies ($\sim 3.5$ eV) being below the typical electron
binding energy threshold ($\sim$ a few tens of eV), our simulations show that
intense laser pulses primarily produce a photoionized plasma environment. This
is facilitated by electron collisions that populate higher energy levels,
leading to subsequent photoionization. In radiation hydrodynamics simulations
for laser-plasma experiments, several methodologies for calculating
ionization, which are strategically chosen for rapid computation, have yet to
consider the importance of these physical phenomena, thereby highlighting
their significance Scott  _et al._ (2022); Frank and Scott (2022); Kluth  _et
al._ (2020).

To investigate the dynamics of the ionization process when a gas target is
subjected to laser irradiation, Non-Local Thermodynamic Equilibrium (NLTE)
calculations Scott and Hansen (2010); Chung  _et al._ (2013); Mashonkina  _et
al._ (2005); Cho  _et al._ (2024) are performed, employing the experimental
parameters from the Omega Laser Facility Milder  _et al._ (2022). In these
experiments, eleven laser beams, each imparting 200 joules of energy, are
concentrated onto argon gas from a supersonic jet to investigate the heat
transport mechanisms within the ensuing plasma. The flat-top laser pulse has a
FWHM duration of 1 ns with rise and fall times near 100 picoseconds, and a
peak intensity of roughly 1.1 x $10^{15}$ $W/cm^{2}$ at the focal point. A
wavelength of 351 $nm$ is used, and the ion density in the plasma is estimated
at $1.8$ x $10^{18}cm^{-3}$. Thomson scattering is utilized to measure
electron temperature and density at the plasma’s core, providing benchmarks
for our simulations.

Ionization calculations are conducted using the code Cretin Scott (2001);
Scott and Hansen (2010), based on the experimentally measured temperatures and
densities. Cretin, extensively utilized in laser plasma experiments,
especially for indirect inertial confinement fusion (ICF) simulations Lindl
_et al._ (2014); Hurricane  _et al._ (2023), proves to be well-suited for the
analysis of this experiment. This code facilitates time-dependent modes
capturing ionization dynamics, which allows comparison with the commonly used
steady-state models in plasmas with $<$10% critical density Milder  _et al._
(2021); Turnbull  _et al._ (2023). Note that the time-dependent mode operates
in two distinct forms. The first involves calculating ionization based on
inputs that reflect temporal changes in plasma conditions (TD-P), such as
electron density ($N_{e}$) and temperature ($T_{e}$). The second method
computes ionization by taking the characteristics of the laser used and the
density of the target as initial conditions (TD-L). In this mode, $N_{e}$ and
$T_{e}$ are calculated self-consistently within the framework of the code.
(See ’Supplementary Material 1’ for more detailed information on NLTE
calculation modes.)

Figure 1:  (a) The evolution of average ionization under various calculation
modes is examined. Ionization assuming the steady-state (SS) condition is
denoted by circles, in contrast to the time-dependent mode with plasma
conditions (TD-P), which is represented by a solid line. The dashed line
represents another time-dependent mode involving laser inputs (TD-L), which
provides simulated $N_{e}$ as blue and $T_{e}$ as red dashed lines in (b).
Measured $N_{e}$ and $T_{e}$ at the center of an Ar gas are plotted together
as circles with experimental error bands, redrawn based on Ref.Milder  _et
al._ (2022). The rapidly changing plasma conditions are marked by a pink
shaded region, and the green shaded region illustrates the quasi-stable state
where $T_{e}$ and $N_{e}$ exhibit minimal variation due to the balance of
incoming and outgoing energy in the system.

Figure 1(a) illustrates the temporal evolution of ionization for Ar under
measured plasma conditions, demonstrating significant lag in ionization
response compared to the steady-state values within the NLTE model, despite
identical plasma input conditions. In the steady-state assumption, the
ionization degree is calculated by assuming plasma equilibrium at each
measured point of temperature and density. Notably, at the initial measurement
interval of 200 ps, the plasma displays an ionization degree of approximately
16, suggesting that all electrons, except those in the K-shell, are ionized.
Conversely, the TD-P calculations, using identical plasma conditions, shows a
persistence of many electrons in the levels with the principal quantum number
(n) $>$ 1, failing to reach an ionization degree of 16 within the observed
timeframe. This phenomenon indicates that in systems undergoing rapid plasma
condition changes ($<$ 500 ps), the rate of ionization cannot match the pace
of these changes, preventing equilibrium attainment, corroborating a model
previously proposed by D. Bishel, et al. Bishel  _et al._ (2023).

More importantly, the differences in ionization is still observed within the
quasi-stable period, the green-highlighted region in Fig.1(a) where key plasma
parameters, $N_{e}$ and $T_{e}$ remain stable. This plasma appears to maintain
steady conditions, suggesting that the system and other dependent physical
quantities might also remain stable. Nonetheless, time-dependent NLTE
calculations indicate ongoing ionization processes within this segment. At
equilibrium, the plasma conditions—specifically $N_{e}$ of about 2.3x$10^{19}$
cm-3 and $T_{e}$ of 600 eV—are sufficient to ionize all electrons except
K-shell electrons. However, given the rapid attainment of these conditions,
the ionization rate fails to stabilize even by 1 nanosecond. In this period,
the collisional ionization serves as the predominant ionization mechanism.
When electron densities stabilize at approximately $2.4\times 10^{19}$ cm-3,
the collisional ionization rate afforded by the Lotz model implemented within
Cretin Lotz (1968), is $\sim$ $10^{10}$ Hz. This rate is not sufficiently
rapid to compensate for previously established disparities in ionization
within this regime. Therefore, the free electrons in the system, during this
temporal range, continue to ionize through collisions, driving the system
toward a steady-state ionization degree under quasi-stable conditions.

For validation, we conduct a time-dependent calculation using the laser
profile from the experiment and maintaining a constant initial ion density
(TD-L). Calculations with a laser intensity of ($3$x$10^{14}$ $W/cm^{2}$) and
initial gas density($1.8$x$10^{18}$ $cm^{-3}$) show agreement with the
measured plasma conditions [Fig. 1(b)] and TD-P ionization states [Fig. 1(b)].
Like the TD-P calculations, the TD-L calculations show an evolution of the
ionization state while the plasma conditions are quasi-stable. Conclusively,
if the system’s plasma conditions evolve rapidly, the ionization degree will
continue to evolve both during and after this rapid transition phase. The
ionization in the later phase (post-500 ps) is slightly lower than in TD-P
calculation, due to the model’s zero-dimensional nature, which does not
account for spatial dynamics such as heat conduction or plasma expansion. This
leads to slightly different $N_{e}$ and $T_{e}$ values that are calculated
self-consistently in this phase.

Unlike simulation modes relying on $N_{e}$ and $T_{e}$ which obscure the
direct influence of lasers on the ionization process, a notable advantage of
the TD-L calculation is its ability to investigate the ionization processes
induced by intense laser fields at the atomic level. Therefore, we analyze the
ionization dynamics during the initial 500 picoseconds which provides good
agreement of $N_{e}$ and $T_{e}$ with experiment, as shown in Fig.1(b), when
the effects of heat conduction and hydrodynamic expansion are negligible.

Figure 2:  (a) Net population flux of three different ionization processes in
time: Radiative ionization process (red), collisional ionization process
(blue), auto ionization process (green) considering their inverse processes.
"Net" denotes the subtraction of the recombination processes (the reverse of
ionization), with the absolute value taken, the dash line indicates the
inverse process is dominant. (b) Detailed time-dependent ionization
calculations for the first 500 ps, while omitting individual transition
processes. This approach enables focused analysis on how each transition
affects the overall ionization. ’w/o PI’, ’w/o CX’, and ’w/o CI’ represents
ionization of TD calculation without photo-aided ionization, collisional
excitation, and collisional ionization, respectively. The inverse of each
process is also turned off simultaneously for each case.

The primary finding of our study is that during the initial 500 picoseconds,
the two-step ionization mechanism, where excitation occurs first and is
followed sequentially by ionization, plays a significant role in ionization.
Figure 2(a) illustrates the net population flux, calculated by multiplying
each ionization rate by the population of the corresponding configuration,
thereby quantifying the transitions attributable to each ionization process
over time. The simulation indicates that the radiative ionization process, or
photo-aided ionization process—meaning ionization driven by photons, including
multi-photon ionization—dominates. This finding is particularly surprising
given that the photon energy in our system, approximately 3.5 eV, is
substantially lower than the ionization potential of Ar plasmas (e.g., the
ionization energy of $\text{Ar}^{6+}$ is greater than 100 eV). The bound-
electron distribution, depicted in Figure 3, provides insight into this
discrepancy. It shows the distribution of bound electrons across principal
quantum numbers ($n$) for each ion at 100 picoseconds, indicating a
significant population of electrons in high-$n$ levels accessible to photo-
aided ionization. This suggests that excitation processes are simultaneously
occurring in the system, populating high-$n$ energy levels with sufficiently
low binding energies to be accessible via a few photons. For the reference,
tunneling ionization rates are not included because the Keldysh parameter is
approximately 1.5 under the system conditions, indicating that multiphoton
ionization is the dominant process and thus primarily considered Keldysh
(2024).

Furthermore, this excitation into high-n levels primarily arises from
collisional processes. Figure 2(b) demonstrates this, as turning off the
collisional excitation process significantly reduced ionization after 70
picoseconds. Compared to the minimal impact observed when collisional
ionization process is disabled, turning off photo-aided ionization process
reduces ionization, emphasizing it as the primary ionization mechanism.
Turning off collision excitation results in the lowest ionization, as it
prevents electrons from being excited to high n-levels, leading to a
significant reduction in both photo-aided ionization and collisional
ionization.

Figure 3:  Population histogram of bound electrons in relation to their
ionization states and principal quantum numbers at 100 picoseconds. The
collisional excitation timescale, which is the inverse of the collisional
excitation rate, depending on the ion stage are depicted on the right axis.
The timescale specifically accounts for the transition from the ground state
to the first excited state of each ion, as this transition typically has the
smallest rate among all excitation processes due to the largest energy
difference.

Physically, the superior efficacy of photons over electrons for ionization of
electrons in high-n levels can be understood by considering the number of free
electrons and photons in the system. The cross-sections for both
photoionization and collisional ionization at these levels, calculated
respectively using Kramers’ formula and Lotz’s formula, are both $\sim$
$10^{-16}$ $cm^{2}$ and do not significantly differ Kramers (1923); Lotz
(1968). Since the photon number flux, approximately $10^{33}$ $cm^{-2}s^{-1}$,
vastly exceeds the electron flux in that range ($\sim$ $10^{28}$
$cm^{-2}s^{-1}$), photo-aided ionization is more likely to occur when the
excited levels are populated. Additionally, despite the presence of a large
number of photons, the collisional process becomes the primary excitation
mechanism due to the high photo-deexcitation rate, which compensates for
photo-excitation.

Additionally, within the two-step ionization mechanism, the collisional
excitation rate governs the ionization time scale due to its substantially
lower rate compared to the radiative ionization process. The time scale of the
collisional excitation, which is the inverse of the transition rate, from the
ground state to the first excited state of each ion is illustrated in Fig.3.
As ionization advances, the energy gap between these states widens, resulting
in increasingly longer collisional excitation times. Consequently, for ions
with a charge state of +7 or higher, the process can extend over tens of
picoseconds, while for ions with a charge state of +12 or higher, it can
extend over hundreds of picoseconds. Given that the rate of the second step,
photo-aided ionization, generally occurs on a femtosecond timescale, this
disparity is the primary factor contributing to the ionization lag.

Due to the complicated interplay that causes ionization lag, predicting the
conditions of the ionization lag with simple physical parameters proves to be
exceedingly challenging and requires time-dependent calculations. Therefore,
we propose an empirical formula that mandates when time-dependent calculations
are needed for various laser intensities and target densities - parameters
readily measurable in experiments:

$t_{eq}=\frac{5e^{-4I_{14}}+3}{N_{18}-ln{(\frac{4}{I_{14}}})}[ns]$ (1)

where $I_{14}$ is the laser intensity with the unit of $10^{14}$ $W/cm^{2}$
and $N_{18}$ is the ion density with the unit of $10^{18}$ $cm^{-3}$. It
elucidates the convergence duration of time-dependent NLTE calculations with
steady-state calculations, informed by a set of 100 Cretin simulation cases.
It is derived by first setting specific laser intensity and ion density
density to perform TD-L calculations, then using the resulting $N_{e}$ and
$T_{e}$ as inputs for SS calculations at each timestep. The convergence
duration refers to the time when the ionization difference between these two
calculation methods converges to within 5$\%$. For this formula, the laser is
characterized by a flat top pulse shape with 1 nanosecond time duration, while
the ion density remains constant over time, allowing only the electron
temperature and density to evolve. (See ’Supplementary Material 2’ for more
detailed information on the derivation process.)

Figure 4:  Convergence time as a function of laser intensity and initial ion
density. Convergence time refers to the duration taken for the time-dependent
calculations to converge to a steady state in terms of ionization. The white
region marks the conditions where the convergence time is longer than 4 ns.
This work is highlighted by the red dot.

This formula demonstrates that an increase in density results in a higher
frequency of collisional processes, thereby reducing the collisional
excitation timescale, which is inversely proportional to the density.
Similarly, an increase in laser intensity raises the temperature and the
number of free electrons, consequently reducing the collisional excitation
timescale as well, which decays exponentially with the intensity in the
numerator of the formula. In summary, the empirical formula suggests that for
systems where the calculated convergence time exceeds the hydrodynamic
timescale, ionization lag will occur. This necessitates the use of a time-
dependent model to accurately predict ionization degrees.

Figure 4 depicts the variation in convergence time derived from this formula.
Our analysis shows that a convergence time of approximately 2 nanoseconds is
necessary for the calculated values from both modes to align, in this study.
This highlights that time-dependent modes inherently yield ionization dynamics
different from those of the steady-state mode, requiring time-dependent
calculations. Conversely, experiments with higher densities and laser
intensities, which involve shorter timescales, can be sufficiently modeled
using the steady-state mode.

For the practical usage of this formula, there are a few caveats: this model
is specific to Ar gas targets, and similar trends will be observed for other
materials, albeit with different corresponding laser intensities and gas
densities Turnbull  _et al._ (2024). Additionally, it does not account for
heat conduction and radiation-driven hydrodynamic expansion, which influence
dynamics on the nanosecond timescale and ultimately extend the convergence
time. Moreover, densities from $10^{18}$ to $10^{20}$ $\text{cm}^{-3}$ and
laser intensities from $10^{13}$ to $10^{15}$ $\text{W/cm}^{2}$ are used in
the derivation, suggesting that the formula’s applicability is confined to
these ranges. Therefore, this formula should be used as an approximate guide
for scenarios where time-dependent NLTE calculations are necessary, especially
in comparison to the timescales of interest to experimentalists.

In conclusion, the ionization dynamics of Ar plasma under intense laser
irradiation, as investigated by a NLTE model, reveal an ionization lag not
only during rapid changes in plasma conditions but also under subsequent
steady-state conditions. The transition process in these conditions is not
faster than this timescale, which causes the lagging and deviations from
steady-state predictions. Additionally, a significant finding is the dominance
of photoionization in the ionization process, challenging the conventional
belief that low photon energy is insufficient for ionization. This enhances
our understanding of ionization mechanisms and informs the strategic
application of NLTE models in plasma studies. The empirical formula derived
from Cretin simulation analyses aids in selecting computational approaches
based on ion density and laser intensity variations.

This work was performed under the auspices of the U.S. Department of Energy by
Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344.
This material is also based upon work supported the Department of Energy
[National Nuclear Security Administration] University of Rochester “National
Inertial Confinement Fusion Program” under Award Number(s) DE-NA0004144.

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## Supplementary Material

###  .1 Three Modes of NLTE Calculation

NLTE model provides population distributions and characterizes the physical
processes occurring in the plasma for a given electron temperature and
density. NLTE model solves sets of rate equations to calculate the number
density of the $i^{th}$ atomic state $n_{i}$ as a function of rates
$A_{j\rightarrow i}$ from the jth atomic state to the $i^{th}$ atomic state,
where 1 $\leq$ i, j $\leq$ m (the maximum number of atomic states) from the
following equation.

$\frac{dN_{i}}{dt}=\sum_{j\neq i}(A_{j\rightarrow i}N_{j}-A_{i\rightarrow
j}N_{i})$ (2)

where $N_{i}$ represents the state population at the state $i$ and $A_{i->j}$
means the transition rates from the state $i$ to $j$, describing how quickly
the transition occurs. They include plasma processes, e.g. photon-excitation
and ionization, collisional-excitation and ionization, autoionization and
their inverse processes obtained by the principle of detailed balance.
Typically the rate equation manifests as a matrix equation of high
dimensionality. To present this more concisely, it can be reformulated in a
matrix representation, drawing upon Eq. 2:

$\frac{d}{dt}\begin{bmatrix}n_{1}\\\ \vdots\\\
n_{M}\end{bmatrix}=\begin{bmatrix}A(n_{1},T_{e},n_{e},J)\\\ \vdots\\\
A(n_{M},T_{e},n_{e},J)\end{bmatrix}\begin{bmatrix}n_{1}\\\ \vdots\\\
n_{M}\end{bmatrix}$ (3)

where the $n_{x}$ is the population of the $x_{th}$ energy level. The solution
to the rate equation is derived by utilizing known parameters of the plasma
state. Specifically, in situations where the plasma is in a steady state, it
exhibits no temporal variations in population levels within the rate equation
framework. As a result, the differential term that represents population
change over time, located on the left side of the equation, is set to zero to
simplify the solution process. Critical inputs at this stage include the
plasma’s electron temperature ($T_{e}$) and density ($n_{e}$), which are used
to calculate the transition matrix. Additionally, the radiation field $J$
includes the external radiation field information $J_{\text{ext}}$, in this
case, the laser information, and the self-generated radiation field
$J_{\text{self}}$, which can be excluded under the assumption of optically
thin plasmas in the calculation of the transition matrix $A$. Note that
although the external radiation field $J_{\text{ext}}$ serves as a main
heating source for the plasmas, when we set $T_{e}$ in the calculation, which
already accounts for the outcomes of laser heating, including $J_{\text{ext}}$
would result in double-counting the incident energy. Therefore, the resulting
population distribution reflects the ion distribution within the plasma,
characterized by a steady state under specific temperature and density
conditions only.

Conversely, in scenarios where plasma temperature and density exhibit temporal
variations, the differential term of the population with respect to time
becomes non-zero. Under these dynamic conditions, the transition matrix $A$
undergoes continuous adjustments in response to the evolving temperature and
density, significantly influencing the resulting population distribution. The
approach to computing electron temperature, $T_{e}$, necessitates the use of
two distinct subroutines. If $T_{e}$ and electron density, $n_{e}$, are
supplied as external, time-dependent inputs, the population is derived
accordingly (TD-P). Thus, the population from the preceding time step serves
as the initial condition for the subsequent step and is subsequently updated
by multiplying with the transition matrix. Conversely, when initial values of
$T_{e}$ and $n_{e}$ are provided along with other variable conditions, such as
the temporal history of the input radiation field (e.g., laser temporal
profile), the population change is initially determined by the radiative
transition rate (TD-L). This, in turn, allows for the self-consistent
calculation of $T_{e}$ and $n_{e}$ based on the principles of energy
conservation. Technical details for NLTE population kinetics is described on
the Ref. Scott (2001); Chung  _et al._ (2005); Scott and Hansen (2010);
Ralchenko (2016); Cho  _et al._ (2018, 2020).

The TD-L mode of the NLTE simulation used in this work lacks spatial
information, thus it cannot fully capture plasma dynamics based solely on
laser intensity and initial ion density data. For instance, it does not
account for theoretical predictions or factors such as laser absorption, heat
and radiation transfer, or plasma expansion. To address this limitation, we
assumed variations in actual laser intensity or initial ion density within an
expected range and aimed for temperature and density calculations that closely
matched experimental observations. This approach is grounded in the use of
density and temperature as primary quantities in the rate matrix for
ionization calculations in the NLTE model.

###  .2 Derivation of the empirical formula

Figure A-1:  Convergence Time as a function of initial ion density in
different laser energy. The legend reveals the scaled intensity with the unit
of $10^{14}$ $W/cm^{2}$ and the initial ion density is also scaled with
$10^{18}$ $cm^{-3}$. This work is marked as the red triangle which is $\sim$
2000 ps.

The convergence time denotes the point at which both the steady-state and
time-dependent modes achieve equivalent ionization levels. Within the steady-
state mode, equilibrium values are attained for given density and temperature
conditions. Under the assumption of constant temperature and density, the
time-dependent mode eventually converges to the steady-state mode in every
case. This convergence time, reflecting the rate at which this transition
occurs, is typically related to the transition rate of the most significant
ionization process. The inverse of the rate roughly corresponds to the
convergence time. For instance, a collision ionization rate of approximately
$10^{12}$ Hz suggests a convergence time of about 1 picosecond.

However, in conventional plasma experimentation, fluctuating temperature and
density parameters pose challenges for theoretically estimating convergence.
To address this, we validated convergence times for both modes through
simulation and introduced a formula for improved comprehension. The Eq.(1) in
the main article incorporates parameters such as target density and laser
intensity, which are observable in experimental settings.

The data acquisition process entails three primary steps: (1) Conducting TD-L
mode simulations for varied laser intensities and initial densities to procure
temporal profiles of temperature and density. (2) Performing steady-state
simulations for each temporal instance with those density and temperature and
compute ionization levels at each instance. (3) Comparing ionization obtained
from both steady-state and time-dependent modes to ascertain convergence time.
The criterion for convergence is set for the time-dependent value to be within
5% deviation from the steady-state value. For Ar, this deviation remains below
one, which typically does not significantly affect other properties like heat
transport.

Results in Fig. A-1 demonstrate that higher ion density leads to faster
convergence to a steady state due to increased collision frequency, while
elevated laser intensity expedites ionization. Fitting ion density and laser
intensity data yields a formula akin to Eq.(1) in the main article, with a
convergence time of approximately 50 picoseconds, calculated as the root mean
square of the dataset.