We calculate the energy spectrum of electrons ejected from a model two-electron atom exposed to an intense laser pulse and follow the spectrum’s evolution in time. For the case of 13-photon double ionization, we see the appearance of above-threshold ionization peaks soon after the laser pulse reaches maximum intensity.
© Optical Society of America
Over the past decade, experiments [1–3] measuring the double-ionization signal of multi-electron atoms using a short, intense laser pulse have obtained results orders of magnitude larger than a single-active electron model predicts. This increased ionization is generally attributed to electron-electron correlation, a factor single-active electron models do not take into account. There is also general consensus that electrons pulled from and then later revisiting the nucleus, so -called “re-scattering” electrons, play an important role in these augmented double-ionization yields. Despite this consensus, definitive results using purely theoretical calculations have not yet materialized due to the complexity of the problem. Given this background, new kinds of information that can be obtained about ionization are especially welcome. We have begun to examine above-threshold ionization (ATI) spectra  in this light, taking a numerical approach.
Numerical simulations are helping to reach a clearer understanding of how a multi-electron atom ionizes, but even for the simplest of two electron systems, solving the fully time-dependent ionization problem is a formidable computational task. Over the past few years, steady progress has been made in numerically examining double ionization of the helium atom , but the complexity of the problem prevents a rapid or flexible search through a wide range of parameters. To examine a large parameter range in a short period of time, we have reduced the dimensionality of the system to one spatial degree of freedom per electron. We can expect qualitative agreement with experiments because an atom’s response to a strong but still non-relativistic laser pulse is essentially one-dimensional. This has been shown to be true in previous investigations  which have used this model to reproduce and analyze the double ionization knee seen in experiments.
Our one-dimensional treatment requires us to soften the Coulomb potential. This is done in the usual way  so that the position dependence of the Coulomb potential is replaced by a form without a singularity.
The laser frequency we use allows us to safely apply the dipole approximation and fix the nucleus at the origin. The Hamiltonian H for this atom in atomic units (a.u.) is
where xn and pn are the position and momentum of the nth particle. The ground state energy of the atom is -2.23 a.u. and the ground state of the one-electron ion is -1.48 a.u. Our laser pulse is modeled as a sine wave with maximum electric field strength Eo modulated by a pulse envelope f(t) :
The maximum laser intensity is the square of the maximum laser field strength (I=(c/8π)) and the phase of the sine wave (ϕ) is chosen such that the vector potential A(t) is zero at the end of the pulse. The pulse envelope profile is trapezoidal with an equal two-cycle turn-on and turn-off and a plateau of one or more cycles. In these units, the vector potential is related to the electric field by cE(t)=-Ȧ(t).
The split-operator method  has been used to solve the Schrödinger equation before and is used again here. To this we add the double-zone algorithm , which splits our integration into “inner” and “outer” regions. For all data shown, the inner region encompasses a box whose boundaries are a distance of ±200 a.u. from the origin and the outer region is four times larger. The grid spacing is Δx=0.4 a.u. for both regions. Electron population is allowed to enter and leave both the inner and outer regions, and repartitioning of the population between the two regions occurs every laser cycle.
We focus our attention here on one feature consistently seen in these simulations. In Fig. 1 we show a movie made of snapshots of the electron energy distribution at times when the vector potential is zero. As the pulse progresses in time the dominant feature that emerges is a grid-like pattern of maxima spaced by one photon of energy. Superimposed is another pattern of diagonal maxima also spaced by one photon of energy. It is the above-threshold ionization (ATI) signal of our model atom. This grid-like pattern has already been reported  for this atom for as many as 39-photon double ionization. It has also been seen recently in numerical simulations of helium without the one-dimensional approximation in the extreme ultraviolet region of one-photon double ionization . There, the grid pattern was explained as sequential ATI originating from the ground state of the helium ion. It arises from a sequential process in which the two electrons ionize independently of each other. The diagonal maxima (appearing as circular arcs in momentum space in ) were explained as non-sequential ionization directly from the helium atom in which energy from an integer number of photons is shared between the two electrons. We will focus here solely on ATI arising from the sequential process by choosing one particular laser frequency and showing how closely the energy spectrum of the ejected electrons mirrors theoretical calculations of ATI.
After running a simulation, we have a wave function distributed in position space ψ(x 1, x 2), or conversely through the use of a Fourier transform (F),distributed in momentum space ψ(p 1, p 2)=F[ψ(x 1, x 2)]. To evaluate the atom’s ATI structure we want the energy distribution and this requires some attention before proceeding further. When transforming from momentum to energy space, we normally lose directional information since a given energy can correspond to either a left- or right-moving electron. We want to keep this information and will include it as the sign of our energy term.
Doing this allows us to distinguish spectra from electrons that were initially emitted preferentially in one direction. This transformation from momentum to energy also causes our grid points to become non-uniformly spaced. To obtain the correct probability distribution, the transformed wave function will need to be properly normalized.
Combining our definition for the electron energy as a function of its momentum and the energy normalization, the properly normalized energy space wave function has the following form:
Ideally, ψ(Em ,En ) would not contain contributions from any bound states. We do not have all of the atom’s high-lying bound-state eigenvectors, and so we cannot project out their contributions when evaluating the energy spectrum. These bound states, as well as singly ionized states, give the possibility of some distortion of ATI peaks very near the zero of energy. An alternative method to deal with this would be to apply a masking to the wave function to approximate fully abound and singly ionized population. Such a procedure has been used to determine ionization yield  as well as investigating ATI . For the frequency used here and in the intensity range shown, this masking finds the double ionization yield is generally an order of magnitude smaller than the single ionization yield with the discrepancy shrinking as plateau length increases. For the six cycle pulse, data illustrating the intensity dependence of single and double ionization yield has previously been published .
In addition, and with the exception of Fig. 1, the plots shown here are of the energy distribution collapsed onto one particle. The system is symmetric with respect to particle interchange, so which is integrated over is not significant.
Collapsing our results allows us to generate two-dimensional plots which are simple to interpret. Finally, as we can see from Fig. 1, spectra emitted in the positive and negative direction will yield essentially the same result. To further focus on the feature of interest, we will plot only the right-moving collapsed wave function with the understanding that plotting both left- and right-moving results give similar patterns.
We begin analyzing the ATI spectrum by first observing the time evolution of a ten cycle laser pulse with a two-cycle plateau and two-cycle ramp on and off. The illustrations shown in Fig. 2 all use a laser with a peak intensity of 5.5×1014W/cm2 and frequency ω=0.1837 a.u., which corresponds to zero-field five-photon single ionization and 13-photon double ionization. The figures all correspond to times when the vector potential of the incident field is zero. Once the laser pulse has spent a cycle at its maximum intensity, a series of ATI peaks are visible. Further time evolution sharpens these peaks. This point is illustrated further in Fig. 3 where we show energy spectra at the end of a laser pulses which are up to ten cycles long. All figures have the same turn-on and turn-off time of two cycles, meaning pulse duration is expanded solely by increasing the time spent at maximum intensity which is fixed at 5.5×1014W/cm2. Plateau length ranges from one to six cycles. It is the time evolution of the energy spectrum of the longest pulse which was shown in the movie in Fig. 1.
To reinforce our interpretation of these results, and show that bound states do not significantly distort the ATI structure for even the lowest energy peaks, we have examined how closely the spectra’s peak-shifting behavior mimics what we would expect from ATI peaks. To do this, we examine the position of the ATI peaks for a range of laser intensities. It is easy to follow the intensity development of the ATI peaks and confirm that they shift in intensity as we would expect. In Fig. 4 we compare peak positions predicted by a Stark shifted ground state and pondermotively shifted continuum states () with peak positions generated by numerical simulations. Looking carefully at the position of the highest energy peaks in Fig. 4, we see that as we increase laser intensity, the movement downward in energy is not entirely smooth, but appears to occur in “steps”. This is the result of a combination of the size of the computational box size and a grid spacing in energy space which becomes coarser at higher energies.
Finally, we show in Fig. 5 energy spectra for a sequence of intensities at the end of laser pulses with differing intensities in order to visualize this movement of ATI peaks with increasing intensities. The pulses used are eight cycles long with a four cycle plateau in contrast to the six cycle pulses with two cycle plateaus used in Fig. 4. The longer pulses produce peaks which are sharper and visually easier to spot. Both Figs. 4 and 5 show the effect of pondermotive threshold shifts.
In summary, we have shown the presence of local maxima in the probability distribution of a two-electron wave function exposed to a short intense laser pulse. These maxima are spaced by one photon of energy and behave as ATI peaks complete with peak shifting and channel closings. We acknowledge receipt, as this note was being prepared, of ATI spectra calculated for helium by Parker and collaborators . Those results were obtained using a three-dimensional treatment of the problem in the extreme ultraviolet region of one-photon single and double ionization. The investigation of a different system in this paper makes a direct comparison impossible. However, the similarity of features in substantially different initial systems as well as laser parameters hints at the wide range in which two-electron ATI should be visible.
The author would like to acknowledge J. H. Eberly for both substantive discussions of these results as well as a critical reading of the manuscript prior to submission. Funding for this research comes from the National Science Foundation through grants PHY-9415583 and PHY-9722079.
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