## Relativistic electron spin motion in cycloatoms

Optics Express, Vol. 8, Issue 2, pp. 51-58 (2001)

http://dx.doi.org/10.1364/OE.8.000051

Acrobat PDF (113 KB)

### 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.

© Optical Society of America

## 1. Introduction

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

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]

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

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]

## 2. Numerical methods

**84**, 3282 (2000). [CrossRef] [PubMed]

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]

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]

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]

**r**,

**p**,t). Here {…}

_{r,p}denotes the Poisson brackets with respect to the phase space variables, V(

**r**)=-(

**r**

^{2}+1)

^{-1/2}is the smoothed Coulomb potential, and c is the speed of light. In this paper we use two different vector potentials, the first one, A(

**r**,t)=-Ec/ω

_{L}sin(ω

_{L}t)

**e**

_{x}+ (Ω

**e**

_{z})×

**r**/2 represents the linearly polarized laser field along the x-direction and the static magnetic field Ω is along the z-direction. The second one, A(

**r**,t)=c E t

**e**

_{x}, represents a static electric field along the negative x-direction.

_{cl}(

**r**, t)=∫d

**p**ρ(

**r**,

**p**, t). None of the phenomena discussed here are very sensitive to the details of the initial state which is chosen as

_{0}.

_{cl}(

**r**,t) will be compared directly with the corresponding quantum mechanical density P

_{qm}(

**r**, t)=

_{i}(

**r**, t)|

^{2}, where the summation extends over the four spinor components. The wave function Ψ(

**r**,t) can be obtained from the corresponding numerical solution to the Dirac equation:

**α**and

**β**denote the 4×4 Dirac matrices [19]. The time-dependent wave function Ψ(

**r**,t) can be calculated on a space-time grid using a recently developed split-operator algorithm based on a fast Fourier transformation that is accurate up to the fifth order in time [15

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]

**r**, t=0)=(2

*π*Δ

^{-3/4}exp[-(

**r**/Δx

_{0})

^{2}/4] Φ

_{x,z}. We used Φ

_{x}=(1,1,0,0)/√2 and Φ

_{z}=(1,0,0,0) to represent initial spin states with averages <S

_{x}(t=0)>=1/2 a.u. and <S

_{z}(t=0)>=1/2 a.u., respectively.

## 3. Quantum analogue of the classical cycloatom electron distribution?

_{L}=80 a.u.. The combined magnetic and laser field accelerates the electron to 44% of the speed of light after 4 laser cycles. We should note that due to the relativistic resonance shifts discussed in [2,3

**84**, 3282 (2000). [CrossRef] [PubMed]

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]

_{i}(x,y,t)|

^{2}, and the right graph shows the solution of the relativistic Liouville equation. The wave packet with an initial width Δx

_{0}=0.1 a.u. develops after a few laser oscillations into a “banana-like” shape that evolves into a ring. The center of this ring-structure follows a circular orbit around the nucleus with the laser period. [17] The agreement between the classical and quantum descriptions is remarkable. The jagged contour lines at the edges of the ring in the classical density are a small numerical artifact due to the discreteness of the individual trajectories.

## 4. Position-dependent spin densities

**A**(

**r**,t)=cEt

**e**

_{x}.

_{x}(t)> is practically constant, whereas the spin perpendicular to the x-axis (<S

_{z}(t)>) decays as the electron’s speed increases. This decay is associated with the Lorentz spin contraction which is different from the Lorentz length contraction in that the perpendicular component rather than the parallel component is affected by the relativistic motion. The spin decay can be understood if we perform a Lorentz transformation into the electron’s rest frame in which the spin remains constant if the negative energy components in the state are not significant. The spin, when observed from the lab frame, (in which the Dirac equation is solved) appears to be contracted by the Lorentz-gamma factor,

**S**(

**r**,t)≡

**Ψ**

^{†}(

**r**,t)

**SΨ**(

**r**,t)/

**Ψ**

^{†}(

**r**,t)

**Ψ**(

**r**,t). This quantity matches well the corresponding spatial spin distribution for a classical ensemble of spins. It is the average value of the spin one would measure if the electron were detected at time t at location r. [25

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]

**S**(t)>=<

**Ψ**(

**r**,t) |

**S**|

**Ψ**(

**r**,t)>=∭ dx dy dz

**S**(

**r**,t) P(

**r**,t), where P(

**r**,t)=

**Ψ**

^{†}(

**r**,t)Ψ(

**r**,t) is the usual spatial probability density, given by the sum of the four squared spinor components of the wave function.

_{z}(x,t) by using different colors. Red corresponds to S

_{z}(x,t)=+1/2 and dark blue to S

_{z}(x,t)=0. The corresponding distribution S

_{x}(x,t) along the direction of propagation remains spatially as well as temporally constant, S

_{x}(x,t)=1/2 a.u.. Initially, the state was in a spin eigenstate, and is not shown since it remains one throughout the time evolution. In other words, the spin operator S

_{x}commutes with the Dirac Hamiltonian in one spatial direction, and the momentum eigenstates with velocities along the x-direction are also spin S

_{x}eigenstates. For a direction perpendicular to the x-direction, the situation is different; the spin is not associated with a “good” quantum number for a wave packet and S

_{z}(x,t) decreases as a function of time and space. It is quite interesting to note that in addition to the overall lowered spin value as time increases, the spatial spin distribution becomes non-uniform. The spins associated with the front-edge of the accelerated wave packet are relatively smaller (more “blueish”), reflecting the fact that the larger velocity components of the wave packet have traveled to the right edge of the quantum state. One can almost view the spin distribution S

_{z}(x,t) as a spatially resolved “speedometer” for the quantum mechanical state in this case.

**r**,

**p**,t) via:

## 5. Spin-densities of cycloatoms

_{z}(r,t).

^{†}(r,t) Ψ(r,t) exceeds 6×10

^{-6}. It has been recorded for the same parameters as Fig. 1. At early times, the packet is non-relativistic and we see the concentric ring-like contour lines reminiscent of those of the non-relativistic orbits. [28] The region around the origin has the smallest velocity contributions and therefore the smallest amount of Lorentz contraction. It is quite remarkable that even at later times (t>0.4 a.u.), when the tail end begins to curve inward towards the origin, the spin contour lines in the front edge of the growing tail still approximately follow the simple concentric circles. At later times when the tail end has closed the distribution to a full circle at time t=0.6 a.u., the front tail again contains very small velocity contributions. This increase and decrease of the Lorentz contraction associated with different spatial parts of the distribution can be directly associated with outward (accelerating) and inward going (decelerating) spiral orbits associated with the classical dephasing model. As a result the spin distribution S

_{z}(r,t) seems to depend mainly on the specific position and not so much on time.

_{x}for an initial state with S

_{x}(r,t=0)=1/2 a.u. It is the result of three independent relativistic effects. The first one is the Lorentz contraction, which restricts the maximum spin value and depends only on the instantaneous velocity in the y-direction v

_{y}. The second effect is due to the relativistic mass-shift, leading effectively to a velocity dependent Larmor frequency; this effect is accumulative in the sense that the entire history of different Larmor frequencies contribute to the phase and the amplitude of the local spin value. A third effect is the well-known Thomas precession, [29,30] whose frequency for a uniformly accelerated system is given by ω

_{T}=(

*γ*-1) a×v/v

^{2}. Approximating the acceleration |a| with vΩ and assuming an average value of v=c/4 for the speed, the Thomas precession frequency |(

*γ*-1)Ω| would amount to Ω/31.5, which is much smaller than the cyclotron frequency and corresponds for our parameters to a time even longer than the total duration of interaction. As the final state is the result of all of these accumulative and non-accumulative effects, we discuss the impact of these effects step by step as the electron becomes relativistic.

_{z}(r,t) the spin contour lines for S

_{x}(r,t) are not concentric circles. If the spin were only affected by the Lorentz effect, we would expect parallel lines. [18] However, in addition to this effect, the faster contributions in the leading tail experience a smaller effective Larmor frequency. As a result, the spin value lags behind the (spin-orbital decoupled) value of 1/2 cos(Ωt). This effect curves the otherwise parallel contour lines. The snapshot at time t=4.08 T nicely illustrates both effects. Here T=2π/ω

_{L}is the laser period. The distribution at the center and the tail end are out of phase but they have the same spin value. As the spins get out of phase in a continuous manner along the distribution one could expect somewhere a maximum spin value of 1/2 a.u. However, in this case the maximum value is associated with the left most part of the distribution [(x,y)≈(-1,0) a.u.] where the velocity v

_{y}is largest. As a result the Lorentz contraction forbids the maximum value of 1/2 in this region. A similar effect can be observed at later times t=9.36 T when the spins of the front end and those close to the origin are completely out of phase. Here the maximum spin value 1/2 a.u. is taken at the upper part of the distribution where the Lorentz contraction is negligible (v

_{y}≈0). We note that the contour lines recorded at the largest time t= 9.36 T are along straight lines; all of which seem to originate at various locations close to the origin.

## Acknowledgements

## References and links

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. , |

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. |

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 |

8. | V.G. Bagrov and D.M. Gitman, |

9. | C. Bottcher and M.R. Strayer, “Relativistic theory of fermions and classical fields on a collocation lattice,” Ann. Phys. NY |

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 |

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 |

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 |

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 |

14. | C. Szymanowski, C.H. Keitel, and A. Maquet, “Influence of Zitterbewegung on relativistic harmonic generation,” Las. Phys. |

15. | J.W. Braun, Q. Su, and R. Grobe, “Numerical approach to solve the time-dependent Dirac equation,” Phys. Rev. A |

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 , |

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, |

19. | B. Thaller, |

20. | R.E. Wagner, Q. Su, and R. Grobe, “High-order harmonic generation in relativistic ionization of magnetically dressed atoms,” Phys. Rev. A , |

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. |

22. | Q. Su, B.A. Smetanko, and R. Grobe, “Relativistic suppression of wave packet spreading,” Opt. Express |

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 |

26. | G.R. Shin, I. Bialynicki-Birula, and J. Rafelski, “Wigner function of relativistic spin-1/2 particles,” Phys. Rev. D |

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. |

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, |

**OCIS Codes**

(000.1600) General : Classical and quantum physics

(020.4180) Atomic and molecular physics : Multiphoton processes

**ToC Category:**

Focus Issue: Quantum control of photons and matter

**History**

Original Manuscript: November 6, 2000

Published: January 15, 2001

**Citation**

Qichang Su, P. Peverly, R. Wagner, P. Krekora, and Rainer Grobe, "Relativistic electron spin motion in cycloatoms," Opt. Express **8**, 51-58 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-2-51

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### References

- 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, 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).
- R.E. Wagner, Q. Su and R. Grobe, "Relativistic resonances in combined magnetic and laser field," Phys. Rev. Lett. 84, 3282 (2000). [CrossRef] [PubMed]
- 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
- 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).
- 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]
- V.G. Bagrov and D.M. Gitman, Exact solutions of relativistic wave equations, (Kluwer Academic, Dordrecht, 1990). [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]
- 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]
- 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]
- 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]
- 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, 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]
- 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).
- 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 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]
- E. Lenz, M. D�rr 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).
- 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).
- 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).
- 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).

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