LPHY 2000 Bordeaux France July 2000 Numerical solution

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

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

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

Maximum speed v/c for each W nonrelativistic w. L W R. E. Wagner, Q. Su and R. Grobe, Phys. Rev. Lett. 84, 3282 (2000)

Non-relativistic Relativistic 0 75 150 y 500 x Orbits stay in phase Time (in 2 p/w. L) Orbits dephase relativistically

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

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

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

Violation of causality ? Causality violation if

Tunneling dynamics under the barrier no spatial localization under the barrier

Time localized state under barrier Spatially resolved tunneling velocity

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) www. phy. ilstu. edu/ILP