^{1,a)}and Anthony F. Starace

^{1}

### Abstract

We present an efficient and accurate grid method for solving the time-dependent Schrödinger equation for an atomic system interacting with an intense laser pulse. Instead of the usual finite difference (FD) method, the radial coordinate is discretized using the discrete variable representation (DVR) constructed from Coulomb wave functions. For an accurate description of the ionization dynamics of atomic systems, the Coulomb wave function discrete variable representation (CWDVR) method needs three to ten times fewer grid points than the FD method. The resultant grid points of the CWDVR are distributed unevenly so that one has a finer grid near the origin and a coarser one at larger distances. The other important advantage of the CWDVR method is that it treats the Coulomb singularity accurately and gives a good representation of continuum wave functions. The time propagation of the wave function is implemented using the well-known Arnoldi method. As examples, the present method is applied to multiphoton ionization of both the H atom and the ion in intense laser fields. The short-time excitation and ionization dynamics of H by an abruptly introduced static electric field is also investigated. For a wide range of field parameters, ionization rates calculated using the present method are in excellent agreement with those from other accurate theoretical calculations.

One of the authors (L.-Y.P.) gratefully acknowledges helpful discussions with D. A. Telnov and S. I. Chu concerning the choice of gauge for multiphoton detachment of and with R. B. Sidje concerning use of the package “EXPOKIT.” This work was supported in part by the U.S. Department of Energy, Office of Science, Division of Chemical Sciences, Geosciences, and Biosciences, under Grant No. DE-FG03-96ER14646.

I. INTRODUCTION

II. COULOMB WAVE FUNCTION DVR

A. DVR using orthogonal polynomials

B. DVR constructed from Coulomb wave functions

III. ONE-ELECTRON ATOMIC SYSTEM IN INTENSE LASER FIELDS

A. Discretization of the spatial coordinates

1. Expansion of the angular part

2. Discretization of the radial coordinate

3. Distribution of the CWDVR grid points

B. Wave function propagation in time

IV. RESULTS AND DISCUSSIONS

A. Choice of gauge

B. Calculation of physical observables

C. Convergence of the CWDVR method

D. Multiphoton ionization of H by intense laser pulses

E. Ionization of H by static electric fields

F. Multiphoton detachment rates for by a strong laser pulse

V. CONCLUSIONS

### Key Topics

- Wave functions
- 41.0
- Ionization
- 25.0
- Multiphoton ionization
- 18.0
- Ground states
- 13.0
- Static electric fields
- 12.0

## Figures

(Color online) Convergence of the CWDVR method for different and values. The natural logarithm of the population as a function of time (in units of the laser period ) is shown for a laser wavelength of and a peak laser intensity of . (a) Results for different CWDVR grids (with and as indicated) are compared to the result of the FD method for time in the range . (b) A magnified version of (a) for time in the range . Note that in both panels the three curves representing the FD result and the two CWDVR results are nearly indistinguishable on the scale of the figure and appear as the lowest curve in each panel.

(Color online) Convergence of the CWDVR method for different and values. The natural logarithm of the population as a function of time (in units of the laser period ) is shown for a laser wavelength of and a peak laser intensity of . (a) Results for different CWDVR grids (with and as indicated) are compared to the result of the FD method for time in the range . (b) A magnified version of (a) for time in the range . Note that in both panels the three curves representing the FD result and the two CWDVR results are nearly indistinguishable on the scale of the figure and appear as the lowest curve in each panel.

The natural logarithm of the decay of the ground state and of populations within different spheres of radius for ionization of atomic H by a laser having frequency , peak intensity , a ramp up, and a flat top . remaining in the ground state, remaining within a sphere of radius , and remaining within the entire grid,

The natural logarithm of the decay of the ground state and of populations within different spheres of radius for ionization of atomic H by a laser having frequency , peak intensity , a ramp up, and a flat top . remaining in the ground state, remaining within a sphere of radius , and remaining within the entire grid,

(Color online) Depletion of the ground state of H by a static electric field having field strength of The natural logarithm of the ground state probability is shown as a function of the field duration. Results calculated by different CWDVR grids for different values ( in each case) are compared against the result calculated by the FD method with Note that the result is indistinguishable from the FD result on the scale of this figure.

(Color online) Depletion of the ground state of H by a static electric field having field strength of The natural logarithm of the ground state probability is shown as a function of the field duration. Results calculated by different CWDVR grids for different values ( in each case) are compared against the result calculated by the FD method with Note that the result is indistinguishable from the FD result on the scale of this figure.

H ground state survival probability, , as a function of time in a static electric field. The field strength is taken to be (a) 0.005, (b) 0.04, (c) 0.06, and (d) These results are in perfect agreement with those by Durand and Paidarova (Ref. 59) and by Scrinzi (Ref. 57) (neither of which are shown here because they are indistinguishable from ours on the scale of the figures).

H ground state survival probability, , as a function of time in a static electric field. The field strength is taken to be (a) 0.005, (b) 0.04, (c) 0.06, and (d) These results are in perfect agreement with those by Durand and Paidarova (Ref. 59) and by Scrinzi (Ref. 57) (neither of which are shown here because they are indistinguishable from ours on the scale of the figures).

## Tables

Comparison of CWDVR grid point distributions for different values of and . Note that equals the difference of the two grid points closest to .

Comparison of CWDVR grid point distributions for different values of and . Note that equals the difference of the two grid points closest to .

Ionization rate for ionization of H by a linearly polarized laser of intensity and frequency . The present results are compared with the results of Chu and Cooper, (Ref. 54), Pont *et al.* (Ref. 55), and Kulander (Ref. 10). Intensities and ionization rates are presented in the form .

Ionization rate for ionization of H by a linearly polarized laser of intensity and frequency . The present results are compared with the results of Chu and Cooper, (Ref. 54), Pont *et al.* (Ref. 55), and Kulander (Ref. 10). Intensities and ionization rates are presented in the form .

Multiphoton ionization rates for H for four different laser electric field strengths, , and seven photon energies . Present results are compared with those of Chu and Cooper (Ref. 54) (who used a nonperturbative non-Hermitian Floquet method).

Multiphoton ionization rates for H for four different laser electric field strengths, , and seven photon energies . Present results are compared with those of Chu and Cooper (Ref. 54) (who used a nonperturbative non-Hermitian Floquet method).

Ionization rate (in a.u.) for ionization of the ground state of H by a static electric field of strength . Results are compared with those of Scrinzi (Ref. 57), Peng *et al.* (Ref. 13), and Bauer and Mulser (Ref. 61).

Ionization rate (in a.u.) for ionization of the ground state of H by a static electric field of strength . Results are compared with those of Scrinzi (Ref. 57), Peng *et al.* (Ref. 13), and Bauer and Mulser (Ref. 61).

Multiphoton detachment rates for for laser wavelengths , 1640, and and 11 intensities (ranging from to ). The present results using the CWDVR method are compared with results of Haritos *et al.* (Ref. 64) and of Telnov and Chu. (Refs. 65 and 66). The detachment rates are given in the form of .

Multiphoton detachment rates for for laser wavelengths , 1640, and and 11 intensities (ranging from to ). The present results using the CWDVR method are compared with results of Haritos *et al.* (Ref. 64) and of Telnov and Chu. (Refs. 65 and 66). The detachment rates are given in the form of .

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