Numerical solution of Dirac equation & its applications in intense laser physics Q. Su Intense Laser Physics Theory Unit Illinois State University LPHY.

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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

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

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

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

Non-relativisticRelativistic Orbits stay in phase Orbits dephase relativistically Time (in 2  L  y x

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

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

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

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

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

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

Violation of causality ? Causality violation if

Tunneling dynamics under the barrier no spatial localization under the barrier

Spatially resolved tunneling velocity Time localized state under barrier

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