# Ultrafast Dynamics of the Rydberg States of CO2: Autoionization and
Dissociation Lifetimes

D. Biswas1, J. K. Wood2, I. Shalaby1, A. Sandhu∗1,2 1Department of Physics,
University of Arizona, Tucson, Arizona 85721, USA  2College of Optical
Sciences, University of Arizona, Tucson, Arizona 85721, USA
<EMAIL_ADDRESS>
###### Abstract

We report the measurement of ultrafast relaxation dynamics of excited states
of carbon dioxide molecule using time-resolved pump-probe photoelectron
spectroscopy. Neutral ground state carbon dioxide is excited to $nd\sigma_{g}$
Henning sharp Rydberg states with an attosecond extreme ultraviolet pulse
train. A time delayed near infrared probe pulse is used to photoionize these
states to their corresponding ionization limit $B^{2}\Sigma_{u}^{+}$. We
obtain differential kinetic energy spectrograms and angular distributions for
photoionization and autoionization channels. We model the competition between
predissociation and autoionization in the Rydberg state dynamics and analyze
differential photoelectron yield as a function of the time delay to extract
previously unknown autoionization and predissociation lifetimes for three
Henning sharp states (n = 4, 5, 6).

##  I Introduction

Ultrafast femtosecond and attosecond light pulses have enabled time-resolved
probing of molecular dynamics. In particular, time-resolved photoelectron
spectroscopy has served as an important spectroscopic tool to study
photoionization and excited state dynamics in complex molecular systems [1, 2,
3]. Specifically, pump-probe time-resolved photoelectron spectroscopy has been
used to gain new insights about autoionization and predissociation in Rydberg
state dynamics [4, 5], non-adiabatic dynamics [6, 7, 8], and vibronic
wavepacket dynamics [9, 10] in molecules.

Here we report an experimental study of the Rydberg state dynamics of carbon
dioxide (CO2) molecule. It is a linear triatomic molecule with several Rydberg
series of interest. Ground state electronic configuration of CO2 is given by
$[core](4\sigma_{g})^{2}(3\sigma_{u})^{2}(1\pi_{u})^{4}(1\pi_{g})^{4}$ with
symmetry $X^{1}\Sigma_{g}^{+}$ [11]. The lowest three ionic continua
$(1\pi_{g})^{-1}X^{2}\Pi_{g}$, $(1\pi_{u})^{-1}A^{2}\Pi_{u}$ and
$(3\sigma_{u})^{-1}B^{2}\Sigma_{u}^{+}$ are at 13.778, 17.314 and 18.077 eV
adiabatic ionization energies, respectively [11, 12, 13, 14, 15]. The Rydberg
series associated with the $A^{2}\Pi_{u}$ continuum is the Tanaka-Ogawa series
[16]. The exact assignments of these series is still not very well-known. The
most acceptable assignments of these states are given as
$(1\pi_{u})^{-1}nd\delta_{g}$ [16]. Tanaka-Ogawa Rydberg series also has seven
known vibrational progressions. All these vibrational levels converge to
corresponding vibrational levels of $A^{2}\Pi_{u}$ continuum [16, 17, 18]. The
$B^{2}\Sigma_{u}^{+}$ continuum has two Rydberg series associated with it,
namely Henning sharp $(3\sigma_{u})^{-1}nd\sigma_{g}$ and Henning diffuse
$(3\sigma_{u})^{-1}ns\sigma_{g}$ series [19, 20, 16].

A number of photoabsorption, photoionization, and photoelectron studies have
been used to study the dynamics of these Rydberg states. Between 17-18 eV
excitation energy, the photoexcitation spectrum is dominated by Henning sharp
and diffuse series. High resolution synchrotron radiation studies have
precisely measured the photoexcitation and photoionization cross-sections of
these states [21, 11, 22, 23]. Recent ultrafast time-resolved studies have
also measured the asymmetry parameters in photoelectron distribution with
branching ratios between different vibrational states [24]. These Rydberg
states can undergo spontaneous preferential autoionization to the
$X^{2}\Pi_{g}$ continuum or they can predissociate to neutral fragments.
Autoionization is an electronic correlation driven process whereas
predissociation dynamics involves both nuclear and electronic motions. It is
well-known from theoretical quantum efficiency calculations as well as
fluorescence emission spectroscopy that once these Rydberg states are excited
there is a competition between autoionization and neutral predissociation
[25]. Despite the extensive interest, there is very little quantitative
information available on the dissociation and autoionization lifetimes of
these Rydberg states. In a recent study by Fidler et al [26] autoionization
lifetimes of a few Henning diffuse [$(3\sigma_{u})^{-1}ns\sigma_{g}$] states
have been measured for the first time using four-wave-mixing spectroscopy and
compared with theoretical calculations from Fano lineshapes. Another recent
theoretical study by Pranjal et al [27] calculates the valence shell resonant
photoionization and molecular frame photoelectron angular distributions for
these states. In our experiment, we use photoelectron spectroscopy to report
previously unknown predissociation and autoionization lifetimes for three
Henning sharp states [$(3\sigma_{u})^{-1}nd\sigma_{g}$, n = 4, 5, 6] to give
the first experimental measurement of the relaxation dynamics of these states.

Figure 1: Schematic of our experimental apparatus showing a $\sim 50$ fs,
$1.5$ mJ linearly polarized near infrared (NIR) beam with central wavelength
$780$ nm split into pump and probe arms. One arm is focused into a Xe-filled
HHG gas cell generating an attosecond XUV pulse train (pump). The other arm
(probe) is time delayed and combined with the pump beam on a flat mirror with
a hole. Both pump and probe are focused onto an effusive gas jet of carbon
dioxide inside of our VMI spectrometer.

##  II Experiment

We use a pump-probe experimental setup (Fig. 1) for our photoelectron
spectroscopy. We start with a titanium-sapphire laser amplifier system to
generate a $\sim 50$ fs near infrared (NIR) Gaussian pulse with a central
wavelength of 780 nm, pulse energy of $\sim 1.5$ mJ and repetition rate of 1
kHz. This pulse is split into two equal parts using a 50-50 beamsplitter. One
half of the beam is focused into a semi-infinite gas cell filled with xenon to
drive high harmonic generation (HHG) and produce attosecond extreme
ultraviolet (XUV) pulse trains. This XUV pulse train is composed of only odd
harmonics of the driving NIR frequency and used as the pump in our experiment.
The XUV spectral content is separately monitored by using a spectrometer
consisting of a grating and an x-ray CCD. Using a toroidal mirror, XUV beam is
focused onto an effusive gas jet of carbon dioxide in a velocity map imaging
(VMI) spectrometer. The other half of the laser pulse is delayed with respect
to the pump and focused using a 50 cm lens to the same interaction point in
the VMI spectrometer. The intensity of the probe pulse at the interaction
region is $\sim 1$ TW/cm2. A flat mirror with a hole in the center is used to
mix the pump and the probe collinearly. Both XUV and NIR pulses are linearly
polarized parallel to each other.

The VMI approach [28] is a well-known and effective approach in the field of
photoelectron spectroscopy. In this technique, the three-dimensional momentum
distribution of the photoelectrons emanating from an extended focal volume is
mapped onto a two-dimensional position-sensitive detector, thus yielding
detailed information about both energy and angular distribution of the
photoelectrons. The VMI spectrometer has an integrated effusive gas jet of 70
$\mu$m diameter at the center of the repeller electrode. The photoelectron
momentum images are recorded using a CCD camera that images the phosphor
screen placed behind the microchannel plates (MCPs). The photoelectron images
at each pump-probe time delay are Abel inverted using polar basis function
expansion (pBASEX) algorithm [29] to obtain 3D photoelectron momentum
distribution. Angular integration of momentum distribution can be used to plot
the kinetic energy distribution at each pump-probe delay to obtain time
resolved spectrogram. The raw VMI images have some asymmetry due to the non-
uniform spatial response of our detector. To address this, we center and
symmetrize each VMI image before reconstruction. VMI spectrometer is
calibrated using photoelectrons generated from the XUV photoionization of
argon. For low energy electrons, small energy shifts were corrected using the
photoionization lines from the Rydberg states of Xenon.

We subtract XUV+NIR spectrogram from XUV-only spectrogram at each time delay
to extract the delay dependent effect of the probe. In the results discussed
below we use this differential photoelectron spectrograms to isolate probe-
induced photoionization arising from the XUV excited Rydberg states of CO2,
while also analyzing the probe induced disruption of autoionization signals.

Figure 2: Experimental approach showing excitation of the ground electronic
state of CO2 to Henning sharp $(nd\sigma_{g})$ and Henning diffuse
$(ns\sigma_{g})$ Rydberg series with $11^{th}$ harmonic XUV pump. Both these
series converge to $B^{2}\Sigma_{u}^{+}$ ionic continuum. Once excited, these
Rydberg states can undergo autoionization to $X^{2}\Pi_{g}$ state (dotted blue
arrows) or predissociate to neutral fragments CO$(A^{1}\Pi)$ \+ O$(^{1}S)$
(dashed brown arrow). Respective timescales associated with these processes
$\tau_{A}$ and $\tau_{D}$ are shown. An NIR probe pulse at time delay $\tau$
photoionizes the Rydberg states causing depletion of both autoionization and
predissociation channels. Expected photoelectron kinetic energies for NIR
photoionization are shown for three Henning sharp states (4d, 5d, and 6d)
observed in our experiment.

The schematic of the XUV excitation and NIR probing in our experiment is shown
in the Fig. 2. The 11th harmonic in the XUV spectrum resonantly excites
neutral carbon dioxide molecule from the ground state ($X^{1}\Sigma_{g}^{+}$)
to a number of neutral Rydberg states between 17-18 eV. The spectrum of the
11th harmonic is peaked around 17.5 eV with a FWHM bandwidth of $\sim$0.4 eV.
As mentioned before, there are three different Rydberg progressions of CO2
that has been reported at this energy range: Henning sharp series
[$(3\sigma_{u})^{-1}nd\sigma_{g}$], Henning diffuse series
[$(3\sigma_{u})^{-1}ns\sigma_{g}$] and, higher vibrational states of Tanaka-
Ogawa series [$(1\pi_{u})^{-1}nd\delta_{g}$)]. Both Henning sharp
($nd\sigma_{g}$) and diffuse ($ns\sigma_{g}$) series converge to
$B^{2}\Sigma_{u}^{+}$ ionic continuum at 18.08 eV. Only these series are shown
in Fig. 2 because they have significantly larger photoabsorption cross-section
than the higher vibrational states of Tanaka-Ogawa progressions. Once excited,
the Rydberg states can undergo spontaneous autoionization to $X^{2}\Pi_{g}$
ionic continuum at 13.78 eV, or they can predissociate to neutral fragments
$CO(A^{1}\Pi)+O(^{1}S)$. In general, autoionization and predissociation decay
pathways compete, and the associated timescales ($\tau_{A}$ and $\tau_{D}$
respectively) have not been fully explored for these states. Measurement of
photoionization quantum efficiency indicates that predissociation is dominant
in sharp series only [16, 11]. We use a probe pulse with photon energy of
$\sim 1.59$ eV at time delay $\tau$, to photoionize both the molecular Rydberg
states as well as the neutral dissociation fragments available at that time
delay. Thus, probe induced photoionization restricts the possibility of
spontaneous autoionization and we isolate the effect of the probe pulse using
differential photoelectron signal as outlined above. We observe that
photoionization and autoionization photoelectron angular distributions (PADs)
are mutually perpendicular. This arises from the preferential autoionization
direction involving change in core ion symmetry, and it allows us to identify
the symmetry of participating Rydberg states. We model the photoelectron yield
due to the competition between autoionization and predissociation to estimate
lifetimes ($\tau_{A}$ and $\tau_{D}$) associated with two decay pathways.
Finally, by fitting the time-resolved differential spectrograms we obtain the
previously-unknown autoionization and predissociation rates for several
members of the Henning sharp Rydberg series of CO2.

##  III Results

###  III.1 Differential Spectrogram

Figure 3: Experimentally observed angle integrated time resolved differential
photoelectron spectrum of the Rydberg states of CO2. Red and blue implies the
NIR probe induced enhancement or depletion of photoelectron yield. Near time
delay zero we see direct electrons from XUV-NIR cross-correlation. At positive
time delay we see long lived Rydberg states. Expected photoelectron kinetic
energies for both autoionization and single photon photoionization are shown
for Henning sharp ($nd\sigma_{g}$) and Henning difuse ($ns\sigma_{g}$) states.

In Fig. 3, the time resolved differential photoelectron spectrogram is shown.
At each time delay, momentum images are recorded, reconstructed, and angularly
integrated, to obtain kinetic energy as a function of time delay. To isolate
the effect of the probe we record both (XUV+NIR) and XUV only spectra at each
time delay and subtract them to get a differential spectrum. The spectrogram
in Fig. 3 shows the time evolution of continuum photoelectron kinetic energies
at pump-probe time delay interval of 10 fs over a delay range of -150 fs – 950
fs. Red and blue color scales correspond to positive and negative differences
between (XUV+NIR) and XUV-only spectrograms, respectively. Thus, red is the
enhancement and blue is the depletion in the photoelectron yield due to the
effect of time delayed probe pulse. At negative time delay, the NIR probe
precedes the XUV pump excitation, showing no dynamics. Around time delay zero,
we see single NIR-photon scattering sidebands of several direct
photoionization bands of the neutral CO2 which correspond to emergence of
electrons in different ionization continua under the action of the XUV pulse
train. These photoelectron bands around time delay zero correspond to XUV-NIR
cross-correlation and gives information about the temporal profile of the NIR
pulse.

In the positive time delay side of the spectrogram, we see long lived red and
blue signals that correspond to the Rydberg state dynamics of the system. Once
the 11th harmonic XUV pump excites neutral CO2 to Henning sharp and diffuse
series, these are further ionized to $B^{2}\Sigma_{u}^{+}$ by the delayed NIR
probe pulse. The long-lived red signals between 0.6-1.5 eV kinetic energies
correspond to photoelectron yield due to NIR photoionization. Correspondingly,
the long-lived blue negative signals between 3-4 eV kinetic energies represent
the depletion of autoionization decay channel by the action of the same NIR
pulse. The differential signal analysis allows us to clearly isolate the time-
delayed Rydberg state dynamics by subtracting out the contribution of direct
XUV photoionization. Expected kinetic energies for both photoionization and
autoionization signals for different Rydberg states are shown with white tick
marks and energy ladders in Fig. 3 for both Henning sharp ($nd\sigma_{g}$) and
diffuse ($ns\sigma_{g}$) series. Blue autoionization signal does not show any
discrete energy structure due to the fact that these Rydberg states can
autoionize to multiple vibrational levels of $X^{2}\Pi_{g}$ ionic continuum
thus producing different kinetic energy electrons from a single Rydberg state.
On the other hand, low energy red photoionization signal shows three discrete
long-lived lines.

To identify the Rydberg states responsible for these photoionization lines, we
use photoabsorption data from the literature [23] as shown in Fig. 4(a). A
typical XUV spectrum of the $11^{th}$ harmonic is also shown. We observe that
Henning sharp and diffuse series dominate the spectrum beyond $17.4$ eV. Below
that energy, some of the Tanaka-Ogawa states with v = 1, 2 and 3 vibrational
levels are visible with cross-sections comparable to the 4d Henning sharp
state. All these series converge to their corresponding ionization limits,
which are also listed in Fig. 4(a). To compare this photoabsorption data with
our work, we do a baseline subtraction followed by multiplication with our XUV
spectral profile, to get the scaled photoexcitation spectra relevant to our
XUV pump. For each resonance in this spectrum, we obtain the expected
photoelectron kinetic energy by adding the energy of single NIR photon and
then subtracting the energy of its corresponding ionization limit. The
expected photoelectron spectrum obtained in this fashion is shown in Fig.
4(c). For comparison, our delay-dependent photoelectron spectrogram for
positive time delays ($\tau>100$ fs) is shown in Fig. 4(b). Comparing the
expected and observed photoelectron kinetic energies between Fig. 4(b) and (c)
it is clear that the three long-lived lines in the spectrogram correspond to
photoionization of 4d, 5d, and 6d states from the Henning sharp series. In
fact, the 4d and 5d lines are clearly isolated from the others. Photoelectrons
from $n=7$ Tanaka-Ogawa states with vibrational levels v = 2 and 3 could
energetically overlap with the 6d line, but as seen from the Fig. 4(c), the
cross-section for 6d photoexcitation is significantly higher than the Tanaka-
Ogawa state. Note that we do not see any Henning diffuse state photoelectron
peaks in the spectrogram beyond 100 fs time delay because of their short
lifetimes (less than 100 fs) [26]. We do not analyze the signals for time
delays shorted than 100 fs, as the data gets contaminated with NIR
modification of the direct XUV photoelectron bands, as discussed before. The
observed 4d, 5d, and 6d Henning sharp lines match the expected positions. To
extract the lifetime of the Henning sharp states, we use narrow energy windows
around the sharp series as shown in Fig. 4(b) using magenta lines and shaded
areas bound by those.

Figure 4: (a) Photoabsorption cross-sections of neutral Rydberg states of CO2
in the energy range 17-18 eV (adapted from [23]). Henning sharp
($nd\sigma_{g}$) and Henning diffuse ($ns\sigma_{g}$) series are shown with
black and green respectively, with their convergence limit
($B^{2}\Sigma_{u}^{+}$) at 18.08 eV. Three Tanaka-Ogawa progressions are also
shown for vibrational levels $v=1$ (magenta), $v=2$ (yellow) and $v=3$
(orange) with corresponding limits ($A^{2}\Pi_{u}$, $v=1,2,3$). A typical
spectrum for $11^{th}$ harmonic XUV pump is shown with the green shaded area
in the background. (b) Differential spectrogram for low energy photoionization
signals at positive time delay ($\tau>100$ fs). Expected kinetic energies for
single photon ionization of Henning sharp and diffuse states are superposed on
the date. Energy windows used for extraction of lifetime and angular
distribution for these states are shown with magenta lines and the bound area
in between (shown with shaded rectangles). (c) Photoexcitation yield is scaled
according to our broadband XUV spectrum and the Rydberg states from panel (a)
are then translated to the expected continuum photoelectron energies.
Photoelectron energies are calculated separately for each series by
considering single NIR photon ionization of the states to the corresponding
ionization limits shown in (a).

###  III.2 Angular distribution

Figure 5: Angular dependence of differential photoionization and
autoionization signal extracted from centered and symmetrized VMI images. (a)
Differential photoelectron VMI image for positive time delay showing three red
photoelectron rings for $nd$ Henning sharp states and autoionization electrons
in blue, (b) Angular distribution of 4d, 5d, and 6d photoionization electrons
at 250 fs time delay.(b) Angular distribution of autoionization depletion
signals integrated from 100-250 fs In each figure, the vertical purple arrow
corresponds to the polarization direction for both XUV and NIR pulses.

The fact that the photoelectron signals correspond to the Henning sharp series
is further corroborated by the angular distributions of the differential
photoionization and autoionization signals. In Fig. 5(a) we show reconstructed
VMI image of the differential change in the angular distribution of the
electrons integrated over positive time delays from 100 – 500 fs. As before,
the red color represents enhancement due to the NIR probe induced
photoionization of 4d, 5d, and 6d Rydberg states. These signals appear as
distinct rings in the VMI image as the photoelectrons emerge along the
polarization direction of the XUV pump and NIR probe, which are parallel to
each other. On the other hand, the blue color represents disruption of the
autoionization pathways, leading to depletion in the yield of autoionization
electrons. It is clear that the autoionization depletion signal peaks
perpendicular to XUV-NIR polarization direction.

Fig. 5(b) shows photoelectron angular distributions for 4d, 5d, and 6d states,
respectively, where each figure contains three sets of data at time delay of
250 fs. We find that all three states show nearly identical angular
distribution which confirms that all signals are arising from states with same
symmetry. We also observed that there is no significant time-dependent change
associated with the angular distribution of 4d, 5d, and 6d states within the
100-500 fs timescale. Fig. 5(c) shows the angular distribution plot for
autoionization depletion integrated from 100-250 fs. Autoionization signal is
weaker and more diffuse due to contribution from different vibrational states
and it also decays much faster than the photoionization signals, leading to
larger error bars.

The parallel (perpendicular) angular distribution of photoelectrons
(autoionizing electrons) elucidates the nature of states excited by the XUV
pump. Essentially, the Henning series states are populated due to parallel
excitation by the XUV pump pulse and they are further ionized to
$B^{2}\Sigma_{u}^{+}$ ionic continuum using a NIR pulse with the same
polarization. Thus we expect a dipolar pattern of photoelectron emission along
the XUV-NIR polarization direction as seen in the angular distributions. These
observations also rule out Tanaka-Ogawa series as the source, which requires a
perpendicular excitation from the ground state. The autoionization angular
distributions further support these conclusions. When autoionization occurs
from $\sigma_{g}$ symmetry Henning states to $X^{2}\Pi_{g}$ ionic continuum,
the electrons emerge with one unit of angular momentum. As the electron decays
into the $X^{2}\Pi_{g}$ states of a molecular ion preferentially aligned along
the XUV polarization, the autoionizing electron comes out perpendicular to the
light polarization direction.

Thus, using both angular distributions and photoexcitation cross-section data
we confirm that the observed photoelectron signals arise predominantly from
the Henning series. Furthermore, beyond 100 fs time delay, the contributions
are solely from the sharp $nd\sigma_{g}$ states which have long lifetimes.
Referring back to Fig. 4(b), we see that the photoelectron yields from these
nd states decay as a function of time delay between pump and probe. Next, we
model the competition between autoionization and predissociation decay to
extract both lifetimes for the three Henning sharp states observed here.

###  III.3 Modeling predissociation and autoionization

To understand the role of autoionization and predissociation dynamics, we used
a model [4] for time-delay dependent photoelectron yield taking both decay
channels into account. Our XUV pulse excites neutral ground state population
to an excited molecular Rydberg state. This population can decay via
autoionization with a rate of $\gamma_{A}$ and via predissociation with a rate
of $\gamma_{D}$. If initial excited state population is $P_{0}$, time-
dependent population in the molecular Rydberg state can be written as

$P_{mol}(t)=P_{0}e^{-\gamma_{A}t}e^{-\gamma_{D}t}.$ (1)

The population going into autoionization or predissociation channels in time t
can be written as

$P_{auto}(t)=P_{0}(1-e^{-\gamma_{T}t})\left(\frac{\gamma_{A}}{\gamma_{T}}\right),$
(2)
$P_{diss}(t)=P_{0}(1-e^{-\gamma_{T}t})\left(\frac{\gamma_{D}}{\gamma_{T}}\right),$
(3)

where $\gamma_{T}=\gamma_{A}+\gamma_{D}$ is the total decay rate of the
molecular Rydberg population $P_{mol}$.

Delayed NIR probe can photoionize the population available in either molecular
Rydberg state or in predissociated neutral fragments. Both channels will
produce photoelectrons of same kinetic energies. Thus, the total population
available for NIR photoionization at time t can be written as

$\displaystyle P_{tot}(t)$ $\displaystyle=P_{mol}(t)+P_{diss}(t)$ (4)
$\displaystyle=P_{0}e^{-\gamma_{T}t}+P_{0}(1-e^{-\gamma_{T}t})\left(\frac{\gamma_{D}}{\gamma_{T}}\right)$
$\displaystyle=P_{0}e^{-\gamma_{T}t}\left(\frac{\gamma_{A}}{\gamma_{T}}\right)+P_{0}\left(\frac{\gamma_{D}}{\gamma_{T}}\right).$

The total photoionization rate with a Gaussian NIR pulse at time delay $\tau$
is

$\frac{dn_{e}}{dt}(t,\tau)\propto P_{tot}(t)e^{-(t-\tau)^{2}/2\sigma^{2}}$ (5)

where $\sigma$ is the width of the Gaussian pulse. By integrating over time t
we get the photoelectron yield as a function of pump-probe time delay.

$n_{e}(\tau)\propto\left(\frac{\gamma_{A}}{\gamma_{T}}\right)\mathcal{G}_{exp}(\tau,\gamma_{T},\sigma)+\left(\frac{\gamma_{D}}{\gamma_{T}}\right)\mathcal{G}_{exp}(\tau,0,\sigma),$
(6)

where $\mathcal{G}_{exp}$ is defined as the exponential convolution of
Gaussian function.

$\displaystyle\mathcal{G}_{exp}(\tau,\gamma,\sigma)=$
$\displaystyle\int_{0}^{\infty}dte^{-\gamma t}e^{-(t-\tau)^{2}/2\sigma^{2}}$
$\displaystyle=$
$\displaystyle\sigma\sqrt{\frac{\pi}{2}}\exp\left(\gamma\tau+\frac{\gamma^{2}\sigma^{2}}{2}\right)$
(7)
$\displaystyle\times\text{erfc}\left(\frac{\gamma\sigma}{\sqrt{2}}-\frac{\tau}{\sigma\sqrt{2}}\right)$

Figure 6: Delay dependent lineouts from the differential spectrogram. (a) XUV-
NIR Gaussian cross-correlation as measured from the NIR dressing of the direct
photoelectrons generated by XUV ionization to $X^{2}\Pi_{g}$ continuum by
$11^{th}$ harmonic around time delay $\tau\sim 0$ fs, (b) NIR photoionization
of XUV excited 4d Rydberg state, (c) NIR photoionization of 5d Rydberg state
and, (d) NIR photoionization of 6d Rydberg state. In these plots, solid blue
lines correspond to photoelectron yields from probe induced photoionization
from molecular as well as atomic Rydberg states, solid green lines correspond
to contributions from probe induced dressing of direct photoelectrons around
time-delay $\sim 0$ fs and, red lines are the sum of both that gives the total
fitting.

###  III.4 Autoionization and predissociation lifetimes

Using the model above we fit experimentally-observed photoelectron yields for
4d, 5d, and 6d Rydberg states to extract autoionization and predissociation
lifetimes. In the discussion of the experimental spectrogram Fig. 3, we
mentioned that there is a contribution to the signal from the pump-probe
cross-correlation around time delay zero. To take that into account, we added
another Gaussian contribution to the photoelectron yield given in Eq. 6.

As a first step, we use this Gaussian term to fit the photoelectron yield in
the NIR dressing sideband near 1.6 eV (Fig. 6(a)). Fitting the XUV-NIR cross-
correlation allows us to extract the width of the probe pulse while also
calibrating the zero delay. The standard deviation ($\sigma$) width of the
Gaussian cross-correlation is obtained to be $29.77$ fs with an estimated
error of $3\%$, while the correction for zero time delay between pump and
probe pulse is obtained within $\sim 1$ fs accuracy. Using these parameters of
the NIR pulse, we fit three Rydberg photoelectron yields as shown in Fig.
6(b)-(d). In each plot, green line represents the XUV-IR cross-correlation
signal, blue line corresponds to the Rydberg state dynamics, while the red
line is a fit to the total signal. Using this method, we extracted the
autoionization and predissociation lifetimes as listed below in Table 1.

Table 1:  Autoionization ($\tau_{A}$) and predissociation ($\tau_{D}$) lifetimes of 4d, 5d, and 6d Henning sharp series. Numbers in the parenthesis account for the range of fitting parameters for 95% confidence interval. State |  $\tau_{A}$ (fs) |  $\tau_{D}$ (fs)
---|---|---
4d | 286 (265, 310) | 1783 (1536, 2124)
5d | 339 (310, 374) | 2574 (2003, 3652)
6d | 427 (388, 474) | 3055 (2079, 5757)

There is no experimental measurements for the lifetimes of these states
available in the literature. Recently, autoionization lifetimes were
calculated for these states in Fidler et al [26]. Our experimental lifetime
for 5d sharp series matches well within the error limits with the calculated
value of 372.6 fs, whereas the lifetime of 6d state does not match with the
calculated value of 1457.5 fs. We suspect that the complexity of involving the
predissociation process with autoionization causes this discrepancy between
the calculation and the experiment. In the experiment by Fidler et al [26],
they observe only Henning diffuse state lifetimes due to two-photon Raman
transitions with nearby dark resonances. Apart from the experimental
measurments of the autoionization lifetimes of Henning sharp states, we have
also obtained the experimental predissociation lifetimes of these states.
Although there is a significantly-large error bar on the predissociation
lifetimes, there is no calculation or experimental measurement of these
lifetimes available in literature.

##  IV Conclusion

In this work we used ultrafast time-resolved photoelectron spectroscopy to
study the Rydberg state dynamics of CO2 molecule. Using delay-dependent
photoelectron kinetic energy and angular distribution measurements we
identified strong signals for three Henning sharp Rydberg states (4d, 5d, and
6d). Using a model that accounts for the competition between autoionization
and predissociation, we report the first experimental measurements of both
autoionization and predissociation lifetimes for these three Henning sharp
states. This XUV-NIR photoelectron spectroscopy method can be generalized to
observe the non-adiabatic relaxation dynamics associated with electronic
correlations and electron-nuclear couplings in a variety of polyatomic
molecules. The knowledge of relaxation rates and dynamics of excited molecules
has implications for understanding the charge and energy redistributions in a
variety of systems.

##  V Acknowledgments

This work was supported by the National Science Foundation under grant numbers
PHY 2207641 and PHY 1919486.

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