Examples of genuinely relativistic phenomena

R. Grobe

ICOMP VIII, Monterey, CA October 1999

Intense Laser Physics Theory Unit

Illinois State University

Postdocs Faculty

J.W. BraunB.A. SmetankoJ.J. CsesznegiP.J. PeverlyR.E. WagnerT. ShepherdS.M. MandelA. Bergquist

Undergraduate students

H. Wanare

P. Krekora

G.H. Rutherford

Q. Su

R. Grobe

Support : National Science Foundation, Research Corporation, Illinois State

Tools to explore phenomena that are

genuinely relativistic

Dirac :

iÝ c p

1

cA(r , t)

c2 V(

r )

Ý [H,] with H c4 c2(p

1

cA)2 V(

r )

Liouville:

A flavor of the numerical work

• Discretize:

r 2563 points

t 103-106 points

• Split operator FFT technique

• Supercomputer

r , t 0 integrate

(r , t)

H cp c2 A(r , t)V(

r )

≈ 4 · 2563 · 106 = 1013 complex numbers

Genuinely Relativistic Phenomena

• ZitterbewegungJ.W. Braun, Q. Su & RG, PRA 59, 604 (1999)

• Klein-Paradox (particle pair production)J.W. Braun, Q. Su & RG, PRA 59, 604 (1999)

• Subnatural wave packet spreadingQ. Su, B.A. Smetanko & RG, Opt. Exp. 2, 277 (1998)J.C. Csesznegi, G.H. Rutherford, Q. Su & RG, Las. Phys. 9, 41 (1999)

• Spin-spatial coupling in magnetic fields G.H. Rutherford & RG, PRL 81, 4772 (1998)G.H. Rutherford & RG, JPA 31, 9331 (1998)

• Chaos J. Kim, and H. Lee, PRE 51, 1579 (1995)

• How good is Liouville?R.E. Wagner, P.J. Peverly, Q. Su & RG, PRA (subm.)

• Counterintuitive enhancement of resonancesR.E. Wagner, Q. Su & RG, PRL (subm.)

• Cycloatoms and dephasingP.J. Peverly, R.E. Wagner, Q. Su, & RG, Las. Phys. (in press)

• Scattered light spectraR.E. Wagner, Q. Su & RG, PRA 60, No.4 (1999)

Schrödinger’s Zitterbewegung

small ∆x neg. energy contrib. “Zitter”

position

spin

Zitterbewegung real? controversial issue...

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.0001 0.0002 0.0003

<S z(t

)> [

a.u.

]

time [a.u.]

x=10c

x=c

x=0.1c

-2 10-5

-1 10-5

0

1 10-5

2 10-5

0 0.0001 0.0002 0.0003

<x(

t)>

[a.

u.]

time [a.u.]

x=10c

x=c

x=0.1c

J.W. Braun, Q. Su & RG, PRA 59, 604 (1999).

Time - resolved Klein Paradox

Interpretation still controversial ...

0.2

0.4

position [nm]

1.61 .61

0-0.5

electron

positron

0

voltage barrier

0.2

0.4

position [nm]

1.0

0-0.5

incomingelectron

0

voltage barrier

J.W. Braun, Q. Su & RG, PRA 59, 604 (1999).

voltage > Ekin + 2c2

Stern-Gerlach separation possible for electrons ???

Pauli/Bohr: Lorentz-force “washes out” separation ??

Dirac solution : Spin separation is possible

G.H. Rutherford & RG, PRL 81, 4772 (1998) and JPA 31, 9331(1998).

Position

0

0.02

0.04

0.06

-20 -10 0 10 20

–+

P±(x,t=120a.u.)

0

0.02

0.04

–+

P±(x,t=0)

Sz

SzSz

B (r ) B

0

0

0

x

inhomog. magnetic field:

Subnatural wave packet spreading

Non-relativistic: Spreading independent of

the center of mass motion

• Spreading is suppressed:

Q. Su, B.A Smetanko and RG, Opt. Exp. 2, 277 (1998)

• Spatial profile becomes asymmetrical :

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1time (a.u.)

x

x NR

y, z

0.00

0.05

0.10

0.15

-40 -20 0 20position (a.u.)

E=1000

E=500

E=2000

x; t

Q. Su, B.A. Smetanko & RG, Las. Phys. 4, 93 (1998)

1

2

3

-10 0 10 20

P(x

,t)

x [a.u.]

t = 0 c

t = 4 c

t = 8 c(b)

0

1

2

3

-10 0 10 20

P(x

,t)

x [a.u.]

t = 0 c

t = 4 c

t = 8 c

(a)

Relativity Induces Chaos

non-relativistic => Liouville = Schrödingerrelativistic => ??? Liouville ≈ Dirac ????

Schrödinger = Newton Dirac ≈ Liouville

What a surprise ...

0

4

8

12

0 20 40 60 80 100

<x

> [

a.u.

]

t [in laser periods]

(a)

2

4

6

8

10

43 44 45 46

<x

> [

a.u.

]

t [in laser periods]

(b)

J. Kim and H. Lee, PRE 51, 1579 (1995):

Ý Ý x 0

2x E sin(t) + relativity => chaos

R.E. Wagner, P.J. Peverly, Q. Su & RG, PRA (subm.)

Relativity enhances resonances

Myth: relativity “heavier mass” slower motion

example: electron in laser and static magnetic field

Maximum speed v/c for each

0.20

0.40

0.60

0.80

1.0

0.003 0.004 0.005 0.006

L 0.0043 a.u.

E0 0.0500a.u.

non-relativistic

relativistic

L

Fact: relativity faster motion

R. Wagner, Q. Su & RG, PRL (submitted)

Novel steady spatial states: Cycloatoms

Non-relativistic Relativistic

Orbits stayin phase

Orbits dephaserelativistically

Time(in 2L

75

150

500

0

y

x

Relativistic dephasing model

relativistic (exact) dephasing model

x(t) x vx

sin t

vy

cos t 1

2 2

cos( t

cos t

y(t) y vy

sin t vx

cos(t) 1

2 2

1

sin t 1

sin t

Time

75

150

500

0

replace (V0)

Q. Su, R.E. Wagner, P.J. Peverly & RG, SPIE (in press)

Steady state spatial electron distributions

Multiple resonances

Fractional resonances

= 3 = 2 =

= 1/2 = 1/3

0

0.4

0.8

1.2

0 0.1 0.2 0.3 0.4 0.5

[a.u.]

= L

= L/2

= 2L

= 3L

= 3L/2

= 2L/3

= L/3

=

L/4

=

L/5

-2.0 104

0.0

2.0 104

-4 104 -2 104 0 2 104

-2.0 104

0.0

2.0 104

-4 104 -2 104 0 2 104

10-8

10-6

10-4

10-2

0 0.05 0.1 0.15 0.210-8

10-6

10-4

10-2

0 0.05 0.1 0.15 0.2

Scattered light spectra

non-relativistic relativistic

x

y

Single orbits

Corresponding spectra

L 0.150 a.u.

E0 1.000a.u.

0.172a.u.

R.E. Wagner, Q. Su & RG, PRA 60, No.4 (1999)

SummaryNumerical solution to the Dirac equation

Relativity leads to new phenomena in the spatial and temporal dynamics

• subnatural spreading

• chaos

• novel resonances => novel experiments

• cycloatoms

• dephasing

• scattered light spectra

www.phy.ilstu.edu/ILP