# Ultrahigh Resolution X-ray Thomson Scattering Measurements
of Electronic Structures

Thomas Gawne<EMAIL_ADDRESS>Center for Advanced Systems Understanding
(CASUS), D-02826 Görlitz, Germany  Helmholtz-Zentrum Dresden-Rossendorf
(HZDR), D-01328 Dresden, Germany  Zhandos A. Moldabekov  Center for Advanced
Systems Understanding (CASUS), D-02826 Görlitz, Germany  Helmholtz-Zentrum
Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany  Oliver S. Humphries
European XFEL, D-22869 Schenefeld, Germany  Karen Appel  European XFEL,
D-22869 Schenefeld, Germany  Carsten Bähtz  European XFEL, D-22869 Schenefeld,
Germany  Victorien Bouffetier  European XFEL, D-22869 Schenefeld, Germany
Erik Brambrink  European XFEL, D-22869 Schenefeld, Germany  Attila Cangi
Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
Sebastian Göde  European XFEL, D-22869 Schenefeld, Germany  Zuzana Konôpková
European XFEL, D-22869 Schenefeld, Germany  Mikako Makita  European XFEL,
D-22869 Schenefeld, Germany  Mikhail Mishchenko  European XFEL, D-22869
Schenefeld, Germany  Motoaki Nakatsutsumi  European XFEL, D-22869 Schenefeld,
Germany  Kushal Ramakrishna  Center for Advanced Systems Understanding
(CASUS), D-02826 Görlitz, Germany  Helmholtz-Zentrum Dresden-Rossendorf
(HZDR), D-01328 Dresden, Germany  Lisa Randolph  European XFEL, D-22869
Schenefeld, Germany  Sebastian Schwalbe  Center for Advanced Systems
Understanding (CASUS), D-02826 Görlitz, Germany  Helmholtz-Zentrum Dresden-
Rossendorf (HZDR), D-01328 Dresden, Germany  Jan Vorberger  Helmholtz-Zentrum
Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany  Lennart Wollenweber
European XFEL, D-22869 Schenefeld, Germany  Ulf Zastrau  European XFEL,
D-22869 Schenefeld, Germany  Tobias Dornheim<EMAIL_ADDRESS>Center for
Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany  Helmholtz-
Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany  Thomas R. Preston
<EMAIL_ADDRESS>European XFEL, D-22869 Schenefeld, Germany

###### Abstract

Using a novel ultrahigh resolution ($\Delta E\sim 0.1\,$eV) setup to measure
electronic features in x-ray Thomson scattering (XRTS) experiments at the
European XFEL in Germany, we have studied the collective plasmon excitation in
aluminium at ambient conditions, which we can measure very accurately even at
low momentum transfers. As a result, we can resolve previously reported
discrepancies between _ab initio_ time-dependent density functional theory
simulations and experimental observations. The demonstrated capability for
high-resolution XRTS measurements will be a game changer for the diagnosis of
experiments with matter under extreme densities, temperatures, and pressures,
and unlock the full potential of state-of-the-art x-ray free electron laser
(XFEL) facilities to study planetary interior conditions, to understand
inertial confinement fusion applications, and for material science and
discovery.

The x-ray Thomson scattering (XRTS) technique Sheffield  _et al._ (2010) has
emerged as a powerful method of diagnosing matter. By probing the electronic
dynamic structure factor $S_{ee}(q,E)$, where $q$ and $E$ are the change in
the momentum and energy of the scattered photon, it is capable of giving
detailed insights into the microphysics of the probed sample Glenzer and
Redmer (2009); Gregori  _et al._ (2003); Dornheim  _et al._ (2023a). This
capability is particularly important for experiments with matter under extreme
densities, temperatures, and pressures Graziani  _et al._ (2014); Drake
(2018); Glenzer and Redmer (2009), as they occur e.g. in astrophysical objects
Benuzzi-Mounaix  _et al._ (2014); Becker  _et al._ (2014); Kritcher  _et al._
(2020), inertial confinement fusion applications Hu  _et al._ (2011); Betti
and Hurricane (2016), and for material science and materials discovery Kraus
_et al._ (2016, 2017); Lazicki  _et al._ (2021). Here, the combination of the
extreme conditions with the highly transient nature of the generated extreme
states in the laboratory Vorberger  _et al._ (2024) renders the unambiguous
diagnosis of plasma conditions challenging.

Since the first observation of plasmons in warm dense beryllium Glenzer  _et
al._ (2007), a number of major developments have helped to establish XRTS as a
reliable method for the study of materials over a vast range of densities and
temperatures. This includes the demonstration of ultrabright seeded x-ray
free-electron laser (XFEL) beams Fletcher  _et al._ (2015), the utilization of
XRTS for the detection and quantification of miscibility Frydrych  _et al._
(2020), and the resolution of ion acoustic modes in single-crystal diamond
Descamps  _et al._ (2020).

The measured XRTS intensity is given by Glenzer and Redmer (2009); Sheffield
_et al._ (2010); Dornheim  _et al._ (2023b)

$\displaystyle I(q,E_{s})=S_{ee}(q,E_{0}-E_{s})\circledast R(E_{s})\ ,$ (1)

i.e., as a convolution of $S_{ee}(q,E)$ with the combined source-and-
instrument function $R(E_{s})$, where $E_{0}$ and $E_{s}$ denote the beam
energy and the energy of the scattered photon. In practice, deconvolving Eq.
(1) is generally numerically unstable; the width of $R(E_{s})$ thus limits the
capability of XRTS to resolve electronic features such as sharp plasmon
excitations. This strongly hampers the model-free diagnostics of parameters
such as the temperature Dornheim  _et al._ (2022a, 2023b) that are important
for equation-of-state measurements Falk  _et al._ (2014); Döppner  _et al._
(2023), and poses a serious obstacle for the benchmarking of theoretical
models against experimental observations Cazzaniga  _et al._ (2011);
Ramakrishna  _et al._ (2021).

Figure 1: Schematic illustration of the set-up modified from Wollenweber  _et
al._ (2021). The x-ray beam is seeded at 7703 eV in the SASE2 undulator
including a SASE pedestal. The beam is passed through a four-bounce Si (111)
monochromator to remove this SASE pedestal before being focused onto an Al
foil. The x-rays are scattered, collected and focused by a spherically bent
diced analyzer crystal onto a Jungfrau detector with asymmetric pixels. Both
the analyzer and detector are mounted onto curved rails to vary the scattering
angle $\theta$. The combination of the monochromated beam and ultra high
resolution spectrometer allows for the high-fidelity measurements of
electronic structure.

In this Letter, we present measurements of the plasmon in Al with an
unprecedented resolution of $\Delta E\sim 0.1\,$eV that allows us to resolve
electronic features from XRTS measurements and reconcile discrepancies in
modelling. Our measurements of Al at ambient conditions, made with a new set-
up at the European XFEL in Germany Tschentscher  _et al._ (2017), are in
excellent agreement with previous electron energy loss spectroscopy (EELS)
measurements Krane (1978); Sprösser-Prou  _et al._ (1989). In addition, the
high resolution of our results allows us to unambiguously benchmark time-
dependent density functional theory (TDDFT) Runge and Gross (1984)
calculations, and to resolve discrepancies reported in previous works
Cazzaniga  _et al._ (2011); Ramakrishna  _et al._ (2021).

This new capability will be of paramount importance for XRTS diagnostics:
first and foremost, it allows for the model-free interpretation of the
measured spectra even for comparably moderate temperatures of $T\sim 1\,$eV,
whereas previous experimental set-ups were limited to $T>10\,$eV Dornheim  _et
al._ (2022a, 2023b). This will unlock the full potential of modern XFEL
facilities for material science and discovery Kraus  _et al._ (2016, 2017);
Lazicki  _et al._ (2021), for the characterization of the initial phase of the
compression path of the fuel capsule in laser fusion experiments, and for the
study of planetary interior conditions Benuzzi-Mounaix  _et al._ (2014). We
note that, despite its comparable resolution, EELS generally cannot be used to
diagnose experiments in this regime due to its stringent requirements for thin
targets and long measurement times Brydson (2020).

In addition, we have demonstrated the capability of our XRTS setup to produce
rigorous benchmark data for first-principles theoretical methods, which is
indispensable for the development of new methodologies to simulate extreme
states of matter Graziani  _et al._ (2014); Bonitz  _et al._ (2020); Dornheim
_et al._ (2023a). Finally, the detailed resolution of electronic features of
different materials constitutes an important end in itself, and promises novel
insights into the behaviour of matter across various density and temperature
regimes.

Experimental set-up. The experiment was performed at the HED instrument
Zastrau  _et al._ (2021) of the European XFEL, Germany. The XFEL beam was
self-seeded to an energy of $E_{0}\sim 7703$ eV, and then passed through a
four-bounce silicon $(111)$ monochromator with an acceptance range of 0.8 eV
to remove the underlying self-amplified spontaneous emission (SASE) pedestal
from the beam. The beam was focused onto a 50$\mu$m thick Al foil using a set
of Be compound refractive lenses Zastrau  _et al._ (2021) located 9 m upstream
to a spot of order 10$\mu$m. The full width at half maximum (FWHM) of the beam
incident on the sample was measured to be $\Delta E\sim 450-530$ meV,
corresponding to a spectral bandwidth of $\Delta E/E\sim 5.8-6.9~{}\times
10^{-5}$, as measured from the quasi-elastic scattering feature. An upstream
gas monitor measured initial beam energies of $146-310~{}\mu$J, however
transmission through the beam line optics ($\sim 70~{}\%$) and after removal
of the large SASE pedestal in the monochromator reduced the energy on target
to $15.5-21.8~{}\mu$J ($\sim 7~{}\%$ transmission), as measured on a second
gas monitor before the target.

The scattered x-rays were collected by a spherically-bent Si $(533)$ diced
crystal analyser (DCA) previously employed to measure phonons Descamps  _et
al._ (2020) and described in more detail by Ref. Wollenweber  _et al._ (2021).
However, in contrast to Ref. Wollenweber  _et al._ (2021) where scattering was
measured in an spectral range of just 0.3 eV, here we measure the signal in a
range of 20–40 eV. The incident photons collected by the DCA are dispersed and
focused onto a Jungfrau detector Mozzanica  _et al._ (2018) employing
asymmetric pixels, with a size of 25 $\mu$m in the dispersive direction and
225 $\mu$m in the non-dispersive direction. It is the combination of
monochromating the incident beam and collection of the scattered photons on
the ultra high resolution DCA spectrometer that allows for high fidelity
measurements of the XRTS signal on electronic energy scales.

The primary source of uncertainty in the plasmon measurement is due to the
finite angular coverage of the DCA. This results in so-called $q$-broadening,
where the inelastic signal measured is integrated over a range of scattering
vectors. To narrow the angular coverage of the DCA, a horizontal slit mask
made from Al was placed on the DCA, which reduced its coverage to $\pm
1.4^{\circ}$, or a $q$ range of $\pm 0.095~{}{\rm\AA^{-1}}$ to $\pm
0.098~{}{\rm\AA^{-1}}$. The DCA reflects a spectral window of 3.5 eV, so it
must be scanned in Bragg angle to collect spectra at different energies. To
mitigate the drop off in reflectivity in the wings of the spectra Wollenweber
_et al._ (2021) the DCA was scanned in small steps so that the spectral
windows overlap.

The recorded spectral window was calibrated using Co Kβ emission and the
position of the quasi-elastic scattering. This was established accurately by
tuning the seeded beam energy to cross the Co K-edge and observing when the
intensity of the elastically scattered photons plateaued relative to the Co
K-shell emission recorded by a spectrometer Preston  _et al._ (2020) viewing
the upstream side of a Co foil. The incident photon beam was determined from
this method to have a central photon energy of $E_{0}=7703.21$ eV for this
study on Al, and the energy dispersion was determined to be 22.54 meV/pixel.

To measure the plasmon at a range of individual scattering vectors, the DCA
setup was moved to different scattering angles between 3.6∘ to 25.6∘,
corresponding to $q=0.245-1.730$ Å-1. This allows the dynamic structure factor
of Al to be probed from the collective regime up into the electron-hole pair
continuum, cf. Fig. 3.

Figure 2: Measured XRTS intensity for five different wavenumbers as a function
of the photon energy loss $E=E_{0}-E_{s}$. The curves are offset vertically
for clarity and show the variation in position, intensity, and shape of the
plasmon in aluminium. Inset: example spectrum of the narrow quasi-elastic
scattering (red) for the highest wavenumber, and a Gaussian fit with
$\sigma=0.19$ eV (black dashed). The intensity of the quasi-elastic scattering
uses the same scale as the main figure.

Results. In Fig. 2, we display the measured XRTS intensity as a function of
the photon energy loss $E=E_{0}-E_{s}$. The inset shows the quasi-elastic
feature around $E=E_{0}$ corresponding to the combined source and instrument
function $R(E_{s})$; it is approximately Gaussian with a full width at half
maximum of $\Delta E=0.46\,$eV ($\sigma=0.2\,$eV). The horizontal dashed lines
indicate the zero intensity base lines for the five considered wavenumbers
$q\in[0.25,1.73]\,$Å-1, and the corresponding intensities are shown as the
coloured curves. Remarkably, the effect of the convolution of $S_{ee}(q,E)$
with $R(E_{s})$ [cf. Eq. (1)] can be neglected in practice due to the high
resolution of our setup. At the same time, we are able to accurately resolve
the plasmon down to below $20\%$ of the Fermi wave number due to the high
brilliance of the XFEL, and its excellent repetition rate. This remarkable
performance might even open up new possibilities to connect XRTS with the
estimation of optical properties such as the dielectric function, opacity, and
electrical conductivity, which are defined for $q\to 0$ Witte  _et al._
(2017); Sperling  _et al._ (2015).

Figure 3:  Plasmon dispersion of aluminium. Crosses: experimental peak
position and associated uncertainty, with red below the pair continuum, and
blue in the pair continuum; red dashed line: the experimental data fitted to
the dispersion of Eq. 2; black stars: previous EELS measurements of the Al
plasmon by Sprösser-Prou et al. Sprösser-Prou  _et al._ (1989); green circles:
TDDFT average plasmon position, with the error bars indicating the standard
deviation of the peak position in the $q$ range; grey squares: TDDFT plasmon
position only at the central $q$ value. The shaded grey area indicates the
pair continuum Giuliani and Vignale (2008), and the dotted vertical line the
Fermi wavenumber $q_{\textnormal{F}}$ in bulk Al with a face-centered cubic
(fcc) lattice Moldabekov  _et al._ (2024).

The resulting dispersion relation of aluminium is shown in Fig. 3, where we
indicate our measurement of the plasmon position as a function of $q$. Due to
the aforementioned $q$-broadening, there are prominent uncertainty bars along
the $q$-axis. In contrast, the accuracy in the energy is substantially
smaller, mainly due to the calibration of the energy axis as we explain in
more detail in the Supplemental Material sup . Identified plasmon peaks are
fitted with the expected quadratic dispersion relation of the form:

$\omega=\omega_{p}+\alpha\frac{\hbar^{2}q^{2}}{m_{e}}\,,$ (2)

following the familiar Bohm-Gross relation Bohm and Gross (1949), where
$\hbar$ is the reduced Planck constant and $m_{e}$ is the electron mass. We
determine a pre-factor of $\alpha=0.37\pm 0.07$ and plasma frequency of
$\omega_{p}=15.06\pm 0.18\,$eV where the error bars represent the maximum
range of fit parameters that reproduce the dispersion within the given
$q$-uncertainty. We find excellent agreement with previous EELS measurements
Sprösser-Prou  _et al._ (1989); Krane (1978), which further substantiates the
high quality of our results. Above the electron–hole pair continuum Giuliani
and Vignale (2008), where the plasmon decays into a multitude of excitations
due to Landau damping, the position of the maximum in $S_{ee}(q,E)$ deviates
from Eq. (2), and the experimental spectrum attains a broad, nontrivial shape
(see the curve in Fig. 2 for highest $q$).

Let us next utilize our new high-resolution XRTS data to assess the accuracy
of _ab initio_ time-dependent density functional theory (TDDFT) simulations
Ullrich (2012); Cazzaniga  _et al._ (2011); Ramakrishna  _et al._ (2021). As a
first step, we carried out linear-response TDDFT calculations employing the
adiabatic local density approximation (ALDA) for the four lowest $q$-values;
see Ref. sup  for technical details. The corresponding peak positions only
considering the central $q$ are included in Fig. 3 and are in reasonable
agreement with the experimental data points for the three smallest probed
wavenumbers, but significantly overestimate the true plasmon position for
$q=1.3\,$Å-1. To get additional insights into the quality of the TDDFT
simulations, we show a detailed comparison of the latter with the experimental
intensity in Fig. 4 for $q=0.92\,$Å-1, clearly demonstrating that TDDFT
underestimates the peak width despite reproducing the peak position with good
accuracy.

Figure 4:  Using high-resolution XRTS measurements (solid red) to benchmark
first-principles TDDFT simulations at $q_{0}=0.92\,$Å-1. The solid green curve
is from a single TDDFT simulation at $q_{0}=0.92\,$Å-1. The blue dashed curves
have been obtained from single TDDFT simulations at different $q$ values in
the experimental range, whereas the solid black line has been averaged over
the 5 individual TDDFT results that are uniformly distributed in the
appropriate wavenumber window. The red area indicates the experimental
uncertainty in the plasmon position.

While the attribution of such deviations between experiment and simulation to
a systematic error due to the employed approximation for the
exchange–correlation functional or exchange–correlation kernel is tempting,
other explanations have to be considered. First, it is noted that the earlier
XRTS experiments had substantially broader source-and-instrument functions
Cazzaniga  _et al._ (2011), which might obscure the true origin of an observed
deviation. The ultrahigh resolution achieved here resolves this conundrum and
rules out any significant effects due to $R(E_{s})$. Second, the finite
angular coverage of the employed DCAs implies that the experimental spectra
are effectively averaged over a finite interval of $q$-values. To rigorously
take into account this effect, we have carried out a series of TDDFT
simulations for a uniformly distributed set of wavenumbers $q\in[-\Delta
q+q_{0},\Delta q+q_{0}]$, and the results are included in Fig. 4 (single $q$).
Interestingly, such slight changes in the wavenumber significantly affect the
peak position, width, shape, and symmetry of the generated XRTS spectra. As
our best prediction of the measured XRTS intensity, these generated spectra
are averaged over the $q$-range with equal weight sup , providing
substantially improved agreement with the experimental observation both with
respect to its form and position (see average in Fig. 4). The same holds for
the other three considered wavenumbers, which are shown in the Supplemental
Material sup . In this way, our novel setup has allowed observed differences
between TDDFT simulations and XRTS experiments to be reconciled, which we
attribute to the mixing of wavenumbers for the present case. For completeness,
we have included the properly $q$-averaged peak positions in Fig. 3 as well as
an indication of the variance of the TDDFT results within the appropriate
$q$-window. The effect of averaging is small for $q\lesssim 1\,$Å-1, but leads
to a substantial improvement for the second largest wavenumber: here the
importance of averaging correctly over the $q$-range is clearly demonstrated.

Discussion. In this Letter, we have presented new ultrahigh resolution XRTS
measurements of electronic features obtained at the European XFEL. Our setup
works over a broad range of wavenumbers and can be implemented in both forward
and backward scattering geometries in future experiments. As a practical
example, the plasmon in aluminium at ambient conditions was studied, which was
resolved with high accuracy even at very small scattering angles. This opens
up the enticing opportunity to connect XRTS measurements with the estimation
of optical properties such as the electrical conductivity.

We are convinced that our work opens up a variety of possibilities for
important future research. First, resolving the electronic structure of
different materials with high resolution is important in its own right and
will give new insights into physical effects such as the predicted roton-type
excitation in low-density hydrogen Hamann  _et al._ (2023); Dornheim  _et al._
(2022b), or very recently reported thermal features in isochorically heated
materials Moldabekov  _et al._ (2024). In addition, the narrow width of
$R(E_{s})$ facilitates the rigorous benchmarking of theoretical models Mo  _et
al._ (2020); Zhang  _et al._ (2024) and first-principles simulations Baczewski
_et al._ (2016); Dornheim  _et al._ (2023a) such as the exchange–correlation
functional and exchange–correlation kernel in TDDFT calculations Moldabekov
_et al._ (2023); Schörner  _et al._ (2023); Cazzaniga  _et al._ (2011);
Ramakrishna  _et al._ (2021). A particularly important field of application is
given by the diagnostics of experiments with matter under extreme conditions.
Specifically, a narrow source-and-instrument function is key to extend the
model-free interpretation of XRTS spectra to temperatures of the order of
$T\sim 1\,$eV, whereas previous efforts where restricted to $T\gtrsim 10\,$eV
Dornheim  _et al._ (2022a, 2023b). This development promises to unlock the
full capability of modern XFEL facilities such as the European XFEL in Germany
Tschentscher  _et al._ (2017) and the Linac Coherent Light Source (LCLS) in
the USA Bostedt  _et al._ (2016) to study matter at planetary interior
conditions Benuzzi-Mounaix  _et al._ (2014), to diagnose material science and
material discover applications, and to characterize states that occur on the
initial stage of the compression path of the fuel capsule in inertial
confinement fusion experiments.

###### Acknowledgements.

We acknowledge the European XFEL in Schenefeld, Germany, for provision of
X-ray free-electron laser beamtime at the Scientific Instrument HED (High
Energy Density Science) under proposal number 3777 and would like to thank the
staff for their assistance. The authors are grateful to the HIBEF user
consortium for the provision of instrumentation and staff that enabled this
experiment. The original datasets can be found here and are available upon
reasonable request: doi:10.22003/XFEL.EU-DATA-003777-00. This work was
partially supported by the Center for Advanced Systems Understanding (CASUS)
which is financed by Germany’s Federal Ministry of Education and Research
(BMBF) and by the Saxon state government out of the State budget approved by
the Saxon State Parliament. This work has received funding from the European
Union’s Just Transition Fund (JTF) within the project _Röntgenlaser-
Optimierung der Laserfusion_ (ROLF), contract number 5086999001, co-financed
by the Saxon state government out of the State budget approved by the Saxon
State Parliament. This work has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2022 research and innovation
programme (Grant agreement No. 101076233, ”PREXTREME”). Views and opinions
expressed are however those of the authors only and do not necessarily reflect
those of the European Union or the European Research Council Executive Agency.
Neither the European Union nor the granting authority can be held responsible
for them. The TDDFT calculations were partly carried out at the Norddeutscher
Verbund für Hoch- und Höchstleistungsrechnen (HLRN) under grant mvp00024, and
on a Bull Cluster at the Center for Information Services and High Performance
Computing (ZIH) at Technische Universität Dresden.

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# Single-photon and multi-photon Fano lines for helium and neon using tRecX-
haCC

Hareesh Chundayil1, Armin Scrinzi2, Vinay Pramod Majety1∗ 1 Department of
Physics and CAMOST, Indian Institute of Technology Tirupati, Yerpedu, India  2
Department of Physics, Ludwig Maximilian University, Theresienstrasse 37,
Munich, Germany<EMAIL_ADDRESS>
###### Abstract

Single-photon and multi-photon ionization of helium and neon atoms by
ultrashort extreme ultraviolet radiation is studied using the tRecX-haCC
package developed by the authors. The tRecX-haCC package solves the time
dependent Schrödinger equation in the presence of a laser pulse and computes
the photoelectron spectra using the iSurf technique to efficiently capture the
slow decay of the resonant states. Wavelength of the radiation is chosen to
expose doubly excited states embedded in the continuum. The lineshapes in the
spectra that arise as a result of these doubly excited states are analyzed.
The computed peak positions and linewidths are in good agreement with the
literature. In addition, the dependence of the Fano $q$ parameter on the order
of the multi-photon process is demonstrated.

##  1 Introduction

Interrogation of electron dynamics in the time domain has become a reality
with the development of attosecond light sources [1]. These sources typically
create a hole in the valence shell and the dynamics can be probed with a
second synchronized pulse (extreme ultraviolet (XUV) or infrared (IR)). In
this regime, single ionization is the dominant process. At certain
photoelectron energies, the ionization probability can exhibit strong
modulations due to the presence of doubly excited states. The characteristic
signature of these doubly excited states is the asymmetric Fano line shape
which arises from the interference of two pathways that lead to the same final
ionized state of the system - direct ionization and double excitation followed
by decay [2]. In a seminal work by Ott et al. [3], it was experimentally shown
that the interference could be controlled using an IR pulse and hence the
lineshapes. This has generated a lot of interest in better understanding
photoionization dynamics in the vicinity of Fano resonances both theoretically
and experimentally [4, 5, 6, 7].

Accurate description of electron correlation is essential to represent these
autoionizing states and it is an important benchmark for any new theory that
aims to model ultrafast electron dynamics. In this work, we employ our newly
developed package, tRecX-haCC [8, 9], to study multi-photon ionization of
atoms - helium and neon. The wavelength of the ionizing XUV pulse is chosen to
expose some of the doubly excited states in the system. We compute the
photoelectron spectra, extract linewidths and compare with literature. We also
numerically demonstrate that the Fano $q$ parameter depends on the number of
photons that the electron absorbs to reach the doubly excited state.

The properties of doubly excited states are traditionally computed using time
independent approaches such as through the diagonalization of the complex
scaled electronic Hamiltonian for two electron systems[10, 11] and using
R-matrix theory [12, 13], multichannel quantum defect theory[14, 15] and the
time dependent configuration interaction theory[16] for many electron systems.
While there is a rich literature on one-photon ionization based on these
methods, modeling multi-photon ionization to arbitrary photon orders for many
electron systems is challenging and is an active area of research [15, 17,
18]. Although the process remains perturbative in nature, the computation of
amplitude and phase of the direct transition matrix elements would require the
use of elaborate multi-photon perturbation theory. Here, we study the single
and multi-photon Fano lineshapes using solutions of the time dependent
Schrödinger equation (TDSE) without making use of perturbation theory. Such a
time dependent approach would be the method of choice for short pulses.

tRecX-haCC solves the TDSE using a hybrid anti-symmetrized coupled channels
discretization of the wavefunction. The photoelectron spectra are computed
using iSurf[19], which is an infinite time extension to the time dependent
surface flux method (tSurff) [20]. In the tSurff method, spectra are computed
by tracking the electron flux through a designated surface instead of a direct
projection of the final state onto the continuum states. The technique allows
us to compute spectra with minimal numerical box sizes. In the iSurf method,
the time integral involved in the tSurff calculation is analytically solved
beyond the pulse duration which allows for efficient computation of spectra at
low photoelectron energies[19] as well as to account for the slowly decaying
doubly excited states[9, 16].

The article is organized as follows: In section 2, we briefly describe the
tRecX-haCC method followed by a description of the tSurff and iSurf methods in
section 3. Finally, we present our results on multi-photon ionization of
helium and neon in section 4.

##  2 tRecX-haCC method

Consider a $N$ electron system in the Coulomb field of a nucleus of charge
$Z$. Starting from the ground state of the multielectron Hamiltonian (in
a.u.),

$\hat{H}_{0}=\sum_{i=1}^{N}\left[-\frac{\nabla_{i}^{2}}{2}-\frac{Z}{r_{i}}\right]+\sum_{i=1}^{N}\sum_{j=1}^{i-1}\frac{1}{|\vec{r}_{i}-\vec{r}_{j}|}$
(1)

the time evolution of the state in the presence of an external light field is
computed by solving the time dependent Schrödinger equation

$i\frac{\partial}{\partial
t}|\Psi\rangle=(\hat{H}_{0}+\hat{H}_{1})|\Psi\rangle.$ (2)

Here, $\hat{H}_{1}$ represents the light-matter interaction operator in the
mixed gauge [21]. The mixed gauge operator in the dipole approximation can be
computed starting from the length form given by

$\hat{H}_{L}=-\sum_{j=1}^{N}\vec{E}\cdot\vec{r}_{j}$ (3)

with $\vec{E}(t)$ being the electric field and applying a gauge transform

$\hat{U}_{g}=\prod\limits_{j=1}^{N}\hat{U}_{g}^{(j)}$ (4)

with

$\hat{U}_{g}^{(j)}=\begin{cases}1&|\vec{r}_{j}|\leq r_{g}\\\
e^{i\vec{A}(t)\cdot\vec{r}_{j}(1-\frac{r_{g}}{|\vec{r}_{j}|})}&|\vec{r}_{j}|>r_{g}\end{cases}$
(5)

and $\vec{A}(t)=-\int_{-\infty}^{t}\vec{E}(t^{\prime})dt^{\prime}$ being the
vector potential.

The wavefunction is discretized using a hybrid coupled channels basis composed
of

  1. 1.

Neutral states ($|\mathcal{N}\rangle$) which are approximate eigenstates of
the $N$ electron system. These are computed using configuration interaction
(CI) theory. The CI states are linear combinations of Slater determinants
constructed using Gaussian based atomic/molecular orbitals resulting from a
Hartree-Fock calculation. We used the COLUMBUS package [22] for this purpose.

  2. 2.

Channel functions which are anti-symmetrized products of ($N-1$) electronic
ionic states ($|I\rangle$) and a numerical one-electron basis. The ionic
states are again computed using COLUMBUS. The numerical one electron basis is
composed of two parts

    1. (a)

Gaussian based atomic/molecular orbital basis ($|i\rangle$)

    2. (b)

A general single centered expansion ($|\alpha\rangle$) constructed using
finite element discrete variable representation (FE-DVR) basis on the radial
coordinate and spherical harmonics on the angular coordinates. In addition,
these are made orthogonal to the molecular orbital basis as
$|\alpha_{\perp}\rangle$ = $|\alpha\rangle-\sum_{i}|i\rangle\langle
i|\alpha\rangle$.

The hybrid anti-symmetrized coupled channels (haCC) expansion for the
wavefunction is written as

$|\Psi\rangle=\underbrace{\sum_{\mathcal{N}}C_{\mathcal{N}}(t)|\mathcal{N}\rangle}_{\text{Neutrals}}+\underbrace{\sum_{I,i}C_{I,i}(t)\mathcal{A}|I,i\rangle+\sum_{I\alpha}C_{I,\alpha}(t)\mathcal{A}|I,\alpha_{\perp}\rangle}_{\text{Channel
functions}}.$ (6)

Here, $\mathcal{A}$ denotes the antisymmetrizer and
$C_{\mathcal{N}},$$C_{I,i}$ and $C_{I,\alpha}$ are the time dependent
expansion coefficients that represent the dynamics. The expansion captures
essential parts of the Hilbert space required for single ionization problems.
The basis respects exchange symmetry, includes correlation in the initial
state that somewhat exceeds the original COLUMBUS description due to the
additional channel functions, and accounts for the ”dynamical correlation”,
i.e. correlation that builds up during the excitation and ionization process.
We label various haCC calculations using the notation haCC(n,i) where the n
and i energetically lowest neutral and ionic states, respectively, are used,
unless indicated otherwise.

Approximating the Hilbert space by a large finite dimensional space
$\\{|\mathcal{N}\rangle\\}\oplus\\{\mathcal{A}|I,i\rangle\\}\oplus\\{\mathcal{A}|I,\alpha_{\perp}\rangle\\}$
converts the time dependent Schrödinger equation into a set of coupled
ordinary differential equations for the time dependent coefficients which are
then solved using adaptive Runge-Kutta methods. Absorbing boundary conditions
are imposed using the infinite range exterior complex scaling method [23].
This results in non-hermitian matrix representation of operators.

The hybrid anti-symmetrized coupled channels (haCC) basis (6) is non-
orthogonal and different operators evaluated in our basis have different
degrees of sparsity. These aspects are efficiently handled using flexible tree
data structures that are part of the tRecX package [24]. We refer the readers
to [24, 9] for further details of implementation of the solver. In the next
section, we will describe the spectral analysis method.

##  3 Spectral analysis using tSurff and iSurf

Consider a photoionization process that ejects a photoelectron with momentum
$\vec{k}$ and leaves the residual ion in state $|I\rangle$. If the final state
is $|X_{I,\vec{k}}\rangle$, the probability for the photoionization process
can be computed as

$P_{I,\vec{k}}=|\underbrace{\langle
X_{I,\vec{k}}|\Psi(T)\rangle}_{b_{I,\vec{k}}}|^{2}$ (7)

where $T>T_{p}$ is any time after the end of the laser pulse at $T_{p}$. The
evaluation of the amplitude $b_{I,\vec{k}}$ is challenging for two reasons:
(1) the wavefunction can spread to large volumes until the end of the pulse at
$T_{p}$ and (2) the calculation of accurate continuum states for multielectron
systems is challenging.

The tSurff method for evaluating $b_{I,\vec{k}}$ avoids both problems by
observing that at large times $T$ only the asymptotic parts of the wave
function $\Psi(T)$ contribute to the amplitude. If one assumes that for
electrons outside a certain radius $R_{c}$ all interactions with the remaining
electrons and with the nuclei can be neglected, in that region the final state
assumes the asymptotic form

$|X_{I,\vec{k}}\rangle\sim\mathcal{A}\left(|I\rangle\otimes|\vec{k}\rangle\right)=:|I,\vec{k}\rangle$
(8)

with a plane wave for $|\vec{k}\rangle$. Picking some large time $T$, where
the unbound electron with coordinate $\vec{r}_{n}$ has moved beyond the radius
$R_{c}$, we approximate the $b_{I,\vec{k}}$ in Eq. (7) as

$\langle X_{I,\vec{k}}|\Psi(T)\rangle\approx
b_{I,\vec{k}}(T)=\sum_{n=1}^{N}\langle
I,\vec{k}|\Theta(r_{n}-R_{c})|\Psi(T)\rangle,$ (9)

where $\Theta(r_{n}-R_{c})$ is the Heaviside function, which is $1$ if the
argument is positive and 0 otherwise.

As described in Refs. [8, 20], the volume integral in Eq. (9) can be converted
to a surface and a time integral, which is computationally more tractable. For
times before the end of the laser pulse, $t<T_{p}$, the action of the laser
field on the emitted electron needs to be taken into account. The time-
evolution of the remote electron is described by the time-dependent Volkov
state

$|\vec{k},t\rangle=\exp\left[-i\int_{0}^{t}\frac{k^{2}}{2}-\vec{k}\cdot\vec{A}(\tau)d\tau\right]|\vec{k},0\rangle$
(10)

which solves the time-dependent Schrödinger equation of the free electron in
velocity gauge

$i\frac{d}{dt}|\vec{k},t\rangle=\left[-\frac{1}{2}\Delta+i\vec{\nabla}\cdot\vec{A}(t)\right]|\vec{k},t\rangle.$
(11)

In the multi-electron case [25] one also needs to include the time-evolution
of the ion due to laser-coupling between the ionic bound states as

$i\frac{d}{dt}|I,t\rangle=\sum_{J}H^{\rm(ion)}_{IJ}(t)|J,t\rangle,$ (12)

where $H^{\rm(ion)}_{IJ}(t)$ is the time-dependent Hamiltonian for the ionic
system and the sum is over all ionic bound states included in the haCC
expansion. The ionic Schrödinger equation (12) generates a unitary time
evolution, which relates the ionic state $|I,t\rangle$ at time $t$ to the
states at time $t=0$ by

$|I,t\rangle=\sum_{J}U^{\rm(ion)}_{IJ}(t)|J,0\rangle.$ (13)

Taking the time-derivative of Eq. (9) and integrating over time, one obtains
the

$b_{I,\vec{k}}(T)=N\sum_{J}U^{\rm(ion)\,\dagger}_{IJ}(T)\int_{0}^{T}dt\langle
J,\vec{k},t|\hat{C}|\Psi(t)\rangle$ (14)

with the notation
$|I,\vec{k},t\rangle:=\mathcal{A}(|I,t\rangle\otimes|\vec{k},t\rangle)$ and
the commutator

$\hat{C}=[-\frac{1}{2}\nabla_{N}^{2}+i\vec{A}(t)\cdot\vec{\nabla}_{N},\Theta(r_{N}-R_{c})].$
(15)

The commutator of the derivatives with the $\Theta$-functions in $\hat{C}$
consists of terms that contain the $\delta$-function at $r_{n}=R_{c}$ and its
derivative. This reduces the evaluation of the matrix element to an integral
over the surface of the sphere with radius $R_{c}$ involving values and
derivatives of the wave functions on the surface. In Eq. (14) the factor $N$
arises from the summation over all electrons and the inverse of the ionic
time-evolution $U^{\rm(ion)\,\dagger}(T)$ accounts for the evolution of
different $|J,t\rangle$ states towards the final $|I,T\rangle$ state. Also,
note that Eq. (11) is in velocity gauge. By our use of mixed gauge, surface
values and derivatives are in length gauge up to $r_{g}$, see Eq. (5). Before
applying Eq. (14) wave function values and derivatives need to be converted to
velocity gauge. We have excluded multiple ionization from our discussion. The
extension of tSurff to multiple ionization is discussed in Ref. [25], but that
has not been implemented in for haCC so far.

To account for all photoelectrons including near zero energy photoelectrons
and those due to delayed emission as in a slowly decaying resonance, one needs
to integrate up to very large times long after the end of the pulse, $T\gg
T_{p}$. If the spectral decomposition of $\hat{H}_{0}$ can be obtained, the
integral beyond $T_{p}$ can be computed analytically, which has been referred
to as iSurf in Ref. [19]: it is the infinite time extension to tSurff. We
review below iSurf for the multielectron case.

Assume we have the spectral decomposition of $\hat{H}_{0}$

$\hat{H}_{0}=\sum_{s}|s\rangle E_{s}\langle\tilde{s}|.$ (16)

where $|s\rangle$ and $\langle\tilde{s}|$ are the right and left eigenvectors
of $\hat{H}_{0}$. As we use absorbing boundary conditions, the $E_{s}$ are
complex in general. The left and right eigenvectors still form a resolution of
identity $1=\sum_{s}|s\rangle\langle\tilde{s}|$, but they are not trivially
related to each other by complex conjugation, see Ref. [23]. The state at the
end of the pulse represented in the haCC basis $|h\rangle$ can be re-expressed
in the spectral basis as

$|\Psi(T_{p})\rangle=\sum_{h}|h\rangle
c_{h}(T_{p})=\sum_{hs}|s\rangle\langle\tilde{s}|h\rangle
c_{h}(T_{p})=:\sum_{s}|s\rangle d_{s}(T_{p}).$ (17)

After the end of the pulse $t>T_{p}$, the time evolution of $|\Psi(t)\rangle$
is

$|\Psi(t)\rangle=\sum_{s}e^{-iE_{s}(t-T_{p})}|s\rangle d_{s}(T_{p}).$ (18)

and the continuum state $|I,\vec{k},t\rangle$ evolves as

$|I,\vec{k},t\rangle=|I,\vec{k},T_{p}\rangle
e^{-i(E_{I}+\frac{1}{2}k^{2})(t-T_{p})},$ (19)

where $E_{I}$ is the energy of the ion and $\frac{1}{2}k^{2}$ the energy of
the free electron.

Using equations (18), (19), the time integral in (14) can be extended to
infinity as:

$\displaystyle b_{I,\vec{k}}(t$
$\displaystyle=\infty)=b_{I,\vec{k}}(T_{p})+\int_{T_{p}}^{\infty}dt\langle
I,\vec{k},t|\hat{C}|\Psi(t)\rangle$
$\displaystyle=b_{I,\vec{k}}(T_{p})-i\sum_{s}\langle
I,\vec{k},T_{p}|\hat{C}|s\rangle\frac{1}{E_{s}-E_{I}-\frac{1}{2}k^{2}}d_{s}(T_{p})$
(20)

where $b_{I,\vec{k}}(T_{p})$ is computed using (14). As the $E_{s}$ are
complex the inverses are well-defined. Note, that in absence of the laser
field there is no coupling between ionic channels and hence no summation over
ionic channels appears in the second term. Eq. (20) can be written as

$b_{I,\vec{k}}(t=\infty)=b_{I,\vec{k}}(T_{p})-i\langle
I,\vec{k}.T_{p}|\hat{C}(\hat{H}_{0}-E_{I}-\frac{1}{2}k^{2})^{-1}|\Psi(T_{p})\rangle.$
(21)

When the spectral decomposition is not available, iterative methods can be
employed to obtain
$(\hat{H}_{0}-E_{I}-\frac{1}{2}k^{2})^{-1}|\Psi(T_{p})\rangle$. Denoting the
photoelectron energy as $E$$(=\frac{k^{2}}{2})$, the photoionization yield
corresponding to emission into channel $I$ is computed by integrating over
directions as

$Y_{I}(E)=\int k^{2}|b_{I,\vec{k}}(t=\infty)|^{2}d\Omega_{k}$

and the total yield for a photoelectron energy is

$Y(E)=\sum_{I}Y_{I}(E)$

##  4 Results

We study photoionization of helium and neon atoms by few cycle linearly
polarized laser pulses. As defined in Ref. [24], the vector potential has the
form

$\vec{A}(t)=\vec{\alpha}(t)\sqrt{\frac{I_{0}}{2\omega^{2}}}\sin(\omega
t-\phi)$ (22)

where $I_{0}$ is the peak intensity of the pulse, $\omega$ is the central
frequency, $\phi$ is the carrier envelope phase and $\vec{\alpha}(t)$ defines
the pulse envelope. We choose a $\cos^{2}$ envelope in our calculations. The
duration is defined in terms of its full width at half maximum (FWHM) which
can be expressed in the units of the number of optical cycles at the central
wavelength [24].

The wavelengths are chosen to expose doubly excited states. At each chosen
wavelength, we present total and channel resolved photoelectron spectra and
analyze select lineshapes resulting from the presence of doubly excited
states. We parameterize the lineshapes using the equation

$Y_{I}(E)=Y_{a}\frac{(q+\epsilon)^{2}}{1+\epsilon^{2}}+Y_{b}$ (23)

Here, $\epsilon=\frac{E-E_{r}}{\frac{1}{2}\Gamma}$, $E_{r}$ is the position of
the resonance, $\Gamma$ is the width, $q$ is the Fano parameter which
characterizes the line profile, $Y_{a}$ and $Y_{b}$ are slowly varying
background yields [26]. Although, the original parametrization was given for
photoionization cross-sections, we employ the same for the yields. This is
justified as our few cycle pulses have a broad bandwidth and the spectral
intensity of the laser pulse can be assumed to remain constant over the width
of a given resonance. Parameters $Y_{a}$ and $Y_{b}$ can be used to construct
the correlation coefficient $\rho^{2}=\frac{Y_{a}}{Y_{a}+Y_{b}}$ which is
equal to 1, if the ionization process involves a single channel [26]. Where
multiple channels are involved, the correlation coefficient deviates from 1
due to a background from the other channels.

###  4.1 Helium

We present here total and channel resolved photoelectron spectra for helium
ionized by 2-cycle XUV pulses with central wavelengths of 20
nm($\lambda_{1}$), 40 nm($\lambda_{2}$), 60 nm($\lambda_{3}$) and 80
nm($\lambda_{4}$) and a peak intensity of $10^{15}$ W/cm2. The wavelengths
were chosen such that a set of doubly excited states can be reached by
absorption of $n=1,2,3\text{ and }4$ photons. The quantum chemical
calculations to construct the ionic and neutral states were performed using
the aug-cc-pvtz basis set. A haCC(1,9) basis set was then constructed and
photoelectron spectra were computed. The ground state energy with the
haCC(1,9) basis is -2.901 a.u. The $1s$ channel energy is -1.999 a.u.. The
ionization potential deviates from the exact value by about 30 meV.

Figure 1 presents the total and $1s$ channel photoelectron spectra at all the
considered wavelengths. We observe multi-photon absorption peaks at positions
defined by $n\hbar\omega-I_{p}$ where $I_{p}$ is the ionization potential. The
width of these peaks is related to the bandwidth of the laser pulse. The
spectra are dominated by the $1s$ channel. We also observe several rapid
modulations around 10 eV, 32-42 eV and 45-50 eV. These features result from
the presence of doubly excited states. In this work, we analyze the features
in the photoelectron energy window 32-42 eV. For this energy window, a partial
wave analysis of the $1s$ channel is presented in figure 2. The states for
which the Fano parameters are tabulated are labelled.

At wavelengths $\lambda_{1}$ and $\lambda_{3}$, the modulations in spectra can
be attributed to the $2snp$ series of doubly excited states that can be
excited upon absorption of one $\lambda_{1}$ photon or absorption of three
$\lambda_{3}$ photons. In addition, at $\lambda_{3}$ wavelength, $2snf$ states
appear. These two series of states corresponding to partial waves
$L=1,\text{and }3$ are shown separately in figure 2. The Fano parameterization
(see Eq. (23)) corresponding to the first four $[001]^{+}$ states that are
part of the $2snp$ series for the one and the three-photon processes is
presented in tables 1 and 2 respectively. We also observe narrow lineshapes
corresponding to $[010]^{-}$ states.

Table 1 presents a comparison of the obtained peak positions, widths and $q$
parameters with literature: experimental works [27, 28, 29] and theoretical
works based on the method of complex rotation using correlated basis sets [10,
11], R-matrix method [12], many body perturbation theory [30], spline-Galerkin
method[31] and Feshbach formalism[32]. The peak positions among various
theoretical studies deviate by about 100 meV. In our study, a deviation of 30
meV is expected due to our first ionization potential. Accounting for this
deviation, our peak positions agree with Refs. [10, 11]. The linewidths and
the $q$ parameters reported from various theoretical studies and experiments
relatively vary by about 10%. As expected for one-photon ionization process
where a single channel is involved, the correlation coefficient
$\rho^{2}\approx 1$.

Table 2 presents the peak positions and widths obtained with the three-photon
ionization process. They agree with those obtained from one-photon process as
they correspond to the same doubly excited states. The $q$ parameter, however,
differs indicating its dependence on the nature of the excitation process. The
correlation coefficient deviates from 1 as the process involves more than one
channel which is seen from the partial wave analysis in figure 2.

At $\lambda_{2}$ and $\lambda_{4}$ wavelengths, the system can be excited to
$2sns$ and $2snd$ series of states through absorption of two and four photons
respectively. In addition, at $\lambda_{4}$, $2sng$ states can be excited. The
relevant partial wave channels are the $L=0,2\text{ and }4$ channels which are
shown in figure 2. Tables 3 and 4 present Fano parameters for the first two
states in the series, namely the ${}^{1}S^{e}$ and ${}^{1}D^{e}$ states. We
compare the peak positions and linewidths with experimental works [33, 34] and
theoretical works based on the method of complex rotation [11, 35], algebraic
variational method[36], Feshbach formalism[37] and the multichannel quantum
defect theory[15]. Our peak positions and widths are in good agreement with
the experimental works. The various theoretical studies differ relatively by
about 10%. In addition, we present the $q$ parameters and correlation
coefficients. In Ref. [37], a complex $q$ was reported for the two-photon
process, however we find that a good fit can be obtained using a real $q$.
This is in line with observation made in Ref. [38] where a real $q$ parameter
was sufficient to fit a three-photon resonance of Argon. For a given doubly
excited state, the $q$ parameter associated with the two-photon and four-
photon processes differ indicating its sensitivity to the excitation process.

Figure 1: Total (solid) and $1s$ channel (dashed) photoelectron spectra for helium computed using haCC(1,9) basis with 2-cycle laser pulses having central wavelengths of 20 nm, 40 nm, 60 nm and 80 nm. Figure 2: Partial wave analysis of $1s$ channel spectra for helium in the photoelectron energy window 32-42 eV. Each sub-plot title indicates the excitation wavelength. Dominant partial waves are presented. State |  | Peak position (eV) | Width (eV) | q | $\rho^{2}$ | Remarks
---|---|---|---|---|---|---
$[001]^{+}$ | n=2 | 35.597 | 0.0406 | -2.63 | 0.999 | haCC(1,9)
|  | 35.626 | 0.0373 | -2.77 |  | Jan M Rost et al. [10]
|  | 35.636 | 0.0374 |  |  | Lindorth [11]
|  | 35.680 | 0.0402 | -2.68 |  | P. Hamacher et al.[12]
|  | 35.674 | 0.0403 | -2.63 |  | S. Salomonson et.al[30]
|  | 35.665 | 0.0399 |  |  | T.Brage et al.[31]
|  | 35.635 | 0.0363 | -2.84 |  | A.K.Bhatia et al.[32]
|  | 35.637$(\pm 0.010)$ | 0.0370$(\pm 0.001)$ | -2.75 $(\pm 0.001)$ |  | M. Domke et al.[39]
|  | 35.623$(\pm 0.015)$ | 0.038$(\pm 0.004)$ | -2.80$(\pm 0.25)$ |  | Madden and Codling[28]
|  | 35.641$(\pm 0.010)$ | 0.038$(\pm 0.004)$ | -2.55$(\pm 0.16)$ |  | Morgan and Ederer[29]
$[001]^{+}$ | n=3 | 39.094 | 0.00927 | -2.41 | 0.999 | haCC(1,9)
|  | 39.134 | 0.00819 | -2.58 |  | Jan M Rost et al.[10]
|  | 39.148 | 0.008 |  |  | Lindorth [11]
|  | 39.157 | 0.00894 | -2.543 |  | P. Hamacher et al. [12]
|  | 39.160 | 0.00896 | -2.43 |  | S. Salomonson et.al[30]
|  | 39.154 | 0.00878 |  |  | T.Brage et al.[31]
|  | 39.151 | 0.009 | -2.60 |  | A.K.Bhatia et al.[32]
|  | 39.148$(\pm 0.010)$ | 0.010$(\pm 0.001)$ | -2.5$(\pm 0.1)$ |  |  M. Domke et al. [39]
|  | 39.145$(\pm 0.007)$ | 0.008$(\pm 0.004)$ | -2.0$(\pm 1.0)$ |  | Madden and Codling[28]
|  | 39.145$(\pm 0.010)$ | 0.0083$(\pm 0.002)$ | -2.5$(\pm 0.5)$ |  | Morgan and Ederer[29]
$[001]^{+}$ | n=4 | 39.897 | 0.00387 | -2.39 | 0.996 | haCC(1,9)
|  | 39.942 | 0.00349 | -2.55 |  | Jan M Rost et al.[10]
|  | 39.957 | 0.00385 | -2.534 |  | Hamacher et al.[12]
|  | 39.961 | 0.00384 | -2.41 |  | S. Salomonson et.al[30]
|  | 39.952$(\pm 0.007)$ | 0.004$(\pm 0.0005)$ | -2.4 $(\pm 0.1)$ |  | M. Domke et al.[39]
$[001]^{+}$ | n=5 | 40.246 | 0.00199 | -2.38 | 0.994 | haCC(1,9)
|  | 40.292 | 0.00179 | -2.54 |  | Jan M Rost et al.[10]
|  | 40.303 | 0.0012 | -2.45 |  | Hamacher et al.[12]
|  | 40.308 | 0.00197 | -2.4 |  | S. Salomonson et.al[30]
|  | 40.303$(\pm 0.007)$ | 0.002$(\pm 0.0003)$ | -2.4 $(\pm 0.1)$ |  | M. Domke et al.[39]
Table 1: Fano parameterization of $2snp$ states following one-photon ionization of helium at 20 nm wavelength. Comparison with earlier theoretical and experimental works is presented. The states are labelled following convention in [10]. State |  | Peak position (eV) | Width (eV) | q | $\rho^{2}$ | Remarks
---|---|---|---|---|---|---
$[001]^{+}$ | n=2 | 35.598 | 0.0405 | -0.002 | 0.498 | haCC(1,9)
$[001]^{+}$ | n=3 | 39.094 | 0.0093 | -0.012 | 0.498 | haCC(1,9)
$[001]^{+}$ | n=4 | 39.897 | 0.0039 | -0.016 | 0.497 | haCC(1,9)
$[001]^{+}$ | n=5 | 40.246 | 0.002 | -0.018 | 0.498 | haCC(1,9)
Table 2: Fano parameterization of $2snp$ states following three-photon ionization of helium at 60 nm wavelength. The states are labelled following convention in [10]. State |  | Peak position (eV) | Width (eV) | q | $\rho^{2}$ | Remarks
---|---|---|---|---|---|---
${}^{1}S^{e}$ |  | 33.253 | 0.131 | 0.0903 | 0.49 | haCC(1,9)
|  | 33.340 | 0.138 | -6.97-i4.03 |  | Sanchez et al.[37]
|  | 33.330 | 0.124 |  |  | Lindorth [11]
|  | 33.339 | 0.125 |  |  | H.Oza[36]
|  | 33.329 | 0.123 |  |  | Ming-Keh Chen[35]
|  | 33.313 | 0.128 |  |  | Y. Wang and C. H. Greene[15]
|  | 33.33$(\pm 0.04)$ | 0.138$(\pm 0.015)$ |  |  | Hicks and Comer[33]
|  | 33.27$(\pm 0.03)$ | 0.138$(\pm 0.015)$ |  |  |  Gelebart et al. [34]
${}^{1}D^{e}$ |  | 35.401 | 0.0689 | 0.0336 | 0.499 | haCC(1,9)
|  | 35.43 | 0.0706 | -15.4-i7.40 |  | Sanchez et al.[37]
|  | 35.39 | 0.064 |  |  | Lindorth [11]
|  | 35.412 | 0.066 |  |  | H.Oza[36]
|  | 35.398 | 0.064 |  |  | Ming-Keh Chen[35]
|  | 35.386 | 0.064 |  |  | Y. Wang and C. H. Greene[15]
|  | 35.400$(\pm 0.03)$ | 0.072$(\pm 0.018)$ |  |  | Hicks and Comer[33]
|  | 35.360$(\pm 0.03)$ | 0.07$(\pm 0.01)$ |  |  |  Gelebart et al. [34]
Table 3: Fano parameterization of ${}^{1}S^{e}$ and ${}^{1}D^{e}$ states following two-photon ionization of helium at 40 nm wavelength. Comparison with earlier theoretical and experimental works is presented. State |  | Peak position (eV) | Width (eV) | q | $\rho^{2}$ | Remarks
---|---|---|---|---|---|---
${}^{1}S^{e}$ |  | 33.251 | 0.132 | 0.057 | 0.49 | haCC(1,9)
${}^{1}D^{e}$ |  | 35.400 | 0.069 | -0.0115 | 0.499 | haCC(1,9)
Table 4: Fano parameterization of ${}^{1}S^{e}$ and ${}^{1}D^{e}$ states
following four-photon ionization of helium at 80 nm wavelength.

###  4.2 Neon

Multi-photon ionization of neon is studied using haCC(1,4) basis set. The
channels include the triply degenerate $1s^{2}2s^{2}2p^{5}(^{2}P^{o})$ and the
non-degenerate $1s^{2}2s^{1}2p^{6}(^{2}S^{o})$ states. The CI wavefunctions
for the ionic and neutral states were derived using the m-aug-cc-pvqz basis.
In our calculations, the first ionization potential is 21.5 eV which is in
good agreement with the value 21.564 eV reported in literature [40]. The laser
pulse is chosen to be a three-cycle pulse with a peak intensity of
$10^{14}$W/cm2 at central wavelengths 27 nm ($\lambda_{5}$) and 81 nm
($\lambda_{6}$).

Figure 3 shows the total and the ${}^{2}P^{o}$ channel photoelectron spectra.
We observe single as well as multi-photon peaks whose peak positions and the
widths are related to the laser frequency, bandwidth and the number of photons
absorbed. Figure 4 presents a partial wave analysis of the ${}^{2}P^{o}$
channel in the photoelectron window 23-28 eV where several rapid modulations
are observed. These features correspond to $2s2p^{6}np$ series of doubly
excited states. These can be excited by absorption of one and three photons at
$\lambda_{5}$ and $\lambda_{6}$ wavelengths respectively. The dominant partial
waves for the one-photon process are $L=0,2$ while for the three-photon
process are $L=0,2,4$. We parameterize the first three peaks using the Fano
formula (23) and the fit parameters are presented in tables 5 and 6.

Figure 3: Total (solid) and ${}^{2}P^{o}$ (dashed) channel spectra for neon computed using haCC(1,4) basis with 3-cycle laser pulses having central wavelengths of 27 nm and 81 nm . Figure 4: Partial wave analysis of ${}^{2}P^{o}$ channel spectra for neon in the photoelectron energy window 23-28 eV. The excitation wavelength is indicated in the title of the sub-plot. Dominant partial wave channels are presented. State | Peak position(eV) | Width(eV) | q | $\rho^{2}$ | Remarks
---|---|---|---|---|---
$2s2p^{6}3p$ | 24.244 | 0.0130 | -1.39 | 0.762 | haCC(1,4)
| 24.021 | 0.0151 | -1.34 | 0.77 | C Marante et al. (velocity)[41]
| 24.021 | 0.0150 | -1.47 | 0.79 | C Marante et al. (length)[41]
| 24.150 | 0.0114 |  |  | Liang et al.[13]
| 24.843 | 0.0139 | -3.69 | 0.514 | M Stener et al.[42]
| 24.147 | 0.0186 | -1.53 | 0.73 | B. Langer et al. (velocity)[43]
| 24.147 | 0.0186 | -1.59 | 0.72 | B. Langer et al. (length)[43]
| 28.315 | 0.013 | -1.4 | 0.77 | Nrisimhamurty et al.[14]
| 24.123 | 0.0349 |  |  | Schulz et al.[44]
| 24.136($\pm 0.008$) | 0.013($\pm 0.002$) | -1.6($\pm 0.2$) | 0.70$(\pm 0.07)$ |  K. Codling et al. [45]
| 24.147 | 0.0132$(\pm 0.0010)$ | -1.58$(\pm 0.1)$ | 0.75$(\pm 0.05)$ |  B. Langer et al. [43]
$2s2p^{6}4p$ | 25.767 | 0.00431 | -1.50 | 0.77 | haCC(1,4)
| 25.532 | 0.00430 | -1.67 | 0.85 | C Marante et al. (velocity)[41]
| 25.532 | 0.00430 | -1.26 | 0.84 | C Marante et al. (length)[41]
| 25.717 | 0.00528 |  |  | Liang et al.[13]
| 25.987 | 0.00386 | -3.95 | 0.505 | M Stener et al.[42]
| 25.701 | 0.0043 | -1.82 | 0.73 | B. Langer et al. (velocity)[43]
| 25.701 | 0.0043 | -1.88 | 0.72 | B. Langer et al. (length)[43]
| 29.908 | 0.007 | -1.35 | 0.63 | Nrisimhamurty et al.[14]
| 25.700 | 0.00665 |  |  | Schulz et al.[44]
| 25.711($\pm 0.005$) | 0.045($\pm 0.0015$) | -1.6($\pm 0.3$) | 0.70$(\pm 0.07)$ |  K. Codling et al. [45]
| 25.701 | 0.057$(\pm 0.001)$ | -1.47$(\pm 0.1)$ | 0.78$(\pm 0.11)$ |  B. Langer et al. [43]
$2s2p^{6}5p$ | 26.327 | 0.0019 | -1.54 | 0.775 | haCC(1,4)
| 26.092 | 0.0017 | -1.78 | 0.86 | C Marante et al. (velocity)[41]
| 26.092 | 0.0017 | -1.35 | 0.86 | C Marante et al. (length)[41]
| 26.287 | 0.00261 |  |  | Liang et al.[13]
| 26.404 | 0.00162 | -4.05 | 0.502 | M Stener et al.[42]
| 26.277 | 0.0018 | -1.87 | 0.75 | B. Langer et al. (velocity)[43]
| 26.277 | 0.0018 | -1.90 | 0.74 | B. Langer et al. (length)[43]
| 30.484 | 0.003 | -1.15 | 0.71 | Nrisimhamurty et al.[14]
| 26.281 | 0.00247 |  |  | Schulz et al.[44]
| 26.282($\pm 0.005$) | 0.002($\pm 0.001$) | -1.6($\pm 0.5$) | 0.70$(\pm 0.14)$ |  K. Codling et al. [45]
| 26.277 | 0.0036$(\pm 0.0018)$ | -1.46$(\pm 0.05)$ | 0.6$(\pm 0.2)$ |  B. Langer et al. [43]
Table 5: Fano parameterization of $2s2p^{6}np$ states following one-photon ionization of neon at 27 nm wavelength. Comparison with earlier theoretical and experimental works is presented. State | Peak position(eV) | Width(eV) | q | $\rho^{2}$ | Remarks
---|---|---|---|---|---
$2s2p^{6}3p$ | 24.244 | 0.0130 | 0.041 | 0.495 | haCC(1,4)
$2s2p^{6}4p$ | 25.767 | 0.0043 | 0.035 | 0.495 | haCC(1,4)
$2s2p^{6}5p$ | 26.327 | 0.0019 | 0.034 | 0.495 | haCC(1,4)
Table 6: Fano parameterization of $2s2p^{6}np$ states following three-photon
ionization of neon at 81 nm wavelength.

For one-photon ionization process, we compare our results with experimental
works [45, 43] and theoretical works based on XCHEM[41], R-matrix method[13,
43, 44], density functional theory[42] and multichannel quantum defect
theory[14]. Our results are in good agreement with the experimental data with
the experimental uncertainty included. Our results deviate with the most
recent theoretical works of Carlos Marante et. al. [41] by about 15%. They
also report a gauge dependence on the same level indicating the accuracy of
their results. Larger deviations are observed with the older theoretical
reports which also exhibit a larger discrepancy with the experiments. For a
given doubly excited state, the $q$ parameter corresponding to the three-
photon process differs from the one-photon process. These are presented in
table 6.

##  5 Conclusion

We presented an application of our newly developed tRecX-haCC package to study
multi-photon Fano lines for helium and neon atoms. iSurf technique was used to
efficiently compute the photoelectron spectra and the modulations arising from
the slow decay of doubly excited states. We analyzed a few lineshapes in the
photoelectron spectra and compared the linewidths and peak positions with the
data available in the literature. We find a good agreement. In addition, we
also reported $q$ parameters for lineshapes on multi-photon absorption peaks.
Although arising from the decay of the same doubly excited state, the
lineshapes characterized by the $q$ parameter are different depending on the
number of photons absorbed. Our study shows that tRecX-haCC is a valuable tool
to study multiphoton processes and opens up a new direction of study on multi-
photon Fano lines with pulse light sources.

## Acknowledgements

VPM acknowledges financial support from Science and Engineering Research board
(SERB) India (Project number: SRG/2019/001169).

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