## Abstract

We discuss the application of quantitatively accurate computational methods to the study of laser-driven two-electron atoms in short intense laser pulses. The fundamental importance of such calculations to the subject area is emphasized. Calculations of single- and double-electron ionization rates at 390 nm are presented.

© Optical Society of America

At sufficiently high laser intensities (*I*>>10^{12} W cm^{-2}) perturbation theory and most other simplified treatments of the Schrödinger equation become useless in the theoretical description of laser-driven helium. At these intensities, even a task as seemingly straightforward as the calculation of accurate ionization rates has been until very recently an unsolved problem. In the past few years, new theoretical methods have been introduced that have proven capable of yielding reliable and quantitatively accurate solutions to the intense field helium Schrödinger equation. The first of these methods, and the subject of this letter, is a time-dependent numerical integration (TDNI) of the full-dimensional Schrödinger equation [1]. In this approach, the problem is treated in its full generality, two-electron dynamics are faithfully modelled without significant approximation. This method is particularly advantageous in the limit of very intense fields, where few or no approximations are likely to be justified. The second method is the *R*-Matrix Floquet (RMF) method [2] [3]. This is a time-independent treatment of the atom-laser interaction capable of generating single-electron ionization rates and Stark shifts. It has The advantage of being easily applied to molecules and many-electron atoms and can be generalized to two-electron excitations. The TDNI and RMF methods have demonstrated agreement to within a few percent in calculations of single-electron ionization rates [4]. More recently, several new full-dimensional treatments of the problem have been introduced [5] [6] based on basis set decompositions of the helium Schrödinger equation.

In this letter we present calculations of single- and double-electron ionization rates in helium at high laser intensities (*I*>10^{14} W cm^{-2}). The results were obtained from a time-dependent numerical integration (TDNI) of the full-dimensional helium Schrödinger equation.

The primary constraint in the design of the TDNI is the requirement that it generate the accurate solution of the full-dimensional helium Schrödinger equation, along with reliable estimates of the absolute error in the solutions. In principle, with sufficiently large expenditure of computer time, the TDNI will generate solutions arbitrarily close to the exact solution of the Schrödinger equation. In practice, with moderate allocations of computer time, and with the program in its standard configuration, single-ionization rates are calculated with errors typically in the 1% to 3% range, and double-ionization rates with errors typically in the 5% to 15% range. In specific cases there is sometimes a requirement for greater accuracy, which is easily obtained by running the program with more stringent parameter settings.

Accurate and reliable solutions of the helium Schrödinger equation have a wealth of important applications in the study of intense field atom-laser interactions. They are a necessary first step in making a quantitative discipline out of the study of atom-laser interactions in the intense field limit. Reasons for the lack of progress toward the goal lie in the experimental, as well as the theoretical, domain. The difficulty experimentalists encounter in measuring laser intensities in the intense field short pulse limit is perhaps not widely appreciated. Over the last decade, typical error estimates in experimental determination of intensities in this limit have been in the ±25% to ±50% range. Errors of this size translate to orders of magnitude errors in absolute determinations of, for example, multiphoton ionization rates. In effect, experimentally ab-initio measurements of intense field ionization rates are not yet possible. On the other hand, if accurate theoretical ionization rates are known, then laser intensities can be much more accurately determined by measuring the ionization yields they produce. Accurate ionization rates have long been known for hydrogen, but (atomic) hydrogen is an experimentally unfriendly gas. The heavier noble gases are experimentally friendly but theoretically unfriendly. The provision of accurate theoretical helium ionization rates for the benefit of experimentalists is therefore one of the most important applications of the TDNI.

Theoretical study of intense field interactions also benefits from the availability of accurate numerical solutions of the Schrödinger equation. The TDNI can play a vital role in the design, test and calibration of simplified models of atom-laser interactions. Reduction of a complex physical system to the simplest possible model is a highly desirable goal, but simplified models are of little value either as a means of calculating physical quantities, or as an aid to understanding, if the model fails to agree with the Schrödinger equation it was designed to approximate. The addition of each new approximation necessarily introduces additional limits in which the simplified model must fail. The TDNI can be used to detect these limits and evaluate the quality of other approximations elsewhere. The TDNI can also be used to calibrate simplified models to improve the degree to which they approximate the Schrödinger equation. We have recently introduced a single active electron (SAE) model of single-electron ionization in helium [7] [8]. The model approximates the Coulomb screening of the inner electron with a pseudo-potential similar to the time-dependent Hartree-Fock pseudo-potential by introducing two parameters based on physical considerations (a Stark shift correction and a dipole moment correction). The model can be calibrated to give near quantitative agreement with the full-dimensional TDNI over a wide range of laser intensities and frequencies. This calibrated model has proved extremely useful in a number of applications. The SAE model is a one-electron model of helium and hence is highly computationally efficient in comparison to the full-dimensional TDNI. Once it is calibrated at a range of intensities it can be used to efficiently generate reliable solutions at nearby intensities. It can for example be used to integrate extremely long laser pulses that are used experimentally. By using extremely large integration volumes and fine finite-difference grids the SAE model can be used to extrapolate TDNI results obtained with coarse grids and smaller integration volumes. The calibrated SAE model can also be used as a more reliable first step in the construction of more sophisticated models of helium-laser interactions.

In the intense field limit where the two electrons are violently driven by the field in the helium Coulomb potential there is no assurance that any simplified assumptions to the full-dimensional Schrödinger equation are justified. In this limit the TDNI will likely remain the only reliable means of exploring the full dynamics of intense field atom-laser interactions. We recently presented the first theoretical observation and description of the process double-electron above threshold ionization (DATI) [9]. DATI is a particularly striking example of novel physics in the intense-field limit. Non-sequential DATI is a resonant process in which the two electrons absorb energy from the field, simultaneously ionize, and along the way interact with each other and share energy in such a way that the sum of their (asymptotic) energies is an integer multiple of *ħω* above the initial state energy. A correct theoretical description of non-sequential DATI will simultaneously exercise the full machinery of the 6-dimensional helium Schrödinger equation, which includes an atomic Hamiltonian, a two-electron interaction Hamiltonian, and the dielectronic electron-electron repulsion term through which the two electrons resonantly absorb energy.

We turn now to a discussion of helium ionization rate calculations at 390 nm. The wavelength 390 nm is typical of frequency-doubled titanium-sapphire lasers and experimental data for helium now exist at this wavelength.

The calculations were performed using the TDNI applied to the full-dimensional Schrödinger equation. In the TDNI method, the wavefunction is represented using a basis set of partial waves and finite-difference grids. The wavefunction is propagated in time by exponentiating the Hamiltonian in the Krylov subspace [1]. A good concise review of the applications of this method to rate calculations is given in [10].

In Figure 1 we present the results of rate calculations for single- and double-electron ionization in helium over the intensity range 3.5×10^{14} to 2.2×10^{15} W cm^{-2}. The single-ionization rates are also presented in tabular form (Table 1) since they are believed to be correct to within 1% to 3%. The estimated error for double-ionization is in the 5% to 15% range.

Ionization rates are calculated by observing the rate at which population crosses boundaries as ionizing wavepackets depart the atomic core. The boundaries are spherical shells enclosing the atomic core at radii of 8, 11, 14 and 17 atomic units. The observed ionization rates are independent of boundary positions, which is not surprising given that ionizing wavepackets cross the boundaries wherever they are placed. In the calculations, pulses are ramped on over 4 field periods and then held at constant intensity. The envelope of the ramp-on is sinusoidal in shape. Additional calculations are performed to verify that results are independent of the length and shape of the ramp-on [4]. Error estimates are made by observing the convergence of the results as parameters in the code are varied. Four parameters in the code govern the degree to which the finite-difference model accurately represents the true Schrödinger equation: the grid point

spacing δr, the maximum angular momentum *L*
_{max} in the basis set, the radius *R* of the integration volume, and the number of terms *N* in the Legendre polynomial expansion of the dielectronic Coulomb term. By making *δr* small and making the other parameters large, solutions of the finite-difference model become arbitrarily close to those of the true Schrödinger equation. In the calculation of double-ionization, if *N* is changed from 3 to 4, then results change by 16% typically. When *N* is changed from 4 to 5, then results change by 2%. We therefore truncate *N* at 5 and accept 2% as a likely estimate of error induced by this truncation. Adding together all such error gives an error estimate of 5% to 10%. At intensities at which resonances are observed (8.0×10^{14} W cm^{-2}) uncertainties can be higher. It would be straightforward to reduce these errors to something like those of the single-ionization rate calculations (1% to 3%). A factor of two or three increase in computer time would be sufficient. If a requirement for double-ionization rates at this level of accuracy were identified, then the expenditure of this amount of CPU time would be routine.

In the case of single-ionization, convergence is faster in all four parameters. For example, full convergence in *N* is observed at *N*=3, rather than *N*=5 as in the double-ionization case. After the full-dimensional Schrödinger equation has been integrated, the results are further improved through extrapolation using the calibrated SAE model described above. The SAE model shares with the full-dimensional integration three of the four parameters: *δr*, *R* and *L*
_{max}. For a particular setting of *δr*, *R* and *L*
_{max} the SAE model is calibrated to give exact agreement with the full-dimensional results. In the SAE model, *R* and *L*
_{max} are subsequently increased by factors of 5 and *δr* is decreased similarly. The difference between the two-electron full-dimensional model results and the extrapolated values varies from 1% to 3%, which we take as an estimate of the error in the calculation. The extrapolation process may be regarded as nothing more than a means of estimating error in the calculation, but there are good reasons to believe that the extrapolated values are the better values. Both the SAE model and the two-electron full-dimensional model appear to converge at the same rate and monotonically in the same direction as *δr*, *R* and *L*
_{max} are varied [8]. Therefore the extrapolated results are given in Table 1. This technique has the added advantage that it allows us to explore potential convergence problems and numerical integration problems with much greater rigour.

In this letter we have reviewed recent progress we have made towards the goal of bringing quantitative discipline to the study of intense field atom-laser interactions. To illustrate our computational methods, we have discussed calculations of single- and double-electron ionization rates in helium at 390 nm. At present, similar calculations are being performed at 780 nm, the most commonly used titanium-sapphire laser wavelength.

## Acknowledgements

This research was supported by grants from the UK Engineering and Physical Sciences Research Council providing support for JSP together with high-performance computer resources at the Computing Service for Academic Research, University of Manchester. LRM acknowledges receipt of a research studentship provided by the Department of Higher and Further Education, Training and Employment.

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