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.

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References

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  1. H.R. Reiss, "Relativistic Strong-Field Photoionization", J. Opt. Soc. Am. B 7, 574 (1990).
    [CrossRef]
  2. H.R. Reiss, "Theoretical Methods in Quantum Optics: S-Matrix and Keldysh Techniques for Strong-Field Processes", Prog. Quantum Electron. 16, 1 (1992).
    [CrossRef]
  3. M.D. Perry and G. Mourou, "Terawatt to Petawatt Subpicosecond Lasers", Science 264, 917 (1994)
    [CrossRef] [PubMed]
  4. K. Boyer and C.K. Rhodes, "Superstrong Coherent Multi-Electron Intense-Field Interaction", J. Phys. B 27, L633 (1994)
    [CrossRef]
  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. 74, 2439 (1995)
    [CrossRef] [PubMed]
  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. 74, 2953 (1995).
    [CrossRef] [PubMed]
  7. "Atoms in Intense Laser Fields", ed. M. Gavrila (Academic Press, Boston 1992).
  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]
  9. 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]
  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. 66, 2601 (1991)
    [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]
  12. 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]
  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]
  14. 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]
  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 48, 554 (1993)
    [CrossRef] [PubMed]
  16. M. Horbatsch, "Magnetic Field Effects in High-Frequency Photoionization by Intense Laser Pulses", Z. Phys. D 25, 305 (1993).
    [CrossRef]
  17. D.P. Crawford and H.R. Reiss, "Stabilization in Relavistic Photoionization with Circular Polarized Light", Phys. Rev. A 50, 1844 (1994)
    [CrossRef] [PubMed]
  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 29, L591 (1996).
    [CrossRef]
  19. J.H. Eberly, Prog. Opt. 7, 359 ed. E. Wolf (Amsterdam: North-Holland 1969);
    [CrossRef]
  20. R. Grobe and M.V. Fedorov, "Packet Spreading, Stabilization, and Localization in Superstrong Fields", Phys. Rev. Lett 68, 2592 (1992)
    [CrossRef] [PubMed]
  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 York 1964).
  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).

Other

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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Figures (2)

Fig. 1.
Fig. 1.

The graphs show the temporal growth pattern of the spatial width obtained from Eqs. (6) Δx(t), Δy(t) and Δz(t) together with the non-relativistic width ΔxNR(t). Superimposed on the graphs for Δx(t), Δy(t) and Δz(t) are the width determined from the time-dependent wave function solution obtained from the full Dirac equation Eq. (2) (dash lines). The two graphs are indistinguishable. [E=1000 a.u., initial quantum state as in Eq. (3) with σ=Δx(t=0)=0.1 a.u.]

Fig. 2.
Fig. 2.

Displayed are the spatial probability distributions P(x, t) ≠ ∑i∫dydz∣Ψi(x, y, z, t)∣2 and P(z, t) ≠ ∑i∫dxdy∣Ψi(x, y, z, t)∣2 in the x- and z direction at time t=0.1 a.u. For comparison, the dashed lines show the corresponding distributions obtained from the non-relativistic Schrödinger time evolution. The initial wave packet was centered initially at r=(-3, 0, 0) for better graphical clarity. The non-relativistic wave packet has moved to x(t=0.1a.u.) =-3a.u. + Et2/2= 2.a.u. [Same parameters as in Fig. 1]

Equations (10)

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A ( t ) = c E t = cEt e x
i ħ t Ψ r t = mc α ( p q c A ( t ) ) + β m c 2 Ψ r t
Ψ ( r , t = 0 ) = [ 2 π σ 2 ] 3 4 ( exp [ ( r 2 σ ) 2 ] , 0,0,0 )
H = m 2 c 4 + c 2 ( p q c A ) 2
x ( t ) = x + 1 qE m 2 c 4 + c 2 ( p + q E t ) 2 1 qE m 2 c 4 + c 2 p 2
y ( t ) = y + c p y qE ln { p x + qEt + m 2 c 4 + c 2 ( p + q E t ) 2 p x + m 2 c 4 + c 2 p 2 }
z ( t ) = z + c p z qE ln { p x + qEt + m 2 c 4 + c 2 ( p + q E t ) 2 p x + m 2 c 4 + c 2 p 2 }
Δ x 2 = ( x x ) 2
Δ x 2 ( t ) = Δ x 2 + 1 q 2 E 2 [ c p x 2 + p 2 + m 2 c 4 ] 1 q 2 E 2 c 2 p 2 + m 2 c 4 2
Δ x 2 ( t ) = Δ x 2 + c 2 p x 2 q 2 E 2

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