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 Δx=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 5×10¹⁴ W/cm²). The period T=110=2π/ω with ω=0.057 corresponds to a wavelength of 800 nm.

 

Fig. 1. Fields E(t) (solid) and E ₀ sin(ωt) (dashed). Position (red solid) and velocity (black short-dashed) for a classical free particle in the field, starting at time t_i =80 (red circle) and returning to x=0 at t ₁=145 with an energy of 3.4, after an excursion with a maximum spatial amplitude of 34 [15, 16].

Download Full Size | PDF

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

 

Fig. 2. (430kB, lower-resolution version 230kB) Probability contours during the first half of a single-cycle laser pulse showing the distorted and singly ionized wavepacket at peak electric field, t=76. The log contour scale is given on the left-hand side.

Download Full Size | PDF

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.

 

Fig. 3. (1.2MB, smaller version 0.5MB) Probability contours during the second half of a single-cycle laser pulse. The first sign of a double ionization jet (in the direction of the arrows) appears at t=130.

Download Full Size | PDF

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.

 

Fig. 4. (1.2MB, smaller version 0.5MB) Probability contours during the second half of a single-cycle laser pulse where the field acts only in the lower and right parts.

Download Full Size | PDF

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 8E 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 =$E_0^2$/(4ω ²) is the ponderomotive energy. The maximum return energy possible for the single-cycle pulse chosen here is 3.7E_p [15]. We choose E₀ =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.

 

Fig. 5. (530kB, smaller version 260kB) Probability contours during the second half of a single-cycle laser pulse, for a weak field of E ₀=0.075, at t=148.

Download Full Size | PDF

3.5 Excursion

 

Fig. 6. (400kB) Probability contours during the second half of a single-cycle laser pulse of field strength E ₀=0.12. The box size is purposefully chosen too small, namely 26, including an absorber of effective absorbing width 6. Left: the first sign of a double ionization jet, at t=126; right: the double ionization yield is further ejected and no further double ionization occurs, after t=148.

Download Full Size | PDF

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.