## Nonequilibrium evolution of strong-field anisotropic ionized electrons towards a delayed plasma-state |

Optics Express, Vol. 20, Issue 3, pp. 2310-2318 (2012)

http://dx.doi.org/10.1364/OE.20.002310

Acrobat PDF (10903 KB)

### Abstract

Rigorous quantum calculations of the femtosecond ionization of hydrogen atoms in air lead to highly anisotropic electron and ion angular (momentum) distributions. A quantum Monte-Carlo analysis of the subsequent many-body dynamics reveals two distinct relaxation steps, first to a nearly isotropic hot nonequilibrium and then to a quasi-equilibrium configuration. The collective isotropic plasma state is reached on a picosecond timescale well after the ultrashort ionizing pulse has passed.

© 2012 OSA

## 1. Introduction

1. T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics **4**, 822–832 (2010). [CrossRef]

2. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. **441**, 47–189 (2007). [CrossRef]

3. Y. Huismans, A. Rouzee, A. Gijsbertsen, J. H. Jungmann, A. S. Smolkowska, P. S. W. M. Logman, F. Lepine, C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker, G. Berden, B. Redlich, A. F. G. van der Meer, H. G. Muller, W. Vermin, K. J. Schafer, M. Spanner, M. Y. Ivanov, O. Smirnova, D. Bauer, S. V. Popruzhenko, and M. J. J. Vrakking, “Time-resolved holography with photoelectrons,” Science **331**, 61–64 (2011). [CrossRef]

## 2. Theoretical approach

*V*=

*q*(

_{e}**E**+

**v**×

**B**) with

*h̄*

**k**=

*m*

_{e}**v**in the drift term leads to the Vlasov equation. Ideally, one would have to treat the full problem of strong-field ionization together with the many-body dynamics at the level of quantum kinetic theory. However, for realistic systems, the numerical solution of the complete process described by Eq. (1) is not feasible with current-day computer resources.

^{23}/

*m*

^{3}the inverse plasma frequency is up to one picosecond. This allows for the separation of the ionization and relaxation dynamics. Furthermore, a typical length scale for electrons should be around the Bohr radius (

*a*≃ 0.05

_{B}*nm*). In contrast typical laser wavelengths (several 100 nm) or filament diameters (around 50

*μm*) are much larger. Therefore, we can restrict the electron dynamics after the ionization to an evolution on the electronic length scale of a quasi-homogeneous electron gas which allows for dropping the space coordinate. Even with these approximations, the numerical solution still remains very time demanding.

4. D. G. Arbo, S. Yoshida, E. Persson, K. I. Dimitriou, and J. Burgdorfer, “Interference oscillations in the angular distribution of laser-ionized electrons near ionization threshold,” Phys. Rev. Lett. **96**, 143003 (2006). [CrossRef] [PubMed]

5. A. Rudenko, K. Zrost, C. D. Schroter, V. L. B. de Jesus, B. Feuerstein, R. Moshammer, and J. Ullrich, “Resonant structures in the low-energy electron continuum for single ionization of atoms in the tunnelling regime,” J. Phys. B **37**, L407–L413 (2004). [CrossRef]

**r**= (

*ρ*,

*z*,

*ϕ*) and

**p**are the coordinate and the conjugate momentum operator of the electron. We assume an exciting linearly in z-direction polarized laser pulse represented by the electric field with

*E*

_{0},

*h̄ω*,

*χ*and

*T*are the peak amplitude, frequency, carrier-to-envelope phase and total duration of the pulse. Since the Hamiltonian in the time-dependent Schrödinger equation (2) is symmetric with respect to rotations over the polarization axis

*z*, the solution Ψ(

*ρ*,

*z;t*) does not depend on the angle

*ϕ*. We discretize the Schrödinger equation on a spatial grid with spacings Δ

*ρ*= Δ

*z*= 0.1 a.u. and use a time step of Δ

*t*= 0.03 a.u. Grid sizes of up to 1200 and 8000 points in

*ρ*and

*z*directions, respectively, are implemented with cos

^{1/6}mask functions at the edges. The grids were large enough to keep the configuration space wavefunction on the grid for analysis of the momentum distributions. We have checked that the results are not influenced by the boundary conditions. The solution Ψ(

*ρ*,

*z;t*) is propagated using the Crank-Nicolson method and the wave function of the (initial) 1

*s*ground state of the hydrogen atom is obtained by imaginary time propagation. The electron momentum distributions are obtained by projection of the full configuration space wavefunction at the end of the pulse onto the analytical solutions of the outgoing continuum wave function, as given in the literature (e.g. [6]), represented on the grid.

*nm*pulses with 3 or 6 cycles, respectively. The assumed densities are in the range of the values reported in the literature, i.e. a few 10

^{23}/

*m*

^{3}[7

7. Z. Sun, J. Chen, and W. Rudolph, “Determination of the transient electron temperature in a femtosecond-laser-induced air plasma filament,” Phys. Rev. E **83**, 046408 (2011). [CrossRef]

8. S. Tzortzakis, B. Prade, M. Franco, and A. Mysyrowicz, “Time-evolution of the plasma channel at the trail of a self-guided IR femtosecond laser pulse in air,” Opt. Commun. **181**, 123–127 (2000). [CrossRef]

^{22}/

*m*

^{3}[9

9. Y. H. Chen, S. Varma, T. M. Antonsen, and H. M. Milchberg, “Direct measurement of the electron density of extended femtosecond laser pulse-induced filaments,” Phys. Rev. Lett. **105**, 215005 (2010). [CrossRef]

4. D. G. Arbo, S. Yoshida, E. Persson, K. I. Dimitriou, and J. Burgdorfer, “Interference oscillations in the angular distribution of laser-ionized electrons near ionization threshold,” Phys. Rev. Lett. **96**, 143003 (2006). [CrossRef] [PubMed]

10. P. Agostini, F. Fabre, G. Mainfray, G. Petite, and N. K. Rahman, “Free-free transitions following 6-photon ionization of xeon atoms,” Phys. Rev. Lett. **42**, 1127–1130 (1979). [CrossRef]

*m*is the mass of the electron [11

_{e}11. M. Wickenhauser, X. M. Tong, D. G. Arbo, J. Burgdoerfer, and C. D. Lin, “Signatures of tunneling and multiphoton ionization in the electron-momentum distributions of atoms by intense few-cycle laser pulses,” Phys. Rev. A **74**, 041402 (2006). [CrossRef]

*I*= 1 × 10

^{14}

*W*/

*cm*

^{2}(Fig. 1) and for

*I*= 5 × 10

^{13}

*W*/

*cm*

^{2}(Fig. 2) the initial distributions in

*k*(black lines in panels (e)) are broad without significant structure in form of maxima and minima. However, the average energy absorbed from the field mainly depends on the wavelength and the intensity of the laser pulse. Therefore, it is approximately the same in Figs. 2(a) and 3(a) (both obtained for an intensity of 5 × 10

^{13}w/cm

^{2}) but it is different in Fig. 1(a) (obtained at 1 × 10

^{14}W/cm

^{2}). Please also note, that in the 3-cycle case the distributions show a strong asymmetry with respect to

*k*= 0. This indicates that the ionization process strongly depends on the carrier-to-envelope phase, which is chosen as

_{z}*χ*=

*π*/2 in the present calculations. In agreement with experimental observations [12

12. G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi, M. Nisoli, S. Stagira, E. Priori, and S. De Silvestri, “Absolute-phase phenomena in photoionization with few-cycle laser pulses,” Nature **414**, 182–184 (2001). [CrossRef] [PubMed]

*k̄*of the photoionized electrons, i.e. a dc current

_{z}*J*∝

_{z}*k̄*. However, this current will be damped, e.g., by the many-body Coulomb scattering. Values of the average momentum

_{z}*k̄*for different times after the ionization are shown in section 3. A detailed investigation of the current strength depending on the pulse parameters can be found in [13

_{z}13. A. A. Silaev and N. V. Vvedenskii, “Residual-current excitation in plasmas produced by few-cycle laser pulses,” PRL **102**, 115005 (2009). [CrossRef]

7. Z. Sun, J. Chen, and W. Rudolph, “Determination of the transient electron temperature in a femtosecond-laser-induced air plasma filament,” Phys. Rev. E **83**, 046408 (2011). [CrossRef]

*m*denote the momentum dependent probability distribution and mass of the electrons (

*λ*=

*e*) and ions (

*λ*=

*i*), respectively. The

*δ*-distribution

*δ*(

*E*) provides the energy conservation during the scattering.

*V*

**is screened using a Debye screening constant**

_{q}*κ*which is obtained as the

_{D}*q*= 0-limit of the static Lindhard formula [14] for the final equilibrium distributions. Due to their large mass the ions hardly contribute to the screening.

**k**| ≃ |

**k**−

**q**| and the ion momentum |

**p**| ≃ |

**p**+

**q**| are conserved. As a consequence, the electron-ion interaction is only relevant for anisotropic initial conditions where the electron-ion interaction contributes to the reduction of the anisotropy and is the only way to change the average momentum of the electron system. Furthermore, in the non-degenerate limit only the total number of ions contributes to the electron dynamics which effectively decouples the ion and electron dynamics.

*Ē*. Even for typical experimental conditions of strong short-pulse ionization – as considered here – we are still in the non-degenerate limit such that

*T*of the final equilibrium electron plasma by the well-known property

*Ē*= 3/2

*k*of Boltzmann distributions.

_{B}T15. C. Jacoboni and P. Lugli, *The Monte Carlo Method for Semiconductor Device Simulation* (Springer-Verlag, 1989). [CrossRef]

16. see, e.g., R. Brunetti, C. Jacoboni, A. Matulionis, and V. Dienys, “Effect of interparticle collisions on energy relaxation of carriers in semiconductors,” Physica B **134**, 369–373 (1985) for self-scattering method for el.-el. Coulomb scattering. [CrossRef]

^{7}particles.

*f*(

*k*,

*θ*) in spherical coordinates using the momentum-histograms of the simulated particles. Taking advantage of the

*ϕ*symmetry we have already dropped any

*ϕ*-angle dependence here and introduce for use in Fig. 1–4 and for comparison its

*θ*-angle integrated version In case of isotropy

*H*(

*k*,

*θ*) will match

*H̄*(

*k*). Taking snapshots of these functions will allow us to follow the isotropization of the initially anisotropic distribution and determine an estimate of the time after which an isotropic plasma like answer of the electrons can be expected.

## 3. Results and discussion

*H*(

*k*,

*θ*) are shown in parts (b), (c), and (d) of Figs. 1–3. Here, part (c) is taken at the inverse plasma frequency – a characteristic timescale for a plasma – and (d) corresponds to a time where the final isotropic equilibrium state is nearly reached. The time development of

*H̄*(

*k*) is plotted in parts (e). We notice that the quasi-discrete MPI structures are smeared out within the first 100

*fs*. The entire distributions relax on the order of picoseconds to nearly isotropic equilibrium distributions which have temperatures of 14000

*K*for the lower intensity case Figs. 2 and 3 as well as 20000

*K*for the higher one in Fig. 1, respectively.

*τ*.

*r*(

*t*), (b) the anisotropy i(t) and (c) the average

*k*momentum for the initial conditions of Fig. 1–3. In all cases, we show the time in units of the inverse plasma frequency

_{z}*f*defines the characteristic timescale this scaling transforms the relaxation dynamics to a similar time frame.

_{pl}*k*momentum. This process – governed by the electron-ion scattering – shows a similar linear dependence in the log plots of Fig. 4, i.e., in (a)

_{z}*τ*= 6.76/

*f*(red),

_{pl}*τ*= 6.06/

*f*(black) and

_{pl}*τ*= 5.3/

*f*(blue), (b)

_{pl}*τ*= 8.17/

*f*(red),

_{pl}*τ*= 6.63/

*f*(black), or in (c)

_{pl}*τ*= 9.15/

*f*(red),

_{pl}*τ*= 6.76/

*f*(black) and

_{pl}*τ*= 6.87/

*f*(blue). A value for the blue line in (b) cannot be extracted. Generally, we see that the inverse plasma frequency indeed yields a rough estimate for the scaling of the time evolution.

_{pl}## 4. Conclusion

## Acknowledgments

## References and links

1. | T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics |

2. | A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. |

3. | Y. Huismans, A. Rouzee, A. Gijsbertsen, J. H. Jungmann, A. S. Smolkowska, P. S. W. M. Logman, F. Lepine, C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker, G. Berden, B. Redlich, A. F. G. van der Meer, H. G. Muller, W. Vermin, K. J. Schafer, M. Spanner, M. Y. Ivanov, O. Smirnova, D. Bauer, S. V. Popruzhenko, and M. J. J. Vrakking, “Time-resolved holography with photoelectrons,” Science |

4. | D. G. Arbo, S. Yoshida, E. Persson, K. I. Dimitriou, and J. Burgdorfer, “Interference oscillations in the angular distribution of laser-ionized electrons near ionization threshold,” Phys. Rev. Lett. |

5. | A. Rudenko, K. Zrost, C. D. Schroter, V. L. B. de Jesus, B. Feuerstein, R. Moshammer, and J. Ullrich, “Resonant structures in the low-energy electron continuum for single ionization of atoms in the tunnelling regime,” J. Phys. B |

6. | L. D. Landau and E. M. Lifshitz, |

7. | Z. Sun, J. Chen, and W. Rudolph, “Determination of the transient electron temperature in a femtosecond-laser-induced air plasma filament,” Phys. Rev. E |

8. | S. Tzortzakis, B. Prade, M. Franco, and A. Mysyrowicz, “Time-evolution of the plasma channel at the trail of a self-guided IR femtosecond laser pulse in air,” Opt. Commun. |

9. | Y. H. Chen, S. Varma, T. M. Antonsen, and H. M. Milchberg, “Direct measurement of the electron density of extended femtosecond laser pulse-induced filaments,” Phys. Rev. Lett. |

10. | P. Agostini, F. Fabre, G. Mainfray, G. Petite, and N. K. Rahman, “Free-free transitions following 6-photon ionization of xeon atoms,” Phys. Rev. Lett. |

11. | M. Wickenhauser, X. M. Tong, D. G. Arbo, J. Burgdoerfer, and C. D. Lin, “Signatures of tunneling and multiphoton ionization in the electron-momentum distributions of atoms by intense few-cycle laser pulses,” Phys. Rev. A |

12. | G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi, M. Nisoli, S. Stagira, E. Priori, and S. De Silvestri, “Absolute-phase phenomena in photoionization with few-cycle laser pulses,” Nature |

13. | A. A. Silaev and N. V. Vvedenskii, “Residual-current excitation in plasmas produced by few-cycle laser pulses,” PRL |

14. | see, e.g. H. Haug and S. W. Koch, |

15. | C. Jacoboni and P. Lugli, |

16. | see, e.g., R. Brunetti, C. Jacoboni, A. Matulionis, and V. Dienys, “Effect of interparticle collisions on energy relaxation of carriers in semiconductors,” Physica B |

**OCIS Codes**

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

(260.5210) Physical optics : Photoionization

(320.0320) Ultrafast optics : Ultrafast optics

(350.5400) Other areas of optics : Plasmas

**ToC Category:**

Atomic and Molecular Physics

**History**

Original Manuscript: November 3, 2011

Revised Manuscript: December 20, 2011

Manuscript Accepted: December 23, 2011

Published: January 18, 2012

**Citation**

B. Pasenow, J. V. Moloney, S. W. Koch, S. H. Chen, A. Becker, and A. Jaroń-Becker, "Nonequilibrium evolution of strong-field anisotropic ionized electrons towards a delayed plasma-state," Opt. Express **20**, 2310-2318 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-2310

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### References

- T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics4, 822–832 (2010). [CrossRef]
- A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep.441, 47–189 (2007). [CrossRef]
- Y. Huismans, A. Rouzee, A. Gijsbertsen, J. H. Jungmann, A. S. Smolkowska, P. S. W. M. Logman, F. Lepine, C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker, G. Berden, B. Redlich, A. F. G. van der Meer, H. G. Muller, W. Vermin, K. J. Schafer, M. Spanner, M. Y. Ivanov, O. Smirnova, D. Bauer, S. V. Popruzhenko, and M. J. J. Vrakking, “Time-resolved holography with photoelectrons,” Science331, 61–64 (2011). [CrossRef]
- D. G. Arbo, S. Yoshida, E. Persson, K. I. Dimitriou, and J. Burgdorfer, “Interference oscillations in the angular distribution of laser-ionized electrons near ionization threshold,” Phys. Rev. Lett.96, 143003 (2006). [CrossRef] [PubMed]
- A. Rudenko, K. Zrost, C. D. Schroter, V. L. B. de Jesus, B. Feuerstein, R. Moshammer, and J. Ullrich, “Resonant structures in the low-energy electron continuum for single ionization of atoms in the tunnelling regime,” J. Phys. B37, L407–L413 (2004). [CrossRef]
- L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 3rd ed. (Butterworth-Heinemann, 1981)
- Z. Sun, J. Chen, and W. Rudolph, “Determination of the transient electron temperature in a femtosecond-laser-induced air plasma filament,” Phys. Rev. E83, 046408 (2011). [CrossRef]
- S. Tzortzakis, B. Prade, M. Franco, and A. Mysyrowicz, “Time-evolution of the plasma channel at the trail of a self-guided IR femtosecond laser pulse in air,” Opt. Commun.181, 123–127 (2000). [CrossRef]
- Y. H. Chen, S. Varma, T. M. Antonsen, and H. M. Milchberg, “Direct measurement of the electron density of extended femtosecond laser pulse-induced filaments,” Phys. Rev. Lett.105, 215005 (2010). [CrossRef]
- P. Agostini, F. Fabre, G. Mainfray, G. Petite, and N. K. Rahman, “Free-free transitions following 6-photon ionization of xeon atoms,” Phys. Rev. Lett.42, 1127–1130 (1979). [CrossRef]
- M. Wickenhauser, X. M. Tong, D. G. Arbo, J. Burgdoerfer, and C. D. Lin, “Signatures of tunneling and multiphoton ionization in the electron-momentum distributions of atoms by intense few-cycle laser pulses,” Phys. Rev. A74, 041402 (2006). [CrossRef]
- G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi, M. Nisoli, S. Stagira, E. Priori, and S. De Silvestri, “Absolute-phase phenomena in photoionization with few-cycle laser pulses,” Nature414, 182–184 (2001). [CrossRef] [PubMed]
- A. A. Silaev and N. V. Vvedenskii, “Residual-current excitation in plasmas produced by few-cycle laser pulses,” PRL102, 115005 (2009). [CrossRef]
- see, e.g. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th ed. (World Scientific Publ., 2009), Chap. 8.
- C. Jacoboni and P. Lugli, The Monte Carlo Method for Semiconductor Device Simulation (Springer-Verlag, 1989). [CrossRef]
- see, e.g., R. Brunetti, C. Jacoboni, A. Matulionis, and V. Dienys, “Effect of interparticle collisions on energy relaxation of carriers in semiconductors,” Physica B134, 369–373 (1985) for self-scattering method for el.-el. Coulomb scattering. [CrossRef]

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