## Relativistic suppression of wave packet spreading

Optics Express, Vol. 2, Issue 7, pp. 277-281 (1998)

http://dx.doi.org/10.1364/OE.2.000277

Acrobat PDF (176 KB)

### Abstract

We investigate numerically the solution of Dirac equation and analytically the Klein-Gordon equation and discuss the relativistic motion of an electron wave packet in the presence of an intense static electric field. In contrast to the predictions of the (non-relativistic) Schrödinger theory, the spreading rate in the field’s polarization direction as well as in the transverse directions is reduced.

© Optical Society of America

1. H.R. Reiss, “Relativistic Strong-Field Photoionization”, J. Opt. Soc. Am. B **7**, 574 (1990). [CrossRef]

3. M.D. Perry and G. Mourou, “Terawatt to Petawatt Subpicosecond Lasers”, Science **264**, 917 (1994) [CrossRef] [PubMed]

8. Q. Su, J.H. Eberly, and J. Javanainen, “Dynamics of Atomic Ionization Suppression and Electron Localization in an Intense High-Frequency Radiation Field”, Phys. Rev. Lett. **64**, 862 (1990) [CrossRef] [PubMed]

11. M.P. De Boer, J. H. Hoogenraad, and R. B. Vrijen, “Indications of High-Intensity Adiabatic Stabilization in Neon”, Phys. Rev. Lett. **71**, 3263 (1993) [CrossRef] [PubMed]

13. J. Grochmalicki, Maciej A. Lewenstein, Martin Wilkens, and Kazimierz Rzazewski, “Beyond Above-Threshold Ionization: Ionization of an Atom by an Ultrashort Laser Pulse Above Atomic Intensity”, J. Opt. Soc. Am. B **7**, 607 (1990) [CrossRef]

19. J.H. Eberly, Prog. Opt.7, 359 ed. E. Wolf (Amsterdam: North-Holland1969); [CrossRef]

20. R. Grobe and M.V. Fedorov, “Packet Spreading, Stabilization, and Localization in Superstrong Fields”, Phys. Rev. Lett **68**, 2592 (1992) [CrossRef] [PubMed]

**E**pointing in the x-direction acting on an electron can be described by the vector potential

**r**,t) is the well-known 4-spinor [21]. A solution to this equation is made possible by discretizing the three Cartesian coordinate axes over a pre-determined volume into 64 to 512 subdivisions. A generalized split-operator Fourier algorithm was used to solve the equation in time. [21]. Our initial state used in the following calculations was:

_{x}, p

_{y}and p

_{z}. From the time derivative of Eqs. (5), one can easily see that y≪ and z≪ go to 0 while x≪ → c. The motion in the plane perpendicular to the polarization axis of the field is altered although there is no force in that plane. This deceleration is a simple consequence of the fact that the speed

**r**,t = 0) = Ψ(

**-r**,t = 0) and ϕ(

**p**, t = 0) = ϕ(

**-p**,t = 0).

_{NR}(t)

^{2}= Δx

^{2}+ Δ

^{2}/m

^{2}for x and similarly for y and z.

_{NR}(t)

^{2}= Δx

^{2}+ Δp

_{x}t

^{2}/m

^{2}. When t=t

^{*}, we obtain Eq. (8) for the relativistic width. Using this reasoning, we could conclude that the Schršdinger theory would roughly agree with the relativistic theory up to t=t

^{*}. We would also expect the final width to be proportional to 1/E because the electron approaches c more quickly and has less time to spread for stronger E fields.

*ħ*= m=1 and c≈137. Our initial state is the first component of (3) and we choose E=1000 a.u. (ca. 5 × 10

^{12}V/cm), meaning t

^{*}=0.137 a.u. The square root expectation values in Eqs. (6) were evaluated numerically in Fourier space.

_{NR}(t) = Δy

_{NR}(t) = Δz

_{NR}(t). The predictions agree for short times but soon begin to diverge. As the electron approaches the speed of light, the spreading rate in all three spatial directions is severely retarded. In fact, spreading in the x-direction approaches the value Δx(t → ∞) = 0.6935 from Eq. (8) and in the transverse directions is reduced logarithmically.

_{x}Δx, however, grows as a function of time as the canonical momentum is conserved under the time-evolution.

## Acknowledgment

## Footnotes

* | undergraduate research assistant |

## References and links

1. | H.R. Reiss, “Relativistic Strong-Field Photoionization”, J. Opt. Soc. Am. B |

2. | H.R. Reiss, “Theoretical Methods in Quantum Optics: S-Matrix and Keldysh Techniques for Strong-Field Processes”, Prog. Quantum Electron. |

3. | M.D. Perry and G. Mourou, “Terawatt to Petawatt Subpicosecond Lasers”, Science |

4. | K. Boyer and C.K. Rhodes, “Superstrong Coherent Multi-Electron Intense-Field Interaction”, J. Phys. B |

5. | C.I. Moore, J.P. Knauer, and D.D. Meyerhofer, “Observation of the Transition from Thomson to Compton Scattering in Multiphoton Interactions with Low-Energy Electrons”, Phys. Rev. Lett. |

6. | P. Monot, T. Auguste, and J. L. Miquel, “Experimental Demonstration of Relativistic Self-Channeling for a Multiterawatt Laser Pulse in an Underdense Plasma”, Phys. Rev. Lett. |

7. | “Atoms in Intense Laser Fields”, ed. M. Gavrila (Academic Press, Boston1992). |

8. | Q. Su, J.H. Eberly, and J. Javanainen, “Dynamics of Atomic Ionization Suppression and Electron Localization in an Intense High-Frequency Radiation Field”, Phys. Rev. Lett. |

9. | M. Dšrr, R.M. Potvliege, and R. Shakeshaft, “Tunneling Ionization of Atomc Hydrogen by an Intense Low-Frequency Field”, Phys. Rev. Lett. |

10. | K.C. Kulander, K.J. Schafer, and J.L. Krause, “Dynamic Stabilization of Hydrogen in an Intense High Frequency, Pulsed Laser Field”, Phys. Rev. Lett. |

11. | M.P. De Boer, J. H. Hoogenraad, and R. B. Vrijen, “Indications of High-Intensity Adiabatic Stabilization in Neon”, Phys. Rev. Lett. |

12. | N.J van Druten, R. C. Constantinescu, and H. G. Muller, “Adiabatic Stabilization: Observation of the Surviving Population”, Phys. Rev. A |

13. | J. Grochmalicki, Maciej A. Lewenstein, Martin Wilkens, and Kazimierz Rzazewski, “Beyond Above-Threshold Ionization: Ionization of an Atom by an Ultrashort Laser Pulse Above Atomic Intensity”, J. Opt. Soc. Am. B |

14. | F.H.M. Faisal and T. Radozycki, “Three-Dimensional Relativistic Model of a Bound Particle in an Intense Laser Field”, Phys. Rev. A |

15. | F.H.M. Faisal and T. Radozycki, “Three-Dimensional Relativistic Model of a Bound Particle in an Intense Laser Pulse. II”, Phys. Rev. A |

16. | M. Horbatsch, “Magnetic Field Effects in High-Frequency Photoionization by Intense Laser Pulses”, Z. Phys. D |

17. | D.P. Crawford and H.R. Reiss, “Stabilization in Relavistic Photoionization with Circular Polarized Light”, Phys. Rev. A |

18. | M. Protopapas, C.H. Keitel, and P.L. Knight, “Relativistic Mass Shift Effects in Adiabatic Intense Laser Field Stabilization of Atoms”, J. Phys. B |

19. | J.H. Eberly, Prog. Opt.7, 359 ed. E. Wolf (Amsterdam: North-Holland1969); [CrossRef] |

20. | R. Grobe and M.V. Fedorov, “Packet Spreading, Stabilization, and Localization in Superstrong Fields”, Phys. Rev. Lett |

21. | Q. Su, B.A. Smetanko, and R. Grobe, “Wave packet motion in relativistic electric fields”, Laser Phys. (in press). |

22. | J.D. Bjorken and S.D. Drell, “Relativistic Quantum Mechanics” (McGraw-Hill, New York1964). |

23. | B.A. Smetanko, Q. Su, and R. Grobe, “FFT Based Split Operator Techniques for Solving the Dirac Equation”, Comm. Comp. Phys. (to be submitted). |

**OCIS Codes**

(270.6620) Quantum optics : Strong-field processes

(350.5720) Other areas of optics : Relativity

**ToC Category:**

Focus Issue: Relativistic effects in strong eectromagnetic fields

**History**

Original Manuscript: October 21, 1997

Published: March 30, 1998

**Citation**

Qichang Su, B. Smetanko, and Rainer Grobe, "Relativistic suppression of wave packet spreading," Opt. Express **2**, 277-281 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-7-277

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### References

- H.R. Reiss, "Relativistic Strong-Field Photoionization", J. Opt. Soc. Am. B 7, 574 (1990). [CrossRef]
- H.R. Reiss, "Theoretical Methods in Quantum Optics: S-Matrix and Keldysh Techniques for Strong-Field Processes", Prog. Quantum Electron. 16, 1 (1992). [CrossRef]
- M.D. Perry and G. Mourou, "Terawatt to Petawatt Subpicosecond Lasers", Science 264, 917 (1994) [CrossRef] [PubMed]
- K. Boyer and C.K. Rhodes, "Superstrong Coherent Multi-Electron Intense-Field Interaction", J. Phys. B 27, L633 (1994) [CrossRef]
- C.I. Moore, J.P. Knauer and D.D. Meyerhofer, "Observation of the Transition from Thomson to Compton Scattering in Multiphoton Interactions with Low-Energy Electrons", Phys. Rev. Lett. 74, 2439 (1995) [CrossRef] [PubMed]
- P. Monot, T. Auguste, and J. L. Miquel, "Experimental Demonstration of Relativistic Self-Channeling for a Multiterawatt Laser Pulse in an Underdense Plasma", Phys. Rev. Lett. 74, 2953 (1995). [CrossRef] [PubMed]
- "Atoms in Intense Laser Fields", ed. M. Gavrila (Academic Press, Boston 1992).
- Q. Su, J.H. Eberly and J. Javanainen, "Dynamics of Atomic Ionization Suppression and Electron Localization in an Intense High-Frequency Radiation Field", Phys. Rev. Lett. 64, 862 (1990) [CrossRef] [PubMed]
- M. Doerr, R.M. Potvliege and R. Shakeshaft, "Tunneling Ionization of Atomc Hydrogen by an Intense Low-Frequency Field", Phys. Rev. Lett. 64, 2003 (1990) [CrossRef]
- K.C. Kulander, K.J. Schafer and J.L. Krause, "Dynamic Stabilization of Hydrogen in an Intense High Frequency, Pulsed Laser Field", Phys. Rev. Lett. 66, 2601 (1991) [CrossRef] [PubMed]
- M.P. De Boer, J. H. Hoogenraad, and R. B. Vrijen, "Indications of High-Intensity Adiabatic Stabilization in Neon", Phys. Rev. Lett. 71, 3263 (1993) [CrossRef] [PubMed]
- N.J van Druten, R. C. Constantinescu, and H. G. Muller, "Adiabatic Stabilization: Observation of the Surviving Population", Phys. Rev. A 55, 622 (1997). [CrossRef]
- J. Grochmalicki, Maciej A. Lewenstein, Martin Wilkens, and Kazimierz Rzazewski, "Beyond Above-Threshold Ionization: Ionization of an Atom by an Ultrashort Laser Pulse Above Atomic Intensity", J. Opt. Soc. Am. B 7, 607 (1990) [CrossRef]
- F.H.M. Faisal and T. Radozycki, "Three-Dimensional Relativistic Model of a Bound Particle in an Intense Laser Field", Phys. Rev. A 47,4464 (1993) [CrossRef] [PubMed]
- F.H.M. Faisal and T. Radozycki, "Three-Dimensional Relativistic Model of a Bound Particle in an Intense Laser Pulse. II", Phys. Rev. A 48, 554 (1993) [CrossRef] [PubMed]
- M. Horbatsch, "Magnetic Field Effects in High-Frequency Photoionization by Intense Laser Pulses", Z. Phys. D 25, 305 (1993). [CrossRef]
- D.P. Crawford and H.R. Reiss, "Stabilization in Relavistic Photoionization with Circular Polarized Light", Phys. Rev. A 50, 1844 (1994) [CrossRef] [PubMed]
- M. Protopapas, C.H. Keitel and P.L. Knight, "Relativistic Mass Shift Effects in Adiabatic Intense Laser Field Stabilization of Atoms", J. Phys. B 29, L591 (1996). [CrossRef]
- J.H. Eberly, Prog. Opt. 7, 359 ed. E. Wolf (Amsterdam: North-Holland 1969); [CrossRef]
- R. Grobe and M.V. Fedorov, "Packet Spreading, Stabilization, and Localization in Superstrong Fields", Phys. Rev. Lett 68, 2592 (1992) [CrossRef] [PubMed]
- Q. Su, B.A. Smetanko and R. Grobe, Wave packet motion in relativistic electric fields, Laser Phys. (in press).
- J.D. Bjorken and S.D. Drell, "Relativistic Quantum Mechanics" (McGraw-Hill, New York 1964).
- B.A. Smetanko, Q. Su and R. Grobe, "FFT Based Split Operator Techniques for Solving the Dirac Equation", Comm. Comp. Phys. (to be submitted).

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