# Dark and bright autoionizing states in resonant high harmonic generation:
simulation via 1D helium model

V. V. Strelkov1,2,∗ 1Prokhorov General Physics Institute of the Russian
Academy of Sciences, Vavilova street 38, Moscow 119991, Russia
2Institute of Applied Physics of the Russian Academy of Sciences, 46 Ulyanov
st., Nizhny Novgorod 603950, Russia
<EMAIL_ADDRESS>
###### Abstract

We study the role of dark and bright autoionizing states (AIS) in
photoionization and high harmonic generation (HHG) using a 1D helium model.
This model allows numerical integration of the time-dependent Schrödinger
equation beyond the singe-electron approximation completely taking into
account electronic correlation. We find the level structure of the system and
the spatial distribution of the electronic density for several states
including AIS. Studying the HHG efficiency as a function of the detuning from
the resonances with AIS we find the HHG enhancement lines. The shapes of these
lines are different from the corresponding Fano lines in the photoelectronic
spectra, in agreement with the experimental studies in helium. Moreover, we
simulate HHG under the conditions when the fundamental frequency is close to
the even-order multiphoton resonance with the dark AIS. We find the enhanced
generation of the neighbouring odd harmonics. The details of the enhancement
lines for these harmonics can be understood taking into account the temporal
delay between the emission of the non-resonant and resonant XUV; this delay is
defined by the AIS lifetime. Finally, our simulations show that the HHG
enhancement due to the dark and the bright AIS is comparable in the studied
system.

## Introduction

Two-electron atoms are probably the simplest systems where the electronic
correlations play crucial role. The Schrödinger equation describing correlated
three-body Coulomb dynamics cannot be resolved analytically; several
approximate yet very accurate analytical methods have been developed to
describe the helium atom in absence of the external field [1, 2]. In the
presence of the intense laser field the numerical integration of the
Schrödinger equation is the most effective tool to study the correlated
dynamics in a two-electron system. This can be done directly [3] for the 3D
helium atom; however, such calculation is feasible for the limited range of
the field parameters and anyway it is very computer-demanding (this makes it
relevant to consider the simplified 3D two-electron system [4]). Moreover,
many aspects of the multi-electron dynamics can be simulated via 1D helium
model [5, 6, 7, 8]. In particular, it can be used to study such dynamics in
the laser field [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19].

The existence of auto-ionizing states (AIS) (quasi-stable atomic states whose
excitation energies exceed the ionization threshold) is one of the striking
multi-electronic phenomena. The role of these states in high-order harmonic
generation (HHG) is actively studied both theoretically [20, 21, 22, 23, 24,
25] and experimentally [26, 27, 28, 29, 30] (for a review see [31] and
references therein).When the one-photon transition from AIS to the ground
state is allowed (such AIS is called a bright one) and the transition
frequency is close to the harmonic frequency, the HHG can be strongly enhanced
(by more than an order of magnitude). This can be attributed to the AIS
population from the continuum in the re-scattering process, and XUV emission
via the AI-ground state transition [22]. Moreover, the HHG enhancement by a
bright AIS dressed by two laser photons was observed [32].

If the transition is forbidden the AIS is called a dark one. In particular,
this is the case when the AI state has the same parity as the ground one. This
AIS can be populated by an even number of laser photons, but the XUV at the
AI-ground state frequency cannot be emitted. However, the dark AIS dressed by
the laser photon can absorb [33, 34] or emit the XUV photon. Very recently the
HHG enhancement due to dressed dark AIS was observed in indium plasma plume
[35].

In this paper we study the resonant properties of HHG using the numerical
integration of the time-dependent Schrödinger equation (TDSE) for 1D helium.
We investigate the level structure of the 1D helium atom and ion with an
emphasis on the AIS properties. Knowledge of the level structure allows
attributing HHG enhancement found in numerical calculations to resonances with
certain AI states. In particular, we investigate the enhancement of two
neighbouring odd harmonics due to a single resonance with the dark AIS.
Calculating intensities and phases of these harmonics we show that the shape
of the resonant harmonic enhancement line can be linked to the attosecond
properties of the resonant XUV emission.

##  I Level structure of 1D helium

We use the following model potential (atomic units are used throughout the
paper):

$V(x,y)=\frac{-2}{\sqrt{x^{2}+a^{2}}}+\frac{-2}{\sqrt{y^{2}+a^{2}}}+\frac{1}{\sqrt{(x-y)^{2}+b^{2}}},$
(1)

where $x$ and $y$ are the electrons’ coordinates, $a$ and $b$ are constants.
We use $a=1/\sqrt{2}=0.707$ and $b=1/\sqrt{3}=0.577$ to reproduce the first
and the second ionization potential of the actual helium, see Appendix for
more details. Note that the similar value of the latter constant was found in
[17], and the former constant was found in Ref [36]. The 2-D time-dependent
Schrödinger equation (TDSE) for the atomic potential given by Eq. (1) is
solved numerically with the approach described in [37]. The solution is done
on a spatial grid of $N\times N$ nodes; we use $N$ from 500 to 1000 depending
on the conditions; the spatial step is $0.2$; 100 notes at the boundaries of
the numerical box are occupied by the (almost non-reflecting) absorbing
layers.

The level structures of the ion and the atom are illustrated by Fig. 1, the
method for calculating the energies of the levels is described in the
Appendix. We present both the binding energies of the levels (multiplied by
$-1$) and the excitation energies with respect to the atomic ground state.

Figure 1: The level structure of the model 1D helium atom and the 1D hydrogen-
like ion. Zero of the excitation energy is the ground state of the atom; zero
of the binding energy is the second ionization threshold.

An atomic level is characterized by a pair of integer quantum numbers $n_{x}$
and $n_{y}$; $n_{x,y}=0,1...$. The state is even if $n_{x}+n_{y}$ is even, and
odd if $n_{x}+n_{y}$ is odd. A quantum number can be understood as describing
the state of one electron111In contrast to the 3D atom, the ground state in
one-electronic 1D system is usually denoted by $n=0$ (so that $n$ gives the
number of wave-function nodes), not $n=1$, see for instance Eq. (A4). Keeping
this tradition, we use $n_{x,y}=0,1,...$ for numbering the states of our
system. So the states $(0,n)$ are the bound atomic states. For $n\gg 1$ these
are the Rydberg states. The state $(1,1)$ is the first doubly-excited state.
Using the modified Pöschl-Teller potential (see Appendix) we can estimate the
energy of this state in zero approximation neglecting the electron-electron
interaction. Within this approximation the energy of the state is twice the
energy $E_{1}$ given by eq. (A4); this gives $2\times-18.6$eV$=-37.2$eV, which
is in a reasonable agreement with the level energy of $-34.0$ eV found
numerically solving the TDSE.

##  II Spatial distribution of the electronic density in the AIS

The wave-functions for several states found via numerical TDSE solution are
shown in Fig. 2, and details of the corresponding calculations are presented
in Appendix, section .2

Let us discuss the properties of the wave-functions shown in Fig. 2. First,
the potential and the wave-functions are symmetric with respect to the line
$y=x$. This is a result of the indistinguishability of the electrons. Note
that we assume that the two electrons can find themselves in the same point;
so we consider the parahelium. Second, in the figure one can see that the
ground (0,0) and dark AIS (1,1) states are even, because the wave-functions do
not change the sign after the inversion of the space coordinates
$x\rightarrow-x$, $y\rightarrow-y$, and the first excited state (0,1) is odd
(see the second and the third columns in Fig. 2). Third, the AIS wave-function
consists of the part which is well-localized near the origin, and the running
waves which ’leak out’ from the atom along the valleys ($x=0$ and $y=0$) of
the potential (see Fig. A2).

Figure 2: Wave-functions of several states in 1D helium atom. Absolute value
squared (left column), real (middle column

and imaginary (right column) part of the wave functions of ground (0,0) (upper
row), first excited (0,1) (middle row) and lowest dark AIS (1,1) (lower row)
states.

##  III Resonances with AI states in one- and two-photon ionization

In this section we study the resonant features appearing in the
photoelectronic spectrum due to bright and dark AI states. To calculate the
photoelectronic spectrum we solve numerically the TDSE for the atom irradiated
with an XUV field, thus finding the wave-function as a function of space and
time $\psi(x,y,t)$. Then we calculate222This operation requires storage of a
big data in computer memory; to make it less demanding, we store the wave
function not at every step of numerical integration $dt=0.05$, but at every
8-th step. Correspondingly, we find only the lowest-energy part of the
photoelectronic spectrum. This is definitely sufficient here, because the
inverse step of the numerical integration is much higher than the reasonable
electronic energies. its spectrum $\psi(x,y,\omega)$ and the energy
distribution for the part of the wave-function, where at least one electron is
free:

$\widetilde{W}(\omega)=\int_{x^{2}+y^{2}>r^{2}}dxdy|\psi(x,y,\omega)|^{2}.$
(2)

We used $r=30$ and checked that results are not sensitive to this parameter
provided that $r>20$. In the latter equation $\omega$ is the binding energy
(negative when the system is in atomic bound states or in ionic bound states +
free electron). It is more convenient to deal with the excitation energy
$\omega-E_{0}$, where $E_{0}=-79.3$eV is the atomic ground state energy. In
the excitation energy range between the first ionization energy (24.9 eV) and
the first excited state of the ion (52.4 eV) the photoionization leads to the
appearance of a free electron and an ion in the ground state. So the energy of
the ion is defined, thus the spectrum (2) directly corresponds to the
photoelectron spectrum. Fig. 3 presents the spectrum as a function of the
excitation energy.

The bright AI states lead to resonant features in the spectrum of the
electrons detached due to one-photon ionization. To find this spectrum we
solve the TDSE for the atom irradiated by a short XUV pulse. The pulse
intensity is $3.5\times 10^{12}$ W/cm2 (the field amplitude is $0.1$ atomic
unit), its duration is 100 as. The AIS leads to a Fano [38] feature in the
photoelectron spectrum. Note that a similar feature was found in the 1D helium
photoelectron spectrum in Ref. [17]. We have found the width and the Fano
parameter of the line, which is shown in the graph.

Figure 3: Normalized photoelectron spectra calculated for the case of
irradiation of the atom by one- (red curve) and two-color (blue and violet
curves) XUV field, see the text for more details. The two-photon resonance
leads to a Fano feature in the total spectrum (violet) due to the interference
of the resonant and the non-resonant terms, whereas the resonant term can be
separated calculating the spectrum over the time interval after the
irradiation (blue). The central frequencies, the FWHM and the Fano parameter
of the peaks are shown in the graph.

To find the resonances with the dark AIS, we solve the TDSE for an atom
irradiated by two-color XUV pulse. The carrier frequencies are 130 eV and 85
eV, the pulse duration is 100 as, the fields’ amplitudes are 1.0 atomic units.
These frequencies of the fields are chosen to have two-photon response in the
studied spectral range (due to Raman-type transition 130-85=45 eV), and no
one-photon response in this range. The short duration of the pulse provides
the wide spectral range covered by the two-photon response (more precise, the
spectrum of the field squared has a wide peak near 45 eV, and the FWHM of this
peak is 27 eV). The TDSE is solved over the time interval of 20 fs thus over
the irradiation time plus a long time after the pulse. For the attosecond
pulse used here the latter time period is important for the correct
photoelectron spectrum calculation because the AI states continue decaying
after the pulse has already passed. Calculating the spectrum over the whole
time we find the spectrum with the Fano feature, see the violet curve in Fig.
3. This feature appears due to the interference of the non-resonant and
resonant contributions to the two-photon ionization; the latter contribution
is due to the dark AIS. Moreover, calculating the spectrum for the time
interval where the field is off, we find only the resonant contribution, see
the blue line.

Note that the authors of Ref. [10] have studied the parameters of the Fano
resonance appearing due to the single-photon ionization from the first atomic
excited state to the dark AIS. The width of the resonance is very close to
that found in our simulations (hence the parameters of the 1D He potential in
Ref. [10] are $a=b=1$). The Fano parameter is also negative but it is much
higher, thus the line shapes of the single-photon transition from the first
excited state to the (1,1) AIS and of the two-photon transition from the
ground state to the (1,1) AIS differ essentially.

##  IV Resonances with AI states in HHG

In this section we study the role of the resonance with the AI states in HHG
simulating the XUV spectrum emitted by our model atom in an external laser
field. The peak pulse intensity is $8\times 10^{14}$ W/cm2, the fundamental
frequency $\omega_{l}$ is about 3 eV (the wavelength is about 400 nm, the cut-
off is approximately at H19-H21, the Keldysh parameter is $\gamma\approx
0.95$); it is tuned near the 14th order resonance with the transition from the
ground to the dark (1,1) AIS and near the 17th order resonance with the bright
(1,2) AIS. The laser pulse intensity increases during 4 cycles, then it is
constant during 8 cycles, than decreases down to zero during 4 cycles; the
slopes of the pulse have $\sin^{2}$ shape.

Figure 4: Intensity (a) and phase (b) of harmonics 15 (blue circles), 17
(violet triangles), and 19 (dark cyan diamonds) in the vicinity of the
17-photon resonance with the bright AIS (1,2).

In Fig. 4 we show the intensities and phases of H15, H17, and H19 as functions
of laser frequency (the horizontal axis above the graph) in the vicinity of
the resonance with the bright AIS. The H17 frequency is shown for reference in
the axis below the graph. We can see the pronounced enhancement of the
resonant harmonic intensity. The ”enhancement line” (enhancement as a function
of the laser frequency) is shifted and broadened with respect to the resonant
line in the one-photon photoelectron spectrum shown in Fig. 3. This can be
attributed to the line modification by the laser field (to the Stark shift of
the levels and photoionization of the AIS). Moreover, there is even
qualitative difference between the shapes of the lines in the photoelectronic
spectrum and harmonic enhancement: in the spectrum the dip is at the high-
energy side of the maximum, in the enhancement line it is at the low-energy
side. This agrees with the experimental studies in helium: in the photo-
absorption spectrum [39] the dip is at the high-energy side, and in the XUV
spectrum [27] it is at the low-energy one. Similar difference was discussed in
[24].

One can also see a pronounced modification of the harmonic phase by the
resonance; such distortion was measured in [40]. Also there is some
enhancement of the harmonic above the resonance (H19) and almost no
enhancement of the harmonic below it (H15). However, the phases of all the
three harmonics are affected by the resonance. The total variation of the
laser frequency in Fig. 4 is small (approx 5%), so the non-resonant
contribution to the generation of a harmonic should not vary much; the
resonant one vanishes at the edges of the considered frequency range. Thus,
the total phase variation over this range for a harmonic is approximately
either $0$ or $2\pi$.

The HHG modification by dark AIS has been studied much less than that for the
bright AIS. Very recently the resonance of an even number of laser photons
with the dark AIS was considered [35] and an enhancement of the harmonic with
an order differing by unity from this number was observed. This phenomenon can
be understood as a resonance between the harmonic and the dark AIS, dressed by
a laser photon.

Our simulations presented in Fig. 5 show that in the case of the 14th order
resonance with the dark AIS the two neighbouring harmonics (i.e. H13 and H15)
are enhanced by the resonance. Note that there are no one-photon resonances
that can cause the HHG enhancement in the considered spectral range (see
Fig.3).

Figure 5: HHG spectra in the vicinity of the 14-photon resonance with the dark
AIS (1,1). The fundamental frequency and the frequency of the 14 harmonic is
shown in the graph.

To study this enhancement in more detail we present in Fig. 6 the intensity of
H13 and H15 as functions of the fundamental frequency in the vicinity of the
14-photon resonance with the dark AIS. Comparing this figure with Fig. 4 we
can see that the enhancements caused by the resonances with dark and bright
AIS are comparable in the considered conditions. The width of the enhancement
lines for H13 and H15 are close to the one in the two-photon photoelectron
spectrum in Fig 3; the center of the HHG enhancement line is slightly shifted
due to the Stark shift of the levels in the laser field. Counterintuitively,
the enhancement lines for H13 (red) and H15 (blue) are different: the peak
enhancement is achieved for different laser frequencies; moreover, zero
enhancement for H13 near 44.6eV does not correspond to any specific feature of
H15 at this frequency.

The difference of the enhancement lines for H13 and H15 can be understood
taking into account that XUV emission time is different under resonant and
non-resonant conditions: one can assume that, similar to the HHG enhancement
with the bright AIS, the emission of the resonant XUV is delayed (with respect
to the non-resonant one) by the AIS lifetime [41, 42]. In more detail, the
difference of the harmonic enhancement lines can be explained as following.

Similarly to the Fano line in the photoionization cross-section, the harmonic
enhancement line originates from the interference of the resonant and non-
resonant terms [24] in the harmonic amplitude. The phase of the resonant term
varies strongly near the resonance as a function of the detuning, whereas the
phase of the non-resonant term does not. Let us denote the phase difference of
these terms for the $q$th harmonic as

$\delta\varphi_{q}=\varphi_{q}^{r}-\varphi_{q}^{nr}$ (3)

.

The difference of H15 and H13 phases $\Delta\varphi=\varphi_{15}-\varphi_{13}$
defines the emission time [43] of the attosecond pulse consisting of H15 and
H13: $t_{e}=\Delta\varphi/(2\omega_{l})$. (Note that $\Delta\varphi$ should
not be confused with $\delta\varphi_{q}$: the former gives the phase
difference between the neighbouring harmonics, whereas $\delta\varphi_{q}$
describes the interference of the resonant and non-resonant terms of the same
harmonic)

Far from the resonance the non-resonant term dominates:
$\varphi_{q}\approx\varphi_{q}^{nr}$, so the emission time is defined by this
term:

$\Delta\varphi^{nr}=\varphi_{15}^{nr}-\varphi_{13}^{nr}=2t_{e}^{nr}\omega_{l},$
(4)

Note that the phases of the non-resonant terms do not vary much within the
considered small detuning interval of the fundamental, in particular, this
equation is valid near the resonance as well.

Near the resonance the resonant term dominates:
$\varphi_{q}\approx\varphi_{q}^{r}$, so the emission time is defined by this
term. This emission time as we mentioned above is $t_{e}^{r}=t_{e}^{nr}+\tau$,
where $\tau$ is the AIS lifetime. So near the resonance

$\Delta\varphi^{r}=\varphi_{15}^{r}-\varphi_{13}^{r}=2(t_{e}^{nr}+\tau)\omega_{l}.$
(5)

From Eqs. (3), (4), and (5) we conclude that near the resonance
$\delta\varphi_{15}-\delta\varphi_{13}=2\tau\omega_{l}$. So the enhancement
line shape defined by the interference of resonant and non-resonant
contributions to the harmonic emission is different for H13 and H15.

From the harmonic phases shown in Fig. 6 we find that
$\Delta\varphi^{nr}\approx 0$ and $\Delta\varphi^{r}\approx 2.8$ rad. From
Eqs. (4) and (5) we find $\tau=300$ as. This agrees with the fact that this
delay is comparable but less than the inverse FWHM of the resonant term in the
two-photon ionization cross-section (blue line in Fig. 3), namely
$1/\Gamma=470$ as.

Note that in Ref. [40] the resonant phase was measured under some detuning
from the resonance because closer to the resonance the intensity difference of
the resonant and non-resonant harmonic make RABBIT measurements problematic.
For HHG enhanced by the dark AIS such measurements can be done near the center
of the resonance, because the two harmonics are enhanced in a similar way.

Figure 6: Intensity (a) and phase (b) of the harmonics 11 (yellow triangles),
13 (red squares), 15 (blue circles), and 17 (violet triangles) in the vicinity
of the 14-photon resonance with the dark AIS (1,1).

Fig. 7 shows the attosecond pulses obtained from the cut-off harmonics
(H19-H23) and harmonics near the 14-photon resonance with dark AIS (H13-H17).
Naturally, the emission time of the cut-off attosecond pulse does not depend
on the detuning from the resonance. So this emission time gives a
straightforward reference for the emission times of the attosecond pulses
obtained from H13-H17. The latter pulses in the off-resonance conditions are
emitted before the cut-off attosecond pulse. (This shows that the short
quantum path dominates in HHG response, see [43].) The H13-H17 pulses are
emitted approximately at the same time for the above-resonance conditions
(laser frequency 3.22eV) and the below-resonance ones (laser frequency 3.07
eV), so the resonance does not affect the attosecond pulses emission. In the
resonant case the attosecond pulse is much more intense and emitted later than
in the off-resonant cases, in agreement with the considerations presented
above. The delay of $310$ as is close to the one found above from the harmonic
phases.

Figure 7: Attosecond pulses produced by the harmonics near the resonance with
the dark AIS (dashed, dotted, and dash-dotted lines) and by the cut-off
harmonics (solid lines) for the three fundamental frequencies shown in the
graph.

##  V Conclusions

In this paper we study the role of the dark and the bright AIS in resonant HHG
using the 1D helium model allowing ab initio numerical TDSE simulations beyond
the singe-electron approximation.

Analytical model using a modified Pöschl-Teller potential allows obtaining a
reasonable estimate of the lowest double-excited state energy. We present the
structure of the (lowest) levels of the studied system. The simplicity of the
system allows straightforward graphical presentation of the wave-functions of
different states, including the AI ones.

We make detailed simulations of the interaction of the atom with
electromagnetic field. The photoelectronic spectra demonstrate pronounced
features due to resonances with AIS. In particular, the resonances lead to
asymmetric Fano peaks in the spectrum of the photoelectrons appearing due to
one- and two-photon ionization.

We simulate the HHG enhancement via the resonance with the bright AIS and find
a pronounced difference between the shapes of the above-mentioned Fano maximum
in the photoelectronic spectrum and the harmonic enhancement line. We find an
essential enhancement of the resonant harmonic, some enhancement of the
harmonic above it and no enhancement of the harmonic below it.

Moreover, we simulate HHG under the conditions when the fundamental frequency
is close to a multiphoton resonance of an even order with the dark AIS. We
find an enhanced generation of the neighbouring harmonics. The shapes of the
enhancement lines for the harmonics are different. The difference can be
understood taking into account the harmonic phase properties defined by a
delay between the emission of the non-resonant and resonant XUV; this delay is
close to the AIS lifetime. Simultaneous enhancement of the two harmonics by
the resonance with dark AIS makes feasible the experimental measurement of
resonance-induced dephasing between them. Finally, our simulations show that
the enhancement due to dark and bright AIS is comparable in the studied
system.

##  VI Acknowledgments

We are grateful to V. Birulia for the TDSE code development, and to M. Singh,
M. A. Fareed, T. Ozaki, A.I. Magunov, A. N. Grum-Grzhimailo for the
discussions on the AI states properties and their role in HHG. We would like
to thank N. Yu. Shubin for his helpful language assistance. The study was
funded by RSF (grant No 22-12-00389).

## Appendix

###  .1 1D hydrogen-like ion

The 1D hydrogen atom or hydrogen-like ion with the Coulomb potential or the
soft-Coulomb potential was studied in an number of papers [44, 45, 46, 47, 48,
49]. In contrast to the 3D system, the bound state of the 1D hydrogen cannot
found analytically, either for the Coulomb, or for the soft-Coulomb potential.
Here we describe our numerical method and approximate analytical approach to
find the eigenstates’ energies and wave-functions.

To study He+ ion we remove the last term in Eq. (1). Then the TDSE describes
two non-interacting ions, and the interpretation of the TDSE solution results
in terms of a single ion is straightforward. The ion’s potential is

$V_{ion}(x)=\frac{-2}{\sqrt{x^{2}+a^{2}}}.$ (A1)

Under $a=1/\sqrt{2}$ the ground state energy of the model ion is equal to the
actual ionization energy of He+ [36]. This soft-Coulomb potential is shown in
Fig. LABEL:Potentialsa.

Solving the TDSE we find the energies of the ground state and several first
excited ones. The energy and wave-functions of the states are found as
following. The initial wave-function of the $n$-th state ($n=0,1$…) is set as
the corresponding wave-function of a harmonic oscillator; then the wave-
function evolution is simulated via numerical TDSE solution. The part of the
initial wave-function that is orthogonal to the bound eigenstates of the ion
spreads in space and finally is absorbed by the absorbing boundary of the
numerical box. So the part of the wave-function bound near the origin
corresponds to the eigenstate of the potential (A1) (more precisely, it
corresponds to a superposition of the eigenstates; however, choosing the
initial wave-function with proper number of zeros guarantees that the weight
of all eigenstates, except one, is very small). Numerical TDSE solution gives
us the wave-function as a function of space and time $\psi(x,y,t)$. We
calculate its spectrum $\psi(x,y,\omega)$ and the following energy
distribution:

$W(\omega)=\int dxdy|\psi(x,y,\omega)|^{2},$ (A2)

(In contrast to Eq. (2), here the integration is done over the whole numerical
box.) The energy of the maximum in this distribution gives the energy of the
found state; this negative energy is the state’s binding energy, multiplied by
$-1$. Other peaks in the energy distribution, if any, are much weaker; the
ratio of the peaks to the main one gives the weights of the other eigenstates
in the found state.

In [44] the approximate equation for the ground states’ energies of the
truncated Coulomb potential was found. The accuracy of the approximation
becomes better as the truncation parameter (roughly analogous to the parameter
$a$) tends to zero. However, under $a=1/\sqrt{2}$ the accuracy for the ground
and first excited state is insufficient. Much better analytical approach can
be developed approximating the soft-Coulomb potential (A1) with the modified
Pöschl-Teller potential:

$V_{PT}(x)=-\frac{\alpha^{2}}{2}\quad\frac{\lambda(\lambda-1)}{\cosh^{2}(\alpha
x)},$ (A3)

where $\alpha$ and $\lambda$ are parameters. The bound states’ energies for
this potential are [50]:

$\begin{array}[]{l}E_{n}=-\frac{\alpha^{2}}{2}(\lambda-1-n)^{2},\\\
n=0,1,...\\\ n\leq\lambda-1,\end{array}$ (A4)

an even $n$ gives the energy of an even state, and an odd $n$ gives the energy
of an odd one.

The parameters $\alpha$ and $\lambda$ are chosen so that:

\- the minimum of the potential (A3) is equal to the minimum of the potential
(A1);

\- the energy of the ground state $E_{0}$ is equal in both potentials.

The found approximating potential is shown in Fig. LABEL:Potentialsb. Using
Eq. (A4) we find the energy of the first excited state $E_{1}=-18.6$ eV.

###  .2 1D helium. Calculation of the electronic density spatial distribution
in the AIS.

Setting $b\approx a$ in potential (1) we consider a two-electron atom. The
parameter $b$ is chosen so that the atom ionization energy is close to the one
of real helium. The found value is $b=1/\sqrt{3}=0.577$. The energy of the
atomic ground state is the energy of the ionic ground state (-54.4 eV) minus
the ionization energy 24.9eV (24.6eV in actual helium); the found energy of
the ground state is -79.3 eV.

The potential given by Eq. (1) under $a=1/\sqrt{2}$ and $b=1/\sqrt{3}$ is
shown in Fig. A2.

To find analytically the energy of (1,1) state in zero approximation one can
neglect the electronic interaction. Within this approximation the energy of
the state is doubled energy $E_{1}$ given by eq. (A4); this gives
$2\times-18.6$eV$=-37.2$eV. This is pretty close to the level energy of
$-34.0$ eV found numerically solving TDSE. This similarity can be explained as
follows: for $n\geq 1$ the 1D soft-Coulomb potential’s levels are lower than
the ones of the modified Pöschl-Teller potential. However, the electronic
repulsion in the atom moves them up. So finally these two ”shifts” partly
compensate each other for $n=1$.

Figure A2: Potential given by Eq. (1) describing the model 1D helium atom.

The wave-functions for several states found via numerical TDSE solution are
shown in Fig. 2.

The wave-functions and the energies of the bound states (the two upper rows in
the figure) are found similarly to the case of the hydrogen-like ion: the
initial wave-function for a state $(n_{x},n_{y})$ is set as a symmetrized
product of the one-electron wave-functions $(n_{x})$ and $(n_{x})$ in the
modified Pöschl-Teller potential; TDSE is solved for a long enough time, so
that the eigenstate survives and the remaining part of the initial wave-
function is absorbed after spreading. The maximum in the energy distribution
(A2) gives the eigenstate energy.

However, this procedure could not be applied directly for an AIS: this state
decays, so, after some time we find zero population in all states. So the
procedure is slightly modified: first we set the initial wave-function (1,1)
as described above, then propagate the TDSE and find the energy $E_{AIS}$ of
the AIS; then, using the wave-function found during the TDSE integration
$\psi(x,y,t)$ we calculate spectrum $\psi(x,y,\omega)$ and finally find the
spectral component with this energy:
$\Psi(x,y)\equiv\psi(x,y,\omega=E_{AIS})$. This function is shown in Fig. 2,
bottom row.

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