## Abstract

The time-dependent Schrodinger equation is solved for a 1D×1D two-electron model helium atom subject to a low-frequency short, intense laser pulse. A half-cycle pulse leads to strong single but no double ionization. A full-cycle pulse leads to double ionization which begins precisely at the classical return time for the first ejected electron. When the excursion range for the first electron is truncated, the double ionization at later times, corresponding to longer excursions, disappears. When the field near the nucleus is turned off during the return of the first electron, double ionization persists.

© Optical Society of America

## 1 Introduction

The theoretical model of a one-dimensional (1D) two-electron atom has been
extensively applied in the study of correlated two-electron systems subject to
external laser fields (see [1–5] and the references therein). In fact, the solution of the
full time-dependent 3D two-electron problem of the helium atom in a laser field is
still a formidable computational challenge [6]. The aim of the present investigation is to gain qualitative
insight into the *mechanism* of double ionization using the numerical
solution of the time dependent Schrödinger equation (TDSE) for the 1D
model. We choose the parameters both for the atom and for the field such as to
reproduce on one hand the binding energies of the helium atom and on the other hand
the parameters of laser fields used in experimental observations. Experimentally, it
was shown [7] that at low laser intensity the double ionization yield from
noble gases is many orders of magnitude larger than the calculated yield from a (two
step) sequential process
A→A^{+}→A^{++},
pointing to a non-sequential mechanism. Present theoretical and experimental
research is in progress [4, 5, 8, 9, 10], investigating this mechanism in more detail. Notably, the
rescattering picture [12], which is so successful in describing high-harmonic
generation, has from the beginning been invoked to explain double ionization at
laser frequencies low compared to the atomic orbital frequency. Recent work,
however, trying to quantify the rescattering theory, has failed to obtain agreement
with the experimental results [13].

## 2 Theory

#### 2.1 Definition of the model

The model atom contains two electrons, moving in one dimension, with coordinates
*x*
_{1} and *x*
_{2}, under the
influence of an external driving field *E*(*t*) :

Here *L*
_{1} and *L*
_{2} are the
Laplace operators in the coordinates *x*
_{1} and
*x*
_{2}, respectively. The nuclear and
interelectronic interaction potentials are smoothed Coulomb potentials

We use atomic units. If one chooses (see [4, 5] and references therein), all smoothing parameters
*a*=1, with a nuclear charge of *Z*=-2 and an
interelectronic repulsion charge in *V*
_{12} of
*Z*=+1 (the latter was taken variable in [4]), one obtains a ground state energy of -2.24. In the
present work, we follow [2], using *a*
^{2}=0.55, which leads
to a ground state energy of -2.90, in close agreement with the 3D helium value
(-2.91). The ground state energy of the singly ionized 1D model atom (the
binding energy of the “second” electron) is -1.92, in
reasonable agreement with the value (-2.0) for He+.

## 2.2 Numerical methods

A finite-difference constant-step 2D
(*x*
_{1},*x*
_{2}) position-space
grid with the three-point approximation for the Laplacian is used. The time
propagation is performed by the standard Peaceman-Rachford
alternating-direction-implicit (ADI) scheme [14], which has been extensively used in the solution of the 2D
(3D with cylindrical symmetry) single-electron TDSE (see references by Kulander and
coworkers in [13]). A smooth mask function absorbs any probability that
reaches the edges of the computational grid. The grid extends to
|*x*
_{1,2}|<*x*
_{max}≈130
with a Δ*x*=0.21 spacing. The resulting linear dimension
of the Hamiltonian matrix is about 1200×1200.

The separation of the singly from the doubly ionized component is strictly possible
only in the asymptotic limit. In wavepacket studies, an arbitrary boundary
*R* is set, defining any probability leaving the region defined by
(*x*
_{1}<*R* or
*x*
_{2}<*R*) as being doubly
ionized. In the present study, we simply identify the probability leaving the
nucleus in two “jets” oriented symmetrically on both sides
near the interelectronic repulsion ridge at
*x*
_{1}=*x*
_{2} as leading
asymptotically to double ionization. This identification has the further advantage
of giving an approximate onset time for double ionization.

## 3 Results

We use a single-cycle pulse, turned on and off smoothly, and centered at
*t*
_{0}=100:

The parameters *b* and *d* are chosen such that the
peak field strength is *E*
_{0} and that the time duration
between the two extrema of *E*(*t*) is
*T*/2=55. We use *E*
_{0}=0.12 (this
corresponds to an intensity of 5×10^{14} W/cm^{2}). The
period
*T*=110=2*π*/*ω*
with *ω*=0.057 corresponds to a wavelength of 800 nm.

All following animations show logarithmic 2D contour plots in the two
electrons’ coordinates *x*
_{1} and
*x*
_{2}. The graphs show only a part of the total
configuration space, bounded by
-64<*x*
_{1}<136 and
-40<*x*
_{2}<40. Positive
*x*
_{1} is towards the right and positive
*x*
_{2} is towards the bottom, so that the line
*x*
_{1}=*x*
_{2} is at
45° from top left to bottom right (the line of reflection symmetry). Most
of the probability remains in the bound part near the nucleus (at
*x*
_{1}=*x*
_{2}=0) throughout the
pulse duration. In order to observe the nonsequential double ionization, it is
important to keep the intensity low enough so as not to saturate the single
ionization [5]. The clock at the bottom gives the relative field strength
through the hand angle *a* from the horizontal:
*E*(*t*)=*E*
_{0}
sin(α).

## 3.1 Half-cycle pulse

Animation Fig. 2 shows the temporal evolution from the pulse turn-on,
through the first field maximum, up to *t*
_{0} (where
*E*(*t*
_{0})=0). The field during this
half-cycle is directed towards the right and towards the bottom (positive
*x*
_{1} and *x*
_{2}). The initial
ground state is centered at the nucleus. At the peak of the field the Stark
distorted ground state is visible, together with a strong single ionization current
along the half-axes (*x*
_{1}=0 and
*x*
_{2}>0) and (*x*
_{2}=0
and *x*
_{1}>0). No double ionization current (into
the lower-right quadrant) is visible, be it direct (originating near the nucleus) or
sequential (originating near the axes, away from the nucleus). At the end, the part
near the nucleus is not distorted. The singly ionized portion is moving away from
the origin with maximum velocity.

## 3.2 Full-cycle pulse

Animation Fig. 3 shows the evolution during the second half of the
laser pulse. The field distorts the atom in the opposite direction (towards the left
and towards the top) and it pulls the electron back that had been ejected during the
first half-cycle. At *t*=130, electronic probability starts to be
ejected into the upper-left quadrant. This double ionization yield does not emerge
exactly on the diagonal because the electrons strongly repel each other [4, 5]. The double ionization begins to appear at the time when the
electron that had been ejected toward the right (or bottom) returns to the vicinity
of the parent ion with sufficient energy. Classically [15], the return time after which double ejection is
energetically allowed is *t*=133. At later times, the double
ionization “jets” propagate into the upper-left quadrant.
Towards the end of the pulse, the field is turned off smoothly, and the double
ionization eventually fades away.

## 3.3 Field-assisted rescattering

In order to investigate the role of the field during the rescattering process of the
first electron on the remaining bound ionic electron [17], the field is turned off in the vicinity of the nucleus
(and in fact in the upper left quadrant, namely by replacing the terms
*E*·*x* in the Hamiltonian by
*E*·max{0, *x*-8}) for the second half only
of the laser pulse.

Animation Fig. 4 gives the probability similarly as in Fig. 3, but with the modified field. Now, at maximum field,
*t*=123, the bound part of the atom near the nucleus is no longer
distorted, but the ejected electron is still driven back towards the parent ion by
the field. The beginning of double ionization is now seen much later, at
*t*=144, because the field does not act over the distance
0<*x*<8 and thus for a comparable excursion
the electron will gain an amount 8*E*
_{0}=1 a.u. less energy.
The result is that the double ionization turns on at significantly later times,
corresponding to higher kinetic energy upon return. The double ejection now proceeds
more slowly because the field is no longer directing it.

## 3.4 Below-threshold double ionization

When the field strength is reduced, we eventually reach the “below
threshold” regime in which the field cannot provide enough energy to the
returning electron to ionize the second (ionic) electron. For a monochromatic field
the maximum return energy is 3.17 *E*_{p}
, where
*E*_{p}
=${E}_{0}^{2}$/(4*ω*
^{2})
is the ponderomotive energy. The maximum return energy possible for the single-cycle
pulse chosen here is 3.7*E*_{p}
[15]. We choose *E*_{0}
=0.075 here, such
that 3.7 *E*_{p}
=1.58. This energy is well below the binding
energy of 2 for the “second”, ionic electron. Thus,
classically, double ionization is energetically forbidden. Animation Fig. 5 shows nevertheless the beginning of a double
ionization current. The time of appearance is compatible with the return time at
which the maximum return energy of the first electron is reached, namely
*t*=153 [15]. At later times, the double ionization current is no longer
visible at the 10^{-12} level of probability density. Thus, the double
ionization trace is much weaker than in Fig. 3, but it is nonzero.

## 3.5 Excursion

In order to further test the importance of the classical excursion trajectory of the
first electron to gain the necessary energy for the rescattering process, the
absorbing wall can be moved closer to the nucleus, cutting off all trajectories with
excursion larger than a given *x*
_{cut}. If we take
*x*
_{cut}=20, we allow only the earliest returns, around
*t*=133, and cut off all longer and more energetic return
trajectories [15]. For *E*
_{0}=0.12, an excursion of
18 leads already to a return energy of 2. Thus, we do not suppress double
ionization, but restrict the double ejection to a small time window.

## 4 Summary and Conclusions

The collinear two-electron model considered here lends itself to an expedient, accurate numerical solution in 2D configuration space. This model cannot address angular effects of electron emission or of laser polarization other than linear. The aim of the present work is not a quantitative simulation of real (3D) experiments, but to exhibit the key qualitative features of the double ionization process by a temporal analysis in configuration space. We consider the low-frequency regime (800 nm on He). For the single-cycle pulse, in the animations one can clearly observe the onset of double ionization without cluttering the wavepacket picture by the strong interference structure arising from previous cycles of the field.

We extract the following information from the test cases considered: (i) A
*half-cycle* pulse leads to no visible double ionization. (ii)
The *starting time* of the double ionization jet occurs precisely at
the minimum return time for a simpleman’s free classical particle [15] with sufficient return kinetic energy to eject the second
electron. (iii) The double ionization yield emerges near the *diagonal*
*x*
_{1}=*x*
_{2}. (iv) The
*laser field near the nucleus* is essential in extracting the
first electron. For the rescattering process itself, however, it is
*not* important: double ionization occurs even when the laser field
near the nucleus is turned off (after ejection of the first electron). (v) At
*lower* intensity, as expected, the double ionization yield fades
away. It is still visible, however, near the return time of maximum energy, even
when the return energy is classically too low. (vi) By reducing the computational
(spatial) box, the longer *trajectories* can be cut off: this is
reflected in the *temporal behaviour* of the double ionization jets.

All results in the present investigation point to the validity of the “simpleman’s” rescattering picture [12] to explain the temporal structure of the ejected double ionization yield. In this light, previous attempts (with negative outcome [13]) to quantify the rescattering ideas should be (and are [11]) reconsidered.

The time delay observed between single and double ionization is compatible with the results of [5] (the latter results did not address the detailed temporal behaviour, the presentation being in “complementary” momentum space). This time delay originates in the classical, large excursion in the field, necessary for the electron to acquire its energy. This contradicts double ionization pictures in which the electron ejection occurs over a shorter timescale, such as the “two-step-one” or the “shake-off” mechanisms (see [10] and references therein).

The present results are also in contrast to some conclusions drawn from calculations that have been performed at much higher frequency (corresponding to 248 nm, or higher [2, 4]). These higher frequencies might be too large to observe the present, genuinely low-frequency effects.

Recent experimental results on the momentum distribution of the ejected charged
particles [9] show that the doubly charged ions’ momentum
distribution is peaked at a nonzero (total) momentum. This is in accord with the
present theoretical model results: since both electrons emerge into the same
direction with approximately equal momenta, the ion must recoil with nonzero
momentum. The total momentum imparted to the nucleus by the field within the
simpleman’s picture is just the sum of the first electron’s
return momentum (at time t*
_{1}, see Fig. 1) plus the drift momentum of a doubly charged particle
released at time t
_{1}. Since the experiment averages over
all returns, the emission and rescattering times can only be extracted within a
rather large uncertainty region. This region is compatible with the rescattering
picture.*

*Acknowledgments*

*The calculation of subsection 3.5 was suggested by Misha Ivanov. Some technical
remarks on the time propagator by Harm-Geert Muller have been helpful. There have
been numerous discussions with colleagues at the MBI, namely Wilhelm Becker, Horst
Rottke and Wolfgang Sandner. MD was supported through a Deutsche
Forschungs-Gemeinschaft Heisenberg fellowship.*

*References and links*

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**15. ***Cf* the programs at http://mitarbeiter.mbi-berlin.de/doerr/mathematica/simplemans.txt

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**17. **K. Burnett, J. B. Watson, A. Sanpera, and P. L. Knight, “Multielectron response to intense laser fields”, Phil. Trans. Roy. Soc. Lond.A356, 317 (1998). This group has also performed a “field-assisted rescattering” study recently [8], using their “crapola” model, and turning off the laser-electron interaction for the second, “inner” electron.