## Photoionization of atoms described by Fermi Molecular Dynamics: toward a firmer theoretical basis

Optics Express, Vol. 8, Issue 7, pp. 401-410 (2001)

http://dx.doi.org/10.1364/OE.8.000401

Acrobat PDF (470 KB)

### Abstract

The application of Fermi Molecular Dynamics (FMD) to the modeling of the photoionization of atoms by a short pulse of long wavelength laser radiation is examined in detail. Depression of the single ionization threshold to values of the electric field strength below the classical over-the-barrier threshold, a common occurrence in FMD, is shown to arise from the preexcitation of bound electrons into a continuum of unphysical low-lying excited states. A connection is made to analogous calculations performed with the quantum Hamilton-Jacobi equation, in which the time-dependent quantum potential (*Q*) plays a role similar to that played in FMD by the so-called Heisenberg potential (*V _{H}
*). Replacement of

*V*by

_{H}*Q*in the FMD equations of motion results in a large reduction in the number of excitations to unphysical bound states, while producing no essential change in the photoionization probability.

© Optical Society of America

## 1 Introduction

1. D. Fittinghoff, P. Bolton, B. Chang, and K. Kulander, “Observations of nonsequential double ionization of helium with optical tunneling,” Phys. Rev. Lett. **69**, 2642–2645 (1992). [CrossRef] [PubMed]

5. P. Lambropoulos, “Mechanisms for multiple ionization of atoms by strong pulsed lasers,” Phys. Rev. Lett. **55**, 2141–2144 (1985). [CrossRef] [PubMed]

6. P. Corkum, “Plasma perspective on strong-field multiphoton ionization,” Phys. Rev. Lett. **71**, 1994–1997 (1993). [CrossRef] [PubMed]

5. P. Lambropoulos, “Mechanisms for multiple ionization of atoms by strong pulsed lasers,” Phys. Rev. Lett. **55**, 2141–2144 (1985). [CrossRef] [PubMed]

*boomeranging*of the first electron to be released in the field of the laser, followed by a collisional ionization of one or more electrons still bound to the target, has gained some acceptance[6

6. P. Corkum, “Plasma perspective on strong-field multiphoton ionization,” Phys. Rev. Lett. **71**, 1994–1997 (1993). [CrossRef] [PubMed]

15. D. Bohm, “A suggested interpretation of the quantum theory in terms of ‘hidden’ variables. I,” Phys. Rev. **85**, 166–179 (1952). [CrossRef]

16. C. Lopreore and R. Wyatt, “Quantum wave packet dynamics with trajectories,” Phys. Rev. Lett. **82**, 5190–5193 (1999). [CrossRef]

## 2 The TDSE and the Quantum Hamilton-Jacobi Equation

*+*

_{real}*i*Ψ

*, and where*

_{imag}*H*(

*t*) is the (generally time-dependent) Hamilton operator. For the problem of one electron moving in the field of a singly charged ion, and acted also upon by an external electric field, the Hamilton can be taken to be

*E*(

_{pulse}*t*) the laser envelope function and

*ϕ*the laser phase, all in atomic units (a.u.); viz.,

*m*=-

*e*=

*ħ*=1.

*R*=

*R*(

*r*⃗,

*t*) and

*S*=

*S*(

*r*⃗,

*t*) are purely real functions of the coordinates and the time, then eqn 1 transforms into the following pair of coupled equations:

*ρ*=

*R*

^{2}is the solution of a continuity equation (eqn 5) while the phase

*S*is the solution of the so-called quantum Hamilton-Jacobi equation (eqn 6). The function

*Q*appearing in eqn 6 has been referred to as the quantum potential[16

16. C. Lopreore and R. Wyatt, “Quantum wave packet dynamics with trajectories,” Phys. Rev. Lett. **82**, 5190–5193 (1999). [CrossRef]

*υ*⃗. It contains, on the right-hand-side, a sum of the purely classical forces and a term -∇

*⃗*

_{r}*Q*which may be thought of as a force of quantum origin.

*s*orbit of atomic hydrogen were: a linearly polarized laser field of peak electric field strength

*E*

_{0}=0.035

*a*.

*u*., for a frequency of

*ω*=0.1

*a*.

*u*., and a laser pulse length of

*t*=1300

_{pulse}*a*.

*u*., including a sine squared turn-on and turn-off ramp of duration

*t*=150

_{ramp}*a*.

*u*. The calculation included an absorbing potential which was zero for all

*r*<20

*a*.

*u*., and peaked up sharply at

*r*=25

*a*.

*u*. The absorber eliminated problems which would otherwise have arisen from outgoing flux, representing ionized states, reflecting from the end of the mesh, located at

*r*=25

*a*.

*u*.

*ℓ*≤7), and on a radial mesh of size Δ

*r*=0.25

*a*.

*u*. We chose these laser parameters in order to minimize the number of partial waves necessary, while at the same time allowing the tunneling and/or the multiphoton regimes to be sampled. Under these conditions, the ionization probability was computed to be

*P*≈0.05.

_{ioniz}*Q*(

*x*,

*t*) are plotted along the verticle axis of this 3D plot as a function of the x-coordinate, which is parallel to the applied electric field, and the time

*t*. For the most part, only positive values of Q are visible in Fig. 1. In Fig. 2, we plot the underside of this surface; i.e., the negative values of Q.

*x*=0) occurs for all t. We will refer to this central peak again, presently. But first, of more interest is the seemingly chaotic line of mountains and very deep valleys which appears for values of x in the region 6<|

*x*|<8

*a*.

*u*., for all times t after the turn-on transient; i.e., for

*t*≥150

*a*.

*u*. The line of very deep valleys appears also to show a spatial periodicity, with a period of Δ

*x*≈7

*a*.

*u*. This period is just twice the quiver amplitude; i.e., Δ

*x*≈2

*α*

_{0}=2

*E*

_{0}/

*ω*

^{2}. The form of the Newton’s equation (eqn 8) when combined with the these features of Q, cries out for some interpretation.

*E*

_{0}=0.035<

*E*=0.0625

_{OBI}*a*.

*u*., so that any ionization which occurs should only be due to tunneling.

*r*-

*r*⃗.

*E*⃗(t) is infiltrated by a collection of shifting fissures, any one of which can be penetrated by the real particle, if the particle arrives at the entrance to a fissure while it is open, with the right velocity, and manages to pass all the way through the fissure before it closes. Hence, whereas in the conventional picture of quantum mechnanics it is the particle which has ghostly or wave-like properties, while the potential experienced by the particle is altogether real, in the Bohmian picture it is the particle which is altogether real while the potential acquires ghostly or unusual properties. Not too surprisingly perhaps, the location of the line of fissures or valleys is at |

*x*|≈2

*α*

_{0}, as has already been noted.

*⃗*

_{r}*S*, in a region around

*x*=0, and for both

*y*~0 and

*z*~0. Since Q is given by eqn 7, one expects that under the conditions of this problem (Ψ(

*t*=0)=Φ

_{1s}=2exp(-

*r*) and

*E*

_{0}<

*E*), and in the vicinity of the origin, Q will be well-approximated by

_{OBI}*t*>0, and will be given by this expression exactly, for

*t*=0; see Fig. 1. Similarly, the total force on the electron will be

*t*=0. Then, from eqn 9, the “velocity” of the electron is

## 3 The Predictions of FMD

*H*(

*t*) for this problem is now written as

*A*and

_{H}*B*(both positive real numbers) are chosen according to a simple prescription[8]; see the remarks following eqn 20. The FMD equations of motion (Hamilton’s equations) are:

_{H}*r*⃗ and

*p*⃗, for a fixed value of the initial laser phase

*ϕ*. All of the relevant laser parameters were taken from the preceding section.

*P*≈0.02. This result might be thought to be somewhat puzzling, however, since the peak applied electric field strength is less than the OBI threshold value; i.e.,

_{ioniz}*E*

_{0}=0.035<

*E*=0.0625

_{OBI}*a*.

*u*. Moreover, one can show that an effect of the purely repulsive Heisenberg potential

*V*(

_{H}*r*,

*p*) must be to raise the OBI threshold, not to lower it. The answer to this puzzle may be found in Fig. 4 (solid curve), where we show the spectrum of final electron total energies computed in the FMD calculation.

*P*is quite reasonable, as judged by the value obtained from the TDSE in the previous section (to within a factor of ~2). We will argue in the next Section that this is more than just a coincidence.

_{ioniz}## 4 FMD and the Quantum Hamilton-Jacobi Equation

*r*⃗(

*Q*-1/

*r*) is exactly zero in the quantum HJ equation, for

*t*=0, and approximately zero in the vicinity of the origin for all

*t*>0 (eqn 12). This condition also holds true for the total force on the electron according to the FMD equation of motion (eqn 18); viz.,

*t*=0, and approximately in the vicinity of the origin, for

*t*>0, by construction[8]. Second, note that the velocity of the electron is also exactly zero for

*t*=0, and approximately zero in the vicinity of the origin, for

*t*>0, in the quantum HJ description (eqn 13). Moreover, the velocity of the electron is identically behaved in the FMD description; i.e.,

*t*=0, and approximately in the vicinity of the origin, for

*t*>0, again by construction[8]. (Here, “by construction” means that the FMD ground-state is a minimum energy time-independent configuration attained when the total force on the electron, and the velocity of the electron, are both zero. These two conditions determine, generally, the values of the constants

*A*and

_{H}*B*in the FMD Heisenberg potential, eqn 15. In terms of these constants and the ground state energy ∊, the values of |

_{H}*r*⃗| and |

*p*⃗| are also determined.)

*V*(

_{H}*r*,

*p*,) behaves similarly to

*Q*(

*r*,

*t*), if individual trajectories obtained from solutions of the FMD equations are used to define a surface of

*V*values labeled by

_{H}*r*and

*t*. Identical remarks can be made about a comparison of values of the function ∇

*⃗*

_{r}*S*(

*r*,

*t*), from the quantum HJ equation, and the function

*p*⃗+∇

_{p}⃗

*V*(

_{H}*r*,

*p*) computed from the FMD equations.

*V*is at best only an approximation to that function which describes the dynamics correctly; i.e.,

_{H}*V*~

_{H}*Q*. It seemed of interest, therefore, to test these notions directly through a new set of FMD calculations, in which all the elements were the same as before, with the exception that

*V*would be replaced by

_{H}*Q*, and

*Q*would be given by the already determined solution of the TDSE. For each trajectory in the new FMD calculation, the value of

*Q*(

*r*,

*t*) was inserted into eqn 18, for values of

*r*and

*t*determined at the previous time-step. No iteration was performed.

*P*≈0.06. Upon comparing the two curves in Fig. 4, it is evident that there has been a suppression of excitations to the unphysical region lying below the

_{ioniz}*n*=2 threshold (at

*∊*=-0.125

*a*.

*u*.), with some promotion to higher energy continuum states. At the same time, no essential change has occurred in the value of

*P*. This seems to be an encouraging result. The source of the increase in the production of high energy continua (for

_{ioniz}*∊*≥0.4

*a*.

*u*.), when

*V*is replaced by

_{H}*Q*, is presently unknown and seems to be an undesirable feature.

## 5 Summary

*V*. Thus, the FMD equation for the total force on an electron has been shown to make predictions for the photoionization probability

_{H}*P*which are similar to those of the quantum HJ equation. Moreover, the behavior of the electron “velocity” predicted by the quantum HJ equation has been shown to be similar to the predictions of the FMD equation for

_{ioniz}*dr*⃗/

*dt*.

15. D. Bohm, “A suggested interpretation of the quantum theory in terms of ‘hidden’ variables. I,” Phys. Rev. **85**, 166–179 (1952). [CrossRef]

## References and links

1. | D. Fittinghoff, P. Bolton, B. Chang, and K. Kulander, “Observations of nonsequential double ionization of helium with optical tunneling,” Phys. Rev. Lett. |

2. | K. Kondo, A. Sagisaka, T. Tamida, Y. Nabekawa, and S. Watanabe, “Wavelength dependence of nonsequential double ionization in He,” Phys. Rev. |

3. | B. Walker, E. Mevel, B. Yang, P. Berger, J. P. Chamberet, A. Antonetti, L. F. Dimauro, and P. Agostini, “Double ionization in the perturbative and tunneling regimes,” Phys. Rev. |

4. | S. Augst, A. Talebpour, S. L. Chin, Y. Beaudoin, and M. Chaker, “Nonsequential triple ionization of argon atoms in a high-intensity laser field,” Phys. Rev. |

5. | P. Lambropoulos, “Mechanisms for multiple ionization of atoms by strong pulsed lasers,” Phys. Rev. Lett. |

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

7. | T. Brabec, M. Ivanov, and P. Corkum, “Coulomb focusing in intense field atomic processes,” Phys. Rev. |

8. | K. J. LaGattuta and James S. Cohen, “Quasiclassical modeling of helium double photoionization,” J. Phys. |

9. | A. Becker and F. H. M. Faisal, “Mechanism of laser-induced double ionization of helium,” J. Phys. |

10. | K. LaGattuta, “Multiple ionization of argon atoms by long-wavelength laser radiation: a fermion molecular dynamics simulation,” J. Phys. |

11. | Th. Weber, M. Weckenbrock, A. Staudte, L. Spielberger, O. Jagutzki, V. Mergel, F. Afaneh, G. Urbasch, M. Vollmer, H. Giessen, and R. Dorner, “Recoil-ion momentum distributions for single and double ionization of helium in strong laser fields,” Phys. Rev. Lett. |

12. | R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C. Schroter, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoffmann, and W. Sander, “Momentum distributions of Ne(n+) ions created by an intense ultrashort laser pulse,” Phys. Rev. Lett. |

13. | A. Becker and F. H. M. Faisal, “Interpretation of momentum distribution of recoil ions from laser induced nonsequential double ionization,” Phys. Rev. Lett. |

14. | K. LaGattuta, “Laser effects in photoionization: numerical solution of coupled equations for a three-dimensional Coulomb potential,” JOSA |

15. | D. Bohm, “A suggested interpretation of the quantum theory in terms of ‘hidden’ variables. I,” Phys. Rev. |

16. | C. Lopreore and R. Wyatt, “Quantum wave packet dynamics with trajectories,” Phys. Rev. Lett. |

17. | D. Wasson and S. Koonin, “Molecular-dynamics simulations of atomic ionization by strong laser fields,” Phys. Rev. |

18. | K. LaGattuta, “Laser effects in photoionization II. Numerical solution of coupled equations for atomic hydrogen,” Phys. Rev. |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(020.4180) Atomic and molecular physics : Multiphoton processes

(260.3230) Physical optics : Ionization

(270.6620) Quantum optics : Strong-field processes

**ToC Category:**

Focus Issue: Laser-induced multiple ionization

**History**

Original Manuscript: February 6, 2001

Published: March 26, 2001

**Citation**

Kenneth LaGattuta, "Photoionization of atoms described by Fermi Molecular Dynamics: toward a firmer theoretical basis," Opt. Express **8**, 401-410 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-7-401

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### References

- D. Fittinghoff, P. Bolton, B. Chang, and K. Kulander, "Observations of nonsequential double ionization of helium with optical tunneling," Phys. Rev. Lett. 69, 2642-2645 (1992). [CrossRef] [PubMed]
- K. Kondo, A. Sagisaka, T. Tamida, Y. Nabekawa, and S. Watanabe, "Wavelength dependence of nonsequential double ionization in He," Phys. Rev. A 48, R2531-R2533 (1993).
- B. Walker, E. Mevel, B. Yang, P. Berger, J. P. Chamberet, A. Antonetti, L. F. Dimauro, and P. Agostini, "Double ionization in the perturbative and tunneling regimes," Phys. Rev. A 48, R894-R897 (1993).
- S. Augst, A. Talebpour, S. L. Chin, Y. Beaudoin, and M. Chaker, "Nonsequential triple ionization of argon atoms in a high-intensity laser field," Phys. Rev. A52, R917-R919 (1995).
- P. Lambropoulos, "Mechanisms for multiple ionization of atoms by strong pulsed lasers," Phys. Rev. Lett. 55, 2141-2144 (1985). [CrossRef] [PubMed]
- P. Corkum, "Plasma perspective on strong-field multiphoton ionization," Phys. Rev. Lett. 71, 1994-1997 (1993). [CrossRef] [PubMed]
- T. Brabec, M. Ivanov, and P. Corkum, "Coulomb focusing in intense field atomic processes," Phys. Rev. A 54, R2551-R2554 (1996).
- K. J. LaGattuta and James S. Cohen, "Quasiclassical modeling of helium double photoionization," J. Phys. B 31, 5281-5291 (1998).
- A. Becker and F. H. M. Faisal, "Mechanism of laser-induced double ionization of helium," J. Phys. B 29, L197-L202 (1996).
- K. LaGattuta, "Multiple ionization of argon atoms by long-wavelength laser radiation: a fermion molecular dynamics simulation," J. Phys. B 33, 2489-2494 (2000).
- Th. Weber, M. Weckenbrock, A. Staudte, L. Spielberger, O. Jagutzki, V. Mergel, F. Afaneh, G. Urbasch, M. Vollmer, H. Giessen, and R. Dorner, "Recoil-ion momentum distributions for single and double ionization of helium in strong laser fields," Phys. Rev. Lett. 84, 443-446 (2000). [CrossRef] [PubMed]
- R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C. Schroter, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoffmann, and W. Sander, "Momentum distributions of Ne(n+) ions created by an intense ultrashort laser pulse," Phys. Rev. Lett. 84, 447-450 (2000). [CrossRef] [PubMed]
- A. Becker and F. H. M. Faisal, "Interpretation of momentum distribution of recoil ions from laser induced nonsequential double ionization," Phys. Rev. Lett. 84, 3546-3549 (2000). [CrossRef] [PubMed]
- K. LaGattuta, "Laser effects in photoionization: numerical solution of coupled equations for a three-dimensional Coulomb potential," J. Opt. Soc. Am. B 7, 639-646 (1990).
- D. Bohm, "A suggested interpretation of the quantum theory in terms of `hidden' variables. I," Phys. Rev. 85, 166-179 (1952). [CrossRef]
- C. Lopreore and R. Wyatt, "Quantum wave packet dynamics with trajectories," Phys. Rev. Lett. 82, 5190-5193 (1999). [CrossRef]
- D. Wasson and S. Koonin, "Molecular-dynamics simulations of atomic ionization by strong laser fields," Phys. Rev. A 39, 5676-5685 (1989).
- K. LaGattuta, "Laser effects in photoionization II. Numerical solution of coupled equations for atomic hydrogen," Phys. Rev. A 41, 5110-5116 (1990).

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