Abstract

We present computer movies of the classical and quantum mechanical time evolution for an atom in a strong static magnetic field and a laser field. The resonantly induced relativistic motion of the atomic electron leads to a ring-like spatial probability density called a cycloatom. We further demonstrate that spin-orbit coupling for a fast moving electron in a cycloatom becomes significant, modifying the time-dependence of the spin even if initially aligned parallel to the static magnetic field direction. We also present several movies on time-evolution of the spin-distribution as a function of the position for a relativistic quantum state. The nature of such a space resolved spin measurement is analyzed.

© 2001 Optical Society of America

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  1. For a review, see e.g. Q. Su and R. Grobe, “Examples of classical and genuinely quantum relativistic phenomena,” in Multiphoton Processes, eds. L.F. DiMauro, R.R. Freeman, and K.C. Kulander (American Institute of Physics, Melville, New York, 2000) p.655 or the website www.phy.ilstu.edu/ILP
  2. Science News, “Ring around the proton,” 157, 287 (2000).
  3. R.E. Wagner, Q. Su, and R. Grobe, “Relativistic resonances in combined magnetic and laser field,” Phys. Rev. Lett.,  84, 3282 (2000).
    [Crossref] [PubMed]
  4. For movies of cycloatoms see Phys. Rev. Focus, “Fast electrons on the cheap”, 5, 15, 6 April (2000) at the web site: http://focus.aps.org/v5/st15.html story
  5. P.J. Peverly, R.E. Wagner, Q. Su, and R. Grobe, “Fractional resonances in relativistic magnetic-laser-atom interactions,” Laser Phys. 10, 303 (2000).
  6. Q. Su, R.E. Wagner, P.J. Peverly, and R. Grobe, “Spatial electron clouds at fractional and multiple magneto-optical resonances,” in Frontiers of Laser Physics and Quantum Optics, eds, Z. Xu, S. Xie, S.-Y. Zhu, and M.O. Scully, p.117 (Springer, Berlin, 2000).
  7. R.E. Wagner, P.J. Peverly, Q. Su, and R. Grobe, “Classical versus quantum dynamics for a driven relativistic oscillator,” Phys. Rev. A 61, 35402 (2000).
    [Crossref]
  8. V.G. Bagrov and D.M. Gitman, Exact solutions of relativistic wave equations, (Kluwer Academic, Dordrecht, 1990).
    [Crossref]
  9. C. Bottcher and M.R. Strayer, “Relativistic theory of fermions and classical fields on a collocation lattice,” Ann. Phys. NY 175, 64 (1987).
    [Crossref]
  10. J.C. Wells, A.S. Umar, V.E. Oberacker, C. Bottcher, M.R. Strayer, J.-S. Wu, J. Drake, and R. Flanery, “A numerical implementation of the Dirac equation on a hypercube multicomputer,” Int. J. Mod. Phys. C 4, 459 (1993).
    [Crossref]
  11. K. Momberger, A. Belkacem, and A.H. Sorensen, “Numerical treatment of the time-dependent Dirac equation in momentum space for atomic processes in relativistic heavy-ion collisions,” Phys. Rev. A 53, 1605 (1996).
    [Crossref] [PubMed]
  12. U.W. Rathe, C.H. Keitel, M. Protopapas, and P.L. Knight, “Intense laser-atom dynamics with the two-dimensional Dirac equation,” J. Phys. B 30, L531 (1997).
    [Crossref]
  13. N.J. Kylstra, A.M. Ermolaev, and C.J. Joachain, “Relativistic effects in the time evolution of a one-dimensional model atom in an intense laser field,” J. Phys. B 30, L449 (1997).
    [Crossref]
  14. C. Szymanowski, C.H. Keitel, and A. Maquet, “Influence of Zitterbewegung on relativistic harmonic generation,” Las. Phys. 9, 133 (1999).
  15. J.W. Braun, Q. Su, and R. Grobe, “Numerical approach to solve the time-dependent Dirac equation,” Phys. Rev. A 59, 604 (1999).
    [Crossref]
  16. U.W. Rathe, P. Sanders, and P.L. Knight, “A case study in scalability: an ADI method for the two-dimensional time-dependent Dirac equation,” Parallel Computing,  25, 525 (1999).
    [Crossref]
  17. P. Krekora, R.E. Wagner, Q. Su, and R. Grobe, “Dirac theory of ring-shaped electron distributions.” Phys. Rev. A, in press.
  18. H. Goldstein, Classical Mechanics, 2nd edition (Addison-Wesley, New York, 1980).
  19. B. Thaller, The Dirac Equation, (Springer, 1992).
  20. R.E. Wagner, Q. Su, and R. Grobe, “High-order harmonic generation in relativistic ionization of magnetically dressed atoms,” Phys. Rev. A,  60, 3233 (1999).
    [Crossref]
  21. For relativistic suppression of wave packet spreading, see,Q. Su, B.A. Smetanko, and R. Grobe, “Wave packet motion in relativistic electric fields,” Las. Phys. 8, 93 (1998).
  22. Q. Su, B.A. Smetanko, and R. Grobe, “Relativistic suppression of wave packet spreading,” Opt. Express 2, 277 (1998), http://www.opticsexpres.org/oearchive/source/2813.htm
    [Crossref] [PubMed]
  23. E. Lenz, M. Dörr, and W. Sandner, Las. Phys., in press.
  24. For a review on Lorentz transformations of 4×4 spin matrices, see, e.g., J.D. Bjorken and S.D. Drell, “Relativistic quantum mechanics,” (McGraw-Hill, 1964); J. Kessler, Polarized Electrons, 2nd edition (Springer Verlag, Berlin, 1985).
  25. For work on the Spin-Wigner function, see, e.g., I. Bialynicki-Birula, P. Gornicki, and J. Rafelski, “Phase-space structure of the Dirac vacuum,” Phys. Rev. D 44, 1825 (1991).
    [Crossref]
  26. G.R. Shin, I. Bialynicki-Birula, and J. Rafelski, “Wigner function of relativistic spin-1/2 particles,” Phys. Rev. D 46, 645 (1992).
  27. For the time-evolution of the spatial width, see, J.C. Csesznegi, G.H. Rutherford, Q. Su, and R. Grobe, “Dynamics of wave packets in inhomogeneous and homogeneous magnetic fields,” Las. Phys. 6, 41 (1999).
  28. P. Krekora, Q. Su, and R. Grobe, “Dynamical signature in spatial spin distributions of relativistic electrons,” Phys. Rev. A, submitted.
  29. L.T. Thomas, Phil. Mag.3, 1 (1927).
  30. J.D. Jackson, Classical Electrodynamics (Wiley, New York, 1999).

2000 (3)

R.E. Wagner, Q. Su, and R. Grobe, “Relativistic resonances in combined magnetic and laser field,” Phys. Rev. Lett.,  84, 3282 (2000).
[Crossref] [PubMed]

P.J. Peverly, R.E. Wagner, Q. Su, and R. Grobe, “Fractional resonances in relativistic magnetic-laser-atom interactions,” Laser Phys. 10, 303 (2000).

R.E. Wagner, P.J. Peverly, Q. Su, and R. Grobe, “Classical versus quantum dynamics for a driven relativistic oscillator,” Phys. Rev. A 61, 35402 (2000).
[Crossref]

1999 (5)

C. Szymanowski, C.H. Keitel, and A. Maquet, “Influence of Zitterbewegung on relativistic harmonic generation,” Las. Phys. 9, 133 (1999).

J.W. Braun, Q. Su, and R. Grobe, “Numerical approach to solve the time-dependent Dirac equation,” Phys. Rev. A 59, 604 (1999).
[Crossref]

U.W. Rathe, P. Sanders, and P.L. Knight, “A case study in scalability: an ADI method for the two-dimensional time-dependent Dirac equation,” Parallel Computing,  25, 525 (1999).
[Crossref]

R.E. Wagner, Q. Su, and R. Grobe, “High-order harmonic generation in relativistic ionization of magnetically dressed atoms,” Phys. Rev. A,  60, 3233 (1999).
[Crossref]

For the time-evolution of the spatial width, see, J.C. Csesznegi, G.H. Rutherford, Q. Su, and R. Grobe, “Dynamics of wave packets in inhomogeneous and homogeneous magnetic fields,” Las. Phys. 6, 41 (1999).

1998 (2)

For relativistic suppression of wave packet spreading, see,Q. Su, B.A. Smetanko, and R. Grobe, “Wave packet motion in relativistic electric fields,” Las. Phys. 8, 93 (1998).

Q. Su, B.A. Smetanko, and R. Grobe, “Relativistic suppression of wave packet spreading,” Opt. Express 2, 277 (1998), http://www.opticsexpres.org/oearchive/source/2813.htm
[Crossref] [PubMed]

1997 (2)

U.W. Rathe, C.H. Keitel, M. Protopapas, and P.L. Knight, “Intense laser-atom dynamics with the two-dimensional Dirac equation,” J. Phys. B 30, L531 (1997).
[Crossref]

N.J. Kylstra, A.M. Ermolaev, and C.J. Joachain, “Relativistic effects in the time evolution of a one-dimensional model atom in an intense laser field,” J. Phys. B 30, L449 (1997).
[Crossref]

1996 (1)

K. Momberger, A. Belkacem, and A.H. Sorensen, “Numerical treatment of the time-dependent Dirac equation in momentum space for atomic processes in relativistic heavy-ion collisions,” Phys. Rev. A 53, 1605 (1996).
[Crossref] [PubMed]

1993 (1)

J.C. Wells, A.S. Umar, V.E. Oberacker, C. Bottcher, M.R. Strayer, J.-S. Wu, J. Drake, and R. Flanery, “A numerical implementation of the Dirac equation on a hypercube multicomputer,” Int. J. Mod. Phys. C 4, 459 (1993).
[Crossref]

1992 (1)

G.R. Shin, I. Bialynicki-Birula, and J. Rafelski, “Wigner function of relativistic spin-1/2 particles,” Phys. Rev. D 46, 645 (1992).

1991 (1)

For work on the Spin-Wigner function, see, e.g., I. Bialynicki-Birula, P. Gornicki, and J. Rafelski, “Phase-space structure of the Dirac vacuum,” Phys. Rev. D 44, 1825 (1991).
[Crossref]

1987 (1)

C. Bottcher and M.R. Strayer, “Relativistic theory of fermions and classical fields on a collocation lattice,” Ann. Phys. NY 175, 64 (1987).
[Crossref]

Bagrov, V.G.

V.G. Bagrov and D.M. Gitman, Exact solutions of relativistic wave equations, (Kluwer Academic, Dordrecht, 1990).
[Crossref]

Belkacem, A.

K. Momberger, A. Belkacem, and A.H. Sorensen, “Numerical treatment of the time-dependent Dirac equation in momentum space for atomic processes in relativistic heavy-ion collisions,” Phys. Rev. A 53, 1605 (1996).
[Crossref] [PubMed]

Bialynicki-Birula, I.

G.R. Shin, I. Bialynicki-Birula, and J. Rafelski, “Wigner function of relativistic spin-1/2 particles,” Phys. Rev. D 46, 645 (1992).

For work on the Spin-Wigner function, see, e.g., I. Bialynicki-Birula, P. Gornicki, and J. Rafelski, “Phase-space structure of the Dirac vacuum,” Phys. Rev. D 44, 1825 (1991).
[Crossref]

Bjorken, J.D.

For a review on Lorentz transformations of 4×4 spin matrices, see, e.g., J.D. Bjorken and S.D. Drell, “Relativistic quantum mechanics,” (McGraw-Hill, 1964); J. Kessler, Polarized Electrons, 2nd edition (Springer Verlag, Berlin, 1985).

Bottcher, C.

J.C. Wells, A.S. Umar, V.E. Oberacker, C. Bottcher, M.R. Strayer, J.-S. Wu, J. Drake, and R. Flanery, “A numerical implementation of the Dirac equation on a hypercube multicomputer,” Int. J. Mod. Phys. C 4, 459 (1993).
[Crossref]

C. Bottcher and M.R. Strayer, “Relativistic theory of fermions and classical fields on a collocation lattice,” Ann. Phys. NY 175, 64 (1987).
[Crossref]

Braun, J.W.

J.W. Braun, Q. Su, and R. Grobe, “Numerical approach to solve the time-dependent Dirac equation,” Phys. Rev. A 59, 604 (1999).
[Crossref]

Csesznegi, J.C.

For the time-evolution of the spatial width, see, J.C. Csesznegi, G.H. Rutherford, Q. Su, and R. Grobe, “Dynamics of wave packets in inhomogeneous and homogeneous magnetic fields,” Las. Phys. 6, 41 (1999).

Dörr, M.

E. Lenz, M. Dörr, and W. Sandner, Las. Phys., in press.

Drake, J.

J.C. Wells, A.S. Umar, V.E. Oberacker, C. Bottcher, M.R. Strayer, J.-S. Wu, J. Drake, and R. Flanery, “A numerical implementation of the Dirac equation on a hypercube multicomputer,” Int. J. Mod. Phys. C 4, 459 (1993).
[Crossref]

Drell, S.D.

For a review on Lorentz transformations of 4×4 spin matrices, see, e.g., J.D. Bjorken and S.D. Drell, “Relativistic quantum mechanics,” (McGraw-Hill, 1964); J. Kessler, Polarized Electrons, 2nd edition (Springer Verlag, Berlin, 1985).

Ermolaev, A.M.

N.J. Kylstra, A.M. Ermolaev, and C.J. Joachain, “Relativistic effects in the time evolution of a one-dimensional model atom in an intense laser field,” J. Phys. B 30, L449 (1997).
[Crossref]

Flanery, R.

J.C. Wells, A.S. Umar, V.E. Oberacker, C. Bottcher, M.R. Strayer, J.-S. Wu, J. Drake, and R. Flanery, “A numerical implementation of the Dirac equation on a hypercube multicomputer,” Int. J. Mod. Phys. C 4, 459 (1993).
[Crossref]

Gitman, D.M.

V.G. Bagrov and D.M. Gitman, Exact solutions of relativistic wave equations, (Kluwer Academic, Dordrecht, 1990).
[Crossref]

Goldstein, H.

H. Goldstein, Classical Mechanics, 2nd edition (Addison-Wesley, New York, 1980).

Gornicki, P.

For work on the Spin-Wigner function, see, e.g., I. Bialynicki-Birula, P. Gornicki, and J. Rafelski, “Phase-space structure of the Dirac vacuum,” Phys. Rev. D 44, 1825 (1991).
[Crossref]

Grobe, R.

R.E. Wagner, Q. Su, and R. Grobe, “Relativistic resonances in combined magnetic and laser field,” Phys. Rev. Lett.,  84, 3282 (2000).
[Crossref] [PubMed]

P.J. Peverly, R.E. Wagner, Q. Su, and R. Grobe, “Fractional resonances in relativistic magnetic-laser-atom interactions,” Laser Phys. 10, 303 (2000).

R.E. Wagner, P.J. Peverly, Q. Su, and R. Grobe, “Classical versus quantum dynamics for a driven relativistic oscillator,” Phys. Rev. A 61, 35402 (2000).
[Crossref]

R.E. Wagner, Q. Su, and R. Grobe, “High-order harmonic generation in relativistic ionization of magnetically dressed atoms,” Phys. Rev. A,  60, 3233 (1999).
[Crossref]

J.W. Braun, Q. Su, and R. Grobe, “Numerical approach to solve the time-dependent Dirac equation,” Phys. Rev. A 59, 604 (1999).
[Crossref]

For the time-evolution of the spatial width, see, J.C. Csesznegi, G.H. Rutherford, Q. Su, and R. Grobe, “Dynamics of wave packets in inhomogeneous and homogeneous magnetic fields,” Las. Phys. 6, 41 (1999).

Q. Su, B.A. Smetanko, and R. Grobe, “Relativistic suppression of wave packet spreading,” Opt. Express 2, 277 (1998), http://www.opticsexpres.org/oearchive/source/2813.htm
[Crossref] [PubMed]

For relativistic suppression of wave packet spreading, see,Q. Su, B.A. Smetanko, and R. Grobe, “Wave packet motion in relativistic electric fields,” Las. Phys. 8, 93 (1998).

P. Krekora, R.E. Wagner, Q. Su, and R. Grobe, “Dirac theory of ring-shaped electron distributions.” Phys. Rev. A, in press.

For a review, see e.g. Q. Su and R. Grobe, “Examples of classical and genuinely quantum relativistic phenomena,” in Multiphoton Processes, eds. L.F. DiMauro, R.R. Freeman, and K.C. Kulander (American Institute of Physics, Melville, New York, 2000) p.655 or the website www.phy.ilstu.edu/ILP

Q. Su, R.E. Wagner, P.J. Peverly, and R. Grobe, “Spatial electron clouds at fractional and multiple magneto-optical resonances,” in Frontiers of Laser Physics and Quantum Optics, eds, Z. Xu, S. Xie, S.-Y. Zhu, and M.O. Scully, p.117 (Springer, Berlin, 2000).

P. Krekora, Q. Su, and R. Grobe, “Dynamical signature in spatial spin distributions of relativistic electrons,” Phys. Rev. A, submitted.

Jackson, J.D.

J.D. Jackson, Classical Electrodynamics (Wiley, New York, 1999).

Joachain, C.J.

N.J. Kylstra, A.M. Ermolaev, and C.J. Joachain, “Relativistic effects in the time evolution of a one-dimensional model atom in an intense laser field,” J. Phys. B 30, L449 (1997).
[Crossref]

Keitel, C.H.

C. Szymanowski, C.H. Keitel, and A. Maquet, “Influence of Zitterbewegung on relativistic harmonic generation,” Las. Phys. 9, 133 (1999).

U.W. Rathe, C.H. Keitel, M. Protopapas, and P.L. Knight, “Intense laser-atom dynamics with the two-dimensional Dirac equation,” J. Phys. B 30, L531 (1997).
[Crossref]

Kessler, J.

For a review on Lorentz transformations of 4×4 spin matrices, see, e.g., J.D. Bjorken and S.D. Drell, “Relativistic quantum mechanics,” (McGraw-Hill, 1964); J. Kessler, Polarized Electrons, 2nd edition (Springer Verlag, Berlin, 1985).

Knight, P.L.

U.W. Rathe, P. Sanders, and P.L. Knight, “A case study in scalability: an ADI method for the two-dimensional time-dependent Dirac equation,” Parallel Computing,  25, 525 (1999).
[Crossref]

U.W. Rathe, C.H. Keitel, M. Protopapas, and P.L. Knight, “Intense laser-atom dynamics with the two-dimensional Dirac equation,” J. Phys. B 30, L531 (1997).
[Crossref]

Krekora, P.

P. Krekora, R.E. Wagner, Q. Su, and R. Grobe, “Dirac theory of ring-shaped electron distributions.” Phys. Rev. A, in press.

P. Krekora, Q. Su, and R. Grobe, “Dynamical signature in spatial spin distributions of relativistic electrons,” Phys. Rev. A, submitted.

Kylstra, N.J.

N.J. Kylstra, A.M. Ermolaev, and C.J. Joachain, “Relativistic effects in the time evolution of a one-dimensional model atom in an intense laser field,” J. Phys. B 30, L449 (1997).
[Crossref]

Lenz, E.

E. Lenz, M. Dörr, and W. Sandner, Las. Phys., in press.

Maquet, A.

C. Szymanowski, C.H. Keitel, and A. Maquet, “Influence of Zitterbewegung on relativistic harmonic generation,” Las. Phys. 9, 133 (1999).

Momberger, K.

K. Momberger, A. Belkacem, and A.H. Sorensen, “Numerical treatment of the time-dependent Dirac equation in momentum space for atomic processes in relativistic heavy-ion collisions,” Phys. Rev. A 53, 1605 (1996).
[Crossref] [PubMed]

Oberacker, V.E.

J.C. Wells, A.S. Umar, V.E. Oberacker, C. Bottcher, M.R. Strayer, J.-S. Wu, J. Drake, and R. Flanery, “A numerical implementation of the Dirac equation on a hypercube multicomputer,” Int. J. Mod. Phys. C 4, 459 (1993).
[Crossref]

Peverly, P.J.

R.E. Wagner, P.J. Peverly, Q. Su, and R. Grobe, “Classical versus quantum dynamics for a driven relativistic oscillator,” Phys. Rev. A 61, 35402 (2000).
[Crossref]

P.J. Peverly, R.E. Wagner, Q. Su, and R. Grobe, “Fractional resonances in relativistic magnetic-laser-atom interactions,” Laser Phys. 10, 303 (2000).

Q. Su, R.E. Wagner, P.J. Peverly, and R. Grobe, “Spatial electron clouds at fractional and multiple magneto-optical resonances,” in Frontiers of Laser Physics and Quantum Optics, eds, Z. Xu, S. Xie, S.-Y. Zhu, and M.O. Scully, p.117 (Springer, Berlin, 2000).

Protopapas, M.

U.W. Rathe, C.H. Keitel, M. Protopapas, and P.L. Knight, “Intense laser-atom dynamics with the two-dimensional Dirac equation,” J. Phys. B 30, L531 (1997).
[Crossref]

Rafelski, J.

G.R. Shin, I. Bialynicki-Birula, and J. Rafelski, “Wigner function of relativistic spin-1/2 particles,” Phys. Rev. D 46, 645 (1992).

For work on the Spin-Wigner function, see, e.g., I. Bialynicki-Birula, P. Gornicki, and J. Rafelski, “Phase-space structure of the Dirac vacuum,” Phys. Rev. D 44, 1825 (1991).
[Crossref]

Rathe, U.W.

U.W. Rathe, P. Sanders, and P.L. Knight, “A case study in scalability: an ADI method for the two-dimensional time-dependent Dirac equation,” Parallel Computing,  25, 525 (1999).
[Crossref]

U.W. Rathe, C.H. Keitel, M. Protopapas, and P.L. Knight, “Intense laser-atom dynamics with the two-dimensional Dirac equation,” J. Phys. B 30, L531 (1997).
[Crossref]

Rutherford, G.H.

For the time-evolution of the spatial width, see, J.C. Csesznegi, G.H. Rutherford, Q. Su, and R. Grobe, “Dynamics of wave packets in inhomogeneous and homogeneous magnetic fields,” Las. Phys. 6, 41 (1999).

Sanders, P.

U.W. Rathe, P. Sanders, and P.L. Knight, “A case study in scalability: an ADI method for the two-dimensional time-dependent Dirac equation,” Parallel Computing,  25, 525 (1999).
[Crossref]

Sandner, W.

E. Lenz, M. Dörr, and W. Sandner, Las. Phys., in press.

Shin, G.R.

G.R. Shin, I. Bialynicki-Birula, and J. Rafelski, “Wigner function of relativistic spin-1/2 particles,” Phys. Rev. D 46, 645 (1992).

Smetanko, B.A.

For relativistic suppression of wave packet spreading, see,Q. Su, B.A. Smetanko, and R. Grobe, “Wave packet motion in relativistic electric fields,” Las. Phys. 8, 93 (1998).

Q. Su, B.A. Smetanko, and R. Grobe, “Relativistic suppression of wave packet spreading,” Opt. Express 2, 277 (1998), http://www.opticsexpres.org/oearchive/source/2813.htm
[Crossref] [PubMed]

Sorensen, A.H.

K. Momberger, A. Belkacem, and A.H. Sorensen, “Numerical treatment of the time-dependent Dirac equation in momentum space for atomic processes in relativistic heavy-ion collisions,” Phys. Rev. A 53, 1605 (1996).
[Crossref] [PubMed]

Strayer, M.R.

J.C. Wells, A.S. Umar, V.E. Oberacker, C. Bottcher, M.R. Strayer, J.-S. Wu, J. Drake, and R. Flanery, “A numerical implementation of the Dirac equation on a hypercube multicomputer,” Int. J. Mod. Phys. C 4, 459 (1993).
[Crossref]

C. Bottcher and M.R. Strayer, “Relativistic theory of fermions and classical fields on a collocation lattice,” Ann. Phys. NY 175, 64 (1987).
[Crossref]

Su, Q.

P.J. Peverly, R.E. Wagner, Q. Su, and R. Grobe, “Fractional resonances in relativistic magnetic-laser-atom interactions,” Laser Phys. 10, 303 (2000).

R.E. Wagner, P.J. Peverly, Q. Su, and R. Grobe, “Classical versus quantum dynamics for a driven relativistic oscillator,” Phys. Rev. A 61, 35402 (2000).
[Crossref]

R.E. Wagner, Q. Su, and R. Grobe, “Relativistic resonances in combined magnetic and laser field,” Phys. Rev. Lett.,  84, 3282 (2000).
[Crossref] [PubMed]

R.E. Wagner, Q. Su, and R. Grobe, “High-order harmonic generation in relativistic ionization of magnetically dressed atoms,” Phys. Rev. A,  60, 3233 (1999).
[Crossref]

For the time-evolution of the spatial width, see, J.C. Csesznegi, G.H. Rutherford, Q. Su, and R. Grobe, “Dynamics of wave packets in inhomogeneous and homogeneous magnetic fields,” Las. Phys. 6, 41 (1999).

J.W. Braun, Q. Su, and R. Grobe, “Numerical approach to solve the time-dependent Dirac equation,” Phys. Rev. A 59, 604 (1999).
[Crossref]

For relativistic suppression of wave packet spreading, see,Q. Su, B.A. Smetanko, and R. Grobe, “Wave packet motion in relativistic electric fields,” Las. Phys. 8, 93 (1998).

Q. Su, B.A. Smetanko, and R. Grobe, “Relativistic suppression of wave packet spreading,” Opt. Express 2, 277 (1998), http://www.opticsexpres.org/oearchive/source/2813.htm
[Crossref] [PubMed]

P. Krekora, Q. Su, and R. Grobe, “Dynamical signature in spatial spin distributions of relativistic electrons,” Phys. Rev. A, submitted.

P. Krekora, R.E. Wagner, Q. Su, and R. Grobe, “Dirac theory of ring-shaped electron distributions.” Phys. Rev. A, in press.

For a review, see e.g. Q. Su and R. Grobe, “Examples of classical and genuinely quantum relativistic phenomena,” in Multiphoton Processes, eds. L.F. DiMauro, R.R. Freeman, and K.C. Kulander (American Institute of Physics, Melville, New York, 2000) p.655 or the website www.phy.ilstu.edu/ILP

Q. Su, R.E. Wagner, P.J. Peverly, and R. Grobe, “Spatial electron clouds at fractional and multiple magneto-optical resonances,” in Frontiers of Laser Physics and Quantum Optics, eds, Z. Xu, S. Xie, S.-Y. Zhu, and M.O. Scully, p.117 (Springer, Berlin, 2000).

Szymanowski, C.

C. Szymanowski, C.H. Keitel, and A. Maquet, “Influence of Zitterbewegung on relativistic harmonic generation,” Las. Phys. 9, 133 (1999).

Thaller, B.

B. Thaller, The Dirac Equation, (Springer, 1992).

Thomas, L.T.

L.T. Thomas, Phil. Mag.3, 1 (1927).

Umar, A.S.

J.C. Wells, A.S. Umar, V.E. Oberacker, C. Bottcher, M.R. Strayer, J.-S. Wu, J. Drake, and R. Flanery, “A numerical implementation of the Dirac equation on a hypercube multicomputer,” Int. J. Mod. Phys. C 4, 459 (1993).
[Crossref]

Wagner, R.E.

P.J. Peverly, R.E. Wagner, Q. Su, and R. Grobe, “Fractional resonances in relativistic magnetic-laser-atom interactions,” Laser Phys. 10, 303 (2000).

R.E. Wagner, Q. Su, and R. Grobe, “Relativistic resonances in combined magnetic and laser field,” Phys. Rev. Lett.,  84, 3282 (2000).
[Crossref] [PubMed]

R.E. Wagner, P.J. Peverly, Q. Su, and R. Grobe, “Classical versus quantum dynamics for a driven relativistic oscillator,” Phys. Rev. A 61, 35402 (2000).
[Crossref]

R.E. Wagner, Q. Su, and R. Grobe, “High-order harmonic generation in relativistic ionization of magnetically dressed atoms,” Phys. Rev. A,  60, 3233 (1999).
[Crossref]

P. Krekora, R.E. Wagner, Q. Su, and R. Grobe, “Dirac theory of ring-shaped electron distributions.” Phys. Rev. A, in press.

Q. Su, R.E. Wagner, P.J. Peverly, and R. Grobe, “Spatial electron clouds at fractional and multiple magneto-optical resonances,” in Frontiers of Laser Physics and Quantum Optics, eds, Z. Xu, S. Xie, S.-Y. Zhu, and M.O. Scully, p.117 (Springer, Berlin, 2000).

Wells, J.C.

J.C. Wells, A.S. Umar, V.E. Oberacker, C. Bottcher, M.R. Strayer, J.-S. Wu, J. Drake, and R. Flanery, “A numerical implementation of the Dirac equation on a hypercube multicomputer,” Int. J. Mod. Phys. C 4, 459 (1993).
[Crossref]

Wu, J.-S.

J.C. Wells, A.S. Umar, V.E. Oberacker, C. Bottcher, M.R. Strayer, J.-S. Wu, J. Drake, and R. Flanery, “A numerical implementation of the Dirac equation on a hypercube multicomputer,” Int. J. Mod. Phys. C 4, 459 (1993).
[Crossref]

Ann. Phys. NY (1)

C. Bottcher and M.R. Strayer, “Relativistic theory of fermions and classical fields on a collocation lattice,” Ann. Phys. NY 175, 64 (1987).
[Crossref]

Int. J. Mod. Phys. C (1)

J.C. Wells, A.S. Umar, V.E. Oberacker, C. Bottcher, M.R. Strayer, J.-S. Wu, J. Drake, and R. Flanery, “A numerical implementation of the Dirac equation on a hypercube multicomputer,” Int. J. Mod. Phys. C 4, 459 (1993).
[Crossref]

J. Phys. B (2)

U.W. Rathe, C.H. Keitel, M. Protopapas, and P.L. Knight, “Intense laser-atom dynamics with the two-dimensional Dirac equation,” J. Phys. B 30, L531 (1997).
[Crossref]

N.J. Kylstra, A.M. Ermolaev, and C.J. Joachain, “Relativistic effects in the time evolution of a one-dimensional model atom in an intense laser field,” J. Phys. B 30, L449 (1997).
[Crossref]

Las. Phys. (3)

C. Szymanowski, C.H. Keitel, and A. Maquet, “Influence of Zitterbewegung on relativistic harmonic generation,” Las. Phys. 9, 133 (1999).

For relativistic suppression of wave packet spreading, see,Q. Su, B.A. Smetanko, and R. Grobe, “Wave packet motion in relativistic electric fields,” Las. Phys. 8, 93 (1998).

For the time-evolution of the spatial width, see, J.C. Csesznegi, G.H. Rutherford, Q. Su, and R. Grobe, “Dynamics of wave packets in inhomogeneous and homogeneous magnetic fields,” Las. Phys. 6, 41 (1999).

Laser Phys. (1)

P.J. Peverly, R.E. Wagner, Q. Su, and R. Grobe, “Fractional resonances in relativistic magnetic-laser-atom interactions,” Laser Phys. 10, 303 (2000).

Opt. Express (1)

Parallel Computing (1)

U.W. Rathe, P. Sanders, and P.L. Knight, “A case study in scalability: an ADI method for the two-dimensional time-dependent Dirac equation,” Parallel Computing,  25, 525 (1999).
[Crossref]

Phys. Rev. A (4)

J.W. Braun, Q. Su, and R. Grobe, “Numerical approach to solve the time-dependent Dirac equation,” Phys. Rev. A 59, 604 (1999).
[Crossref]

R.E. Wagner, P.J. Peverly, Q. Su, and R. Grobe, “Classical versus quantum dynamics for a driven relativistic oscillator,” Phys. Rev. A 61, 35402 (2000).
[Crossref]

K. Momberger, A. Belkacem, and A.H. Sorensen, “Numerical treatment of the time-dependent Dirac equation in momentum space for atomic processes in relativistic heavy-ion collisions,” Phys. Rev. A 53, 1605 (1996).
[Crossref] [PubMed]

R.E. Wagner, Q. Su, and R. Grobe, “High-order harmonic generation in relativistic ionization of magnetically dressed atoms,” Phys. Rev. A,  60, 3233 (1999).
[Crossref]

Phys. Rev. D (2)

For work on the Spin-Wigner function, see, e.g., I. Bialynicki-Birula, P. Gornicki, and J. Rafelski, “Phase-space structure of the Dirac vacuum,” Phys. Rev. D 44, 1825 (1991).
[Crossref]

G.R. Shin, I. Bialynicki-Birula, and J. Rafelski, “Wigner function of relativistic spin-1/2 particles,” Phys. Rev. D 46, 645 (1992).

Phys. Rev. Lett. (1)

R.E. Wagner, Q. Su, and R. Grobe, “Relativistic resonances in combined magnetic and laser field,” Phys. Rev. Lett.,  84, 3282 (2000).
[Crossref] [PubMed]

Other (13)

For movies of cycloatoms see Phys. Rev. Focus, “Fast electrons on the cheap”, 5, 15, 6 April (2000) at the web site: http://focus.aps.org/v5/st15.html story

For a review, see e.g. Q. Su and R. Grobe, “Examples of classical and genuinely quantum relativistic phenomena,” in Multiphoton Processes, eds. L.F. DiMauro, R.R. Freeman, and K.C. Kulander (American Institute of Physics, Melville, New York, 2000) p.655 or the website www.phy.ilstu.edu/ILP

Science News, “Ring around the proton,” 157, 287 (2000).

V.G. Bagrov and D.M. Gitman, Exact solutions of relativistic wave equations, (Kluwer Academic, Dordrecht, 1990).
[Crossref]

Q. Su, R.E. Wagner, P.J. Peverly, and R. Grobe, “Spatial electron clouds at fractional and multiple magneto-optical resonances,” in Frontiers of Laser Physics and Quantum Optics, eds, Z. Xu, S. Xie, S.-Y. Zhu, and M.O. Scully, p.117 (Springer, Berlin, 2000).

P. Krekora, R.E. Wagner, Q. Su, and R. Grobe, “Dirac theory of ring-shaped electron distributions.” Phys. Rev. A, in press.

H. Goldstein, Classical Mechanics, 2nd edition (Addison-Wesley, New York, 1980).

B. Thaller, The Dirac Equation, (Springer, 1992).

E. Lenz, M. Dörr, and W. Sandner, Las. Phys., in press.

For a review on Lorentz transformations of 4×4 spin matrices, see, e.g., J.D. Bjorken and S.D. Drell, “Relativistic quantum mechanics,” (McGraw-Hill, 1964); J. Kessler, Polarized Electrons, 2nd edition (Springer Verlag, Berlin, 1985).

P. Krekora, Q. Su, and R. Grobe, “Dynamical signature in spatial spin distributions of relativistic electrons,” Phys. Rev. A, submitted.

L.T. Thomas, Phil. Mag.3, 1 (1927).

J.D. Jackson, Classical Electrodynamics (Wiley, New York, 1999).

Supplementary Material (4)

» Media 1: MOV (2242 KB)     
» Media 2: MOV (956 KB)     
» Media 3: MOV (1234 KB)     
» Media 4: MOV (1211 KB)     

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Figures (3)

Fig. 1
Fig. 1

The formation of cycloatoms. Please click to play the movie. [2.1 MB]

Fig. 2
Fig. 2

Spatial and spin density for an electron wave packet accelerated by a static electric field. [960 k]

Fig. 3
Fig. 3

Spatial spin distribution in relativistic cycloatoms. [2.3 MB] [Media 4]

Equations (4)

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ρ ( r , p , t ) t = { c 4 + c 2 ( p + 1 c A ( r , t ) ) 2 + V ( r ) , ρ ( r , p , t ) } r , p
i t Ψ ( r , t ) = [ c α ( p + 1 c A ( r , t ) ) + β c 2 + V ( r ) ] Ψ ( r , t )
S ( v ) = [ 1 ( v c ) 2 ] 1 2 S ( v = 0 )
S z class ( r , t ) = d p 1 2 1 ( v ( p ) c ) 2 ρ ( r , p , t )

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