## Phase-space analysis of double ionization

Optics Express, Vol. 8, Issue 7, pp. 411-416 (2001)

http://dx.doi.org/10.1364/OE.8.000411

Acrobat PDF (163 KB)

### Abstract

We use the Wigner transformation to study the electronic center-of-mass motion in phase-space for double ionization in a strong laser field. The rescattering mechanism is clearly visible in the evolution of the fully correlated two-electron system. In a mean-field calculation, on the other hand, the signatures of rescattering are missing. Some properties of the Wigner function in two-particle systems are reported.

© Optical Society of America

## 1 Introduction

## 2 Model

24. R. Grobe and J. H. Eberly, “Single and double ionization and strong-field stabilization of a two-electron system,” Phys. Rev. A **47**, R1605 (1993). [CrossRef] [PubMed]

*z*

_{1}and

*z*

_{2}are the electron coordinates along the direction of the laser polarization, and

*E*(

*t*)=

*E*

_{0}sin

*ωt*is the electric field of the laser. The coordinates can assume positive and negative values, so that the electrons may pass by the nucleus. Numerically, we represent the time-dependent two-electron wave function Ψ(

*z*

_{1},

*z*

_{2},

*t*) on a two-dimensional grid extending at least 300 a.u. in each dimension. The separation between the grid points is 0.2 a.u. The initial state is the singlet ground state which we obtain by propagation of an arbitrary symmetric wave function in imaginary time. The time evolution under the influence of the laser field is calculated through numerical integration of the time-dependent Schrödinger equation, using the split-operator method [25

25. M. D. Feit, J. A. Fleck Jr., and A. Steiger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. **47**, 412 (1982). [CrossRef]

^{15}W/cm

^{2}is employed. The classical oscillation amplitude for these parameters is

*α*=

*E*

_{0}/

*ω*

^{2}=49 a.u., i.e., the grid is large enough to account for recollision events. In our previous work, we have demonstrated [9

9. M. Lein, E. K. U. Gross, and V. Engel, “On the mechanism of strong-field double photoionization in the helium atom,” J. Phys. B **33**, 433 (2000). [CrossRef]

*φ*is governed by the one-particle Schrödinger equation

*v*given by

_{s}## 3 The Wigner transformation for a two-electron system

*φ*(

*x*) is given by

*w*

^{(1)}over the momentum

*p*correctly yields the coordinate probability distribution |

*φ*(

*x*)|

^{2}(except for a prefactor 2

*π*). Likewise, the integration over

*x*yields the momentum distribution. Therefore, and despite the fact that the Wigner transform may assume negative values, it is usually interpreted as the probability distribution in the one-particle phase space spanned by the coordinate

*x*and the momentum

*p*.

*z*

_{1},

*z*

_{2}), the Wigner transformation may be carried out for

*each*coordinate. The number of dimensions is thereby doubled. As in [20

20. M. Lein, E. K. U. Gross, and V. Engel, “Intense-Field Double Ionization of Helium: Identifying the Mechanism,” Phys. Rev. Lett. **85**, 4707 (2000). [CrossRef] [PubMed]

*Z*=(

*z*

_{1}+

*z*

_{2})/2; we do not transform with respect to the relative coordinate

*z*=

*z*

_{2}-

*z*

_{1}:

*z*to obtain the phase-space distribution for the center of mass:

*w*(

*Z*,

*P*), it is more efficient (yet equivalent) to first integrate over

*z*and then carry out the Fourier transformation. In this way, we do not need to handle the complicated integrand in Eq. (6), which depends on four variables.

*w*

^{(1)}(

*x*,

*p*) is the one-particle Wigner function as defined in Eq. (5). From Eq. (8) we see that wHF exhibits some peculiarities. Obviously,

*w*

^{HF}(

*Z*,

*P*) is always greater or equal to zero, contrary to a general Wigner function. This is a nice feature since it simplifies the interpretation. Consider now the case that

*φ*(

*x*) is a superposition of two wave packets, centered at

*x*=±

*a*, i.e.,

*φ*(

*x*)=

*φ*(

_{a}*x*)+

*φ*-

*a*(

*x*). The Wigner function

*w*

^{(1)}(

*x*,

*p*) then contains maxima at

*x*=±

*a*. Therefore, by Eq. (8), the two-particle quantity

*w*

^{HF}(

*Z*,

*P*) contains maxima at

*Z*=±

*a*as well, corresponding to the case where both electrons are located at -

*a*or where both electrons are located at +

*a*. However, the HF wave function Ψ

^{HF}(

*z*

_{1},

*z*

_{2})=

*φ*(

_{a}*z*

_{1})

*φ*(

_{a}*z*

_{2})+

*φ*(

_{a}*z*

_{1})

*φ*

_{-a}(

*z*

_{2})+

*φ*(

_{-a}*z*

_{1})

*φ*(

_{a}*z*

_{2})+

*φ*

_{-a}(

*z*

_{1})

*φ*

_{-a}(

*z*

_{2}) also contains the possibility that the electrons are located on opposite sides, corresponding to phase-space density around

*Z*=0. By Eq. (8), the one-particle distribution then contains density at

*x*=0; thinking classically, this should not be the case. The solution to the paradox is that the one-particle distribution exhibits rapid oscillations between positive and negative values in the region at

*x*=0. These oscillations would average to zero if the Wigner function was convoluted with some smoothing window function. By applying the square in Eq. (8), we obtain a positive function which cannot average to zero.

*φ*

_{1},

*φ*

_{2},

*w*

_{φ1}and

*w*

_{φ2}are the Wigner distributions for

*φ*

_{1},

*φ*

_{2}. The second term in Eq. (10) arises from the symmetrization of the wave function.

## 4 Results

*T*/2 where

*T*is one optical cycle, and the still images in Figs. 1, 2 are the final frames at

*t*=3

*T*/2. Both movies include audio tracks explaining the time-dependent processes. We look at the exact calculation first. The system starts out from the ground state which, on the scale of our figures, has a narrow phase-space density around

*Z*=0 extending from about

*P*=-2 a.u. to

*P*=+2 a.u. In the first half optical cycle, the external electric field is positive so that the electrons are accelerated towards negative coordinates. At

*t*=

*T*/2, a very broad wave packet has appeared in the region

*Z*<0 and

*P*<0. Since double ionization is negligible during the first half cycle [9

9. M. Lein, E. K. U. Gross, and V. Engel, “On the mechanism of strong-field double photoionization in the helium atom,” J. Phys. B **33**, 433 (2000). [CrossRef]

10. M. Dörr, “Double ionization in a one-cycle laser pulse,” Opt. Express **6**, 111 (2000), http://www.opticsexpress.org/oearchive/source/19114.htm. [CrossRef]

*P*at the lower end of the wave packet which agrees well with the maximum momentum that one electron can receive classically, namely the momentum after a half cycle of free acceleration in the electric field, |

*P*

_{max}|=2

*E*

_{0}/

*w*=5.8 a.u. This means that the wave packet must be interpreted as one electron at

*z*=2

*Z*with momentum

*p*=

*P*, while the second electron remains close to

*z*=0,

*p*=0. Part of this density can return to the core after acceleration into the opposite direction during the second half cycle. When the electron collides with the core, i.e., when the density crosses the vertical axis

*Z*=0, we find that structures appear in the wave packet. See, e.g., the oscillations in 0<

*Z*<20 a.u., 3 a.u.<

*P*<5 a.u. at the time

*t*=

*T*. They are due to the superposition of scattered and unscattered density. At lower

*P*we observe a strong second single-ionization wave packet. The fast oscillations for

*Z*>20 a.u. result from the overlap of the two single-ionization wave packets. The snapshot at

*t*=

*T*clearly shows why the phase-space pictures are much more informative than snapshots in configuration or momentum space: There is very little overlap between the wave packets corresponding to ionization at different times; they can be clearly separated. Projection either onto the

*Z*or

*P*axis would lead to considerable overlap, and we could not observe the time evolution of the individual wave packets. With increasing time, we find that tails evolve out of the scattered density. For

*T*<

*t*<3

*T*/2 the field accelerates the electrons towards negative coordinates. The tails are accelerated more strongly than the broad remainder of the wave-packet. A close examination reveals that the acceleration is about twice as large, indicating that this is density corresponding to two electrons freely accelerated by the laser. At

*t*=3

*T*/2, the rescattering process is completed and the double-ionization wave packet at (

*Z*,

*P*)~(-80,-7.5) a.u. has become essentially isolated from the rest. The ponderomotive momentum shift that the two electrons will receive when the laser field is adiabatically switched off amounts to

*E*

_{0}/

*ω*=2.9 a.u. per electron. Therefore, a center-of-mass momentum of

*P*=-7.5 a.u. at

*t*=3

*T*/2 leads to a final recoil-ion momentum of -(-7.5a.u.+2×2.9a.u.)=1.7 a.u.. This is in good agreement with experiment [11

11. Th. Weber*et al*., “Recoil-Ion Momentum Distributions for Single and Double Ionization of Helium in Strong Laser Fields,” Phys. Rev. Lett. **84**, 443 (2000). [CrossRef] [PubMed]

15. A. Becker and F. H. M. Faisal, “Interpretation of momentum distribution of recoil ions from laser induced nonsequential double ionization,” Phys. Rev. Lett. **84**, 3456 (2000). [CrossRef]

## Acknowledgment

## References and links

1. | 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. |

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

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

4. | D. Bauer, “Two-dimensional, two-electron model atom in a laser pulse: Exact treatment, single-active-electron analysis, time-dependent density-functional theory, classical calculations, and non-sequential ionization,” Phys. Rev. A |

5. | J. B. Watson, A. Sanpera, D. G. Lappas, P. L. Knight, and K. Burnett, “Nonsequential Double Ionization of Helium,” Phys. Rev. Lett. |

6. | D. G. Lappas and R. van Leeuwen, “Electron correlation effects in the double ionization of He,” J. Phys. B |

7. | W.-C. Liu, J. H. Eberly, S. L. Haan, and R. Grobe, “Correlation Effects in Two-Electron Model Atoms in Intense Laser Fields,” Phys. Rev. Lett. |

8. | D. Dundas, K. T. Taylor, J. S. Parker, and E. S. Smyth, “Double-ionization dynamics of laser-driven helium,” J. Phys. B |

9. | M. Lein, E. K. U. Gross, and V. Engel, “On the mechanism of strong-field double photoionization in the helium atom,” J. Phys. B |

10. | M. Dörr, “Double ionization in a one-cycle laser pulse,” Opt. Express |

11. | Th. Weber |

12. | R. Moshammer |

13. | Th. Weber |

14. | Th. Weber |

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

16. | H. W. van der Hart, “Recollision model for double ionization of atoms in strong laser fields,” Phys. Rev. A |

17. | H. W. van der Hart, “Sequential versus non-sequential double ionization in strong laser fields,” J. Phys. B |

18. | R. Kopold, W. Becker, H. Rottke, and W. Sandner, “Routes to Nonsequential Double Ionization,” Phys. Rev. Lett. |

19. | B. Feuerstein, R. Moshammer, and J. Ullrich, “Nonsequential multiple ionization in intense laser pulses: interpretation of ion momentum distributions within the classical ‘rescattering’ model,” J. Phys. B |

20. | M. Lein, E. K. U. Gross, and V. Engel, “Intense-Field Double Ionization of Helium: Identifying the Mechanism,” Phys. Rev. Lett. |

21. | J. Chen, J. Liu, L. B. Fu, and W. M. Zheng, “Interpretation of momentum distribution of recoil ions from laser-induced nonsequential double ionization by semiclassical rescattering model,” Phys. Rev. A |

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

23. | M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. |

24. | R. Grobe and J. H. Eberly, “Single and double ionization and strong-field stabilization of a two-electron system,” Phys. Rev. A |

25. | M. D. Feit, J. A. Fleck Jr., and A. Steiger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. |

**OCIS Codes**

(020.4180) Atomic and molecular physics : Multiphoton processes

(260.5210) Physical optics : Photoionization

(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**

Manfred Lein, Volker Engel, and E. Gross, "Phase-space analysis of double ionization," Opt. Express **8**, 411-416 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-7-411

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### References

- 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]
- B. Walker, B. Sheehy, L. F. DiMauro, P. Agostini, K. J. Schafer, and K. C. Kulander, "Precision measurement of strong field double ionization of helium," Phys. Rev. Lett. 73, 1227 (1994). [CrossRef] [PubMed]
- A. Becker and F. H. M. Faisal, "Mechanism of laser-induced double ionization of helium," J. Phys. B 29, L197 (1996). [CrossRef]
- D. Bauer, "Two-dimensional, two-electron model atom in a laser pulse: Exact treatment, single-active-electron analysis, time-dependent density-functional theory, classical calculations, and nonsequential ionization," Phys. Rev. A 56, 3028 (1997). [CrossRef]
- J. B. Watson, A. Sanpera, D. G. Lappas, P. L. Knight, and K. Burnett, "Nonsequential Double Ionization of Helium," Phys. Rev. Lett. 78, 1884 (1997). [CrossRef]
- D. G. Lappas and R. van Leeuwen, "Electron correlation effects in the double ionization of He," J. Phys. B 31, L249 (1998). [CrossRef]
- W.-C. Liu, J. H. Eberly, S. L. Haan, and R. Grobe, "Correlation Effects in Two-Electron Model Atoms in Intense Laser Fields," Phys. Rev. Lett. 83, 520 (1999). [CrossRef]
- D. Dundas, K. T. Taylor, J. S. Parker, and E. S. Smyth, "Double-ionization dynamics of laser-driven helium," J. Phys. B 32, L231 (1999). [CrossRef]
- M. Lein, E. K. U. Gross, and V. Engel, "On the mechanism of strong-field double photoionization in the helium atom," J. Phys. B 33, 433 (2000). [CrossRef]
- M. Doerr, "Double ionization in a one-cycle laser pulse," Opt. Express 6, 111 (2000), http://www.opticsexpress.org/oearchive/source/19114.htm. [CrossRef]
- Th. Weber et al., "Recoil-Ion Momentum Distributions for Single and Double Ionization of Helium in Strong Laser Fields," Phys. Rev. Lett. 84, 443 (2000). [CrossRef] [PubMed]
- R. Moshammer et al., "Momentum Distributions of Ne^n+ Ions Created by an Intense Ultrashort Laser Pulse," Phys. Rev. Lett. 84, 447 (2000). [CrossRef] [PubMed]
- Th. Weber et al., "Sequential and nonsequential contributions to double ionization in strong laser fields," J. Phys. B 33, L127 (2000). [CrossRef]
- Th. Weber et al., "Correlated electron emission in multiphoton double ionization," Nature (London) 405, 658 (2000). [CrossRef]
- A. Becker and F. H. M. Faisal, "Interpretation of momentum distribution of recoil ions from laser induced nonsequential double ionization," Phys. Rev. Lett. 84, 3456 (2000). [CrossRef]
- H. W. van der Hart, "Recollision model for double ionization of atoms in strong laser fields," Phys. Rev. A 62, 013407 (2000). [CrossRef]
- H. W. van der Hart, "Sequential versus non-sequential double ionization in strong laser fields," J. Phys. B 33, L699 (2000). [CrossRef]
- R. Kopold, W. Becker, H. Rottke, and W. Sandner, "Routes to Nonsequential Double Ionization," Phys. Rev. Lett. 85, 3781 (2000). [CrossRef] [PubMed]
- B. Feuerstein, R. Moshammer, and J. Ullrich, "Nonsequential multiple ionization in intense laser pulses: interpretation of ion momentum distributions within the classical `rescattering' model," J. Phys. B 33, L823 (2000). [CrossRef]
- M. Lein, E. K. U. Gross, and V. Engel, "Intense-Field Double Ionization of Helium: Identifying the Mechanism," Phys. Rev. Lett. 85, 4707 (2000). [CrossRef] [PubMed]
- J. Chen, J. Liu, L. B. Fu, and W. M. Zheng, "Interpretation of momentum distribution of recoil ions from laser-induced nonsequential double ionization by semiclassical rescattering model," Phys. Rev. A 63, 011404(R) (2001).
- P. B. Corkum, "Plasma perspective on strong field multiphoton ionization," Phys. Rev. Lett. 71, 1994 (1993). [CrossRef] [PubMed]
- M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, "Distribution functions in physics: fundamentals," Phys. Rep. 106, 121 (1984). [CrossRef]
- R. Grobe and J. H. Eberly, "Single and double ionization and strong-field stabilization of a two-electron system," Phys. Rev. A 47, R1605 (1993). [CrossRef] [PubMed]
- M. D. Feit, J. A. Fleck, Jr., and A. Steiger, "Solution of the Schroedinger equation by a spectral method," J. Comput. Phys. 47, 412 (1982). [CrossRef]

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