## Non-sequential double ionization of helium and related wave-function dynamics obtained from a five-dimensional grid calculation.

Optics Express, Vol. 8, Issue 7, pp. 417-424 (2001)

http://dx.doi.org/10.1364/OE.8.000417

Acrobat PDF (332 KB)

### Abstract

Numerical integration of the time-dependent Schrödinger equation for two three-dimensional electrons reveals the behavior of helium in the presence of strong 390 nm and 800 nm light. Non-sequential double ionization is seen to take place predominantly at times when the electric-field component of the light reaches its peak value. Double ionization starts only in the second cycle of a flat-top pulse, and reaches a stable value only after many cycles, showing that recollision, sometimes through very long trajectories, must be involved.

© Optical Society of America

*IP*increases even for ‘equivalent’ electrons roughly proportional to final-state ionic charge

*Z*, successive electrons are more tightly bound, and therefore difficult to remove. Optical frequencies are already quite small compared to the ionization potential of neutral noble-gas atoms, and this is even more true for the higher charge states. As a consequence, the instantaneous ionization rates do not differ too much from those for ionization by a DC field[1

1. N.B. Delone and V.P. Krainov, “Tunneling and barrier suppression ionization of atoms and ions in a laser radiation field,” Phys.-Uspekhi **41**, 469 (1998). [CrossRef]

^{1}. This makes the intensities required to drive successive ionization steps so different, that the earlier step proceeds to completion before the next one becomes measurable[3

3. P. Lambropoulos, “Mechanisms for multiple ionization of atoms by strong laser pulses,” Phys. Rev. Lett. **55**, 2141 (1985). [CrossRef] [PubMed]

6. P. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. **71**, 1994 (1993). [CrossRef] [PubMed]

*e*, 2

*e*) event. Alternatively, the ‘shake-off’ mechanism[7

7. D.N. Fittinghoff, P.R. Bolton, B. Chang, and K.C. Kulander, “Observation of nonsequential double ionization of helium with optical tunneling,” Phys. Rev. Lett. **69**, 2642 (1992). [CrossRef] [PubMed]

*e*, 2

*e*) process. Classical trajectory calculations show that this return energy is limited[6

6. P. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. **71**, 1994 (1993). [CrossRef] [PubMed]

*U*

_{P}

^{2}. For instance, in the case of 390-nm light at an intensity 6.5·10

^{14}W/cm

^{2}, the ponderomotive energy

*U*

_{P}is only 0.36 Hartree, and the maximum return energy is barely half of what is needed to dislodge the remaining 1s electron of He

^{+}(which is bound by 2 Hartree).

^{-4}of that of the single ionization, or 30% of the value the double/single ratio has at saturation of the single ionization[8

8. J.S. Parker, L.R. Moore, D. Dundas, and K.T. Taylor, “Double ionization of helium at 390 nm,” J. Phys. B **33**, L691 (2000). [CrossRef]

9. E.S. Smyth, J.S. Parker, and K.T. Taylor, “Numerical integration of the time-dependent Schrödinger equation for laser-driven helium,” Computer Phys. Comm. **114**, 1 (1998). [CrossRef]

10. K.J. Schafer and K.C. Kulander, “Energy analysis of time-dependent wave functions: Application to above-threshold ionization,” Phys. Rev. A **42**, 5794 (1992). [CrossRef]

*r*

_{1}<

*R*

_{out}; 0<

*r*

_{2}<

*R*

_{in}, where in

*r*

_{1}=0 and

*r*

_{2}=0 the boundary condition prescribed by the Coulomb singularity in

*V*is enforced. At the other boundaries (nearly) reflectionless absorption of the wave function takes place by specially tailored boundary conditions[14], that keep track of the amount of probability they absorb. In the present calculation,

*R*

_{in}is chosen very small (8 Bohr), and current absorbed at the

*r*

_{2}=

*R*

_{in}boundary for

*r*

_{1}>

*R*

_{in}is counted as double ionization. The strip is much larger in the

*r*

_{1}dimension (

*R*

_{out}≈36 Bohr, plus an absorbing region of 11 Bohr), allowing electron 1 to perform its full dynamics of laser acceleration and recollision. Only when

*r*

_{1}gets so large that there is no hope of future collisions it meets the absorber. Current leaving the strip at

*r*

_{1}=

*R*

_{out}is counted as single ionization without examining its dependence on

*r*

_{2}, since any population that did not have electron 2 in the ground state would be pulled by the laser over the

*r*

_{2}=

*R*

_{in}boundary long before

*r*

_{1}could reach

*R*

_{out}. Exchange symmetry (corrected for the gauge difference between the electrons) can be enforced on the boundary

*r*

_{1}=

*r*

_{2}[11].

*r*

_{2}direction, together with the use of the length gauge in this dimension, only few angular momenta are required to represent electron 2. In the current calculations,

*l*

_{2}running upto 4 was enough to converge all quantities except angular distributions to better than a percent. (The latter require 2 more

*l*

_{2}.) This also limits the theoretically required number of

*m*to 4, but in practice including

*m*≥2 did not affect any of the presented results visibly where tried

^{3}. Due to the high spatial order of the finite-differences employed, good convergence is already obtained for

*δr*=0.25 Bohr, provided that the ionization potentials are tuned to their limit values by tweeking the boundary condition[15

15. J.L. Krause, K.J. Schafer, and K.C. Kulander, “Calculation of photoemission from atoms subject to intense laser fields,” Phys. Rev. A **45**, 4998 (1992). [CrossRef] [PubMed]

9. E.S. Smyth, J.S. Parker, and K.T. Taylor, “Numerical integration of the time-dependent Schrödinger equation for laser-driven helium,” Computer Phys. Comm. **114**, 1 (1998). [CrossRef]

*r*

_{i}=0 and at

*r*

_{1}=

*r*

_{2}. The number of

*l*

_{1}required varies strongly with laser parameters, and a typical value is 20. This brings the grid requirements to 188×32×20×5×2, or about 1.2 million points.

*iHt*)/(1+

*iHτ*). Thus all partial propagators are exactly unitary, and their (local) discretization error is

*O*(

*τ*

^{3}). Just like in the predecessor single electron code on which this work builds[12, 13

13. H.G. Muller, “Numerical solution of high-order ATI enhancement in argon,” Phys. Rev. A **60**, 1341 (1999). [CrossRef]

*τ*

^{2}convergence is maintained despite the splitting by first applying them in a certain order for half the step size, and then in the reverse order to complete a full step. The total hamiltonian is split into atomic contributions for each of the electrons (employing potentials -2/

*r*

_{2}and -(1+(1-

*r*

_{1}/2)

^{4}

_{[r1<2]})/

*r*

_{1}for the inner and outer electron, respectively), the laser-interactions

**E**(

*t*)·

**r**and

**A**(

*t*)·

**p**, and the first

*N*components from the multipole expansion of the electron repulsion 1/

*r*

_{12}. For efficiency the latter is usually only included up to the dipole moment (

*N*=2), although convergence tests have shown that results sometimes change up to 10% after inclusion of the higher multipoles. As a compromise, sometimes the higher multipoles are included only in the

*m*=0 subspace (which carries almost all population). The atomic propagators naturally factorize into tri-diagonal operators, and the other operators are split further until they do[12]. This makes propagation quite efficient, around 3.6 seconds per time step (1 optical cycle per hour) for the mentioned grid size on a 333 MHz PC.

**E**(

*t*)=

*E*

_{0}

*z*ẑ sin

*ωt*, (

*t*>0), immediately preceeded by a half-cycle sine-square turn-on

^{4}. The frequency

*ω*was chosen as 0.11683, corresponding to a laser wavelength of 390 nm. This wavelength is interesting because it results in a fair amount of double ionization while the maximum return energy is still below the impact energy required for an (

*e*, 2

*e*) process.

*R*

_{in}boundary as a function of time, for various intensities around 800 TW/cm

^{2}. This current peaks quite strongly slightly after the time where the

*E*(

*t*) reaches its maximum value. This is in agreement with earlier calculations for this process[8

8. J.S. Parker, L.R. Moore, D. Dundas, and K.T. Taylor, “Double ionization of helium at 390 nm,” J. Phys. B **33**, L691 (2000). [CrossRef]

*R*

_{in}this suggest the corresponding electrons leave the atom at these field maxima. An interesting observation, made possible by the very steep turn-on of the pulse, is that the first burst of double ionization does appear only one full cycle after the first time a field maximum was reached. The absence of double ionization during the first cycle shows that shake-off only plays a minor role under these conditions, and points to a recollision mechanism.

6. P. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. **71**, 1994 (1993). [CrossRef] [PubMed]

^{5}. This inner electron leaves the atom mainly in the direction where the laser field pushes it at the time. The corresponding charge distribution of the outer electron is shown in Fig. 3b. For this calculation exchange symmetry was not enforced (for Fig. 1 it was). This treats the electrons on unequal footing; electron 1 is allowed to venture far from the atom to return later, while electron 2 is immediately and irrevokably absorbed when it leaves the close vicinity of the atom. Processes triggered only by the return of an electron will thus be underestimated (by a factor two).

19. J.B. Watson, A. Sanpera, K. Burnett, D.G. Lappas, and P.L. Knight “Double ionization of helium beyond the single electron active electron approximation,” in P. Lambropoulos and H. Walther, (eds.) “*Multiphoton Processes: ICOMP VII, 7th International Conference*,” CS154, p. 132 IOP, Bristol, UK (1997).

^{2}) this return energy is enough to kick out the inner electron if the first electron gives up more energy than it has, to end up in a He

^{+}bound excited state (from which it is ionized immediately afterwards). Fig. 5 plots the charge-distribution of the non-driven electron when it is the inner one. This shows that this electron still prefers to leave in the direction of the field it can not feel.

*e*, 2

*e*) process. The calculation is also much more demanding: because of the larger quiver amplitude

*R*

_{out}has to be chosen much larger (116 Bohr), and 30 to 50

*l*

_{1}-components are required. Fig. 6 shows the double ionization at 505 TW/cm

^{2}, where

*U*

_{P}=1.1 Hartree (which makes maximum return energy 3.5 Hartree). The double ionization is now much less localized to the E-field maxima, and switches on at the zero crossing 0.75 cycle after the first maximum. Nevertheless, there is still a distinct peaka t the E-field maxima, that starts to dominate in later cycles. Apparently very long trajectories also play an important role here, but due to the expensive nature of the calculation no systematic intensity scan for locating resonances could yet be undertaken (a 5-cycle run now takes over 2 days).

^{2}. At this intensity

*l*

_{1}has to run upto 50 and 4000 steps per optical cycle are required to obtain convergence. The double-ionization current also shows a strongly directional component at this intensity, that peaks around the field maxima. Unlike in the other movies shown, this contribution originates mainly from situations where the outer electron is very far away. It seems quite independent of where the outer electron exactly is: from

*r*

_{1}=50 to

*r*

_{1}=

*R*

_{out}=127 Bohr the current of the second electron leaving the ion at the time of an E-field maximum is practically constant. In other words, the contribution near these field maxima seems to correspond to sequential double ionization.

*r*

_{1}<50 Bohr). In Fig. 7 this current can be seen to be much more isotropic, leaving in all directions at the same time. (Note that this does not reveal anything about the final momentum of the electrons, which is mainly determined by the phase of the laser at the time of emission.)

## Acknowledgement -

## Footnotes

1 | ^{1}Ignoring level shifts this happens at I=Z
^{2}, where Z is the final ion charge; [22. S. Augst, D. Strikland, D. Meyerhofer, S.L. Chin, and J. Eberly, “Tunneling ionization of noble gases in a high-intensity laser field,” Phys. Rev. Lett. m
_{e}=ħ=e=1 unless stated differently.) |

2 | ^{2}Here U_{P}
=ω
^{2} denotes the cycle-averaged kinetic energy a free electron would get due to the quiver motion forced upon it by the laser. |

3 | ^{3}Note that a grid with only two m values is topologically equivalent to that needed to describe an atom of two 2-dimensional electrons (m taking the role of parity with respect to the polarization axis). |

4 | ^{4}Even half an optical cycle is slow for the atomic ground state, so that it adapts adiabatically if the switching is smooth. |

5 | ^{5}For best representation of both on-axis and off-axis probability, this and later movies plot 18. H.G. Muller, “Identification of states responsible for ATI enhancement in argon by their calculated wave functions,” Opt. Express |

## References and links

1. | N.B. Delone and V.P. Krainov, “Tunneling and barrier suppression ionization of atoms and ions in a laser radiation field,” Phys.-Uspekhi |

2. | S. Augst, D. Strikland, D. Meyerhofer, S.L. Chin, and J. Eberly, “Tunneling ionization of noble gases in a high-intensity laser field,” Phys. Rev. Lett. |

3. | P. Lambropoulos, “Mechanisms for multiple ionization of atoms by strong laser pulses,” Phys. Rev. Lett. |

4. | B. Walker, B. Sheehy, L.F. DiMauro, P. Agostini, K.J. Schafer, and K.C. Kulander, “Precision measurement of strong field double ionization,” Phys. Rev. Lett. |

5. | A. l’Huillier, A. Lompre, G. Mainfray, and C. Manus, “Multiply charged ions induced by multi-photon absorption in rare gases at 0.53 |

6. | P. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. |

7. | D.N. Fittinghoff, P.R. Bolton, B. Chang, and K.C. Kulander, “Observation of nonsequential double ionization of helium with optical tunneling,” Phys. Rev. Lett. |

8. | J.S. Parker, L.R. Moore, D. Dundas, and K.T. Taylor, “Double ionization of helium at 390 nm,” J. Phys. B |

9. | E.S. Smyth, J.S. Parker, and K.T. Taylor, “Numerical integration of the time-dependent Schrödinger equation for laser-driven helium,” Computer Phys. Comm. |

10. | K.J. Schafer and K.C. Kulander, “Energy analysis of time-dependent wave functions: Application to above-threshold ionization,” Phys. Rev. A |

11. | H.G. Muller, “Solving the time-dependent Schrödinger equation in five dimensions,’ in L.F. Di-Mauro, R.R. Freeman, and K.C. Kulander (eds.)” |

12. | H.G. Muller, “An efficient propagation scheme for the time-dependent Schrödinger equation in the velocity gauge,” Laser Phys. |

13. | H.G. Muller, “Numerical solution of high-order ATI enhancement in argon,” Phys. Rev. A |

14. | H.G. Muller, “Calculation of double ionization of helium,” in B. Piraux et al. (eds.)“Super-Intense Laser-Atom Physics VI proceedings,” NATO series B (2001) in print. |

15. | J.L. Krause, K.J. Schafer, and K.C. Kulander, “Calculation of photoemission from atoms subject to intense laser fields,” Phys. Rev. A |

16. | R. Kopold, W. Becker, and D.B. Milosevic, “Quantum orbits: a space-time picture of intense-laser-induced processes in atoms,” Comments At. Mol. Phys. (2000) to be published. |

17. | H.G. Muller and F.C. Kooiman, “Bunching and Focusing of Tunneling Wave Packets in Enhancement of High-Order ATI,” Phys. Rev. Lett. |

18. | H.G. Muller, “Identification of states responsible for ATI enhancement in argon by their calculated wave functions,” Opt. Express |

19. | J.B. Watson, A. Sanpera, K. Burnett, D.G. Lappas, and P.L. Knight “Double ionization of helium beyond the single electron active electron approximation,” in P. Lambropoulos and H. Walther, (eds.) “ |

**OCIS Codes**

(020.0020) Atomic and molecular physics : Atomic and molecular physics

(270.6620) Quantum optics : Strong-field processes

**ToC Category:**

Focus Issue: Laser-induced multiple ionization

**History**

Original Manuscript: February 5, 2001

Published: March 26, 2001

**Citation**

Harm Muller, "Non-sequential double ionization of helium and related wave-function dynamics obtained from a five-dimensional grid calculation.," Opt. Express **8**, 417-424 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-7-417

Sort: Journal | Reset

### References

- N.B. Delone and V.P. Krainov, "Tunneling and barrier suppression ionization of atoms and ions in a laser radiation field," Phys.-Uspekhi 41, 469 (1998). [CrossRef]
- S. Augst, D. Strikland, D. Meyerhofer, S.L. Chin, J. Eberly, "Tunneling ionization of noble gases in a high-intensity laser field," Phys. Rev. Lett. 63, 2212 (1989). [CrossRef] [PubMed]
- P. Lambropoulos, "Mechanisms for multiple ionization of atoms by strong laser pulses," Phys. Rev. Lett. 55, 2141 (1985). [CrossRef] [PubMed]
- B. Walker, B. Sheehy, L.F. DiMauro, P. Agostini, K.J. Schafer and K.C. Kulander, "Precision measurement of strong field double ionization," Phys. Rev. Lett. 73, 1227 (1994). [CrossRef] [PubMed]
- A. l'Huillier, A. Lompre, G. Mainfray, and C. Manus, "Multiply charged ions induced by multiphoton absorption in rare gases at 0.53 µm," Phys. Rev. A 27, 2503 (1983). [CrossRef]
- P. Corkum, "Plasma perspective on strong field multiphoton ionization," Phys. Rev. Lett. 71, 1994 (1993). [CrossRef] [PubMed]
- D.N. Fittinghoff, P.R. Bolton, B. Chang, and K.C. Kulander, "Observation of nonsequential double ionization of helium with optical tunneling," Phys. Rev. Lett. 69, 2642 (1992). [CrossRef] [PubMed]
- J.S. Parker, L.R. Moore, D. Dundas and K.T. Taylor, "Double ionization of helium at 390 nm," J. Phys. B 33, L691 (2000). [CrossRef]
- E.S. Smyth, J.S. Parker, K.T. Taylor, "Numerical integration of the time-dependent Schroedinger equation for laser-driven helium," Computer Phys. Comm. 114, 1 (1998). [CrossRef]
- K.J. Schafer and K.C. Kulander, "Energy analysis of time-dependent wave functions: Application to above-threshold ionization," Phys. Rev. A 42, 5794 (1992). [CrossRef]
- H.G. Muller, "Solving the time-dependent Schroedinger equation in five dimensions,' in L.F. DiMauro, R.R. Freeman, and K.C. Kulander (eds.) "Multiphoton processes: ICOMP VIII, 8th International Conference," CP525, p. 257 AIP, Melville NY (2000).
- H.G. Muller, "An efficient propagation scheme for the time-dependent Schroedinger equation in the velocity gauge," Laser Phys. 9, 138 (1999).
- H.G. Muller, "Numerical solution of high-order ATI enhancement in argon," Phys. Rev. A 60, 1341 (1999). [CrossRef]
- H.G. Muller, "Calculation of double ionization of helium," in B. Piraux et al. (eds.) "Super-Intense Laser-Atom Physics VI proceedings," NATO series B (2001) in print.
- J.L. Krause, K.J. Schafer and K.C. Kulander, "Calculation of photoemission from atoms subject to intense laser fields," Phys. Rev. A 45, 4998 (1992). [CrossRef] [PubMed]
- R. Kopold, W. Becker, and D.B. Milosevic, "Quantum orbits: a space-time picture of intense-laser-induced processes in atoms," Comments At. Mol. Phys. (2000) to be published.
- H.G. Muller and F.C. Kooiman, "Bunching and Focusing of Tunneling Wave Packets in Enhancement of High-Order ATI," Phys. Rev. Lett. 81, 1207 (1998). [CrossRef]
- H.G. Muller, "Identification of states responsible for ATI enhancement in argon by their calculated wave functions," Opt. Express 8, 44 (2001). http://www.opticsexpress.org/oearchive/source/26830.htm [CrossRef] [PubMed]
- J.B. Watson, A. Sanpera, K. Burnett, D.G. Lappas and P.L. Knight "Double ionization of helium beyond the single electron active electron approximation," in P. Lambropoulos and H. Walther, (eds.) "Multiphoton Processes: ICOMP VII, 7th International Conference," CS154, p. 132 IOP, Bristol, UK (1997).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

### Supplementary Material

» Media 1: MOV (740 KB)

» Media 2: MOV (1268 KB)

» Media 3: MOV (1028 KB)

» Media 4: MOV (1067 KB)

» Media 5: MOV (1137 KB)

» Media 6: MOV (277 KB)

« Previous Article | Next Article »

OSA is a member of CrossRef.