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

(r,p,t)  H p p H
t
H  c 4  c 2 (p  A / c)2  V(r)
P. Peverly, R. Wagner, Q. Su and R. Grobe, Las Phys. 10, 303 (2000)
Dirac

2
i   ic    A V(r)  c 
t J. Braun, Q. Su and R. Grobe, PRA 59, 604 (1999)
Laser
Magnetic field
Bo || Br ,t 
Maximum speed v/c
for each W
1.0
nonrelativistic
relativistic
0.80
0.60
0.40
0.20
0.003
W
L
0.004
0.005
0.006
 L 0.0043a.u.
E0 0.0500a.u.
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 2p/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
2
i t   icx  x   c    W(x) 
  1,2 ,3, 4 
65,536 grid pts,
1,500,000 pts in time
J. Braun, QS, R. Grobe, PRA 59, 604 (1999)
200
Dirac
c
100
Schrödinger
0
0
0.02
0.04
w [a.u.]
Dirac: + exact
Schrödinger: o exact
larger v for Dirac
subluminal
ve [a.u.]
superluminal
Dirac & Schrödinger => v > c possible
0.06
- stat. phase approx.
- stat. phase approx.
SPA best for
broad packets
Superluminal speeds = Pulse reshaping effect
.. .. .. .. ..
.. ..
IQ Tunnel
Center
.. .. .. .. ..
Center
Center
.. .. .. .. ..
Center
No violation of causality
Violation of causality ?

Causality violation if
2

 (x,t)   (x  ct,t  0)
x
x
P(x)
10-1
violation of causality
10-2
10-3
10-4
995
Schrödinger
Dirac
light front
1005
x [a.u.]
1015
2
Tunneling dynamics under the barrier
P(x)
P(x)
T=0
0
10
T=0.050
100
10-2
T=0.045
T=0.035
10-2
10-4
T=0.040
T=0.030
10-4
P(x)
100
T=0.05 a.u.
10-2
10-6
10-4
0
10
T=0.025
T=0.055
barrier
T=0.065
P(x)
100
10
T=0.1 a.u.
-2
T=0.060
T=0.070
10-2
10-4
10-4
-8
T=0.075
0
x [a.u.]
8
10-6
-0.04
0
x [a.u.]
0.04
no spatial localization under the barrier
P (x=0)
0.02
Tp
Time localized state
under barrier
0.01
T
x [a.u.]
0.015
0
1
2
w/2
0.005
barrier
-w/2
-0.005
1.0000
1.0001
Tp [a.u.]
1.0002
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