1. Numerical solution of Dirac equation & its applications in intense laser physics Q. Su Intense Laser Physics Theory Unit Illinois State University LPHY 2000Bordeaux FranceJuly 2000 Support: NSF, Research Corporation, NCSA J. BraunP. Krekora P. PeverlyR. Grobe R. Wagner

2. Classical phase space approach valid for Non-linear systems of relativistic particles? Quantum cycloatoms Relativistic theory of tunneling Superluminal speeds Goals

3. Numerical techniques Dirac Liouville J. Braun, Q. Su and R. Grobe, PRA 59, 604 (1999) P. Peverly, R. Wagner, Q. Su and R. Grobe, Las Phys. 10, 303 (2000) Laser Magnetic field

4. Maximum speed v/c for each  non- relativistic LL  R.E. Wagner, Q. Su and R. Grobe, Phys. Rev. Lett. 84, 3282 (2000)

5. Non-relativisticRelativistic Orbits stay in phase Orbits dephase relativistically Time (in 2  L  75 150 500 0 y x

6. Dirac Liouville Confirmed: Dirac Cycloatoms P. Krekora, R. Wagner, Q. Su and R. Grobe, PRA, submitted

7. Summary 1 - Phase space approach valid in relativistic regime - Quantum cycloatom confirmed R.E. Wagner, Q. Su and R. Grobe, Phys. Rev. Lett. 84, 3282 (2000) P. Krekora, R. Wagner, Q. Su and R. Grobe, PRA, submitted

8. Questions about tunneling  Dirac theory predict superluminal speeds?  Violation of causality? If v > c  Instantaneous speed inside the barrier? A.M. Steinberg, P.G. Kwiat and R.Y. Chiao, Phy. Rev. Lett. 71, 708 (1993) C. Spielmann, R. Szipöcs, A. Stingl and F. Krausz, Phys. Rev. Lett. 73, 2308 (1994) V. Gasparian, M. Ortuno, J. Ruiz and E. Cuevas, Phys. Rev. Lett. 75, 2312 (1995) L. Wang, private communications

9. Theoretical Model Dirac J. Braun, QS, R. Grobe, PRA 59, 604 (1999) 65,536 grid pts, 1,500,000 pts in time

10. Dirac & Schrödinger => v > c possible Dirac: + exact - stat. phase approx. Schrödinger: o exact - stat. phase approx. larger v for Dirac SPA best for broad packets

11. .............. Center IQ Tunnel Center.................... Superluminal speeds = Pulse reshaping effect No violation of causality

12. Violation of causality ? Causality violation if

13. Tunneling dynamics under the barrier no spatial localization under the barrier

14. Spatially resolved tunneling velocity Time localized state under barrier

15. Summary 2  Dirac + Schrödinger theories predict superluminal effects  Causality non-violation for Dirac theory  Instantaneous tunneling velocity defined www.phy.ilstu.edu/ILP P.Krekora, QS, R.Grobe, Phys. Rev. Lett. (submitted)